Advances in Applied Mechanics, Volume 24

Advances in Applied Mechanics Volume 24

Editorial Board T. BROOKEBENJAMIN Y. C. FUNG PAULGERMAIN RODNEYHILL L. HOWARTH C.-S. YIH(Editor, 1971-1982)

Contributors to Volume 24 G. BERTIN M. A. BIOT WEI-ZANGCHIEN TAKAOINUI C. C. LIN JOHN MILES HIDEAKIMIYATA STUARTB. SAVAGE

ADVANCES IN

APPLIED MECHANICS Edited by Theodore Y. Wu

John W. Hutchinson DIVISION OF APPLIED SCIENCES HARVARD UNIVERSITY CAMBRIDGE, MASSACHUSEITS

ENGINEERING SCIENCE DEPARTMENT CALIFORNIA INSTITUTE OF TECHNOLOGY PASADENA, CALIFORNIA

VOLUME 24

1984

ACADEMIC PRESS, INC. (Harcourt Brace Jovanovich, Publishers)

Orlando San Diego San Francisco New York London Toronto Montreal Sydney Tokyo

COPYRIGHT @ 1984, BY ACADEMIC PRESS,INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED I N ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

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United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London N W I 7DX

LIBRARY OF CONGRESS CATALOG CARD NUMBER:48-8503 ISBN 0-12-002024-6 PRINTED IN THE UNITED STATES OF AMERICA

84 85 86 87

9 8 7 6 5 4 3 2 1

Contents xi ix

CONTRIBUTORS PREFACE

New Variational-LagrangianIrreversible Thermodynamicswith Application to Viscous Flow, Reaction-Diffusion, and Solid Mechanics

M . A . Biot I. Introduction

2

11. Restructured Thermodynamics of Open Systems and the Concept of

6 9

Thermobaric Transfer 111. New Chemical Thermodynamics

IV. Homogeneous Mixtures and Reformulation of the Gibbs-Duhem Theorem V. Nontensorial Virtual Work Approach to Finite Strain and Stress VI. Thermodynamic Functions of Open Deformable Solids VII. The Fluence Concept VIII. The Nature of Entropy -Production and Its Evaluation IX. The Principle of Virtual Dissipation X. General Lagrangian Equations XI. Dynamics of Viscous Fluid Mixtures with Reaction-Diffusion and Radiation Pressure XII. Dynamics of Solids with Elastoviscous Stresses and Heat Conduction, and Thermoelasticity XIII. Inhomogeneous Viscous Fluid with Convected Coordinates and Heat Conduction XIV. Lagrangian Equations of Heat Transfer and Their Mechanical Interpretation, and a Mass Transfer Analogy xv . Deformable Solids with Thermomolecular Diffusion and Chemical Reactions XVI. Thermodynamics of Nonlinear Viscoelasticity and Plasticity with Internal Coordinates and Heredity XVII. Dynamics of a Fluid-Saturated Deformable Porous Solid with Heat and Mass Transfer XVIII. Linear Thermodynamics near Equilibrium XIX. Linear Thermodynamics of a Solid under Initial Stress xx. Linear Thermodynamics and Dissipative Structures near Unstable Equilibrium XXI. Thermoelastic Creep Buckling XXII. Lagrangian Formulation of Bifurcations Y

13 16 19

21 23 30 35 36 44 51

56 59

62

64 68 13 78 81 82

Contents

vi

XXIII. Generalized Stability Criteria for Time-Dependent Evolution Far from Equilibrium XXIV. Creep and Folding Instability of a Layered Viscous Solid xxv. Coupling of Subsystems and the Principle of Interconnection XXVI. Completeness of the Description by Generalized Coordinates. Resolution Threshold and Lagrangian Finite Element Methods XXVII. Lagrangian Equations in Configuration Space. Internal Relaxation, Order-Disorder Phenomena, and Quantum Kinetics References

82 85 86 88 89 89

Incompatible Elements and Generalized Variational Principles Wei-Zang Chien I.

Introduction

94

11. Generalized Variational Principle Related to Incompatible Elements of

Small-Displacement Linear Elasticity

94

111. Generalized Variational Principle of Incompatible Elements for the Plane

Problems in Elasticity IV. Generalized Variational Principle for Plate Elements of Bending V. Conclusions References

118 127 152 153

Galactic Dynamics and Gravitational Plasmas C . C . Lin and G . Bertin 1. Introduction 11. Observations 111. Density Wave Theory of Spiral Structure

IV. V. VI. VI1.

Dynamic Mechanisms Theory of Discrete Spiral Modes Dynamic Approach to Classification of Galaxies Concluding Remarks References

156 158 164 17 I 175 182 185 187

Strange Attractors in Fluid Dynamics John Miles I. Introduction The Spherical Pendulum Lorenz’s Convection Model The Howard-Malkus-Welander Convection Model Mathematical Routes to Turbulence

11. 111. IV. V.

189 193 199 202 204

Contents VI. Conclusions References

vii 212 212

Nonlinear Ship Waves Hideaki Miyata and Takao lnui I. Introduction 11. Nonlinear Waves Generated by Ships

Ill. IV. V. VI.

Characteristics of Waves around Wedge Models Modified Marker-and-Cell Method Computed Waves around Wedge Models Concluding Remarks References

215 218 242 264 214 287 287

The Mechanics of Rapid Granular Flows Stuart B . Savage

Flows in Vertical Channels and Inclined Chutes Rheological Test Devices and Experiments Theories for Rapid Granular Flows Concluding Remarks References

290 292 302 32 1 335 358 359

INDEX

361

SU0JECT INDEX

313

I. Introduction 11. Preliminary Discussion of Some Granular Flow Regimes

Ill. IV. V. V1.

AUTHOR

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Contributors Numbers in parentheses indicate the pages on which the authors’ contributions begin.

G. BERTIN(155), Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, and Scuola Normale Superiore, Pisa, Italy M. A. BIOT( l ) , Royal Academy of Belgium, Brussels, Belgium WEI-ZANGCHIEN*(93), Tsing Hua University, Beijing, People’s Republic of China TAKAOINUI(215), Department of Mechanical Engineering, Tamagawa University, Tokyo, Japan C. C. LIN ( 1 5 9 , Massachusetts Institute of Technology, Cambridge, Massachusetts 02 139 JOHNMILES(189), Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California 92093 HIDEAKI MIYATA(215), Department of Naval Architecture, University of Tokyo, Bunkyo-ku, Tokyo 113, Japan STUARTB. SAVAGE (289), Department of Civil Engineering and Applied Mechanics, McGill University, Montreal, Quebec, Canada H3A 2K6

* Present address: Shanghai Technical University, Shanghai, People’s Republic of China. ix

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Preface This volume includes expository surveys of the present state of knowledge and of new research in several diversified fields of applied mechanics. In Professor Biot’s article, new concepts are elaborated to explain the new variational-Lagrangian theory of irreversible thermodynamics for open systems, which has been developed gradually with great care; the result is noted for its interdisciplinary generality and its broad scope of applications. With equal care Professor W.-Z. Chien has worked out a series of generalized variational principles which are presented here to illustrate their use and value for various theoretical and numerical investigations. In the realm of astrofluid mechanics, Professor Lin and Professor Bertin give us an up-to-date, in-depth discussion of the main ideas and the current level of understanding of the dynamics of spiral galaxies. Professor Miles’s article, “Strange Attractors in Fluid Dynamics,” draws convincing arguments on how effective these concepts can be in providing new insight into physically important problems involving the transition from regular to chaotic motions. On the topic of nonlinear ship waves Professors Miyata and Inui bring forth meticulous and valuable experimental data, which hopefully will stimulate further studies of the difficult problem of the bow wave field. In addition, we have from Professor Savage a very comprehensive exposition on the important subject of granular flow. Warmest thanks are due these authors for their outstanding contributions. This series, bearing the heritage from the era of its founding editors, is intended for students, scientists, and engineers who are interested in acquiring from active cultivators in the field their learned views and mentor philosophy about major advances in areas of current and increasing importance. Leading contributors are encouraged to share in this series their foresight and conviction on fruitful new directions for future development. In setting up an example for the diversified kind of approach, Professors von Karman and von Mises wrote in the Preface of Volume 1, “It is a well known fact that the more research in mechanics expands, the more interconnections of seemingly far distant fields become apparent.” It will be our endeavor to maintain this spirit. Finally, I join my fellow Editor, John W. Hutchinson, in thanking our predecessor, Chia-Shun Yih, for his continuing assistance and advice, in many capacities, for the Advances.

THEODORE Y. Wu xi

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ADVANCES I N APPLIED MECHANICS, VOLUME

24

New Variational-Lagrangian Irreversible Thermodynamics with Application to Viscous Flow, Reaction-Diffusion, and Solid Mechanics M. A. BIOT Royal Academy of Belgium, Brussels, Belgium

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Restructured Thermodynamics of Open Systems an Thermobanc Transfer. . . . . . . . . . . . . . . . . . . . 111. New Chemical Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . IV. Homogeneous Mixtures and Reformulation of the Gibbs-Duhem

..........................

ch to Finite Strain and Stress VI. VII. VIII. IX. X.

Thermodynamic Functions of Open Deformable Solids. . . . . . . . . . . . . The Fluence Concept. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Nature of Entropy Production and Its Evaluation The Principle of Virtual Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . General Lagrangian Equations . . . . . . . .... ...

XI. Dynamics of Viscous Fluid Mixtures with Reaction-Diffusion and Radiation Pressure. . . . . . . . . . . . . . . . . . . . . . ........ XII. Dynamics of Solids with Elastoviscous Stresses and Heat Conduction, and Thermoelasticity . . . . . . . . XIII. Inhomogeneous Viscou Conduction. . . . . . . . ............ XIV. Lagrangian Equations of Heat Transfer and Their Mechanical Interpretation, and a Mass Transfer Analogy . . . . . . . . . . . . . . . . . . . XV. Deformable Solids with Thermomolecular Diffusion and Chemical Reactions. . . ............................. XVI. Thermodynamics of Nonlinear Viscoelasticity and Plasticity with Internal Coordinates and Heredity . . . . . . . . . . . . . . . . . . . . . . . . . . XVII. Dynamics of a Fluid-Saturated Deformable Porous Solid with Heat and Mass Transfer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......... XVIII. Linear Thermodynamics near Equilibrium XIX. Linear Thermodynamics of a Solid under Initial Stress. . . . . . . . . . . . .

2 6

9 13 16 19

21

23 30 35 36 44

51

56 59

62

64 68 73

I Copyright 0 1984 by Academic Press, lnc. All rights of reproduction in any form reserved. ISBN 0-12002024-6

2

M . A . Biot

XX. Linear Thermodynamics and Dissipative Structures near Unstable Equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXI. Thermoelastic Creep Buckling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXII. Lagrangian Formulation of Bifurcations. . . . . . . . . . . . . . . . . . . . . . . XXIII. Generalized Stability Criteria for Time-Dependent Evolution Far from Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . YXIV. Creep and Folding Instability of a Layered Viscous Solid. . . . . . . . . . . XXV. Coupling of Subsystems and the Principle of Interconnection . . . . . . . . XXVI. Completeness of the Description by Generalized Coordinates. Resolution Threshold and Lagrangian Finite Element Methods . . . . . . . . . . . . . . . XXVII. Lagrangian Equations in Configuration Space. Internal Relaxation, Order-Disorder Phenomena, and Quantum Kinetics . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78 81

82 82 85

86

88 89 89

I. Introduction Our purpose here is to present a comprehensive view of a distinct approach to the thermodynamics of irreversible processes that is based on a principle of virtual dissipation. It represents a generalization of d’Alembert’s principle of classical mechanics to irreversible thermodynamics and leads to equations of evolution of thermodynamic systems. They are derived directly either as field equations or as Lagrangian equations. This work originated in the years 1954-1955, first in the context of linear phenomena, and developed gradually along with its applications during the next 25 years. During this period it became evident that the concepts and methods are interdisciplinary. They extend far beyond pure mechanics and in particular should include chemistry. It therefore became necessary to reexamine some of the foundations of classical thermodynamics. This was accomplished more recently and has led to a restructuring of the thermodynamics of open systems. The approach is based on physical “thought experiment” procedures, in contrast with current standard formalism. It provides new definitions for the energy and entropy of open systems that bypass the fundamental difficulties of the classical Gibbs approach without recourse to quantum statistics. The well-known Gibbs paradox is avoided, and the Gibbs chemical potential is replaced by a “convective potential” that does not involve undetermined constants. The Gibbs-Duhem theorem is reformulated accordingly. Results in thermochemistry introduce the concept of “intrinsic heat of reaction,” which is more representative of chemical energy than the concepts derived from standard definitions. It leads to a new expression for the affin-

New Variational-Lagrangian Irreversible Thermodynamics

3

ity and to a rigorous and complete generalization of the Kirchhoff formula. Thus the work involves two distinct developments, constituted on the one hand by a new approach to classical thermodynamics and on the other by a variational-Lagrangian formulation of irreversible processes. The first provides a sound basis for the second. The same logical sequence will be followed here, and many applications to mechanics and coupled chemical reaction-diffusion will be presented. Attention should be called to some of the more fundamental aspects of the concept of virtual dissipation and the associated Lagrangian formulation. The concept itself does not postulate any particular kinetics of the phenomena involved. This provides great flexibility in the applications, because one can introduce either a kinetics based on Onsager’s principle or a more general one represented, for example, by nonlinear chemical reactions or a non-Newtonian viscosity. The extreme generality of this viewpoint opens a wide domain that does not exclude quantum kinetics. The variational principle represents essentially a probing of the system in the vicinity of a frozen state of evolution. It is accomplished by applying virtual changes that obey constraints of mass and energy conservation and by evaluating the virtual entropy of an equivalent closed and adiabatic system that represents the virtual entropy produced. In this process the virtual work of the inertial forces is taken into account. This implies a generalization of d’Alembert’s principle of mechanics to include the thermal energy. A fundamental aspect of the principle of virtual work in mechanics seems to have been overlooked. It is not merely a formal tool but embodies a physical law, sometimes called the third law of mechanics, namely that action and reaction forces are equal and opposite. This leads to the vanishing of virtual work between frictionless or adherent surfaces and the disappearance of interfacial forces from the equations. Its generalization involves the continuity of mass and energy flux and leads to a principle of interconnection of subsystems whereby the generalized interfacial forces, mechanical as well as thermodynamic, of interacting subsystems are eliminated from global equations. The formulation of the principle was based on the introduction of a thermodynamic function that was shown by the author to be fundamental in irreversible thermodynamics. It was initially referred to as “generalized free energy.” A formally similar expression was later given the name “exergy” in the literature. We have retained this term for the more precise definition required by the present context, using the concept of “hypersystem” as a basic model in the thermodynamics of open systems. The viewpoint here is completely different from the purely formal conventional procedures that start from the differential field operators and

4

M . A . Biot

then derive corresponding variational principles. This implies a prior knowledge of the equations that govern the continuous field, and each particular case must then be treated separately with its own boundary conditions. In this context the deeper unitary physical insight is lost. By contrast, the principle of virtual dissipation introduces fundamental physical invariants that lead at the same time to field differential equations of evolution of the continuous system as well as to Lagrangian equations of collective evolution with generalized coordinates. In some problems new physical terms whose existence has been overlooked are thus obtained and are revealed essentially by the variational approach. Prior knowledge of the field differential equations is thus not required. One may analyze very complex systems made up of components of physically very different natures. Lagrangian equations are then derived for each subsystem by simplified analysis, and the subsystems are coupled together using a principle of interconnection yielding Lagrangian equations for the global system from which the interfacial coupling forces have been eliminated. This includes modal synthesis. Essentially the procedure also provides the foundation of a large uariety of finite-element methods without requiring the use or knowledge of the field equations of the continuum. The ideas embodied in this way of thinking may be considered as the natural extension to thermodynamics of those developed by Lagrange, d’Alembert, and Rayleigh in the more restricted domain of mechanics. The use of global generalized coordinates to describe the evolution of complex mechanical structures has been common practice, particularly among aeronautical engineers. A classic example is also provided by the Lagrangian equations of motion of a rigid solid immersed in a perfect incompressible fluid. The generalized coordinates are fundamentally more general than those obtained by a choice of basis in functional spaces. This is exemplified by the concept of “penetration depth” in diffusion where, starting from a linear problem, a nonlinear equation is obtained that is easier to solve than the original one. In analogy with Lagrangian mechanics, the generalized coordinates are chosen so that the mechanical constraints as well as global or local conservation of mass and energy are satisfied. This already solves part of the problem by the very choice of coordinates and gives the method a remarkable accuracy with very modest calculation requirements and rough approximations for assumed distributions of the intensive variables. One may add that the Lagrangian formulation is another procedure for achieving what has been one of the objectives of “computer algebra,” whereby complicated analytical expressions describing physical systems are manipulated by a computer and simplified by elimination of negligible and physically nonrele-

New Variational-Lagrangian Irreversible Thermodynamics

5

vant tp-ms. The Lagrangian equations directly provide such a formulation leadi J to simplified analytical results and improved physical insight. This c h a cter of simplicity is important, particularly in industrial problems, where a good grasp of physical reality as well as reliability and low cost are essential. The Lagrangian equations also lead naturally to general methods of bifurcation and related stability analyses. New stability criteria of evolution are obtained that are simpler and more general than those currently in use. Some general remarks are in order regarding the use of spatial derivatives. In continuum theories of matter, because of the discontinuous molecular structure of matter, spatial derivatives are of only conventional significance. From the viewpoint of the physicist, it is just as rigorous to treat macroscopic problems dealing with matter by considering finite elements described by a finite number of coordinates, without using field differential equations. The size of the finite element simply must not be situated below a resolution threshold where the macroscopic laws break down because of fluctuations and molecular scale effects. On the other hand, the use of continuum models to represent material physical systems constitutes an extrapolation beyond the validity of physical laws. Furthermore, they create spurious mathematical difficulties by forcing the introduction of the concept of measure and associated properties of completeness of representation that have no physical relevance. In fact, it has been shown that much of the current fashionable continuum mechanics and thermodynamics represents purely formal exercises without physical foundation. These difficulties are eliminated if we use a description based on finite elements of suitable size to represent the macroscopic physics correctly. From this viewpoint, the corresponding Lagrangian equations provide a rigorous description of the physical evolution. We should add that purely formal methods often tend to mask the more profound unitary aspect of the physical principles involved. It may even be said that it is sometimes in the context of physical applications that abstract generalizations are discovered. As an example, we may cite the extension of variational methods in terms of symbolic Heaviside operators and corresponding convolutions, which has played the role of catalyst in ulterior formal developments. The same remark can be made regarding the Lagrangian variational formulation, which has instigated much formal work on finite-element methods. The general ideas discussed stand in contrast with methods based on prior knowledge of the field equations of evolution that derive variational properties for each particular case by manipulation in the context of functional spaces. Such formal methods have their place, but their role should not be exclusive. Removed from their physical context they may lead to

6

M . A . Biot

serious errors and stand in the way of intuitive understanding of physical problems as well as their analysis by simpler and more direct methods.

11. Restructured Thermodynamics of Open Systems and the Concept of

Thermobaric Transfer A new conceptual approach to the thermodynamics of open systems has been developed (Biot, 1976a, 1977a) that avoids the difficulties inherent in standard procedures without recourse to the axioms and ponderous methods of quantum statistics. It replaces the classic formalism by an operational approach using thought experiments on a physical model called a hypersystem. This model is constituted by primary cells Z Cp (primary system), supply cells Z CSk, and a thermal well TW. The supply cells are large and rigid and contain pure substances denoted by k , all at the same pressure po and temperature T o ,whereas TW is a large rigid isothermal reservoir at a temperature To assumed for convenience to be the same as in the supply cells. For the present we consider an open primary cell Cp constituted by a fluid mixture. Its state is determined by its volume u, its temperature T , and the masses M k of each pure substance k added to it, starting from a given initial state. An infinitesimal change of state may be obtained as follows. We extract a mass dMk of pure substance adiabatically from the ~ compress and heat it gradually and reversibly to the supply cell C S and temperature T of Cp and pressure pk so that it is in equilibrium with Cp through a semipermeable membrane. We then inject it reversibly into Cp through the membrane. In addition we also inject into Cp a quantity of heat dh. This operation is called a thermobaric transfer. This type of thermobaric transfer does not involve the thermal well TW. Another type of thermobaric transfer that does involve TW will be considered later in connection with the concept of exergy. We may perform this operation for each pure substance and also increase the volume of the cell by the amount dv. These operations produce an increase of energy of the collective system Cp + Z CSk equal to

We denote by p the total pressure acting on Cp, and Ek is given by

N e w Variational-Lagrangian Irreversible Thermodynamics

7

where p i , p i , T ' , and d.Fi are, respectively, the pressure, density, temperature, and specific entropy differential of the mass dM" along the path of integration. The value of Ek includes the work of extraction of d M k from C Sand ~ injection into Cp, and T' dfl is the heat injected into d M k at every step of the thermobaric transfer. Similarly, the increase of entropy of the subsystem Cp + Cskis

c

k

We called P k the injection pressure and E k and sk, respectively, the injection enthalpy and injection entropy. From Eq. (2.2) we derive the important differential relation for each substance d E k = dpklpk

+ T dSk,

(2.4)

where P k is the density at the temperature T and pressure P k . By this thermobaric transfer process we may bring the cell Cp to any desired temperature T and increase the masses of each substance in the cell by arbitrary amounts M k . Starting from a given initial state, the variables v, Mk,and T define the state of Cp. Because the M k are the masses extracted from the supply cells, they also determine the state of the sup~ its increase of ply cells. Hence the state of the subsystem Cp + C Sand collective energy (U and entropy Y in the thermobaric transfer are determined by the same variables v , Mk,and T . As a consequence, we may adopt (U and Y as definitions of the energy and entropy of the primary cell Cp, keeping in mind that they refer to collective concepts. To determine their values, we write the heat added d h in the form d h = hkT d v

+

c hiT d M k + k

Cur,d T .

(2.5)

We call hiT the heat of mixing at constant temperature and volume; whereas Cumis the heat capacity of Cp at constant volume and composition. The differential coefficients hkT and hiT may be evaluated without calorimetric measurements by generalizing a procedure leading to the classic Maxwell relations (Biot, 1982a). The result is obtained by substituting the value of d h from Eq. ( 2 . 5 ) into Eqs. (2.1) and (2.3) and noting that dQ and d Y are exact differentials. We obtain the relations

(2.7)

8

M . A. Biot

Elimination of dC,,ldu

between these equations yields

The subscript urn indicates that the derivative is for u and M k constant. Similarly, we obtain the relations (2.9) (2.10)

Elimination of dCUrn/dMk between these equations yields (2.11)

Taking into account the differential relation (2.4), we derive (2.12)

The derivatives in Eqs. (2.8) and (2.12) are obtained from relations (2.13) (2.14)

which may be considered as generalized equations of state derived experimentally or from kinetic theories. For perfect gas mixtures the injection pressure is (2.15)

Pk = Pyk,

where Yk is the molar fraction of substance k in the mixture. With the value of d h from Eq. (2.5), the differential coefficients of d% and dY are now known functions of u , M k , and T . We may then integrate Eqs. (2.1) and (2.3) along any path, putting Q = Y = 0 for the initial state. We thus obtain % = %(u, M k , T ) ,

Y

=

Y ( u ,M k , T )

(2.16)

for the collective energy and entropy of the open cell Cpin terms of u , M k , and T . By eliminating dh between Eqs. (2.1) and (2.3), we obtain (2.17)

New Variational-Lagrangian Irreversible Thermodynamics

9

where c$k

=

(2.18)

i5k - T.Tk

is the convective potential (Biot, 1976a, 1977a). Relation (2.17) is analogous to the Gibbs equation with cpk replacing the chemical potential. In contrast with the classic procedure, Eq. (2.17) is not used to define & but constitutes a theorem. In addition, &, Ek, rk do not involve undetermined constants as in the classic case. Note that these undetermined constants are not eliminated by taking differentials, because we then obtain d$k

=

dZk - T d.Fk

-

Sk

dT,

(2.19)

where, if we follow standard procedures, an undetermined constant still remains for nonisothermal transformations in the coefficient Lfk of dT. This difficulty was already recognized by Gibbs himself (Gibbs, 1906) as well as by others (see, e.g., Hatsopoulos and Keenan, 1965). Gibbs’s paradox is also eliminated, as shown by Eq. (2.3), because for identical substances the injection pressures Pk and hence also the injection entropies Fk are the same, with the result that in Eq. (2.3) entropies become additive. It is important to note that, when two components become identical, Eq. (2.15) for the injection pressure in terms of the molar fraction loses its validity. This is consistent with the physical fact that in that case a semipermeable membrane loses its ability to distinguish between components.

111. New Chemical Thermodynamics The concept of thermobaric transfer has provided a new approach to chemical thermodynamics (Biot, 1976a, 1977a). The first step is to introduce the new concept of intrinsic heat of reaction defined as follows. A chemical reaction is measured by a reaction coordinate 6 such that the masses of pure substances dmk produced by a given chemical reaction d [ are given by dmk =

uk

dt,

(3.1)

where Yk are constants characteristic of the reaction that satisfy the condition & Vk = 0 of mass conservation. The coefficients are algebraic, so that the masses produced may be positive or negative. Consider a reaction d6 occurring in a rigid cell in such a way that the temperature is maintained constant while the products of the reaction are extracted reversibly through semipermeable membranes. Because the temperature and com-

10

M . A . Biot

position of the cell remain unchanged, its state and hence its pressure and volume do not vary. The constancy of the temperature is obtained by injecting into the cell a quantity of heat hTd t . This defines the intrinsic heat of reaction hT. How this is related to standard definitions can be shown by reinjecting into the cell the products of the reaction either at constant volume or at constant pressure, with T constant. We write (3.2a) (3.2b) where huTand hpTare the standard heats of reaction at constant temperature and, respectively, at constant volume and pressure. The heat of mixing htT at constant T and u is given by Eq. (2.12) and h:T denotes the heat of mixing at constant temperature and pressure. By a method similar to the previous derivation of hiT, we showed (Biot, 1982a) that the value Of h:T iS (3.3) The derivative dlaT is for constant pressure and composition. For a perfect gas mixture, substitution of the value forpk from (2.15) yields hiT = 0. Hence in this case hiT = 0, i.e., the standard and intrinsic heats of reaction coincide. Note that the intrinsic heat of reaction is more representative of chemical properties than standard concepts, because it does not involve the heat of mixing or external work. For this reason its value is obtained by a very general formula relating the two heats of reaction for two different states. Consider a rigid cell in state 1 with values pi’), T I and another in state 2 with values p f ’ , T 2 .A reaction d t occurs in cell 1 and - d t in cell 2, while products of reaction are extracted from I and injected into 2 by thermobaric transfer. The temperatures of the cells are kept constant by injecting amounts of heat h:) d t and -h‘,2’ d t , respectively, into each cell. Conservation of energy implies the relation (Biot, 1977a)generalizing Kirchhoff’s formula (3.4) Hence if we know the intrinsic heat of reaction for a single state, we may derive its value (as well as the standard values huTand hpT)for any other state of arbitrary composition and temperature.

New Variational-Lagrangian Irreversible Thermodynamics

11

In the preceding section we showed how to evaluate the energy and entropy of an open cell by thermobaric transfer. The procedure may be extended to the case where a chemical reaction also occurs in the cell. This is accomplished by adding to the hypersystem a chemical supply cell c c h in which the reaction takes place and is in equilibrium at the temperature Teq,with injection pressures Pkeq and an intrinsic heat of reaction h"Tg. We consider a reaction d t to occur in C c h while the products ukdt are extracted and injected into the primary cell Cp by thermobaric transfer. The temperature Teqof C c h is also kept constant by injecting the heat h$ d t . In this process the state of c c h does not vary. At the same time we proceed as in the previous section by injecting masses dMk from CSk into Cp, adding an amount of heat dh, and changing its volume by dv. The change of state of the system through the reversible process of thermobaric transfer can be made to be the same as i f a reaction d t had occurred in the primary cell in addition to the other changes. The increase of energy and entropy of the subsystem C c h + CCSk + Cp is

+

Ek dMk k

+ dh,

(3.5)

with

These results are obtained by adding to Eqs. (2.1), (2.3), and (2.5) the terms due to the masses vk d( transferred and the heat hFq d t injected into C c h . We have also taken into account the fact that C c h is in equilibrium, so no change of entropy occurs in the system due to the chemical reaction in cch

The differentials (3.5) and (3.6) may be integrated from a given initial state, yielding values % = %(u,

5, M k , T ) ,

Y

=

Y ( v , 6, M k , T ) ,

(3.8)

functions of u , 5, M k ,and T , considered as state variables of the primary cell Cp. The values in Eq. (3.8) define the energy and entropy of Cpin the presence of a chemical reaction. Note that, starting from a given initial

M . A . Biot

12

state 8 = 0, M k = 0, the increase of masses of the various substances in the cell is

Mk. (3.9) However, the energy and entropy of the cell as defined by Eqs. (3.5) and (3.6) will vary euen ifmk remains constant. This is due to the collective definition of % and Y ,which involves the subsystem C,,, + X C S +~ Cp. On the other hand, in the equations of state, mk = vk8

mk, T ) ,

(3.10)

Pk = P k ( U , mk, T )

(3.11)

P

= P(U,

are functions of only u , mk, and T . Elimination of dh between Eqs. (3.5) and (3.6) yields d%= -pdu - A d [

+ Ck +kdMk + T d Y ,

(3.12)

where

Equation (3.12) generalizes Eq. (2.17) in the presence of a chemical reaction. When a reaction occurs adiabatically in a rigid closed cell we put d% = du = dMk = 0, and Eq. (3.12) yields A d t = T dY.

(3.14)

This corresponds to De Donder's formula (1936) and shows that A is the affinity whereas dY is the entropy produced by the reaction and defined here precisely in a new way as a collective concept. We may write the affinity in a way formally identical to a standard expression by putting (3.15) (3.16) The integrals at the lower limit are evaluated by extrapolating gaseous classic properties of E k and Fk to absolute zero, with Ek(0) and Sk(0)constants assumed to be characteristic of the substance and independent of any particular chemical reactions. We may consider these assumptions as basic axioms. Substitution of the values from Eqs. (3.15) and (3.16) into Eq. (3.13) yields

New Variationul-Lugrangian Irreversible Thermodynamics

13 (3.17)

where PA =

dEi

-

T

dfi

+ E/,(O)

-

Ts/,(O).

(3.18)

The constants ~ ~ (and 0 ) S X ( 0 ) are determined experimentally from chemical reactions. This provides a novel definition of the chemical potentials pk*

The values of ~ ( 0 and ) sk(0) may also be defined and evaluated by introducing the axioms of quantum statistics and physical properties of matter in the cryogenic range. However, in practice, in most cases this leads to many difficulties, as shown by Fowler and Guggenheim (1952). According to expression (3.13) and the value of PI from Eq. (3.1 I ) , the affinity A

=

A(u, mA, T )

(3.19)

is a function of u, m k , and T. With the value of mk from Eq. (3.9) it also becomes a function of u, 5, M A ,and T : A

=

A ( v , 5, M k , T).

(3.20)

We considered a single chemical reaction, but the results are readily generalized to multiple simultaneous reactions by adding a set of chemical supply cells Cchp,one for each reaction p . The masses produced by the reactions are then (3.21) With an affinity A, for each reaction, Eq. (3.12) becomes d~=--pdv-CA,d5,+C~kdMkT +dY. P

(3.22)

k

IV. Homogeneous Mixtures and Reformulation of the Gibbs-Duhem Theorem The new restructured thermodynamics by thermobaric transfer provides a novel evaluation of the thermodynamic functions of a cell containing a homogeneous mixture of fluids. To evaluate the thermodynamic functions for this case we apply Eqs. (3.5) and (3.6) by writing them in the form

14

M . A . Biot

where we have put

(4.4)

and dmk = V k d( + d M k . For simplicity we assume a single reaction, but the results may be readily generalized to multiple reactions. We start with a cell of zero volume. Its volume v is then increased gradually while maintaining constant the pressure p , the temperature T, and the concentrations of the various substances. Hence, PkEk and sk also remain constant. The heat that must be injected is

dh =

Ck h$

dmk,

(4.5)

where hiT is the heat of mixing at constant pressure and temperature. Substitution of the value of dh from Eq. (4.5) into Eqs. (4.1) and (4.2) yields

d% = - p dv dY

=

Y e ,d [

+ Qeq +

dt

+

%k k

dmk,

c Yk dmk, k

(4.6) (4.7)

with %k

= Ek

+ hET,

yk

=

sk + h;T/T.

(4.8)

During the transformation considered here, the value of hiT remains constant, hence also the values of %k and Ypk as well as p and Qeq. Thus integration of Eqs. (4.6) and (4.7) yields % = -PV

+ %eq[ + Ck Qkmk,

9 = Yeqt +

Ykmk.

(4.9) (4.10)

k

These values are the energy and entropy of a homogeneous cell containing masses mk = v k t + M kof each substance.

New Variational-Lagrangian Irreversible Thermodynamics For a pure gas mixture h:T we obtain

=

15

0, and without chemical reactions (5

ou + pv = 2 Ekmk, k

Y

=

c skmk,

=

0)

(4.11)

k

showing that in this case the enthalpy of the mixture is the sum of the enthalpies of the individual components, and likewise for the entropy (Biot, 1982a). We note again that for identical components the injection pressures Pk are the same, hence also the values of sk, so that Gibbs’s paradox is avoided. T o reformulate the Gibbs-Duhem theorem in the present context, we differentiate Eqs. (4.9) and (4.10). Taking into account the values of these differentials from Eqs. (4.6) and (4.7), we derive (4.12)

Hence we may also write

(4.14) Using Eqs. (2.18) and (4.8) we also write and Eq. (4.14) becomes

v dp

=

2 mk d4k + Y c dT,

(4.16)

k

where (4.17) We call Y c the convective entropy (Biot, 1982a),because it represents the value of the entropy obtained by pure convection of the masses mk into the cell without chemical reaction. Equation (4.16) constitutes a reformulation of the Gibbs-Duhem theorem (Biot, 1982a). It is expressed here in terms of & and the convective

M . A . Biot

16

entropy Y c instead of the chemical potentials p.k and the entropy used in the classical form. The present formulation avoids the basic d$jculties of the classical treatment, which involves undetermined constants in &, and in the entropy. Expressions (4.9) and (4.10) are readily generalized for multiple reactions as

where k

k

The superscript p refers to values for the particular reaction p .

V. Nontensorial Virtual Work Approach to Finite Strain and Stress The irreversible thermodynamics developed here is based on a principle of virtual dissipation. It constitutes a generalization of the traditional method of virtual work in classical mechanics. These methods in the field of solid mechanics have been used extensively by the author since 1934 (see Biot, 1965b); more recently, it was also emphasized by Washizu (1975). Because of the currently fashionable trends, the power and simplicity of this method have generally been overlooked. The method provides an approach to finite strain that is more general than that derived from tensor definitions and contains the latter as a particular case. At the same time, it is flexible and ideally suited to describe physical properties of anisotropic media such as laminated and fiber composites of technological importance. We start from a homogeneous deformation defined by the linear transformation of the initial coordinates xi of the material points to Ti =

(aij + aij).xj,

(5.1)

where 6" is the unit matrix and aij are nine coefficients. Summation is implied for repeated tensor indices. We may consider the transformation to be equivalent to the following process (Biot, 1965b). First, a pure solid rotation is described by three degrees of freedom and written as

New Variational-Lagrangian Irreversible Thermodynamics x! = R..x. U J'

17

(5.2)

It is then followed by another transformation, xi =

(80

+

(5.3)

&ij)Xi',

where the coefficients are not independent but satisfy three constraints, so that they contain only six degrees of freedom. For example, we may adopt the three constraints &..

U

= &..

(5.4)

JI

and call E~ the six finite strain components defining the deformation (5.3). In this case, the transformation (5.3) is such that principal directions of strain remain invariant (Biot, 1965b). Other choices may be made. For example, the transformation (5.3) may be chosen so that the material on the x i axis remains on that axis while the 1 E31 = 8 3 2 material in the x i x i plane remains in that plane. In this case, ~ 2 = = 0, and the six remaining coefficients define the strain components. Other definitions may be chosen suitable to the particular physical properties of the material (Biot, 1973a). For the method to be complete we must be able to express the six strain components E~ in terms of the nine coefficients aO.We have shown that this can easily be done to any order by a systematic procedure (Biot, 1965b). For example, to the second order, for the choice (5.4), we have

where

In two dimensions, for a choice of strain components such that E~~ = 0, the three strain components to the second order are (Biot, 1973b) 1

~ 2 = 2 e22

-

- 2 a21(2a12+ ~ 2 1 1 ,

all).

(5.7)

(5.8)

The second-order approximation is sufficient in a vast group of problems. We may also choose Green's definition of finite strain, E.. V

=

e.. + -21a W.aW.' U

(5.9)

M . A . Biot

18

This has the advantage of providing the exact functional dependence in terms of aij. However, in practice this leads to difficulties because the strain components are nonlinearly related to extension ratios and introduce spurious purely geometric nonlinearities in the physical description. The virtual work method readily provides the definition of six stress components 7" associated with the strain. A cubic element of unit size oriented along the coordinate axes is subject to the transformation (5.3) by forces applied on the six faces. A virtual deformation 68, produces a virtual work 6W

(5.10)

= TijSEQ

with six coefficients rij defining stress components conjugate to E ~ The . quantity 6W is a physical invariant, but the factors rijand 6eijneed not be tensors. The indices i a n d j indicate a summation to all six independent terms whatever their nature. If E~ is expressed as a function of a,, we may write 6W

= rPyde,,lda,

6aij = To 6a,,

(5.11)

where

(5.12)

To = rpVaE,/aa,

are the nine components of the Piola tensor. We may also consider the stress cr, per unit area after deformation, referred to the initial axes x i , A virtual transformation after deformation may be written in terms of final coordinates Xi as 6Xi

= Xj

aa,,

(5.13)

where 6Zij are nine suitable coefficients to be determined. The unit cube after deformation has become a parallelepiped, and the virtual work of the forces 'T, in the virtual transformation (5.13) is 6 W = J0-U 6a,,

(5.14)

where J is the Jacobian of the transformation of xi to X i . To derive 6 8 g , consider the virtual transformation in terms of initial coordinates by substituting into Eq. (5.13) the value of Xj from Eq. (5.1). We obtain 6Xi =

CjkXk

6ii0,

cjk

=

6jk

+

ajk

;

(5.15)

and from Eq. (5.1), 6Xi = x k 6aik.

(5.16)

Since Eqs. (5.15) and (5.16) are identical transformations, we derive

New Variational-Lagrangian Irreversible Thermodynamics 6a.rk = c. Jk

aa.. r~

19

(5.17)

*

With this value, Eq. (5.11) becomes

6w

=

TikCjk 6ajj.

(5.18)

Comparing Eqs. (5.14) and (5.18) yields ~

i

=j

(I/J)Tikcjk.

(5.19)

Since the virtual work (5.14) must vanish for a pure rotation 6aij= -6Zjj, the tensor is symmetric and satisfies the three relations (T.. U

=

(5.20)

(T.. JI

which also expresses the conditions that the moments due to the nine stress components To vanish. Substitution in Eq. (5.19) of the values (5.12) of To yields =

(1 /J)7,,Cjk ar,,/aaik

.

(5.21)

This expression is valid for all definitions of 7,,, and the summation is extended to all six components of the stress 7pyas defined above (Biot, 1981).

For a continuous deformation field, we write Xi = xi + uj(xJ, where the displacement ui is a function of the initial coordinates xi. The local infinitesimal transformation dxi =

(so + aldi/axj) d-yj

(5.22)

is the same as Eq. (5.1) provided we write auj/axj = aij.

(5.23)

With aii representing the nine displacement gradients, the foregoing definitions of stress and strain remain valid as local values for a nonhomogeneous deformation.

VI. Thermodynamic Functions of Open Deformable Solids The collective energy and entropy per unit initial volume of a deformable solid based on virtual work and nontensorial concepts were derived earlier (Biot, 1981). The results may be readily extended to an open solid with chemical reactions by proceeding as in Section 111. We consider initially a cube of unit size. A homogeneous deformation is defined by the

M . A . Biot

20

six strain components eii under the corresponding stresses rii. The solid contains substances in solution. The increment of mass of each substance is

drnk

= Uk

d.$ + d M k ,

(6.1)

where V k d.$ is due to a chemical reaction and dMk is the mass acquired from an external source. For simplicity we assume a single reaction, because results are readily generalized for multiple reactions. An infinitesimal change is defined by d q , d l , dMk and the temperature change dT. The corresponding changes in energy and entropy are obtained by thermobaric transfer and their values are derived from Eqs. (3.5) and (3.6) by ~ yields . replacing -p du by rij8 ~This

The value from Eq. (3.7) must also be changed to

dh = hiT d q

+

c htT k

(uk

d.$ + d M k ) + C,, dT.

(6.4)

The terms h i Tdeijrepresent the heat that must be injected when applying a deformation deijunder the constraint drnk = dT = 0. Similarly, htTis the heat of mixing, where htT dMk is the heat injected when adding a mass dMk under the constraint d q = dT = 0. The heat capacity C,, is for dEij= dmk = 0. We may show that h i T and h$ may be determined without calorimetric measurements when we know the equations of state

i.e., when we know the stress ru and the injection pressures Pk as functions of the strain E " , the masses r n k of each substance added by chemical reaction and transport, and the temperature T. From the conditions that Eqs. (6.2) and (6.3) are exact differentials, proceeding exactly as before in deriving Eqs. (2.8) and (2.12), we obtain

where the subscripts indicate that the partial derivatives are for eijand mk constant.

New Variational-Lagrangian Irreversible Thermodynamics

2I

The coefficients of the differentials in expressions (6.2) and (6.3) of d% and dY are now known as functions of EU,, mk = V k t + M k , and T. Integration along any arbitrary path with initial conditions % = Y = 0 yields the values

6 , M k , TI, Y = Y ( ~ i j6,, M k , T ) as functions of the state variables e i j , 6, M k , and T. %=

%(~ij,

(6.9) (6.10)

Replacing - p du by rijdeij, Eq. (3.22) is generalized to

d%

= rij

dEij -

2 A , d t , + 2 4k d M k + T dY. P

(6.11)

P

WI. The Fluence Concept In a fixed coordinate system x i , a rate of mass flow or mass flux of a . component given substance is represented by a Cartesian vector & f fThe is the mass flux across a unit area normal to x i : conservation of mass is expressed by

nik

=

-a&f:tax,.

(7.1)

The total mass entering a domain R is then (7.2) where n, is the unit normal to the boundary A. Since M: represents a time derivative, we may integrate the relations with respect to the time with zero initial values. Hence ML

=

-aMf/ax,,

(7.3)

where M: is called the massfluence. It constitutes a Cartesian vector. In deformable systems, a more general definition of the fluence has been found extremely useful (Biot, 1977b). We consider an area initially equal to unity and normal to the x, axis. This area is then transported and deformed by the coordinate transformation 3, = X,(x,, t ) .

(7.4)

We now call M: the mass flux across this transformed area. Obviously the mass flowing into a transformed element initially of unit volume is given

22

M. A. Biot

by (7.1), where M frepresents the more general mass flux. Hence Eqs. (7.1)-(7.3) retain exactly the same form with this generalized definition of the mass flux M fand mass fluence M:. In this case, however, &f,"is not a Cartesian vector. We denote by M : h the corresponding Cartesian components, i.e., &f:" represents the mass flux per unit area in the space XI.An initial surface A is transformed into A ' by transformation (7.4). The mass flux through it may be expressed in two different forms as M f n ,dA =

/A,

M,"C,!ln:dA',

(7.5)

where the second integral is expressed in terms of the coordinates Fi and

C; is the cofactor of dxj/dFjin the Jacobian J'

=

detIaxi/aZjI.

(7.6)

Obviously, in the second integral of Eq. (7.5) the integrand A?:C;i is the Cartesian component M l k of the mass flux in the space X i . Hence

Mi" = 11;16C!.* J

(7.7)

JI

This relation may be given another useful form by considering the linear transformation of dXi into dxi and solving for dXi. We find

azj/aTj = cyj/Ji.

(7.8)

nilh = J I M ; axi/axj.

(7.9)

Hence

Multiplying this equation by dxjld.fj, we obtain (Biot, 1977b)

M,;= J M ( axj/axi, ~

(7.10)

where J

=

IIJ'

=

det(aX;/a.q(.

(7. I I )

These results show the mass flux, as defined above for a deformable coordinate system, to be a relative contravariant vector (Sokolnikoff, 1951, see pp. 58, 72). We shall refer to A?," and M f , respectively, as the contravariant mass flux and the contravariant mass Jluence. Similarly, consider the local Cartesian heat flux at X i and a contruvariunt heat flux hi defined as the rate of heat flow across a transformed surface initially equal to unity and initially normal to the xi axis. They satisfy the same relations as Eqs. (7.9) and (7.10), namely HI

=

J ' H , axitaxi,

H,j = J H ;

aTj/axi.

(7.12)

New Variational-Lagrangian Irreversible Thermodynamics

23

The time integral of f i i is the contravariant heatfluence H i . We may write a conservation condition analogous to Eq. (7.3) as h = -aHi/dxi,

(7.13)

where h is the heat acquired by a deformed element of unit initial volume. The entropy flux across an area is defined as (Biot, 1977b) (7.14) Its time integral is the entropy fluence Si.They are either Cartesian or contravariant components. The fluence fields M fare state variables, because they determine M k by Eq. (7.3). However, the fluence fields H i and Siare not state variables. Other fluence fields that are state variables may be introduced by putting

Y=

-as:/axi,

% = -a9:/axi,

(7.15)

where S: is called a pseudo-entropy fluence and %: a pseudo-energy fluence. Using such concepts, the state of a system may be partially described by fluence vectors as further clarified below in several applications.

VIII. The Nature of Entropy Production and Its Evaluation In order to formulate the principle of virtual dissipation in the next section, we must evaluate the entropy produced by an irreversible process. Entropy production is due to various phenomena, in particular thermomolecular diffusion, viscosity, and chemical reactions. We shall first consider the entropy produced by diffusion, using a derivation that brings out its physical significance (Biot, 1982b). Consider a medium that may or may not be deformable. The entropy increase per unit initial volume as given by Eq. (2.3) is written in variational form

where 6h is an amount of heat added reversibly, so as to produce the same change of state as the actual irreversible process. This amount of heat may be written

M . A. Biot

24

where -a 6H;lax;is the amount of heat added by the actual irreversible heat flow. We assume that this amount is added reversibly along with an additional heat ahP so that the total generates the actual irreversible change ofstate. The heat 6hPdefines the heat produced by the irreversibility and corresponds to Clausius’ uncompensated heat. From Eq. (7.3) we also write 6 M k = -a 6MFldx;.

(8.3)

In these equations, H; and M t are fluence concepts either Cartesian or contravariant, as defined in Section VII, and are valid in a deformable system in terms of initial coordinates xi. We substitute the values (8.2) and (8.3) into Eq. (8.1). The result may be written in the form

where

is the variation of entropy fluence as defined by Eq. (7.14), and

Equation (8.4), when integrated over a volume R with n; 6H; = n; 6Mi = ni 6s; = 0 at the boundary, yields

is the virtual entropy produced per unit initial volume. It has Hence asNTM three terms. The first is due to the masses convected 6MF, the second is due to the heat 6hPproduced by the irreversibility, and the third is due to the heat flow 6H; across a temperature gradient. It remains to evaluate the heat produced ahP.This is obtained by considering energy conservation. Using the value (8.2) of ah, the variation of energy (2.1) per unit initial volume with 8u = 0 is 8% =

a Ck Ek 6 M k + 6hP - 6H;. dXi

Substitution of Eq. (8.3) for 6 M k and integration over a domain R yields

New Variational-Lagrangian Irreversible Thermodynamics

25

6% d R =

+ 23axi6 M ; + ahP] d R .

(8.9)

k

On the other hand, conservation of energy is expressed by

+ C(%, - a ; ) 6M1] dil. I

(8.10)

This expression is based on the significance of (& E k 6Mf + 6H,)n, as representing the variation of diffusive energy flowing across a surface, and while 9Al is the body force per unit mass and a! the acceleration of each substance. Note that Eq. (8.10) is actually based on a generalized d'Alembert principle, where -a," plays the role of a body force per unit mass. Comparing the values (8.9) and (8.lo), which are valid for an arbitrary domain 0, we derive for the heat produced 6hP = E ( B I- a;

- dEk/ax,)

6Mf.

(8.11)

h

This provides a physical interpretation of 6hPas the heat produced by the irreversible mechanical work. When it is substituted in the variational relation (8.6), taking into account relation (2.4), we obtain T aseTM=

2 X:

6M:

+ X,'

6H,,

(8.12)

k

where (8.13)

are dissipative disequilibrium forces. They are functions of the mass and the heat flux fi;. We write fluxes k/( Xi"

=

%i"(k!, J fi.) J '

X#T =

a:(&!,hi).

(8.14)

It should be understood that these relations also depend on the local state variables, although this is not formulated explicitly. They embody the irreversible kinetics of the system and may be obtained either experimentally or theoretically from molecular kinetics. With the values (8.14),

M . A . Biot

26

equations (8.12) may be written (8.15)

si

In many problems it is convenient to use the entropy flux instead of the heat flux H i . Solving Eq. (7.14) for H i and Eq. (8.5) for 6 H i , and substituting these values into Eq. (8.13, we obtain T

=

2

6M:

k

+

6Si,

(8.16)

where

s.) a! - Tf

afk(j$f!, J J =

k

a‘r i r

%,?T(M/,S j ) = TaT.

(8.17) (8: 18)

The present derivation of entropy production is fully general and does not assume linearity or any dependence on Onsager’s (1930, 1931) principle. For example, we may include nonlinear diffusion of a non-Newtonian fluid through a porous medium. on the fluxes and in the In the case of a linear dependence of absence of a temperature gradient we obtain from (8.14) the linear relations

%T(n;i/, H j ) = 0,

(8.19)

where hiis the coupled heat flux due solely to the mass flux. If we solve constitute I,” what is generally these equations for hi,the coefficients of & called the heat of transport. Consider now the entropy produced in a primary cell by a chemical reaction. This is the entropy increase of a cell for du = dMk = d% = 0. In this case Eq. (3.22) is applicable and is written in variational form as (8.20)

where is the entropy produced by the reactions the affinities are functions of u , mk, T , A,

=

A,(u, mk, TI,

atp.The values of (8.21)

c,

where mk = V k p t p + M k is the total mass of each substance added to the cell by reaction and transport. An expression similar to Eq. (8.14) is obtained by introducing the kinetics of the various reactions. The rates of reaction 1, are functions of u , m k , and T ,

4,

=fp(u,

mkr T ) .

(8.22)

New Variational-Lagrangian Irreversible Thermodynamics

27

Using Eqs. (8.21) and (8.22) it is easily shown that A, is of the form A,

=

aP(tp, v , mh, T ) .

(8.23)

The affinities A, are similar to the disequilibrium forces Xi"and Xi'and in analogy with (8.14); they are expressed by a, in terms of the irreversible kinetics. Equation (8.20) becomes (8.24) P

The total entropy production 6s* due to simultaneous diffusion and chemical reactions, obtained by adding Eqs. (8.16) and (8.24), is given by

This expression is applicable to a deformable cell with fluence concepts for M! and S;. It yields the virtual entropy production per unit initial volume. For an actual irreversible transformation we replace the variations by time derivatives and Eq. (8.25) becomes (8.26)

This expression represents the rate of dissipation per unit initial volume and is positive. Note that Eq. (8.26) is a consequence of relation ( 8 . 2 3 , which is more general and concerns a virtual change, whereas the actuul change in Eq. (8.26) is a particular case. When in addition to being linear the relations between the rates and the dissipative forces satisfy Onsager's reciprocity relations (1930, 193 I ) , they may be written

a,= a 9 b ' h / a [ , , where

=

agTM/aMf,

%.ST

=

a9TM/aSj, (8.27)

aCh is a quadratic form in t pand BTM a quadratic form in A#

and

S;, with coefficients functions of the local state.

Applying Euler's theorem, we write Eq. (8.26) as

(8.28)

where 9 = gCh + BTMis a combined dissipation function for chemical reactions and thermomolecular diffusion. Other types of entropy production, such as that due to viscous effects, will be discussed below in the applications.

M . A. Biot

28

A fundamental entropy balance equation may be expressed in terms of entropy production by adding 6iSchin Eq. (8.4). The total entropy variation is then

6Y =

-

a

+ as*,

- 6s; axj

6s* = 6s*TM

+ 6s*Ch.

(8.29)

We replace the variations by time derivatives and write

9 = -aS,/ax, + S * .

(8.30)

If we integrate this equation with respect to time, assuming zero initial values, we obtain the entropy balance equation

Y

=

-asi/axi

+ s*.

(8.31)

It should be understood that in problems where we choose a nonzero initial Y , we must simply subtract this value of Y in Eq. (8.31). The entropy production has been expressed here in terms of 8, ,Mf, and S;. While 8, and M f are state variables, the entropy fluence S; is not. In order to avoid this difficulty, it was shown (Biot, 1981 , 1982b) that we may introduce the pseudofluence concepts (7.15) by proceeding as follows. We note that because Eq. (8.10) is valid for an arbitrary domain, we have (8.32)

Using Eqs. (8.13) and (8.14) and the definition (7.15) of 9?, this yields - -a 6 f3.;T + -- - -

ax;

a ax;

(2 Ek GMf+ 6 H ; ) +

6Mf

(8.33)

with (8.34)

From Eqs. (8.14), we may obtain H ; in terms of XTand M:.Substituting these values of H iinto 3;determines the latter in terms of local state variables, their gradients and k;. Thus in the quantity

=hA(nij")

(8.35)

the only fluence fields are the state variables M!. With g ; ( x j , x:) representing a fluence field due to a unit concentrated source at xi+, Eq. (8.33) may be written

N e w Variational-Lagrangian Irreversible Thermodynamics

29

where the integral is extended to the space R+ of x:. Considering the value (8.5) of SS, and (2.18) of 4 k , this may be written T SS,

=

8%; -

c 4~SM: +

\fl+

I

g , ( x / ,x,?)

Ah SM:

d a + . (8.37)

h

With time derivatives, this becomes TS, = $: -

2 4~h,"+

g , ( x ~x,?) ,

\cl+

I

cJ;'hn;r," dR+.

(8.38)

h

Thus SS, and S, are now expressed in terms of the fluence vectors 9;and M:, which are state variables. An important simplification occurs in problems where the effects of inertia and body forces are negligible, and more generally ifxk is negligible. In that case, withx' = 0, Eq. (8.33) yields (8.39)

where

or

Under these conditions 9: coincides with the actual physical energy flux 9idue to molecular and heat diffusion. Relations (8.37) and (8.38) become (8.42) TS; =

$i

-

c +AM;.

(8.43)

I

In many problems where Jk remains small these simplified results are applicable. Another possibility is to introduce the pseudo-entropy fluence S: defined by Eq. (7.15). Equation (8.29) becomes - -a 6s: axi

Hence

a

= - - 6Si

axi

+ as*.

(8.44)

M . A . Biot

30 6Si = 6s:

+

g i ( x / ,$1

6s* d R +

(8.45)

dR+.

(8.46)

and

Si = $7

+ jn+ gi(x,, x:)S*

In many problems we may neglect the entropy s * produced ~ ~ by thermomolecular diffusion. In that case, we write Eqs. (8.45) and (8.46) as (8.47) (8.48)

Thus 6s;and 3; are now expressed in terms of state variables S7 and 5, . Another important simplification also applies for problems where the entropy produced does not contribute significantly to the total value of the entropy 9.In that case we may write

Y

=

-asi/axi,

(8.49)

and Siitself becomes a state variable. This is the case for quasi-reuersible processes and in linear thermodynamics.

IX. The Principle of Virtual Dissipation Consider again a hypersystem constituted by a collection of primary cells (a primary system), a collection of supply cells including chemical cells, and a thermal well. In the process of thermobaric transfer as defined above, the thermal well did not play any role, because the source of the heat injected in the primary cells and the transferred masses is not specified. We now introduce a modified thermobaric transfer, in which the heat injected is provided by reversible heat pumps extracting heat from the thermal well. The mechanical work necessary to inject an amount of heat dh into an element at the temperature T is dw = BIT dh,

(9.1)

B=T-To

(9.2)

where and To is the thermal well temperature. A thermobaric reversible transformation of the hypersystem is now accomplished entirely through mechan-

New Variational-Lagrangian Irreversible Thermodynamics

31

ical work performed on the system, without any exchange of matter or heat with the environment. For such a transformation, the increase of energy of the hypersystem is equal to the mechanical work dW performed on the system. This may be written dU

-

To dS = dW,

(9.3)

where dU is the energy of the primary system defined above as a collective concept that includes the energy of the supply cells. The term -To 3 is the energy acquired by the thermal well; its increase of entropy -dS, because of reversibility, must be equal and opposite in sign to the entropy increase dS of the primary system. We remember that dS is defined as a collective concept that includes the supply cells. We may write Eq. (9.3) as dV = d W ,

(9.4)

V = U - ToS

(9.5)

where is a thermodynamic function shown by the author (Biot, 1954, 1955) to be the key concept that plays a fundamental role in irreversible thermodynamics. It was initially referred to as the “generalized free energy” because it coincides with the Helmholtz definition for the particular case of isothermal transformations. The term “exergy” was later introduced by others to designate a formally similar expression without defining more precisely the significance of U and S in the general case of open systems in terms of collective concepts, as we have done here. However, we shall keep the name exergy for the more general concept represented by Eq. (9.5). Further physical insight is provided by considering the exergy of a single primary cell: T- = 6 u

-

TOY.

(9.6)

In order to simplify the writing, and without loss of generality, we may consider a rigid cell without chemical reactions. Using the values (2.1) and (2.3), with dv = 0, we may write the differential of the exergy as (Biot, 1976a, 1977a) dSr

=

$h

dMh + BIT dh,

(9.7)

h

where $h

= Fh -

TOFA,

(9.8) (9.9)

M . A . Biot

32

In a reversible transformation, dSr is the increase of energy of the hypersystem, and (O/T) dh is the work accomplished by the heat pump in order to inject a quantity of heat dh into the primary cell. Hence the remaining term $k dMk is the work required to bring the mass to equilibrium with the primary cell and inject it into the cell. We have called $k the thermobaric potential. In terms of the convective potential (2.18), we write =

$k

&

+ OSL.

(9.10)

Note that $k is dejined purely in terms of mechanical work, which is not the case for c$k, which involves the concept of entropy. Until now we have been dealing with reversible transformations. However, the value of V is determined by the state variables, whether they follow a reversible or irreversible evolution. We therefore consider variations of the state variables for an irreversible change of state. These variations are arbitrary except for one condition. They must satisfy the constraint that the flow of heat and matter between cells is continuous. For a continuous system considered as a collection of infinitesimal cells, this means that the heat and mass fluence field must satisfy the constraint of continuity. At the boundary of the primary system, heat and matter may be injected into it by a modified thermobaric transfer within the hypersystem. Hence the variation of the hypersystem occurs without exchange of matter and heat with the environment. As a consequence, conservation of energy for the hypersystem is expressed by

6u -I- To 6 S ~ w= 6w,

(9.11)

where asTw is the variation of entropy of the thermal well and To 6STwits variation of energy. This relation differs from Eq. (9.3) in that the transformation is irreversible and we may not put 6STw = -6s. With the value (9.5) of V we may write Eq. (9.1 1) as 6V

+ T 6S*

= 6W,

(9.12) (9.13)

is the variation of entropy produced in the hypersystem. In terms of the local entropy produced as*, as evaluated in Section VIII, it is expressed by TO6S*

=

To

1,

6s* d o .

(9.14)

An important step results from a generalized interpretation of the value of 6W, the virtual work of the external forces acting on the hypersystem.

New Variational-Lagrangian Irreversible Thermodynamics

33

It represents the work 6 WM of the mechanical forces acting on the primary system plus the work 6 WTH accomplished in the thermobaric transfer of heat and mass injected at the boundary. In addition, we generalize d' Alembert's principle by including in the external work the virtual work -&Z, 6q, of the reversed inertial forces due to mass accelerations in the primary system at any particular instant of the evolution. The generalization involved here implies the validity of d'Alembert's principle where U is a thermodynamic energy involving heat and not simply a mechanical potential as in classical mechanics. Accordingly, we may write 6W = -

2 z, 6q, + 6WM + 6 W H ,

(9.15)

I

where 6WTH = -

I,(c&

6M:

+ O/T SH,)n, dA

(9.16)

represents the virtual work due to the thermobaric transfer of heat 6H,n, and masses 6M;kn, at the boundary. Note that for a deforming boundary, M," and H , are the contravariant mass and heat fluences as defined in Section V11, whereas the surface integral is evaluated at the initial boundary A . Note also that Eq. (9.16) may be expressed in terms of the entropy fluence S, at the boundary by substituting into Eq. (9.16) the value of SH, extracted from Eq. (8.5). This yields 6WTH = -

i(T & 6 M f +

O 6Si)nid A .

(9.17)

We have called 6WTH the virtual work of thermodynamic forces at the boundary. With the value (9.15) of 6W substituted in Eq. (9.12), we obtain Z; 6qi

+ 6V + TQ6s" = 6WM + 6WH.

(9.18)

i

This constitutes the generalized principle of virtual dissipation (Biot, 1955,1975,1976b, 1982b) valid for arbitrary variations, provided continuity of heat and mass flux is preserved. If the system is in a potential field such as a gravity field, we denote by G the potential energy in this field. By including -6G in the mechanical work of external forces, we may write Eq. (9.18) as

2I j 6 q j + 6 9 + T Q 6 S * = 6 W M + 6WCiTH,

(9.19)

i

where 9 = V + G (Biot, 1975, 1976b) is called the mixed collective potential, while 6WM denotes the work of forces other than gravity acting

34

M. A. Biot

directly on the primary system. The term (9.20) now includes the work against the gravity field in the boundary thermobaric transfer. We have put pk =

-k %,

d'k

(9.21)

where % is the gravity potential field per unit mass chosen so that the supply cells are on the surface % = 0. We have called Vk the mixed convective potential. With time derivatives instead of variations, Eq. (9.14) yields the total rate of dissipation as To,$*

=

loS * d f l

TO

(9.22)

It is interesting to compare To&* and T i * . For example, if dgk/dxi = dT/dxi = 0, writing Eq. (8.6) with time derivatives yields Ti*

= hP,

(9.23)

where hP A t is the heat produced by the irreversibility in the time interval A t . If we have a thermal well at a lower temperature T o , this heat is not entirely lost, because we may recover the mechanical work T - To hp A t T

(9.24)

through a heat engine. The actual loss of useful energy is therefore (To/T)hPA t

=

TOS*A t .

(9.25)

For this reason we have called TS* the intrinsic dissipation, and To&*the relative dissipation (Biot, 1975, 1976b). In the modified thermobaric transfer heat is pumped from the thermal well and injected into the various elements and cells of the hypersystem. The process involves only mechanical work (given by Eq. 9.1) and provides purely mechanical definitions of &, as well as the exergy and 6WTH. The pump may use matter, subject to a Carnot cycle. However, it is of interest to point out (Biot, 1976a) that use o f t h e Carnot cycle may be avoided by using pure heat as blackbody radiation extracted from the thermal well and compressed adiabatically to the required temperature of injection.

New Variational-Lagrangian Irreversible Thermodynamics

35

X. General Lagrangian Equations Consider a system going through an irreversible evolution. We shall assume that its state may be described at every instant by a finite set of parameters q;, unknown functions of time called generalized coordinates. In particular, this may be accomplished by the use of fluence fields as state variables either exactly or approximately. Such fluence fields, along with material displacements and reaction coordinates, are then considered to be given functions of the initial coordinates and generalized coordinates q; to be determined. Arbitrary variations 6qi generate field variations that may be called holonomic and may satisfy identically the condition of field continuity. We may write (10.1) To 6S*

=

2 R; 6q;,

(10.2)

2 Qi6q,

(10.3)

I

6WM + 6Cl/rH =

I

Substitution of these values into Eq. (9.191, which expresses the principle of virtual dissipation, considering that 6q; is arbitrary, yields 1;

+ m / a q ; + R; = Q;,

(10.4)

where I ; are generalized inertia forces, R; are generalized dissipative forces, and Q; are driving forces of a mixed mechanical and thermodynamic nature due to environmental conditions (Biot, 1975, 1976b). These equations are the general Lagrangian equations of evolution of irreversible thermodynamics. For systems that are quasi-reversible such that local states do not deviate much from a local equilibrium satisfying Onsager's (1930, 1931) principle, the total virtual dissipation may be expressed in terms of a dissipation function. For example, when the entropy production is due to thermomolecular diffusion and chemical reactions near local equilibrium, Eqs. (8.25), (8.27), and (9.14) yield (10.5)

If

&,

M;, and S; are expressed as functions of q ; , this may be written (10.6)

36

M . A . Biot

where D is the total dissipation function

D

=

I,$)

9d f l

=

I 2

- 7,J*

(10.7)

and 9 is given by Eq. (8.28). Expression (10.7) is a quadratic form in 4; with coefficients depending on 9 i . The Lagrangian equations (10.4) for quasi-reversible evolution become I;

+ d 9 / d 9 ; + dD/dq, = Q;.

(10.8)

This form of the Lagrangian equations was initially derived in the linear context (Biot, 1954, 1955, 1956a). In many problems it is possible to express the generalized inertial forces in the classical form (10.9) where 3 denotes the kinetic energy expressed as a function of q;and 4;. In this case the variational principle may be written in the Hamilronian form (10.10)

For a quasi-reversible evolution we may write Eq. (10.8) as dDld4; = X ; ,

(10.11)

with X i = Q; - I; - d 9 / d q i . This equation shows that in any given state, when we consider all possible generalized velocity fields 4; satisfying the constraint

C X;Q; = const.,

(10.12)

i

the actual velocity of evolution minimizes the dissipation function D.In view of relation (10.7), the rate of total entropy production is also a minimum (Biot, 1955, 1976b). As a simple physical illustration, consider a fluid under gravity seeping through a porous medium. At a given instant, of all possible velocity fields with the same rate of descent of the center of mass, the actual one minimizes the dissipation.

XI. Dynamics of Viscous Fluid Mixtures with Reaction-Diffusion and Radiation Pressure Field and Lagrangian equations of viscous fluid mixtures with chemical reactions, thermomolecular diffusion, and radiation pressure have been

New Variational-Lagrangian Irreversible Thermodynamics

37

derived directly from the principle of virtual dissipation (Biot, 1979, 1982a). The results provide new and powerful methods in a large variety of technological problems as well as in stellar dynamics, in particular in the analysis of stability and oscillations of self-gravitating bodies. The coordinate system xi is Cartesian and fixed. The fluid flow through it is described by a Cartesian mass flux M: of each substance in the mixture. In addition, we consider a Cartesian entropy flux fii.The fluid is in a gravity field. Because the mixture is homogeneous we may apply the concepts of Section X111, where a cell of unit volume is created by thermobaric transfer from an initial state of zero volume at constant temperature T and pressure p. The energy % and entropy Y per unit volume are then obtained from Eqs. (4.9) and (4.10) by putting u = I . Actually, to conform with our general procedure, we add suitable constants so that % = Y = 0 at the initial time t = 0. The values mk represent the masses of each substance per unit volume in the mixture. They may be written as (11.1)

where mOkare the initial masses at t = 0. We also put 5, = 0 at t = 0 and assume the chemical reactions to be initiated at that instant, whereas M k are the masses per unit volume added by convection for t > 0. As before, we may write the mass conservation equation (7.3) as M h = -aM:/dxi,

( 1 1.2)

where M: is the Cartesian mass fluence. In the present case, the entropy production is due to three causes: the chemical reactions, the thermomolecular diffusion, and the viscosity. The first two have been evaluated and expressed by Eq. (8.25). In order to evaluate the entropy produced by the viscosity we consider the viscous stresses due to strain rates. The strain rate of a mixture is expressed in terms of the barycentric velocity ui defined by ( 1 I .3)

where p is the density of the mixture. The viscous stresses are then written rrjj

= V,(Ujj

+ V j j ) + 712 8 j j U / / ,

ujj = aui/axj,

(11.4)

where q l and q2 are viscosity coefficients depending on the local state of the mixture. These stresses may be written in the form where

M . A . Biot

38

(11.6) the viscous dissipation function, is a quadratic form in uij. When varying the mass fluence 6 M f , we produce a virtual displacement derived from Eq. (1 1.3) as (11.7) and a virtual work

ahP = uija6ui/axj.

( 1 1.8)

The variation ahP is the virtual heat produced and is analogous to the value in Eq. (8.2), namely, the heat to be added reversibly in order to obtain the same change of state as was due to the irreversible process. Hence the virtual entropy produced by the viscous stresses is given by T SS*V = 6hP = asui/axj. (11.9) We shall assume that the mass and heat flux obey locally linear laws and Onsager's principle. In that case the virtual entropy production due to thermomolecular diffusion as derived from Eqs. (8.26) and (8.27) is given by

T

aQTM a%lTM an;r," 6M: + -6s; asi

=

(11.10)

with a dissipation function gTM to be specified below. For the dissipation due to the chemical reactions we retain the general form of Eq. (8.24). The total virtual entropy production is then obtained by adding the values of Eqs. (8.241, ( 1 1.9), and ( 1 1.10). We write

For a domain R under gravity the mixed collective potential is

9=

J-p + p % ) dlR

(11.12)

where (8 is the gravity potential field and lf the exergy per unit volume. The fixed-coordinate system defines cells whose volume is not varied. Hence from Eqs. (3.22) and (9.6) with 6u = 0, we obtain

i W

=

6% - To 6 9

= -

C A,, S t , + CI P

6 M h + 0 6 9 (11.13)

New Variational-Lagrangian Irreversible Thermodynamics

Also 6p

=

39

& 6 M h , and we may write

where ' P k is the mixed convective potential (9.21). The virtual work of the inertial forces is evaluated by assuming that all accelerations are the same and equal to the barycentric value a; = av;/at

+ vj av,/axj

(11.15)

as determined by v;. This amounts to neglecting the inertial forces due to the relative velocities of diffusion. We write (11.16)

We now apply the principle of virtual dissipation (9.19), considering variations inside the domain 0. In this case 6 W M = 6WTH = 0. With the values (11.14) and (11.16) and the value (8.29) for 6 9 , the principle of virtual dissipation becomes (1 1 .17)

We then introduce the value (1 1.11) of T 6s* and the value (1 1.2) for M k . After integration by parts we equate to zero the coefficients of the arbitrary variations and obtain ( 1 1.18)

aT/axi + aWM/aSi = 0,

( 1 1.19)

+ 3,= 0.

(1 1.20)

-A,

These are the field dynamical equations of the viscous fluid mixture with reaction-diffusion. They bring out a fundamental coupling between diffusion and the viscous stress gradient a q l a x j . They contain the unknowns t,, M:, and Si.According to Eqs. (8.31) and (11.21, they determine the state of the system as a function of time, if we may neglect the contribution of the entropy produced s* to the value Yof the entropy. However, if this is not the case, we may determine s* by adding the auxiliary equation Ti*

=

c

%,&

+ 29" + 29TM,

(11.21)

P

which expresses the rate of dissipation and is obtained from Eq. (1 1.11) by replacing the variations by time derivatives.

40

M . A. Biot

Use of s* and the auxiliary equation ( I I .21) may be avoided by using pseudofluence vectors such as 9;or 9'; discussed in Section VIM as state variables. For example, in the present case, if we use 9:,Eq. (8.33) must be completed to take into account an additional heat source mij d8ui/axj corresponding to the energy of viscous dissipation. Hence expressions (8.37) and (8.38) for T 6s and TS; remain valid, provided we add terms mii aSui/axj and mij au;/axjin the volume integrals. We obtain T6Si=S%:-C&6M! k

With these values the field equations become integro-differential equations, except if the integral terms in Eqs. (1 l .22) and (1 1.23)are negligible, which is usually the case. An energyflux theorem was derived (Biot, 1979, 1982a) from the field equations (11.18), (11.19), and (11.20). This may be shown by adding these equations after multiplying the first set by h!, the second set by Si, and the third by t p We . obtain aFi 1 a + - - (PUiVi) axi 2 at

-

+ % + b% = 0,

(1 1.24)

where

1 Fi = 5 ujuj

- ujuu +

2 & l / ( E k + 93) + kli

(11.24a)

k

represents the total energy flux. Note that in this expression the quantity & M / E k + ki is the diffusive energy flux (8.41). We have not yet mentioned the important fact that the dissipation funcfor thermomolecular diffusion must be invariant under translation 9TM tion. It was shown (Biot, 1982a) that this condition is satisfied if we put ( I I .25) where

New Variational-Lagrangian Irreversible Thermodynamics

41

and Y c is equal to the value (4.17). The coefficients CIkand Ckare functions of the local state, and the values of Ck are chosen such that the coefficient of st is T/2k where k is the local thermal conductivity of the mixture. The dissipation function (1 I .25) satisfies the identity ( 1 1.27)

Using this relation, we may verify that the field equations (11.18) and ( 1 1.19) satisfy the total momentum balance (Biot, 1982a) by multiplying Eq. (1 1.18) by m k and Eq. (1 1.19) by Yc. Adding the results and taking into account the identity ( 1 1.27) and the modified Gibbs-Duhem theorem (4.16) for v = 1 , we obtain pa; - ac,/axj

+ aplax; + p a%laxj = 0.

( 1 1.28)

This result expresses the total momentum balance. As shown in a detailed discussion (Biot, 1982a), the effect of radiation pressure may be taken into account in the present analysis by including it in the total pressure p and the injection pressures Pk as generalized equations of state. It was pointed out that kinetic and radiation pressures may not be additive in dense mixtures because the radiation group velocity in dense matter is smaller than the velocity of light (Brillouin, 1930). The foregoing results neglect the accelerations due to diffusion. Equations that avoid this simplification and introduce partial viscous stresses were also derived (Biot, 1979) in the absence of chemical reactions. The latter condition is required if relative accelerations due to diffusion are to retain any physical meaning, because chemical reactions imply a form of impact and coalescence between molecular flows that are not taken into account in the present theory. Lagrangian equations of evolution are obtained directly from the principle of virtual dissipation. The state of the system may be described by 5, and the fluence fields Mi" and 9 : expressed in terms of generalized coordinates qi as ( I 1.29) (11.30)

(11.31)

We derive the Lagrangian equations using the general procedure of Section X. Instead of ST we may also use the fluence fields B i , S,? , or Si according to the particular approximations involved (see Section VII). The virtual work of the inertial forces is

42

M . A . Biot (11.32)

Hence the generalized inertial force is

(1 1.33) The virtual dissipation is

TO6S*

=

h, To 6s* dfl

=

R; 6 q i ,

(11.34)

where 6s* is given by Eq. (11.11). Hence Ri = R:h + RY

+ RTM.

(11.35)

The term ( 11.36)

is the chemical dissipative force. It can be shown that the viscous dissipative force is (11.37) where Bv has the value in Eq. ( I 1.6). The dissipation function Dv is a is the dissipative force due to quadratic form in 4,. The value of thermomolecular diffusion. It is obtained by writing the variational relation

replacing 6s; and 3; by the values obtained from Eqs. (8.37) and (8.38). This provides the exact value of RTMas a nonlinear function of q i. However in most problems we may neglect the integral in the values (8.37) and (8.38). In this case we may write

aDTM RTM = aq; '

DTM =

In T gTM dfl

(1 1.38a)

where DTMis a quadratic form in 4; representing the total dissipation of thermomolecular diffusion. Determination of the generalized driving force Qiis obtained from Eq. (10.3), written in the form

New Variational-Lagrangian Irreversible Thermodynamics

c Q;6q;

=

6WM-

lA(T

pk

SMi" + 0 6Si)n;dA,

43

(11.39)

where A is the boundary of R . On the right is the virtual work of all the mechanical forces. The integral represents the work due to thermobaric transfer at the boundary considered as fixed, whereas 6 WMrepresents the virtual work of all other forces. Special care must be exercised here by noting that in the absence of viscous stresses 6 W M = 0. In this case the virtual work due to boundary displacement 6u; against the local equilibrium pressure p is already included in cpk6M/ of the surface integral. Hence the remaining work 6 W Mis due only to the viscous stresses and is written 6WM=

I A

crjnl

6uj dA.

(11.40)

This is also verified using a lengthier procedure (Biot, 1979, 1982a). We substitute this value into Eq. (1 1.39) and use the values in Eqs. (1 I .22) and ( I I .23) for 6s; and 3;. Again we may generally neglect the volume integrals in these expressions. Coefficients of 6qi yield Q ; . With these results the Lagrangian equations are written I;

+ R ; ~+ dD1dq; + a 9 / a q j = Q ; ,

(11.41)

where D = D v + DTMis the dissipation function due to viscous and thermomolecular dissipation. For a quasi-reversible evolution including chemical reactions we may neglect s* in describing the system and use the entropy fluence Si as a state variable instead of 9;.In that case the dissipation is expressed by a single dissipation function and the Lagrangian equations assume the simpler form [Eq. (10.8)]. It was also shown (Biot, 1979, 1982a) that the inertial force may be expressed in terms of the kinetic energy

(1 1.42) as

where (11.44) and

M . A . Biot

44

(1 I .45)

It is interesting to note that djUi =

c Aiq;

0,

=

i

0.

(11.46)

Hence hirepresents a generalized inertial force normal to the velocity, corresponding to a Magnus effect.

XII. Dynamics of Solids with Elastoviscous Stresses and Heat Conduction, and Thermoelasticity The dynamics of a deformable solid will now be considered in the particular case where rate-dependent viscous stresses are present in addition to elastic stresses, with simultaneous heat conduction across the deforming medium. The problem was treated in detail earlier (Biot, 1976b, 1981) and we shall present here a short and simplified version of the results. The field is described by mass point displacements ui such that xi =

xi

+ u;(x;, t ) ,

(12.1)

where a solid mass point with initial coordinate xi is displaced to a point with coordinate X i . We have seen (Section V) that the local deformation may be defined as an affine transformation relative to rotating axes by six components e i j ,which are assumed to be known functions of the displacement gradients dui/ax,. The corresponding stress components T~ are then derived from the principle of virtual work. In the absence of molecular diffusion and chemical reactions, the energy 0% and entropy .cP of the solid per unit initial volume are obtained from Eqs. (6.2) and (6.3) as

+ h f ) dEii + C , dT,

(12.2)

( I / T ) ( h f d q+ C , d T ) ,

(12.3)

dOU

= (7:

dY

=

where 7; is the elastic part of the stress for reversible deformations and C , is the heat capacity of the solid per unit initial volume with constant strain. The value of h! is given by Eq. (6.7) without calorimetric measurements as

h$

=

-T(a7:/dT),,

(12.4)

N e w Variational-Lagrangian Irreversible Thermodynamics

45

where E E 70 - Ty ( E p ,

(12.5)

T, XI)

represents the local equations of state of the solid with the elastic stresses expressed in terms of strain and temperature. The subscript E in Eq. (12.4) indicates constant strain. Since the solid may be intrinsically nonhomogeneous, r i may depend also on the initial coordinates x i . Integration of Eqs. (12.2) and (12.3) yields %, Y , and V = % - TOYas functions of .zij. T and x i . The entropy production arises from the viscous stresses and the thermal conduction. This provides another example of two distinct types of entropy production, the first being due to an irreversible production of heat while the second is not. In order to apply the principle of virtual dissipation we must evaluate the virtual entropy production. We therefore replace the differentials by variations in Eqs. (12.2) and (12.3). They yield 8% =

8 ~ +u T 6 Y .

(12.6)

For an irreversible transformation, conservation of energy is expressed by 8% = (TF + ‘TY) 8 E i j - d 8Hj/dXi, (12.7) where r{ is the additional viscous stress and 8Hi is the variation of contravariant heat fluence. We recall that hi is the heat flux across a deforming material area initially normal to xi and equal to unity, whereas Hi is its time integral. Equating (12.6) and (12.7) yields

w = as* - a 8si/axi,

(12.8)

8 ~” dT/dxi 8Si,

(12.9)

where T as*

= T{

6s; = 8H;/T.

We recognize the term (-aTldxi) 8Si already derived in (8.6) for thermal diffusion. The first term may be interpreted as due to heat production (Biot, 1981, 1982b) by considering an adiabatic transformation where conservation of energy requires 8% = (r;

+ rjy) 6 E j . j .

( 1 2.10)

Obviously this is equivalent to a reversible transformation where an amount of heat 6hP = T & ~ ~is added in order to reproduce the same change of state as in the irreversible process. The heat corresponds to the concept of uncompensated heat of Clausius and illustrates the physical

M . A . Biot

46

difference between the two types of entropy production in Eq. (12.9). The same distinction was discussed above in Section VIII. As in the case of thermomolecular diffusion (see Section VIII) we now introduce the kinetics of irreversibility. We write v-

Tij

-

v

T u ( E , w ~&v,

T, Xi),

-8TldX; = AgHj,

(12.11) (12.12)

where

A.. rJ = A..(e rJ

PLY1

T, Xi)

(12.13)

is the thermal resistivity of the deformable solid expressed in terms of the contravariant heat flux H j . It is a function of the local state variable E~~ and T. If the material is nonhomogeneous, it is also a function of the initial coordinates x i . The viscous stress T;, as a function of the rate of deforma, the kinematics of the mechanical irreversibility. Note tion i Uembodies that T; and - l / T dT/dxi = AgSj,

(12.14)

with 3; = HJT, play the same role as the dissipative disequilibrium forces % f k , and $%f' in Eq. (8.25) for the case of thermomolecular diffusion with chemical reactions. This can be seen by writing the virtual dissipation (12.9) in terms of rate variables in the form T as* = T; 8 ~ + " TAUS; SSj,

( 1 2.15)

where T; is the rate function (12.11). Comparing with Eq. (8.25), we see that the viscous stress 7; may be considered as the tensor equivalent of the ufinity aP. Field equations may be obtained readily for the dynamics of the deformable solid by applying the principle of virtual dissipation (9.19). We vary the displacement ui and the entropy fluence S;. The virtual work of the inertial forces is

2 z; 6q; = I,,PUJ 6Uj dR,

(12.16)

i

where the volume integral is for the initial domain fi of the space xi and p is the initial density. The variation of 9 is 69

=

\c,{ 6"lr + p d % ( x ; ) l d f ; 6 U i } dR,

where %(xi)is the gravity potential field per unit mass at the point u ; . The exergy variation per unit initial volume is

(12.17) = xi

+

New Variational-Lagrangian Irreversible Thermodynamics 67'" = 6%

-

TO 6Y

= T;

~ S E ~ ,+ , 8 6Y

47 (12.18)

(8 = T - TO).The variations are applied only inside the domain Q , and with the values (12.16), (12.17), and (12.18), the variational principle (9.19) is written Ia(pii; 6u;

+ r; 8 . z ~+ 8 6Y + p d%ldFi 6ui + TOas*) dR = 0.

(12.19)

Equation (8.29) is quite general, and with the value (12.15) it yields 6~ = -a 8s;/dxi + as*.

(12.20)

Hence Eq. (12.19) becomes ( p i ; 6ui

+ rii

a

- 8 - 6s;

axj

a%

+p6ui + ay;

T as*) dQ = 0. (12.21)

We replace T 6s* by its value (12.15) and write 6.zILY= ds,,/aa, 6aU,

6aO = dSui/dxj.

(12.22)

After integration by parts, we equate to zero the coefficients of the arbitrary variations 6ui, 6s;and obtain (12.23)

dT

+ TA$, dXi

= 0.

(12.24)

These are the field equations for the deformable solid with elastic and nonlinear viscous stresses and with thermal conduction. In solving the problem we may consider ui and Si as unknowns that determine the state through Eq. (8.30) if we also determine the entropy produced s*. This may be done by adding to the field equations the auxiliary equation . . (12.25) Ti* = pi.. u u + TA..S.S. U l J derived from Eq. (12.15). It extends Meixner's (1941) result to deformable solids with viscosity. Another procedure in analogy with the use of Eq. (8.36) for thermomolecular diffusion is to introduce a pseudoenergy : defined by Eq. (7.15). The energy balance equation (12.7) is fluence 9 then written -a6%:/axi

Hence

= ($

+ T;)

6~~ - a 6 H j / d x j .

(12.26)

M . A . Biot

48

This becomes

With this value of si Eqs. (12.23) and (12.24) become integro-differential equations in ui and !T:, which are now state variables. A similar procedure has also been discussed in an earlier paper (Biot, 1981). If the irreversible process is quasi-reversible, we may neglect the contribution of S * to the value of Y and use Si as a state variable. In that case Onsager's (1930, 1931) reciprocity relations are valid, and we may write where 5?hv is a quadratic form in iijwith coefficients that are functions of the local state variables eij and T, and 5?hT

=

21 TAijSisj,

for A, = Aj;,

(12.30)

is a purely thermal dissipation function. The virtual dissipation (12. IS) now becomes

T GS*

=

a9v/ai,

GE,

+ a%T/aSi asi

(12.31)

and leads to the field equations

aT axi

a5?hT asi

- + - = 0.

(12.33)

The contravariant thermal resistivity A" may be expressed in terms of the local Cartesian thermal conductivity k, as follows. With the Cartesian resistivity [hijl

= [kijl-',

(12.34)

we write the heat conduction law in the form -aT/aii = hijHj,

(12.35)

where h/is the Cartesian heat flux in the X i coordinates. By using Eqs. (7.12) we introduce the contravariant heat flux and obtain -aT/aii = J ' X , H ~aq/ax,.

(12.36)

New Variational-Lagrangian Irreversible Thermodynamics

49

Multiplying both sides by aX;ldx,, this yields (1 2.37)

Comparing with Eq. (12.12), we see that

(12.38) showing the covariant nature of A,. Note that A,, is a function only of the strain E,, T , and x i , and hence represents an intrinsic physical property, whereas A, is referred to the fixed axes xi and depends also on the rotation of the material. Lagrangian equations are immediately obtained from the principle of virtual dissipation. For simplicity, we shall assume that the entropy produced s* does not contribute significantly to the state variables, so that the system may be described in terms of generalized coordinates qi by the displacement field ui and the contravariant entropy fluence Si . We write

(12.39) (12.40) We apply arbitrary variations 69;. The variation of the mixed collective expressed as a function of 9;. potential is obtained from its value 9(9;) By a classical procedure the virtual work of the inertia forces yields

(12.41) where 1

3 =3

Ind f l pU;U;

(12.42)

is the kinetic energy, and .fl the initial domain. We shall assume that the thermal conduction satisfies the reciprocity relations A, = Aji while the viscous stress remains a general nonlinear function T:(E,,, E,,, T). In this case, according to Eqs. (12.15) and (12.29), the virtual dissipation is

T as*

= T:

+ (daT/aS;)GSi.

(12.43)

For the whole system we write

(12.44)

M . A . Biot

50

where (12.45)

The total thermal dissipation function DT is a quadratic form in coefficients being functions of 4;. It is obtained by writing

By noting that aSj/aqi =

with

dS,/ag,, we derive (12.47)

where

DT =

larTo 9' dR.

(12.48)

Finally we express the virtual work of the mechanical and thermodynamic forces applied at the boundary A . This is 8WM + S v H =

Q; 6q; = \A

(A 6ui - 0 SSj n,)

dA.

(12.49)

I

The integral is extended to the boundary A in the initial space x i , whileJ is the boundary force per unit initial area at the deformed boundary. Values of 8 and SSj are also at the deformed boundary. The generalized mechanical and thermodynamic forces at the boundary are therefore au.

Qi =

as,

[ if-! dq; - O -dq;n j ) A

dA.

(12.50)

With S9(qi) and the values (12.41), (12.44), and (12.50), the principle of virtual dissipation (9.19) leads to the Lagrangian equations (12.51)

The term RY is the generalized dissipative force due to the viscous stresses; it embodies the most general case where these stresses are nonlinear functions of the strain cij and the strain rate E i j If these stresses are linear functions of the strain rate and satisfy the Onsager reciprocity relations e

a7,,/aEij

=

ar,/ai,, ,

(12.52)

New Variational-Lagrangian Irreversible Thermodynamics

51

they may be expressed in terms of a dissipation function 9"that is a quadratic form in E g . This leads to the Lagrangian equations (12.53) with a total dissipation function

D

=

I,,+ (9"+

9 T ) dfl,

(12.54)

which represents the combined viscous and thermal dissipation. We have simplified the Lagrangian formulation by neglecting s* as a state variable. The accuracy may be improved in several ways. One way is to evaluate s* as a function of time from the first approximation and introduce it in the value (8.31) of Y,which is now written

Y=

- d(Si

+ A S i ) / d ~+i s*,

(12.55)

whereas the displacement is ui + Aui. The new unknowns are the corrections ASi and Aui to the first approximations Si and ui. The corrections are expressed in terms of generalized coordinates as Eqs. (12.39) and (12.40), and the corresponding Lagrangian equations contain the time explicitly. Another procedure is to use the pseudo energy fluence 9? and Eqs. The procedure is analogous to the one (12.27) and (12.28) for 6Si and explained in Section VIII and discussed more extensively elsewhere (Biot, 1981). If there are no viscous stresses the solid is rhermoelastic. The field equations are obtained from Eqs. (12.23) and (12.24) by putting T; = 0. The Lagrangian equations are obtained from Eq. (12.51) by putting RY = 0. In this case the exergy r/. becomes the thermoelastic potential (Biot, 1973a).

si.

XIII. Inhomogeneous Viscous Fluid with Convected Coordinates and Heat Conduction The general case of a deforming solid, analyzed in the previous section, has a number of interesting applications that are better discussed separately. In particular, we shall consider the case of a compressible viscous fluid, Newtonian or non-Newtonian, with thermal conduction. For the homogeneous fluid, this may also be treated from an Eulerian viewpoint as in Section XI for the more general case of a mixture. The purpose here

M . A . Biot

52

is to develop equations using convected coordinates P; = xi + u; for a fluid, which may be inhomogeneous. The energy % and entropy Y per unit initial volume are functions

9' = Y ( J , T , x;)

% = % ( J , T , x;),

(13.1)

of the temperature T and the Jacobian J (7.1 I), which represents the ratio of final to initial volume of each fluid element. If the fluid is nonhomogeneous, % and Y are also functions of the initial coordinates xi. The thermal dissipation function per unit initial volume is derived from Eq. (12.30) as follows. The thermal conductivity

k

=

(13.2)

k ( J , T , xi)

remains isotropic and is a function of J , T , and xi. The Cartesian thermal resistivity is hij

=

6,Ik.

(13.3)

By substituting this value in the covariant thermal resistivity (12.38) we obtain (13.4) The thermal dissipation function per unit initial volume (12.30) is therefore (13.5) in terms of the contravariant entropy flux S;. In order to derive the dissipation due to the fluid viscosity for a Newtonian fluid we consider the stress components uijin terms of the velocity gradients in the Pi space. They are (13.6) where 7 1 = ~ I ( JT,, Xi),

=

~ ( 5T , xi)

are viscosity coefficients, functions of the local state variables J , T and the initial coordinates xi. If there is no bulk viscosity we put 2

7711

+ 72

=

0.

(13.7)

New Variational-Lagrangian Irreversible Thermodynamics

53

The virtual dissipation due to the viscosity per unit initial volume is T 6 ~ ="Ju;,, ~ a6ui/ax, = Jui, axjlax, 6aU

(13.8)

6aii = a6u;/axj.

(13.9)

where We may express the derivative dxi/aFj in terms of axj/axjby writing the affine transformation from dxj into dTi and solving the linear equations for dx;. The result is similar to Eq. (7.8). We obtain axj/azj=

c~;/J,

(13.10)

where Cii is the cofactor of aXi/axj in the Jacobian J . Hence the virtual dissipation (13.8) becomes T ~ S * V=

c

.u.6a.. U = TVU 6a.. U'

W J C

(13.11)

The physical significance of this expression is brought out by comparison with the virtual work (5.12). Obviously, (13.12)

TVU = CW.(T.W

is the Piola viscous stress, i.e., the Cartesian components of the viscous forces acting on areas initially equal to unity and normal to x j . Similarly, we may express the velocity gradients in Eq. (13.6) in terms of partial derivatives with respect to x i . We note that hi = ui and write (13.13)

where U" = ahi/axj.

(1 3.14)

We substitute the value (13.13) into expression (13.6) and changej into p , thus obtaining up =

(qi/J)(CpU;U+ Ci&J + (q2/JPiwC/&u.

(13.15)

With this value, the Piola stress (13.12) becomes (13.16) T; = (ql/J)(CpuCfijUiu + CPjC;,Ufiu)+ (~2/J)6;pCpjC/uU/u.

This may be written as

+

+ ( ~ 2 / J ) C ~ C p , U p ,(13.17) .

= (q1/J)(CluC,j6jp CfijC;,)UPu

By putting BY = (ql/J)(C/,C,j6ip+ C&jU)

+ (q,/J)CjjC,,,

(13.18)

M . A . Biot

54 the stress (13.17) becomes

TVU = BPk lLVr

for BY

=

B&.

(13.19)

Hence, Onsagers reciprocity relations BY = B$, are verified for T i . This is as should be because it is an invariant property, already verified by the Newtonian stresses (13.6) (Biot, 1976b). The rate of dissipation per unit initial volume is obtained from Eq. (13.11) by replacing the variations by time derivatives. We obtain ,. Tj*V = TVh,. r J U = 29V = B?"k 0 lLu aU'

(13.20)

where '?bv is a quadratic form in 6,. Because of the reciprocity property (13.19), we may write T: = a%V/ak.. U'

(13.21)

The exergy per unit initial volume is 'V = ' V ( J , T, xi);

(13.22)

it is a function of the Jacobian J , the temperature T, and the initial coordinates x i . Considering 'V as a function of J and Y , its variation is

8-v = a"lr/aJ 8.1

+ 0 w.

(13.23)

Obviously, the local equilibrium pressure is P = 8clrlaJ

( 1 3.24)

for a reversible slow deformation. This result corresponds to equation (12.6). We may also write Sclr = T; 6uO

+ 0 SS,

(13.25)

where T: = P aJlaa,,

( 13.26)

is the Piola stress for the elastic pressure P . By applying the principle of virtual dissipation with arbitrary variations Sui inside a domain R and proceeding as in the preceding section for the solid, we obtain the field equations piii

a -(T: axj

a3 + Ti) + p = 0, azj az-/ax, + aBT/aSi= 0 ,

(13.27) ( 13.28)

where Yi is the gravity potential, '?bT is given by Eq. (13.51, and p is the initial density. An auxiliary equation for i* is

New Variational-Lagrangian Irreversible Thermodynamics

Ti* = 2(9"

+ 9T).

55

(13.29)

Under the assumption that S; may be used as a state variable, the Lagrangian equations have the same form as Eq. (12.53), where the values (13.20) and (13.5) are now substituted in Eq. (12.54) for the dissipation function. For a non-Newtonian fluid, the viscous stress aijis expressed as uu

=

+ F2eh + F3e,!keij,

F,60

( 13.30)

where

ek = -1 (avi/azj+ auj/axi) 2

(13.3 1)

and F , , F 2 , F3 are functions of the three invariants e&, eieh, ehejkeLi. In the general case they are also functions of T and xi if the fluid is nonisothermal and nonhomogeneous. A very simple proof of this formula has been given (Biot, 1976b). The Piola viscous stress T: in this case cannot generally be derived from a dissipation function as for the Newtonian case (13.21). However, it is given by the same formula (13.12) in . field equations are then the same as Eqs. terms of the stress T ~ The (13.27) and (13.28). However, the Lagrangian equations become (13.32) where m

derived from Eq. (13.1 l), is the generalized non-Newtonian viscous force and (13.33) is the thermal dissipation function. A special case of practical interest is that of a viscous incompressible solid undergoing slow deformations under isothermal conditions ( T = To). Inertial forces are negligible, and we put 9 = 0. The material may be inhomogeneous, either continuous or composed of different adherent solids. The potential 9 depends only on gravity, and its value is

9=

I,,

p%(X;) dfl.

(13.34)

M . A . Biot

56

The dissipation function D depends only on the viscosity. By assuming Newtonian viscosity and putting T = T o , we derive (13.35)

The displacement field uiis represented by generalized coordinates qi as uj

= uj(q,,q 2 r

***,

x/)-

(13.36)

The dissipation function (13.35) is then a quadratic form ( 13.37)

where the coefficients bUare functions of the generalized coordinates qi . The Lagrangian equations become a9/aqi + aD/agi = Q ; .

(13.38)

The generalized driving forces Qi are obtained by putting 8 = 0 in the surface integral (12.50), where 5 are the forces at the boundary per unit initial area.

XIV. Lagrangian Equations of Heat Transfer and Their Mechanical Interpretation, and a Mass Transfer Analogy The particular case of pure heat conduction is obtained by considering a rigid solid ( E = ~ 0). An important feature here is that the heatjluence H i becomes a state variable. The heat content, or energy per unit volume, is then ‘IL = h =

-aHi/axi.

(14.1)

The total exergy of the domain R is now V=

In

7f

dfl,

7f =

Ih 0

BIT dh,

(14.2)

where 7 f is a function of h. The dissipation function is To D =2

h.. In $ HiHj dfl.

(14.3)

New Variational-Lagrangian Irreversible Thermodynamics

57

The thermal time history of the domain is determined by the fluence field Hi in terms of a finite number of generalized coordinates whose evolution is governed by the Lagrangian equations

+ aD/aGi = ei.

av/aqi

(14.4)

The thermal generalized force Q; is obtained as a particular case of Eq. (12.50) by putting 6ui = 0 so that

Q; =

-

1T A

8 dHj n,i dA. aq;

(14.5)

--

It is worth noting the physical significance of these equations. The purely thermal exergy Y. represents the work to be accomplished mechanically by thermobaric transfer to bring a unit volume of solid to a given temperature. The heat is transferred by a heat pump from the thermal well at a temperature TOto the solid at a different temperature T. Similarly, the generalized thermal force (14.5) is the work accomplished mechanically in order to inject heat at the boundary by the same heat pumping process. On the other hand, the dissipation function (14.3) represents the rate of loss of mechanical availability of the thermal energy due to thermal conduction. Thus the Lagrangian equations (14.4) for heat conduction are given a purely mechanical interpretation as related to an availability balance. An important formal simplification is immediately evident in the linear theory where 8 is small and T = TO.In this case IITOis eliminated from the equations as a common factor. It was found that the factor IITOmay also be eliminated in the general nonlinear case. This is shown by applying the principle of virtual dissipation with arbitrary variations inside the domain R. This leads to

la(:6h + ~2TO h,H; 6Hj) d R

=

0.

(14.6)

After integration by parts using the relation d(0IT)d.x;= - ( T O / T 2 )dB/dx, and taking out the factor TO/T2,we obtain

(ax, ae +

hgfij) 6Hi d R

=

0.

(14.7)

Another integration by parts then yields (14.8)

which is the variational principle derived earlier (see Hot, 1970). It leads to the Lagrangian equations (14.4) with a thermal potential V , a dissipa-

M . A . Biot

58

tion function D,and a generalized thermal force Q i , which are now expressed as h I v = n dfl\ 0 0 dh, D = A u H i H j d f l ,

/

(14.9) Applications of this Lagrangian approach to heat transfer were developed in great detail in a monograph (Biot, 1970). The Lagrangian equations for conduction and simultaneous convection in an incompressible fluid of prescribed motion were also obtained. The heat flux in this case is

Hi = Ji + uih,

(14.10)

where ui is the velocity of the medium. The dissipation function is now written

D

=

1

j

AijJiJjdfl.

(14.11)

This result may be derived directly (Biot, 1970) or also as a particular case of the general treatment of Section XI when restricted to pure heat transfer for a given velocity field of the fluid. Attention should be called to special formulations such as that of ussociutedfiuence j e l d s , which lead to the use of scalar temperature fields as unknowns instead of H i , and the treatment of boundary heat transfer to a moving fluid using the concept of a trailing function (Biot, 1970). The latter eliminated the inconsistencies of standard methods based on local heat transfer coefficients. Collective analysis by Lagrangian equations is ideally suited to the unified treatment of heat transfer in mixed systems constituted by solids and moving fluids. A highly useful concept that was derived from the Lagrangian formulation is that of penetration depth, which yields immediately the heat fluence due to sudden temperature rise at the boundary. It can be used as a basic tool simplifying the formulation of very complex problems. Application of the Lagrangian equations has brought simplification and physical insight in many problems of heat transfer (Lardner, 1963, 1967; Prasad and Agrawal, 1972, 1974; Chung and Yeh, 1975; Yeh and Chung, 1977). Many types of finite element methods may also be derived directly from the Lagrangian equations. For example, we may treat as generalized coordinates the fluence vectors located at the vertices of a grid, using linear or polynomial interpolations of these values to represent the complete

New Variational-Lagrangian Irreversible Thermodynamics

59

field. The method is systematic and should be highly accurate, because heat flux continuity is preserved (see Section XXVI). It should also be noted that the Lagrangian equations are derived directly from the principle of virtual dissipation, using the dissipation function as a basic invariant, without requiring any prior knowledge of the field differential equations. This is in contrast with formal methods based on functional space theories. In addition, the variational principle yields directly the field differential equations in any coordinate system. The invariance of the variational Lagrangian formulations also brings to light unifying fundamental properties of the governing equations (Lonngren and Hsuan, 1978). The Lagrangian variational equations of heat transfer are completely isomorphic to those of pure isothermal mass transfer (Biot, 1970). The heat fluence field Hiis simply replaced in all equations by the mass fluence Mi.Field and Lagrangian equations for nonlinear problems of mass transfer are thus readily obtained, as exemplified by the case of a moving boundary (Senf, 1981).

XV. Deformable Solids with Thermomolecular Diffusion and Chemical Reactions As another example we shall consider the deformation of a nonhomogeneous solid, which contains a number of substances in solution. Thermal and molecular diffusion of the substances are induced by the deformation as well as their concentration and thermal gradients. Simultaneous chemical reactions between the dissolved substances may also occur. For simplicity we shall neglect the inertial forces and assume creeping deformation. The energy % and entropy Y per unit initial volume are obtained by integrating Eqs. (6.2) and (6.3) along an arbitrary path. Although they are formulated for a single reaction, the results are readily generalized to multiple reactions t,, as was done for the value (6.11). We thus obtain (15.1) (15.2)

as functions of the deformation e i j , the chemical coordinates tprthe masses Mk added by diffusion, the temperature T, and the initial coordinates xi. The latter dependence represents the inhomogeneity. From

M . A . Biot

60

Eq. (15.2) we may obtain Tas a function of Y,E exergy

V=%

-

~

t,,, ~ ,M L , and x i . Thus the

TOY

(15.3)

per unit initial volume becomes a function of these variables. In a gravity field %(Xi) the mixed collective potential of a solid occupying the initial domain R is

where rno is the initial mass per unit initial volume and m is the mass added by the contravariant fluences M f . We recall that It,?; is the mass flux of the dissolved substance k through a deforming solid area initially equal to unity and initially normal to the xi axis. From Eqs. (6.11) and (15.3) we obtain 6~

= T~ $Eii

+ Ck +k

+

6 ~ k e 69 -

C A, st,.

(15.5)

P

This result extends the value (12.18) to the solid with molecular diffusion and chemical reactions. The virtual dissipation is due only to the chemical reactions and the thermomolecular diffusion. It has a form similar to Eq. (8.25).

where A@ and Si are now contravariant fluxes. The affinity a, is a function of t,, t,,,E U , M k , T, and xi. The dissipation function '2JTMfor thermomolecular diffusion is a quadratic form in M: and Si whose coefficients are E ~M , k , T, and x i . functions We now apply the principle of virtual dissipation with arbitrary variations, at,, 6ui, 6 M f , 6s; inside the domain R of the solid. Proceeding as for the solid in Section XI1 we derive the field equations

ep,

(15.7)

+ a%TM/aIt,?: = 0, dT/axi + a % T M / a S j = 0 , -A, + a,, = 0 ,

aqk/aXi

(15.8) (15.9) (15.10)

where is the mixed convective potential (9.21). The symmetric structure and simplicity of these equations is worth noting in view of the

New Variational-Lagrangian Irreversible Thermodynamics

61

complexity of the physics. We may consider f p , ui, M!, and Si as the field unknowns, describing the evolution of the state of the system. However if the contribution from s* is significant in determining the state of the system we may add an auxiliary equation such as Eq. (1 1.21) for the rate of dissipation or proceed as in Sections XI and XI1 by introducing a pseudo-energy fluence %,f in the field equations. By neglecting s* in first approximation we may describe the system in terms of generalized coordinates qi as uj

= uj(qt 9 q2r

XI),

i@ = MjYq, 42, 9

Sj = sj(q1, 92, 5p

= tp(q17 q2r

*..,X I ) ,

(15.1 1) (15.12)

..., X I ) ,

(15.13)

XI).

(15.14)

*'.9

By applying the principle of virtual dissipation with arbitrary variations 6qi, we derive the Lagrangian equations a9Iaq; + R; + aDTM/aqi= Q ; ,

(15.15)

where (1 5.16)

is similar to Eq. (1 1.36) and represents the generalized chemical dissipative force or affinity. The dissipation function for thermomolecular diffusion is

and the generalized driving force

is due to forces jj applied at the boundary per unit initial area and to thermodynamic forces of the environment on the open system. This expression contains the particular case [Eq. (12.50)] derived above. For the case where chemical reactions are not far from equilibrium, the evolution is quasi-reversible, and we may introduce a chemical dissipation function DCh=

71

TO

91ch dfl

(15.19)

62

M . A . Biot

where gchis a quadratic form in

&, such that

clip = 8 9 C h / 8 t P .

(15.20)

In this case the Lagrangian equations (15.15) become a9/aqi

+ a ~ / a q =, ei,

(15.21) (15.22)

is the total dissipation function. Note the formal identity of (15.21) with (13.38) for the creeping viscous solid.

XVI. Thermodynamics of Nonlinear Viscoelasticity and Plasticity with Internal Coordinates and Heredity A general theory of linear viscoelasticity based on the Lagrangian equations of irreversible thermodynamics was established (Biot, 1954, 1955, 1956a, 1958). Its characteristic feature is the use of the concept of internal coordinates to represent heredity properties. The same Lagrangian formulation provides a natural extension to nonlinear viscoelasticity (Biot, 1976b). We shall consider the linear case in a subsequent section dealing with general linear phenomena and present here the thermodynamic theory of nonlinear viscoelasticity. In order to formulate the stress-strain relations of the solid we apply a finite homogeneous deformation defined by six strain components cijto a solid element that is initially a cube of unit size. The six strain components may be defined by any of the various ways described in Section V . As already pointed out, they are not necessarily tensor components. The six stress components T~ are corresponding forces applied to the faces of the solid element and defined by the principle of virtual work. Obviously the solid element represents a domain fl as considered in Sections XI1 and XV, where cijplays the role of generalized coordinates qi while T~ plays the role of generalized forces Qi applied externally to the system. In addition to external degrees of freedom represented by q ,there may be internal ones represented by internal coordinates q s , for which there are no external driving forces (Qs = 0). Such will be the case if, for example, the solid contains substances in solution that react chemically. Internal coordinates may also correspond to internal fluence fields due to thermomolecular diffusion if the material possesses a microstructure of inhomogeneities.

New Variational-Lagrangian Irreversible Thermodynamics

63

Nonlinear viscoelasticity may be defined by the property that the system is quasi-reversible, i.e., that the irreversibility is represented by a dissipation function that is a quadratic form in the time derivatives of the generalized coordinates. The stress-strain relations are then obtained by writing the Lagrangian equations of the system. We write the exergy of the solid element as

v = V ( q r9, q s ) ,

(16.1)

where Y is the entropy of the solid element and qs represents a very large number of internal coordinates. The dissipation function

D

=

D(&ijrE i j . Y , 3, q s , ris)

is a quadratic form in iu,3, and element are

cis.

(16.2)

The Lagrangian equations of the

av/aEij + aD/ai,, = Tl,,

(16.3)

+ aDla9 = 8 , av/dq, + aD/aq,7= 0.

(16.4)

dV/dY

(16.5)

By referring to Eq. (8.31), we note that because of the assumption of quasi-reversibility, S* is negligible; hence this equation, integrated over the domain R of the element, shows that Y is the surface integral of the fluence Si at the boundary. Hence 9’is a state variable whose conjugate generalized force is 8. As a consequence, T dY represents the amount of heat provided reversibly to the solid element. Equations (16.5) are linear in cis. They may be integrated for q 5 ,which then become functionals qs

= %s[Eij(f)9

W)l

(16.6)

of E i j ( t ) and Y(r). When these functionals are substituted into Eqs. (16.3) and ( I 6.4), these equations provide the stress-strain relations for nonlinear thermoviscoelasticity. Note that this result takes into account the thus yielding a temperature change and heat injection at the rate heredity of the specific heat. Plasticity may be handled in the same way (Biot, 1976b). However in this case the deformation is not quasi-reversible, and we may not assume the existence of a dissipation function. The internal coordinates for a plastic material are represented by dislocation slips q ; . The double subscript defines the orientation of the dislocation and the slip direction. For simplicity we assume constant temperature T = T O .The exergy

e,

v = V ( E i j , qfj)

(16.7)

is a function of the deformation and the internal coordinates. The virtual dissipation is

64

M . A . Biot To as*

=

R!. lJ aq!. lJ

7

(16.8)

where R ; is the dissipative force acting on the dislocations. It may be expressed as a nonlinear rate function R.F.=RS.(&.. rJ U UY &.. I J ? q!. 4;)' rJ7

(16.9)

The corresponding Lagrangian equations

av/aeij = T i i ,

(16.10)

+ R; = o

(16.11)

evlaq;

represent the stress-strain relations for plastic deformation with internal coordinates q; and heredity. Comparison with Eq. (15.16) shows that R ; is the tensor equivalent of the affinity. Strain hardening is represented by freezing an increasing number of internal coordinates qi as the deformation proceeds. For a material that becomes weaker as the deformation increases, the opposite procedure is used by releasing an increasing number of internal coordinates. This property may be assimilated to internal failures. It is of particular interest for fiber composites. For a viscoelastic continuum we may also derive field equations from the variational principle (Biot, 197613). The result is entirely similar to Eqs. (15.7) and (15.9). However for a solid that is not centrosymmetric we and must take into account possible coupling terms of the type of iuip in the dissipation function. This leads to a coupling between the temperature gradient and the rate of deformation. These more general results are easily obtained by straightforward application of the variational procedures described above.

XVII. Dynamics of a Fluid-Saturated Deformable Porous Solid with Heat and Mass Transfer The principle of virtual dissipation has been used to develop the field and Lagrangian equations for the dynamics of a porous solid with fluid saturation of the pores and heat transfer by convection and conduction. We shall follow the procedures developed earlier for the isothermal noninertial case (Biot, 1972) and later for the nonisothermal dynamic case (Biot, 1977b). For the noninertial case the problem is similar to that of a solid with a single substance in solution, and in many ways it may be considered as a particular case of the one treated in Section XV. However in the present

65

New Variational-Lagrangian Irreversible Thermodynamics

case we shall also take into account the inertial forces due to the motion of the porous frame and the motion of the fluid relative to the frame. We start with a unit initial volume of the material, initially of total mass mo. As the material is deformed, the exergy per unit initial volume is 'Y

where

cij are

" V ( E ~m , , T, XI).

=

(17. I )

the six strain components as defined in Section V, and

m

=

-aMi/axi

(17.2)

is the mass of fluid added in the pores per unit initial volume. The contravariant fluence M i is the mass of fluid that has been flowing through a material surface of the porous frame initially equal to unity and initially normal to x i . A material point of the solid frame initially of coordinates xi is displaced to a point of coordinates X i = xi + u i . If the porous material, which is approximated as a continuum, is inhomogeneous, Sr is also a function of the initial coordinates x i . Using procedures similar to those in Section XV, we may evaluate the entropy Y of the element in terms of q , m , T , and XI. Hence we express Sr replacing T by Y as one of the state variables. We write Sr

=

' Y ( E Um , , Y ,XI),

(17.3)

with the property analogous to Eq. (15.5):

(17.4) where 0 = T - To.The fluid pressure pr in the pores in the present case is equal to the injection pressure, which was denoted by pk for the case of a substance in solution. The corresponding injection enthalpy Ef and injection entropy Sf are given by Eqs. (2.2) and (2.4) as

where p i , pi T ' , and F; are the pressure, density temperature, and specific entropy of the fluid along the path of thermobaric transfer from a supply cell of fluid at the pressure po and temperature To. The convective potential is

(17.6) and the mixed collective potential is now

9=

[Sr

+ (mo + m)%(X)]di2,

where % is the gravity potential.

(17.7)

M . A . Biot

66

We may write the rate of dissipation per unit initial volume as Ti* = 2 9 ,

(17.8)

+ C;MiSj + (TI2)AuSiSj

(17.9)

where the dissipation function 1

9 = -2 CrMiMj

is a quadratic form in A& and Si with coefficients functions of the local state variables eu and T and the initial coordinates xi. The contravariant entropy flux is S

=

&Mi + HJT.

(17.10)

It embodies the convective heat transfer due to hi and the heat flux Hi due to conduction. For Mi = 0 the dissipation is due entirely to heat conduction and the dissipation reduces to the term 4TA,SiSj already obtained above (12.30) with a covariant thermal resistivity Au. For hi= 0 the dissipation function involves only Mi and leads to Darcy's law generalized to a nonisotropic deformable medium. In order to evaluate the inertial forces, we consider the momentum of the masses of solid and fluid of an element initially of unit volume and mass mo. This momentum is

A,= (mo + m)lii + M / J ,

(17.11)

where A/ is the mass flux of pore fluid relative to the solid frame per unit final area in the space Xi. The Jacobian (7.11) is the volume J of the element. We may express M/ in terms of the contravariant mass flux Mi using relations (7.9) and (7.11): (17.12)

Hence ./ui

=

(mo+ m)ui + Mj a7;ldxj.

(17.13)

Because the element of volume J is not of constant mass, the resultant of all inertial forces is aMilat plus an additional term due to the rate of flow of momentum out of J . If we neglect the square of the relative velocity of the pore fluid then the resultant of the inertial forces is 41;

=

a.&/at

+ a(uiMj)/axj.

(17.14)

The virtual work of the inertial forces for the porous solid, when we vary 6ui and 6 M i , is

New Variational-Lagrangian Irreversible Thermodynamics

67

where fi is the domain R in the coordinates X i . In evaluating the second integral we have assumed that the acceleration of the fluid particles may be approximated as iii. On the other hand, in variational form (7.12) is written

J 6 ~ =/ axi/axj 6

~

~

.

(17.16)

Hence (7.15) becomes

(17.17)

By varying 6ui and 6Mi inside the domain R and applying the principle of virtual dissipation, we proceed as in Sections XI1 and XV. This leads to the field equations (17.18) aQ/aXi

+ a%/anii = - U j

aT/axi +

axj/axi,

a%/aSi = 0,

(17.19) (17.20)

where (O = a"lr/am + % = 4 + %, rpy= a"lrla&,,, and aii = aui/axj. The equations are for the field components Mi and S i . If we wish to take into account the entropy produced s* as contributing to the state variables we add the auxiliary equations (17.8). Lagrangian equations are also obtained directly from the principle of virtual dissipation by describing the fields ui , M i , and Si in terms of generalized coordinates, as in Eqs. (15.1I), (15.12), and (15.13), assuming Si to be a state variable as an approximation. The Lagrangian equations are written Ii

+ aD/aqi + a9/aqi = Q i ,

(17.21)

with the generalized inertial force

(17.22) the dissipation function D = TI / n To r9dR, and the generalized driving force at the boundary A

(17.23)

M . A . Biot

68

The forces J;. are applied at the boundary per unit initial area. A semilinear theory of deformation of porous solids was also developed for the particular case where linear dependence is assumed between fluid pressure and microscopic volume changes while bulk deformations remain nonlinear (Biot, 1973~). The problem of fluid flow through a rigid porous solid with heat transfer was also discussed in detail (Biot, 1978), including phase changes from vapor to liquid and multiple diffusion channels due to surface adsorption. The results lead to a large variety of possible applications in geothermal and aquifer problems with heat and fluid flow including associated subsidence.

XVIII. Linear Thermodynamics near Equilibrium Perturbations of a system near equilibrium are described by small changes of the state variables. We denote by 8 the first-order increase of the initial uniform temperature To.Other first-order perturbations are M k , the mass increase of each molecular species per unit volume due to convection, the entropy Y ,and the reaction coordinates t,,with initial zero values. The affinities also vanish at equilibrium, and their perturbations A , are first-order quantities. There are two important simplifications in the linear theory. First consider the linearized value of the first-order entropy obtained from Eq. (23, (18.1) where h is the heat acquired per unit volume and Fok is the initial value of the injection entropy in the initial equilibrium state. Since 9’and Mkare state variables, the heat injected h is also a state variable. Another simplification, which is a consequence of the first, is the use of the thermobaric potential I/Jk instead of the convective potential Pk. For example, we may write Eq. (15.5) for the more general case of a solid, in differential form, as =

Tii

dEij

-

2P A, d t p + 2 I/Jk dMk + (8/To)dh, k

where dh is now a state variable. The heat fluence Hi defined by

(18.2)

New Variational-Lagrangian Irreversible Thermodynamics h

=

-aHj/axj

69 (18.3)

is now also a state variable. By introducing the constant factor l/To we may use as state variable the thermal entropy sT.We write sT = hITo

=

ST

-aST/dx;,

=

H;/To,

(18.4)

with a thermal entropy fluence ST. These quantities were introduced in the earlier developments of the theory (Biot, 1956b). Finally, we may of course use the total entropy fluence (18.5) The fact that it is a first-order state variable is also a consequence of Eq. (8.31), namely, Y = -as,/ax, + s*. (18.6) Since the entropy produced s* is now of the second order, it may be neglected in a first-order description of the state of the system. Another point of importance in the linear theory is the dual role played by the energy (4L. and entropy Y.This may be illustrated in the simple case of linear thermoelasticity (Biot, 1956b, 1981). For this case the linear equations of state are 7rJ. . = C?"& rJ Fw - PiP? (18.7) where C;" are the isothermal moduli of linear elasticity. The energy and entropy per unit volume are obtained by applying Eqs. (6.2) and (6.3) for the particular case d t = d M k = 0. They become d%

= rg

dY

=

dEg + dh,

dhIT.

(18.8) (18.9)

By using the equations of state (18.7) in (6.7), the value (6.4) of dh becomes dh

=

T& dejj

+ c do,

(18.10)

where c is the heat capacity per unit volume at constant strain, which we shall assume constant. We integrate the values (18.8) and (18.9) along a path, first for 0 = 0 and then for dEg = 0. Keeping only first- and secondorder terms, we derive

+

with

(4L. = ~ C ~ " ETosT, ~ ~ E ~ ~

(18. I I )

Y = S~

(18.12)

-

A(cO2/T;),

M . A . Biot

70

TosT = h = T & E ~

+ ce.

(18.13)

We see that sT is a first-order state variable, but it does not represent the correct entropy to the second order. The thermoelastic potential or exergy becomes (Biot, 1981) Q

=

+

(4.L - TOY = & ( C ; ” E ~ ~ Ecf12/To). ~,,

(18.14)

It is interesting to note that in this expression the first-order terms drop out. Hence the entropy plays here a dual role, first as a linearized state variable sT and second as a value 3’ that includes second-order terms. A general discussion of the linear theory is particularly fruitful in Lagrangian form. Small perturbations from equilibrium may be described by expressing the fluence and displacement fields as linear functions of generalized coordinates qi . For example, we may represent materia! point displacements as (18.15)

where uji(xJ are fixed displacement fields. Fluence fields Mjk and Sj as well as &, are represented similarly. Application of the principle of virtual dissipation then yields the linear Lagrangian equations for the perturbations q i . They are (18.16)

with a kinetic energy (18.17)

a dissipation function (18.18)

and a mixed collective potential (18.19)

The constant coefficients are symmetric:

mV. . = mP. ’.

bV. . = bJI. .

while 3 and D are positive definite.

aV. . = aJI. .

(18.20)

New Variational-Lagrangian Irreversible Thermodynamics

71

Note that the generalized coordinates constitute a complete and accurate representation of the physical system and are not essentially “trial functions.” This can be seen by using as generalized coordinates values of the fields at the vertices of a grid system of finite elements sufficiently small so that macroscopic laws are still valid while fluctuations at the molecular scale do not yet enter into plciy (see Section XXVI below). The Lagrangian equations (18.16) thus govern and unifL a vast domain of linear physics obeying a single universal mathematical formalism. In particular, in the absence of inertial and gravitational forces we may write av/aqi + avlaq,

a m g , = ei,

(1 8.2 1)

+ ~ D I ~ G= .0,,

(18.22)

where qs are a large number of internal coordinates, while Q; are external driving forces. Consider the case where the equilibrium is stable. Then V is also nonnegative. We then solve Eqs. (18.22) for qs and substitute these values into Eq. (18.21). The forces Qi are then expressed as (Biot, 1954) (18.23) with (18.24) where Db, D, and Di are nonnegative symmetric matrices and rs 2 0. The quantity p must be interpreted as p = io for the case where qi varies proportionally to the harmonic function of time exp(pt). For a nonperiodic dependence on time starting at t = 0 in a quiescient system, it is easy to show that Z , may be interpreted as an operator where P f ( t ) = e-rf P+V

1;

df e“’ - d t ’ , dt’

z.

p f ( t ) = df

(18.25)

This interpretation is completely general provided we introduce generalized Dirac functions (Biot, 1970). The linear Lagrangian equations (18.21) and (18.22) were applied to derive the stress-strain relations of linear viscoelasticity with heredity (Biot, 1954, 1955, 1956a). By proceeding as in the more general nonlinear case in Section XVI, we consider the driving forces Qito represent stresses rii applied to a unit cube of solid, and E , to represent the associated response q i . Equations (18.23) in this case are written 7.. !I =

cev, !/ Irv ’

(18.26)

M . A . Biot

72 with the operators

(18.27)

e’ff“

epu

These operators satisfy the symmetry properties = elW’ J = JI = @‘. I/ They are the same as the symmetry relations satisfied by the elastic moduli in classical linear elasticity. They also formally coincide with the elastic moduli obtained for various cases of geometric symmetry, such as isotropy, cubic symmetry, etc. As a consequence, a principle of viscoefastic correspondence was obtained (Biot, 1954, 1955, 1956a, 1958) whereby all formulas of linear elasticity are immediately applicable to linear viscoelasticity by simply replacing the elastic moduli by the operators (18.27). Another important application of linear thermodynamics is to porous media including viscoelastic behavior of the solid component. The stresses ru and the fluid pore pressure pf are considered as driving forces Q l ,while the response q1is the bulk solid strain cIJand the volume of fluid 5 which has entered the pores. Its value is 5 = m / p f , where m is the mass of fluid added in the pores per unit initial volume of the bulk material and pr is the fluid density. By applying the general solution (18.23) we obtain the stress-strain relations (Biot, 1962) rpU

Pr

= =

Arc,+ &IJ(?

(18.28)

+ML

(18.29)

where the operators A;”, M u ,and M are of the type (18.27). Note that the heredity properties are due not only to viscoelasticity of the solid itself but also involve interactions of a very general relaxation type between the fluid and the solid, such as fluid squirting in microcracks and between grains, microthermoelasticity, mutual solubility, adsorption, and surface diffusion as well as chemical reactions. A principle of viscoelastic correspondence is also valid for porous media whereby all formulas of the purely elastic theory are valid for the viscoelastic case if one simply replaces the elastic coefficients by the corresponding operators of the stress-strain relations (18.28) and (18.29) (Biot, 1962). The linear thermodynamic theory has also been applied to piezoelectric crystals with thermal dissipation (Mindlin, 1961, 1974) by adding suitable electrostatic terms.

New Variational-Lagrangian Irreversible Thermodynamics

73

XIX. Linear Thermodynamics of a Solid under Initial Stress The theory of small perturbations of a system initially in thermodynamic equilibrium is quite general and is applicable to the case where the system is in equilibrium in an initial state of stress. However the state of initial stress must be taken into account in the evaluation of the thermodynamic functions. The mechanics of initially stressed continua including elastic and viscoelastic properties was treated extensively in a monograph (Biot, 1965b). The theory was later extended on the basis of thermodynamics to include thermomolecular diffusion and chemical reactions (Biot, 1977~). We shall present a summary of this completely general case. Attention is called to its formal applicability to porous media under initial stress insofar as we may identify seepage with thermomolecular diffusion. The problem of porous solids under initial stress was also treated earlier in a somewhat different context (Biot, 1963). The initial Cartesian stress components are denoted by Sy. It is important to note that the system may lie in a gravity field, so that equilibrium does not imply uniformity of the initial injection pressure P O k . As a consequence, the initial values Fok, E O k , and $0k of the injection entropy, the injection enthalpy, and the thermobaric potential may vary from point to point. The initial equilibrium temperature To is of course uniform. When the system is disturbed from equilibrium, the stress becomes (19.1) The stress is defined here as in Section V on the basis of virtual work and may be nontensorial, so that ty is the increase of T~ per unit initial area. The thermobaric potential becomes $k

= $Ok

+ A$k.

(19.2)

The perturbation involves a small displacement field uj of the solid corresponding small strain components e0-

= +(ay

+ aji),

ay = aui/dxj.

(19.3)

An important point here is that the linear strain components (19.3) are not sufficient to establish a linearized theory under initial stress. Actually we must develop the actual strain eij to the second order. We shall write (19.4)

M. A. Biot

74

where qijis a second-order quantity in a i j .For example, in the case ( 5 3 , we obtain qij =

f(eifiwpj

+ cjfiofii+ oifiojfi),

(19.5)

and in the two-dimensional case (5.7) 711

=

fa:h

722 = -ia2,(2a12

7 1 2 = BU2l(U22

+ a21), (19.6)

- all).

The perturbation from equilibrium is described by the displacement field ui of the medium, the masses Mk of various substances acquired by diffusion per unit initial volume, small chemical reactions t,, and temperature increments 8 . As already pointed out, in the case of the general linearized theory we may use instead of 8 the quantity sT = h/Toas a state variable, where h is the heat acquired per unit initial volume. According to Eq. (18.4), we also introduce the thermal entropy fluence ST as a state variable. Another simplification is also to use the thermobaric potential $k instead of +L. We shall first evaluate the exergy r/' per unit initial volume. Its differential (18.2) for the solid under initial stress is

or (19.8)

where dW = tij dsij -

A, d t ,

+

P

k

A$k dMk + 8 dsT

(19.9)

The exergy @ isIaf function of c i j ,t,,,Mh,and sT. Because So dcij + $Oh dMk is an exact differential, dW is also an exact differential. Furthermore we may consider aijas independent state variables. In a linear theory we need evaluate .Ir only to the second order, hence we may replace cijby e!, in the value of dW. We write dW

= to

de,

A, dt,

-

+

P

Integration yields the quadratic form

in e", t,, M k , and sT with the property

A$h dMk 1

+ 8 dsT.

(19.10)

New Variational-Lagrangian Irreversible Thermodynamics -A,

t , = dWlde,,

=

aW/a&,,

A@

=

dW/dMA,

75

8 = dW/dsT. (19.12)

The potential energy in the gravity field is

G = In(mO +

2 M k ) % ( T i )d f l ,

(19.13)

k

where %(Ti) is the gravity potential at the displaced point Ti = xi + u; and mo is the initial density. We develop %(Ti) to the second order in ui and write %o

%(xi)=

+ %;U; + i%ijUjl4j,

(19.14)

with %o

1

%(xi),

%i

= d%(~;)/d~i,

(8jj = d'%(x;)/dxi d x j . (19.15)

Hence to the second order except for a constant G

c M k + 5I mo%,uiuj]dSZ.

/o[mo%iui+ (%jo+ '8;~;)

=

(19.16)

h

We now apply the principle of virtual dissipation (9.19). With the value (19.8) of dSr it becomes i

Ii 6q; +

I

(6W

11

c

+ S, a&, +

JlOk

6 M k ) dSZ

h

+ 6G + T" 6S*

where AJ is the increment of forcesJ per unit initial area at the boundary ST is the thermal entropy fluence (18.5). When applied to the initial state of equilibrium under initial stress, the variational principle (19.17) yields

I,(s, 6eij + 2

+Ok

6

~

k

=

IA[f; c 6ui -

k

(+ok

+ kmo%;6u;)

+ %o)

ni a

~ ; ]d

~ .

(19.18)

We now subtract equation (19.18) from (19.17) and obtain

2 I; 6qi + 69 + To 6S* i

=

IAIAf; 6ui -

(Aqk k

+ Yiiui)ni 6M;

-

Oni SST] dA (19.19)

76

M . A . Biot

where

\

9 = (W

+ S"qU + %;ui ck M k + 21 mo%Uuiuj)d R

(19.20)

plays the role of an incremental mixed potential already encountered in the quoted monograph (Biot, 196513) for the less general case. The variational principle (19.19) represents the particular form of the principle of virtual dissipation for a system under initial stress. In a uniform gravity field %" = 0, and 93; is a constant equal to the acceleration of gravity. The virtual work of the inertial forces is

+ 2k mok(iii+ a:) 6ufJ d R ,

(19.21)

where mok is the initial mass of substance k per unit volume of solid and a! its acceleration relative to the solid. We denote by 6uf the virtual displacement of the substance associated with 6 M f . We have in the linear case a/ = MFimOk,

6uf = 6M//mOk.

(19.22)

With these values, (19.21) may be written in the form

where

(19.24) is the kinetic energy per unit volume. Finally we must evaluate the virtual dissipation. In the linearized theory we may assume the validity of Onsager's principle. Hence the dissipation is derived from a dissipation function. We write

where

9= 9 c h

+ 9TM.

(19.26)

The first term 5JChis a quadratic form in p', and represents the dissipation due to chemical reactions. The second term represents the dissipation due

New Variational-Lagrangian Irreversible Thermodynamics

77

to thermomolecular diffusion. It is a quadratic form in M,"and $7 analogous to (17.9) with constant coefficients. We now substitute the values (19.9), (19.20), (19.23), and (19.25) into the variational principle (19.19) and vary u i , t p ,M,", and ST arbitrarily inside the domain. Proceeding as before in the nonlinear problems (Sections XI, XII,XV),using integration by parts, we obtain the field equations (19.27)

aT/axi + -A,

awaST

=

0,

(19.29)

+ a9/a&

=

0,

(19.30)

where (19.31) A'PA

A$A

=

+ YiIul.

(19.32)

The quantity represents the increase of a mixed thermobaric potential $A+ Yi at the displaced point, as already considered in earlier work (Biot, 1963) for the initially stressed porous solid. The unknowns in the field equations are i d I , t,, M:, and Sy. They determine the state of the system to the first order as functions of time. Lagrangian equations are also obtained directly from the principle of virtual dissipation (19.20). Using representations of the fields u I ,t,, M,k, and S,' as linear functions of generalized coordinates ql of the type (18.15), we derive ( 1 9.33)

where

I,,

% dR

(19.34)

D = III 9 d R

(19.35)

3

=

is the total kinetic energy and

is the dissipation function. The generalized driving force is

M . A . Biot

78 Qi = IA(Af;

au. $ - 7

aMh a4i

nj

-

8

asT a4i

nl) dA.

(19.36)

Note that it is an incremental quantity defined by the increments A&, and 8 at the boundary A. The values of 3 , D,and 9 are quadratic forms with constant coefficients, formally the same as (18.17), (18.18), and (18.19). The Lagrangian equations (18.16) and (19.33), with or without initial stress, are identical, as they should be, because they both govern the same fundamental physics of perturbations of a thermodynamic system near equilibrium. The difference lies in the particular evaluation of 9 and Qi for each case. A difference also appears in the nature of the equilibrium state, which may be stable or unstable, as will be discussed in the next section. As already mentioned, the theory is directly applicable to fluid saturated porous media under initial stress. The only modification is in the kinetic energy, which is written

where u is a factor taking into account the distribution of the microvelocity field of the fluid in the pores. The same microvelocity field is also taken into account in the evaluation of the dissipation function associated with the viscous fluid seepage (Biot, 1963). A further generalization of the concept of generalized inertial forces is obtained by the introduction of viscodynamic operators (Biot, 1962, 1976b).

XX. Linear Thermodynamics and Dissipative Structures near Unstable Equilibrium Thermodynamic equilibrium may be stable or unstable. There are many cases in nature where the equilibrium is unstable. In particular this may be the case for systems under initial stresses, which may be due to gravity or externally applied forces. Such systems are governed by the linear Lagrangian equations (18.16). They exhibit important physical properties of a very general nature, which do not seem to have been recognized. One of these is the appearance of regular spatial distribution of the unstable perturbations that may be called dissipative structures and are not bifurcations. The erroneous notion that such structures require the system to

New Variational-Lagrangian Irreversible Thermodynamics

79

be nonlinear and far from equilibrium has been propagated by some currently fashionable schools. Another important property of linear instability is its nonoscillatory character. The properties of such unstable systems near equilibrium may be derived in complete generality by the linear Lagrangian equations (18.16) after putting (9; = 0, i.e., assuming no perturbations of the applied mechanical or thermodynamic forces at the boundary. The instability is governed by the Lagrangian equations (20.1)

or, explicitly, C ( mY. q. J”+ . b U. q. J‘ .+ a,. U q J.) = 0.

(20.2)

j

Solutions of these equations are of the type exp(pt) where the p are characteristic roots of the system. The roots p are either real or complex conjugate and the system is stable if the real part of each root is negative. However if there are roots with positive real parts the system is unstable. A fundamental theorem has been established that states that unstable roots are all real, hence that the instability is always nonoscillatory (Biot, 1965b, 1974). To show this, we consider a root p and its complex conjugate p * . The roots satisfy the equations

We multiply Eqs. (20.3) by q? and Eqs. (20.4) by 4; and add the results. Taking into account the symmetry properties mij = mj;,bij = bjj,and aij = uii, we obtain

Cu b 2 q+ pbij + aij)qjq? = 0, C ( p * 2 m i j+ p*b-Y

+ arJ, .qJq? ) . = 0.

(20.5) (20.6)

rJ

The difference of these two equations yields ( P - P*)[(P + P*)

2 mijqjq? + 2 bvqjq?] = 0. 1J

ij

(20.7)

M . A . Biot

80

By their physical nature the kinetic energy 9and dissipation function D , as represented by the quadratic forms (18.17) and (18.18), are positive definite. Hence (20.8)

If the solution is unstable, P

+ P* > 0,

(20.9)

and Eq. (20.7) cannot be verified unless p = p*, i.e., unless the root is real. Thus near unstable equilibrium the perturbations in the linear range are nonoscillatory and proportional to increasing exponentials. Note that this includes dynamical systems with inertial forces. Another property is derived by assuming an unstable solution with real p . Equation (20.5) shows that in this case we must have (20. lo)

Hence no instability is possible if 9 = &Cjjaiiqiqj is positive definite. This constitutes a fundamental stability criterion. An important example of linear instability is provided by a layer of viscous medium resting on a rigid base and surmounted by another viscous medium of higher density. Due to gravity forces, the system is under initial stress. The interface is unstable and shows a wavy structure of given wavelength and amplitude growing exponentially with time (Biot, 1965b) as illustrated in Fig. 1. The appearance of such dissipative structures in linear thermodynamics near equilibrium is quite general. In the context of geophysics and the formation of salt domes, such structures have been analyzed in detail (Biot and Ode, 1965; Biot, 1966).

2

'

FIG. 1 . Viscous layer ( I ) surmounted by a denser viscous layer (2) under gravity. Unstable waviness of increasing amplitude ( 3 ) appears at the interface.

New Variational-Lagrangian Irreversible Thermodynamics

81

XXI. Thermoelastic Creep Buckling Physical insight into the thermodynamics of systems near a state of unstable equilibrium is provided by the case of a purely thermoelastic continuum under initial stress. As a simple example, consider a straight elastic rod initially under axial compression. It will tend to buckle. If the thermal conductivity is very small, the buckling load will be determined by the adiabatic elastic coefficients. On the other hand, if thermal conductivity is very large, the buckling load will be determined by the isothermal elastic coefficients. Hence we may distinguish between isothermal and adiabatic buckling. In the case where the axial load is between the isothermal and adiabatic value the thermal conduction will determine the rate at which the buckling instability appears. According to the general theorem of the previous section, the buckling that includes the effect of the inertial forces will be nonoscillatory , and all buckling modes will exhibit exponentially increasing amplitudes. If the axial load barely exceeds the isothermal value, the buckling amplitude will increase slowly and constitute a form of creep motion. In this case, the rate of creep is limited and dominated by the thermal conduction. We are dealing here with an example of creep instability that does not involve any viscosity and is entirely of thermodynamic nature. When we increase the axial load the rate of buckling increases until it is dominated by the inertial forces and becomes a dynamic buckling. By idealizing the case for a massless material we see that creep buckling will occur between two critical buckling loads, a lower one for isothermal buckling and a higher one for adiabatic buckling for which the rate of buckling becomes infinite. In the range between these two loads the massless rod exhibits a finite rate of creep. These effects were brought to light in some earlier work and discussed in the context of a complete analogy with the buckling of a porous elastic medium saturated with a massless fluid (Biot, 1963, 1964). The two phenomena are isomorphic and belong to the same underlying theory of instability of linear thermodynamic systems governed by the general Lagrangian equations (18.16). I n the thermoelastic case, the dissipation function is due to thermal diffusion, whereas in the case of a porous medium it is due to viscous forces generated by fluid seepage between pores obeying Darcy’s law. The phenomenon was analyzed in more detail in later work in the context of general three-dimensional thermoelasticity. In particular, it was pointed out that for an isolated system the instability may be considered as occurring because the unstable state of equilibrium corresponds to a minimum value of the entropy of the whole system (Biot, 1973a, 1974).

82

M . A . Biot

XXII. Lagrangian Formulation of Bifurcations By their very nature, the generalized coordinates may be used to describe departures from a given time-dependent evolution of a system. For example, we may write the displacement field of a solid as ui = ui(q1, q 2 9 .*.,XI,t)i

(22.1)

where 41= q2 = ..*= 0 corresponds to a given time-dependent evolution. Note that in principle the dependence may be chosen arbitrarily, so that the case qi = 0 does not necessarily represent a solution of the equations of evolution. The principle of virtual dissipation is applicable to this more general case, where variations 6qi are applied to any state of the system as frozen at a particular instant t. Hence Lagrangian equations may be derived that govern the departures from a given arbitrary evolution as measured by the generalized coordinates q i . Of considerable interest is the case where qi = 0 represents a solution of equations governing the system and corresponds to an actual physical evolution. It is immediately evident that the Lagrangian equations in this case provide a clue to solutions that represent branching or bifurcations away from the case qi = 0. In particular, this provides a powerful method of testing the stability of a given evolution. There are many advantages associated with the Lagrangian formulation. One is the possibility of studyingfinite departures from a given evolution in contrast to linearized methods of infinitesimal perturbations. Another is that a finite number of generalized coordinates may be used, with resulting simplification in the numerical or analytical treatment. To be mentioned also is the probing of the accuracy of a given solution by testing the magnitude of possible departures.

XXIII. Generalized Stability Criteria for Time-Dependent Evolution Far from Equilibrium The Lagrangian equations may be applied to provide very general stability criteria for the time-dependent physical evolution of a system that is not near equilibrium and for which linear thermodynamics is not applicable. In order to illustrate the method we consider the case of a system for which the inertial forces are negligible. Putting Zi = 0, the Lagrangian equations (10.4) become

New Variational-Lagrangian Irreversible Thermodynamics a9/aqj + R; = Q;,

83 (23. I )

where Ri is a generalized dissipative force and Q; the generalized driving forces. Such equations govern (for example) coupled thermomolecular diffusion and chemical reactions in a gravity field. We denote by 4; =

(23.2)

cpiw

a time-dependent solution; a perturbed solution is given as qi = ~ i ( t + ) Aqi,

(23.3)

where Aqi is a small perturbation. The driving force Qi is given and maintained unperturbed. Substitution in Eqs. (23. l), neglecting higher-order terms, yields

a29

7-

Aqj

+ AR; = 0.

(23.4)

The perturbed dissipative force is (23.5)

The linear differential equations (23.4) in Aqi determine the time-dependent evolution of the perturbations Aq;. The coefficients of these equations are generally functions of time. If the perturbations Aq; tend to zero with time the evolution is stable. The mathematical theory of linear differential equations provides stability criteria for the solutions of the perturbation equations (23.4). When considered in the context of irreversible thermodynamics, special stability criteria may be obtained. Consider, for example, the steady-state evolution already discussed (Biot, 1976b). We write the perturbation equations (23.4) in the form (23.6) where a29

aR; aqj

d.. = -+-, r~

aq; aqj

dR; a,.=-. u

aqj

In addition, we assume that the system is quasi-reversible. In this case the dissipative forces are derived from a dissipation function (23.7)

M . A . Biot

84

which is a positive quadratic form in 4; with coefficients bij depending on q i . We derive

3,. rJ = b.. 1J = b.. JI

Ri = dD/dq,.,

9

(23.8)

and Eqs. (23.6) become (23.9)

By multiplying these equations by Aqi and adding the results, we obtain

2ii do Aqi A4j + 2 bu A4; Aq,j = 0 .

(23.10)

IJ

We note that for a steady state evolution bu is constant. Hence we may write (23.11)

with the positive quadratic form (23. 2)

Equation (23.10) is now d 2 d" Aqi Aq, + dt ( A D ) = 0. ij

(23. 3)

If we assume doto be a positive definite matrix, i.e., if

2ii

dij Aqi

Aqj

0,

(23.14)

then Eq. (23.13) shows that AD must decrease with time. Because it is positive definite, the values of A4; must also tend to zero. Hence the inequality (23.14) constitutes a fundamental stability criterion. If the inequality (23.14) is not verified, instability may arise. It may be due to the negative nature of d29/dqidqj, in which case it is analogous to the instability of a linear system near equilibrium considered in Section 20. The instability may also be due to the negative nature of (dRi/dqj+ dRj/t)4;),which arises essentially from the nonequilibrium state of evolution. Such a case of instability was illustrated by the example of an embedded viscous layer (Biot, 1976b), as recalled in more detail in the next section. In addition, in this example du = dji,which implies that the instability is nonoscillatory.

New Variational-Lagrangian Irreversible Thermodynamics

85

The stability criterion (23.14) differs fundamentally from those presently in vogue (Glansdorff and Prigogine, 1971) by its generality and simplicity as well as by the physical insight provided.

XXIV. Creep and Folding Instability of a Layered Viscous Solid Consider a solid viscous layer embedded in a large solid viscous medium of much lower viscosity. We assume incompressible media. A strain rate is imposed upon this system corresponding to uniform compressive strain parallel to the axis of the layer. For example, it may be compressed by two rigid frictionless planes normal to the layer, whose distance decreases with time. Obviously if the geometry is perfect, the layer will remain straight and be uniformly compressed. However, it is known (Biot, 1965b, 1976b) that the evolution is unstable, and if there are small initial perturbations of the geometry, they will grow and develop into a sinusoidal buckling of wavelength L

=

21rh

m,

(24. I )

where h is the layer thickness, -ql its viscosity, and -q the viscosity of the embedding medium. The result was verified experimentally. The regular pattern of wavelength L is obviously a dissipative structure due to an unstable state of evolution away from equilibrium. It has important implications in geology. In the context of the general stability theory in the previous section, the folding represents a perturbation of a state of evolution. The unperturbed state of evolution is a uniform compression where the layer remains perfectly straight. We have assumed isothermal deformation and neglected gravity forces. The Lagrangian equations of creeping motion of the solid in this case become extremely simple. They are aDlaq;

=

0,

(24.2)

where D is a quadratic form in 4 ; with coefficients depending on 4;.With

9= 0, the stability criterion is given by Eq. (23.14). It can be shown that it is not verified in this case for perturbations A4; from the uniform compression. The instability and Eq. (24. I ) for the buckling wavelength have also been derived by this method of perturbation of the Lagrangian equations (Biot, 1976b). Another example is provided by the internal folding of a stack of viscous layers of alternately high and low viscosity. When the system is

M . A . Biot

86

-IH P

- ” FIG.2. Stack of alternate layers of large and small viscosity between rigid planes subject to a compression P. Note the appearance of internal folding of wavelength 15.

compressed in the direction of the layers, an internal folding develops as shown in Fig. 2, with a wavelength L = 1.9 K H ,

(24.3)

where h is the thickness of the more viscous layers and H the total thickness of the stack (Biot, 1965a, 1967). These instability problems may also be considered from the standpoint of bifurcation, as discussed in Section XXII. In this case the uniform compression is represented by a steady state where the generalized coordinates are known functions of time q; = cpi(t). Any deviation from this steady state may then be represented by new generalized coordinates qi that represent departures from the steady state. We then evaluate the dissipation function D in terms of qf and qi.It will be a function that is linear and quadratic in qi with coefficients functions of q; and q i . The Lagrangian equations are of the same form as Eq. (24.2) with coefficients that may now be functions of time. We note that they express minimum dissipation under the constraint qf = cpi(i). A final remark is in order here in connection with Helmholtz’s theorem, which states that under creeping flow conditions a viscous fluid tends to a stable steady-state flow (see Lamb, 1932, p. 619). This is not in contradiction with our results, because Helmholtz’s theorem applies only to a fluid of uniform viscosity. This is not the case for an embedded layer or a stack of layers with two different viscosities.

XXV. Coupling of Subsystems and the Principle of Interconnection In many problems, we deal with complex systems made up of separate components that differ from each other by their physical nature. Each of

New Variational-Lagrangian Irreversible Thermodynamics

87

these components or subsystems may be analyzed separately, and Lagrangian equations may be obtained that govern its behavior. The analysis of each subsystem may usually be achieved by simple methods adapted to its particular physical nature in terms of a small number of generalized coordinates that determine the field distribution of mass and energy fluence, material displacements, and reaction coordinates. The evolution of each subsystem (s) is governed by Lagrangian equations

Ijs) + Rjs) +

&JJ(s)/dqi

=

Qjs)int +

Qjslext

(25. I )

where Qf'"''are the driving forces on the subsystems at a coupling interface while Qjs)ext are the driving forces external to the combined system. We may add Eqs. (25.1) for all subsystems. In this process the interfacial forces Qfs)intmay be grouped in pairs where they are equal and opposite in sign at each interface. Hence (25.2) and

(25.3) We thus obtain unified Lagrangian equations for the combined system from which interfacial forces have been eliminated. The method constitutes a generalization of a process of elimination of interfacial forces in classical mechanics by the method of virtual work. We note that in this classical context it is nothing but the expression of the third law of mechanics whereby action is equal and opposite to reaction. Its extension to thermodynamic forces is evident from expression (9.17), in which the entropy and mass fluences are continuous at the boundary while the outward normal directions ni of the subsystems are equal and opposite at the interface. A special remark is in order for mechanical systems that are not adherent at interfaces. In this case the interfacial virtual work vanishes only if there is no friction. However, in the context of the more general thermodynamic formulation we may consider the solids to be adherent by considering that one of the surfaces is constituted by a thin, deformable, adherent skin where the shear deformation generates the friction forces and entropy production. By this artijice, Coulomb friction may be included in the general formulation. The principle of interconnection is applicable to a wide variety of problems. It was discussed in the particular context of heat transfer (Biot, 1970). There are also a number of problems that have been treated in the past without realizing that the methods involved are particular cases of

88

M . A . Biot

such a broad unifying principle. For example, in aeronautical structural analysis and aeroelasticity during the years 1942-1945 it became common practice to consider normal modes of subsystems as generalized coordinates and interconnect the subsystems by modal synthesis. Another example is in classical mechanics, where Lagrangian equations are obtained for the motion of rigid solids in a perfect incompressible fluid (see Lamb, 1932, p. 160). These equations may be derived by using the interconnection principle. Lagrangian equations are obtained separately for the motion of the solids and that of the fluid due to generalized interfacial forces that are equal and opposite. By interconnection, dynamical equations are obtained for the solids that embody implicitly the dynamics of the surrounding fluid. Many more general problems suggest themselves here. Among others we may cite those of interaction between elastic solids and compressible fluids with or without viscosity.

XXVI. Completeness of the Description by Generalized Coordinates. Resolution Threshold and Lagrangian Finite Element Methods The present treatment of irreversible thermodynamics emphasizes the description of a complex system as an assemblage of cells. From a fundamental viewpoint, the size of these cells may be extremely small while remaining above a resolution threshold, below which the statistical average definition of temperature and entropy breaks down and fluctuations enter into play. The cells are finite in number and determined by a finite number of generalized coordinates, which, as pointed out earlier (Biot, 1970), provide a complete physical description from the macroscopic viewpoint. As a consequence, the corresponding Lagrangian equations also describe rigorously the evolution of the system. It is important to note that use of continuum models is an extrapolation, beyond the validity of pl- ysical laws, which introduces spurious difficulties in terms of completeness in the context of the mathematical concepts of measure, continuous sets, and functional space theories. Recent work by Woods (1981) has demonstrated the lack ofphysical validity of much of the current fashionable formalism of continuum mechanics and thermodynamics. The Lagrangian equations also provide the foundation of a large variety of finite element methods where the state of finite cells is described by

New Variational-LagrangianIrreversible Thermodynamics

89

generalized coordinates as values of scalar and vector fields at grid vertices, linear or quadratic interpolation providing values in the cells in terms of these generalized coordinates. The corresponding Lagrangian equations for the discrete variables are then obtained directly without prior knowledge of the jield differential equations.

XXVII. Lagrangian Equations in Configuration Space. Internal Relaxation, Order-Disorder Phenomena, and Quantum Kinetics The concept of threshold minimum size of cells, as described above for physical space, may be extended to subspaces in the abstract multidimensional configurational thermodynamic space (Biot, 1982b). For example, we may consider the translational and vibrational degrees of freedom of gas molecules as constituting distinct subspaces with their own entropy and temperature. The state of an assembly of such subsystems is then determined by mass and energy fluence between them. Following the same procedures as used for cells in physical space, completely general Lagrangian equations of evolution may be obtained with exchanges represented by internal fluence coordinates. In particular, this approach is applicable to internal relaxation effects in gases. This procedure with Lagrangian equations and internal coordinates is also implicit in the general thermodynamic theory of relaxation and heredity in viscoelasticity (Biot, 1954). The concept of subspace cells may be extended to quantum levels with their own temperatures and entropy. The kinetics of exchanges obeyed by fluence coordinates is then obtained from transition probabilities in quantum kinetics and statistics. The same procedure may also be used for order-disorder phenomena in metal alloys where the order-disorder state is described by internal fluence coordinates.

REFERENCES Biot, M. A. (1954). Theory of stress-strain relations in anisotropic viscoelasticity and relaxation phenomena. J . Appl. Phys. 25, 1385-1391. Biot, M. A. (1955). Variational principles in irreversible thermodynamics with application to viscoelasticity. Phys. Rev. 97, 14. Biot, M. A. (1956a). Variational and Lagrangian methods in viscoelasticity. I n “Deforma-

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M . A . Biot

tion and Flow of Solids” (IUTAM Colloquium, Madrid, 1959, pp. 251-263. Springer, Berlin. Biot, M. A. (1956b). Thermoelasticity and irreversible thermodynamics. J . Appl. Phys. 27, 240-253. Biot, M. A. (1958). Linear thermodynamics, and the mechanics of solids, Proc. Third U . S . Nut. Cong. Appl. Mech.. pp. 1-18. ASME, New York. Biot, M. A. (1962). Generalized theory of acoustic propagation in porous dissipative media. J . Acoust. SOC.A m . , 34, 1254-1264. Biot, M. A. (1963). Theory of stability and consolidation of a porous medium under initial stress. J. Math Mech. 12,521-542. Biot, M. A. (1964). Theory of buckling of a porous slab and its thermoelastic analogy. Trans. ASME, Ser. E 31, 194-198. Biot, M. A. (1965a). Further development of the theory of internal buckling of multilayers. Bull. Geol. SOC.A m . 76, 833-840. Biot, M. A. (1965b). “Mechanics of Incremental Deformations.” Wiley, New York. Biot, M. A. (1966). Three-dimensional gravity instability derived from two-dimensional solutions. Geophysics 31, 153-166. Biot, M. A. (1967). Rheological stability with couple stresses and its application to geological folding. Proc. R. SOC.Ser. A , 298,402-423. Biot, M. A. (1970). “Variational principle in heat transfer.” Oxford Press. Biot, M. A. (1972). Theory of finite deformations of porous solids, Indiana Uniu. Math. J . 21, 597-620. Biot, M. A. (1973a). Nonlinear thermoelasticity, irreversible thermodynamics and elastic instability. Indiana Uniu. Math. J . 23, 310-335. Biot, M. A. (1973b). Buckling and dynamics of multilayered and laminated plates under initial stress. Int. J . Solids Struct. 10, 419-451. Biot, M. A. (1973~).Nonlinear and semilinear rheology of porous solids. J . Geophys. Res. 78,4924-4937. Biot, M. A. (1974). Thermoelastic buckling. An unstable thermodynamic equilibrium at minimum entropy. Bull. CI. Sci. Acad. R . Belg. 60, 116-140. Biot, M. A. (1975). A virtual dissipation principle and Lagrangian equations in nonlinear irreversible thermodynamics. Bull. CI. Sci. Acad. R . Belg. 61,6-30. Biot, M. A. (1976a). New chemical thermodynamics of open systems. Thermobaric potential, a new concept, Bull. CI. Sci. Acad. R . Belg. 62,239-258; Erratum 62,678. Biot, M.A. (l976b). Variational-Lagrangian irreversible thermodynamics of nonlinear thermorheology. Q. Appl. Math. 34, 213-248. Biot, M. A. (1977a). New fundamental concepts and results in thermodynamics with chemical applications, Chem. Phys. 22, 183-198. Biot, M. A. (1977b). Variational-Lagrangian thermodynamics of nonisothermal finite strain mechanics of porous solids and thermomolecular diffusion. Int. J. Solids Struct. 13, 579-597. Biot, M. A. (1977~). Variational-Lagrangian irreversible thermodynamics of initially stressed solids with thermomolecular diffusion and chemical reactions. J. Mech. Phys. Solids 25, 289-307; Errata 26,79. Biot, M. A. (1978). Variational irreversible thermodynamics of heat and mass transfer in porous solids: new concepts and methods. Q. Appl. Math. 36, 19-38. Biot, M. A. (1979). New variational-Lagrangian thermodynamics of viscous fluid mixtures with thermomolecular diffusion, Proc. R. SOC.London, Ser. A . 365, 467-494. Biot, M. A. (1981). Generalized Lagrangian thermodynamics of thermorheology J . Thermal Stresses 4. 293-320.

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Biot, M. A. (1982a). Thermodynamic principle of virtual dissipation and the dynamics of physical-chemical fluid mixtures including radiation pressure, Q. Appl. Marh. 39, 5 17540. Biot, M. A. (1982b). Generalized Lagrangian equations of nonlinear reaction-diffusion, Chem. Phys. 66, 11-26. Biot, M. A., and Ode, H. (1965). Theory of gravity instability with variable overburden and compaction. Geophysics 30,213-227. Brillouin, L. (1930). “Les Statistiques Quantiques et Leurs Applications.” Presses Universitaires de France, Paris. Chung, B. T. F., and Yeh, L. T. (1975). Solidification and melting of materials subject to convection and radiation. J . Spacecr. Rockets U ,329-333. De Donder, T. (1936). “L’AffinitC.” Gauthier-Villars, Paris. Fowler, R., and Guggenheim, E. A. (1952). “Statistical Thermodynamics.” Cambridge Univ. Press, England. Gibbs, J. W. (1906). “Thermodynamics I , ” Longmans, London. Glansdorff, P., and Prigogine, I. (1971). “Structure, StabilitC et Fluctuations.” Masson, Paris. Hatsopoulos, G. N.. and Keenan, J . H. (1965). “Principles of General Thermodynamics.” Wiley, New York. Lamb, H. (1932). “Hydrodynamics.” Dover, New York. Lardner, T. J. (1963). Biot’s variational principle in heat conduction. AIAA J. 1, 196-206. Lardner, T. J. (1967). Approximate solutions to phase change problems, AIAA J . 5 , 20792080. Lonngren, K. E., and Hsuan, H. C. S . (1978). A consequence of the invariance of Biot’s variational principle in thermal conduction. J. Math. Phys. 19, 357-358. Meixner, J. (1941). Ziir Thermodynamik der Thermodiffusion. Ann. Phys. 39, 333-356. Mindlin, R. D. (1961). On the equations of motion of piezoelectric crystals. In “Problems in Continuum Mechanics,” pp. 282-290. SOC.Ind. Appl. Math. Philadelphia, Pennsylvania. Mindlin, R. D. (1974). Equations of high frequency vibrations of thermopiezoelectric crystals. Int. J . Solids Struct. 10, 625-637. Onsager, L. (1930). Reciprocal relations in irreversible processes I. Phys. Reu. 37,405-426. Onsager, L. (1931). Reciprocal relations in irreversible processes 11. Phys. Rev. 37, 22652279. Prasad, A., and Agrawal, H. C. (1972). Biot’s variational principle for a Stefan problem. AIAA J . 10, 325-327. Prasad, A., and Agrawal, H. C. (1974). Biot’s variational principle for aerodynamic ablating melting solids. AIAA J . l2,250-252. Senf, L. (1981). A special case of diffusion with moving boundary. Int. J . Hear Muss Transfer 24, 1903-1905. Sokolnikoff, I. S. (1951). “Tensor Analysis.” Wiley, New York. Washizu, K. (1975). “Variational Methods in Elasticity and Plasticity.” Pergamon, Oxford. Woods, L. C. (1981). On the local form of the second law of thermodynamics in continuum mechanics. Q. Appl. Math. 39, 119-126. Yeh, L. T., and Chung, B. T. F. (1977). Phase change in a radiating medium with variable thermal properties. J . Spacecr. Rockets, 14, 178-182.

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ADVANCES IN APPLIED MECHANICS, VOLUME

24

Incompatible Elements and Generalized Variational Principles WEI-ZANG CHIEN" Tsing Hcra University Beijing, People's Repcihlic of Chino 1. Introduction

............................................

11. Generalized Variational Principle Related to Incompatible Elements of Small-Displacement Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Small-Displacement Elasticity Problems

.......................

94 94 94

B. Generalized Variational Principle Used for Compatible Elements Derived from the Minimum Potential Energy Principle . . . . . . . . . . . . . . . . . . .

96

C. Generalized Variational Principle for Incompatible Displacement Elements Derived from the Minimum Potential Energy Principle . . . . . . D. Generalized Variational Principle of Hybrid Incompatible Elements Derived from the Minimum Potential Energy Principle . . . . . . . . . . . . . E. Global Generalized Variational Principle Derived from the Minimum Complementary Energy Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Generalized Variational Principle of Compatible Elements Derived from the Minimum Complementary Energy Principle . . . . . . . . . . . . . . . . . . G. Generalized Variational Principle for Incompatible Elements Derived from the Minimum Complementary Energy Principle . . . . . . . . . . . . . . 111. Generalized Variational Principle of Incompatihle Elements for the Plane Problems in Elasticity. . . . . . . . . . . . . . . . . . . ...........

103 108

I 11

I I2 I IS I I8

A . Generalized Variational Principle of Displacement-Incompatible Elements

Derived from the Minimum Potential Energy Principle

........

I I8

B. Generalized Variational Principle of lncompatib the Minimum Complementary Energy Principle

I24

IV. Generalized Variational Principle for Plate Element

I27

A. Minimum Potential Energy Principle of Plates and the Related

H. C.

D. E.

Generalized Variational Principle for Compatible and Incompatible Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Matrix Equations of Deflection-Incompatible Elements . . . . . . . . . . . . . Generalized Variational Principle Derived from the Minimum Complementary Energy Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . (ienerdlized Variational Principle o f Incompatible Elements Derived from ........... the Minimum Complementary Energy Principle . . . . The Matrix Formulation of Incompatible Elements Thin Plate. . . . . . . . . . . . . . . . . . . . . . . . . . . ...........

V. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

*

I27 131 I39 I45 I47

IS2 153

Present address: Shanghai Technical University. Shanghai, People's Republic of China. 93 Copyright 0 1984 by Academic Press, I n c . All nghts of reproduction in any form reserved. ISBN 0-12-002024-6

94

Wei-Zang Chien

I. Introduction The modified variational principles (or generalized variational principles) have been studied in detail by Tong (1970) and Pian and Tong (1972) for the purpose of formulating incompatible hybrid finite elements, but the continuity conditions of field variables on the interelement boundaries are represented in general by means of Lagrangian multipliers. In the process of variation, Lagrangian multipliers are considered to be undetermined variables. Thus in the matrix computations of finite elements, we have to use higher degrees of freedom and consequently a more complicated form of rigidity matrix. Chien (1980) indicated that, by means of Lagrangian multipliers, the conditional principles of variation can be reduced to nonconditional principles of variation, and, by means of the stationary condition of variational functionals, the unknown physical nature of these multipliers can be determined; that is, these unknown multipliers can be determined in terms of the original physical quantities. Thus the number of degrees of freedom can be reduced to the original number, and simplification of the incompatible-element calculation can be achieved. In this chapter this method is illustrated by means of static problems in elasticity. There is, however, no difficulty in using this method in other fields.

11. Generalized Variational Principle Related to

Incompatible Elements of Small-Displacement Linear Elasticity A. SMALL-DISPLACEMENT ELASTICITY PROBLEMS In Cartesian coordinates xi (i = 1, 2, 3), the small-displacement elasticity problems can be formulated by means of the following three sets of equations: (I)

Equations of stress equilibrium:

-

m IJ.-J

,

+ Fi= 0,

?

in r, i 3.1, 2, 3.

(2.1)

(2) Stress-strain relations:

,, ',.. L I .

c.. = ,/u>' p !./ . =

h.. ,,u0-11,

i, .i = 1 , 2. 3;

(2.24

2. 3.

(2.2b)

i,j

= 1.

Incompatible Elements: Generalized Variational Principles

95

(3) The strain-displacement relation: "jj

= HM;,;

+ u;,;),

(2.3)

in which M;.,; = arr;/dx; with respect to a Cartesian coordinate system, 'T;; are the components of stress tensor, ev the components of strain tensor, ui the components of displacement, F j the components of body force, uiiAl the rigidity constants of elasticity, b, the flexibility constants of elasticity, and aqkland bjjk/satisfy the following symmetrical relations: a$,/ = a;;Al =

aijlA =

a!,/;;

(2.4a)

b.. !Ik/ = b.. ,lrhl

b.. rJlA

hA/;;.

(2.4b)

=

=

7

On the boundary surfaces of an elastic body there are boundary conditions to be satisfied. The boundary surfaces can be divided into two parts: (1) on S,, the external force pi is given, and (2)on S , , the displacement i& is given:

s = s, + s,

(2.5)

and

q n j = pi, -. uI. = u I ,

on S,, on

s,,

(2.6a) (2.6b)

where nj are the direction cosines of the outward normal to the surface. From the preceding relations, we see that the equilibrium problems of an elastic body are in fact boundary value problems. There are altogether fifteen unknowns, that is, six components of stress tensor f l u , six components of strain tensor el;,and three components of displacement ui.There are fifteen equations [Eqs. (2.1)-(2.3)] for the determination of these fifteen unknowns under the boundary conditions (2.6a) and (2.6b). Let A ( e O )be the strain energy density of an elastic body. It is a function of strain components. We have by definition (2.7) Similarly, the complementary energy density B ( q ) of an elastic body is a function of stress components. By definition, we have A ( e j j )= &aijkleijekl.

B(@g) = &bjjk[gjjck/.

(2.8)

Thus we have the well-known minimum principle of potential energy and the minimum principle of complementary energy for a linear elastic body in small-displacement deformation. Variational Principle I (the minimum principle of potential energy for small-displacement deformation and linear elasticity): Among all per-

96

Wei-Zang Chien

missible strains ev and ui that satisfy the small-displacement relation of strain and displacement (2.3) and the boundary conditions of given displacement (2.6b), the actual solution of eij and ui minimizes the total potential energy of the elastic body

n, = / / / [ A ( e j i )- F ; r c i ] d~ -

/I

pirr; dS.

(2.9)

s,,

7

That is to say, the functions eg and ui that minimize the functional HI in (2.9) under the conditions of (2.3) and (2.6b) also satisfy the equation of equilibrium (2.1) and the boundary conditions for given external forces (2.6a). For this principle, A(eg)is defined to be faijk,evek,.The proof of this principle is well known. Variational Principle I1 (the minimum complementary energy principle for small-displacement deformation and linear elasticity): Among all permissible stresses uvthat satisfy the equilibrium condition (2.1) and the boundary condition of given external force (2.6a), the actual set of uij minimizes the complementary energy 1111of this elastic body. ~

I =I

/ J / [ B ( U ( j ) dT ] 7

//

u;CT;jtZ,j

dS.

(2.10)

.&,

That is to say, the functions uijthat minimize n,, under the equilibrium condition (2.1) and the boundary condition of given external forces (2.6a) also satisfy the boundary conditions of given displacement (2.6b). For this principle, B(u& is defined to be i i b i j k p p k , , and the strain-stress (2.2b) and displacement-strain relations (2.3) are used. The proof is also well known. Variational Principles I and I1 are conditional principles of variation. The conditions for the minimum potential energy principle are (2.3) and (2.6b), while the conditions for the minimum complementary energy principle are (2.1) and (2.6a). In the former, the stress-strain relation (2.2a) is used, while in the latter, the strain-stress relation (2.2b) and displacement-strain relation (2.3) are used. The kinds of relations used in the process of the proof can also be considered as subsidiary conditions of variation.

B . GENERALIZED VARIATIONAL PRINCIPLE USED COMPATIBLE ELEMENTSDERIVED FROM THE MINIMUMPOTENTIALENERGYPRINCIPLE

FOR

Let us use the Lagrangian multiplier method to reduce the minimum potential energy principle with two variational conditions (2.3) and (2.6b)

Incompatible Elements: Generalized Variational Principles

97

from conditional Variational Principle I to a nonconditional variational principle. Let A, and pi be two Lagrangian multipliers which will be determined later. Then from Eq. (2.9), the functional of the nonconditional variational principle can be written as

Regarding e,, i d ; , A,, and pi as independent variables, the stationary condition of variation of IIf gives 6II: = 0, where

- A,614;*

-

Fj6uj}dr

+ / I [ p ; 6 / t ;+ ( I / ; - M;)8pi] dS .L

11

pj614; dS.

(2.12)

S.,

By using Green’s formula, we get

111

Au6ui.j dr

=

11

AUni6iii dS -

s,, i s,,

7

111

Aij.,i6ii; dr.

(2.13)

T

in which ni is the direction cosine of the outward normal from the surface S,, + S,,. Substituting (2.13) into (2.12), we obtain after some rearrangement

6IIT

=

(a,k/ek/ + h,)6e,

r

-

1\(A,ni

I

+ 1e, - -2 (u;. + u,.;)

1

6h,

+ pj)6rii dS.

(2.14)

.S<,

Since the variations of functions 6e,, ah,, and 614; in r , 614; and 6pi in S,,, and 6rr; in S, are all independent, the stationary condition of 6n;F = 0 gives A,.!I = - ~ i !./ k./ p !.I? e!I- = $ ( I / ; . , + uj.;), A,.U. I. - F . z 0, pI. = X..n. rJ .I 14; -

M; = 0

in r

(2.15a) (2.15b) (2.15~)

on S,,

(2.15d) (2.15e)

98

Wei-Zang Chien Acnj

+ ij; = 0

on S,.

(2.150

Equations (2.15b,c) are the original conditions of integration. Equations (2.15a,d) give the unknown Lagrangian multipliers in terms of specific physical quantities; that is, A,.rJ =

-0.. rJk/ ek / = -ujj

p,.I = A..n. r J J = -u..n

in

7,

on S,.

(2.16a) (2.16b)

Substitution of (2.16a) into (2.1%) gives the equation of equilibrium (2.1). Similarly, substitution of (2.16a) into (2.150 gives the boundary condition of given boundary forces (2.6a). Consequently, the functional of the nonconditional generalized variational principle is obtained by substituting Ajj and pi from (2.16a) and (2.16b) into (2.11),

S,,

.%<

Vuriutionul Principle I* (generalized variational principle of small-displacement elasticity derived from the minimum potential energy principle): Among all possible sets of functions u i , e 6 , and u j j ,the set that makes the functional IIf stationary is the solution of (2. I), (2.2a), (2.3), (2.6a), and (2.6b). It should be noted that in this variational principle the functions u ; , ejj and uv are all treated as independent quantities. If the quantities eu and ujj are not considered to be independent variables but are taken as functions equal to t(u;,j + uj,;)and aijklexl, respectively, then (2.17) can be reduced to a functional with variable ui only. In fact, this functional can be expressed as

in which ejjis, by definition, equal to

H u ; , , ~+ ui.;).

Vuriutional Principle I*A (generalized variational principle, with displacement M as the only variable, derived from the minimum potential energy principle): Under the combined actions of body force Fi in r , the given surface force pi on S,,, and the given boundary displacement rT; on S,,, the displacement u; makes functional IIfA in (2.18) stationary. In

Incompatible Elements: Generalized Variational Principles

99

(2.18). ell is, by definition, $(ill + uJ.,).and the stress ullis given by Eq. (2.2a). Let us now assume that the region T is divided into N finite elements and that the displacement 11, is continuous over interelement boundaries. We call these kinds of finite elements compatible elements. We denote the displacement M, of the nth finite element by ,)'(i and e l lrJ! r )

-

{ (I // p +

(2.19)

/{I;)).

Then the functional (2.18) can be simplified into the form

Variational Principle I**A (generalized variational principle of displacement-compatible elements derived from the minimum potential energy principle): Under the combined actions of the body force Fi in r , the given surface force jTj on S, , and the given boundary displacement Ej in S, , the displacement ulm)( m = 1 , 2 , ..., N ) in the displacement-compatible finite element makes the functional II&* stationary. In this functional, e!,m)is the abbreviation of (2.19). The proof is as follows: The variation of Eq. (2.20) gives

(2.21)

By means of Eq. (2.191, we can prove that

111

U U k / d [ ~ ) & ' ~ dT ~) =

111 11f,

<'Ly)8/41~i~' dT

UJ1,i/

r,,,

T,,,

=

Ilk1 ~ I~ /~

dS

" l ~ ~ / ( ~ ! l l ~ n ~ l l l ~

L, - //1(u,,k/e$')),

8 ~ : ' "dT )

(2.22)

T,a

and also that, for two elements ( m ) and (m') having common interelement boundaries, we have on their common interelement boundaries the relations

Wei-Zang Chien

100

Equation (2.21) reduces to the following form:

Since Suj"') in T and Sp),8 ~ : ' " ~on' ) interelement boundaries S('n'n'),and n'm'"' k l on $,""are all independent variations, the stationary condi!,k/ = 0 not only give the following relations tions of LI,,

(2.26a) (2.26b) (2.26~)

but also the relation a..

&kld n ( jm ) +

Ukl e(m')n(m') k/ j =

0

(2.27)

on interelement boundaries ,Wm'). It should be noted that the stationary condition of n&,*(i.e., the variational principle I**A) not only gives the equations of equilibrium for all the elements, but also the given surface force boundary condition (2.26b) and the given displacement boundary condition (2.26~)for all the elements situated on the boundary surface of this body. Besides these, the solution is continuous not only for the displacement over the interelement boundaries (this is actually the assumed condition to be satisfied by the displacement vector), but also for the stress vectors agk/.eiy)ny)= c%z!m) U J over the interelement boundaries (i.e., they are equal in magnitude and opposite in direction). See Fig. 1.

101

Incompatible Elements: Generalized Variational Principles

FIG.I . The stress vector on interelement boundary surface S'mnt'l= S(m' = S'"'''.

Let us formulate the matrix equation of compatible finite elements. Let and q(m)represent, respectively, the displacement vector and the generalized displacement matrix in the m th element: ,,(tn)T = [u OI ) l ) , u:"', u 3 , (2.28a) q(m)T =

[qp,q y ,

...) qjm)],

(2.28b)

and N(") be the matrix of interpolation functions: N ( VI t ) . . N:") .

Nj'")

.

N ( Im )

.

.

NI")

.

N\m)

N(m) =

.

. .

NIm)

.

.

NIm)

...

[ .

.

*I

.

NI"' (2.28~)

in which t denotes the number of degrees of freedom assigned to each of our finite elements. Thus we have U(m)

= N ( m )q(4.

(2.29)

Stress-strain relation (2.2a) is u

=

ae

(2.30)

in which 0'

= [ullr u 2 2 r

u 3 3 3 f l l 2 1 c 2 3 , c311,

eT = [ e l l , e22, e33, e12, e23, e d . The strain-displacement relation (2.3) can be written as e

=

Du

(2.31a) (2.3 1b) (2.32)

Wei-Zang Chien

102 in which

(2.33)

D =

Substitution of this result into (2.20) gives (2.34)

in which

-

j~(n(ftflaDN(f~t))Tii dS ,

(2.3Sb)

where drn) is the direction cosine of the outward normal to S(r’7).Let the outward normal n, be ( n l , nz, nd. Then (2.36a)

Furthermore, we use

FT = [ F , , F ? . F31,

(2.36b)

Incompatible Elements: Generalized Variational Principles

103

(2.36~) (2.36d) K")')in (2.35a) is the element generalized rigidity matrix. ~q("'~TKi'")q"''~ is the element generalized strain energy density, and Q"")is the element generalized external force. Equation (2.34) can be assembled into a global matrix equation in the usual manner:

n&*= 12 9TKq - q'Q

(2.37)

where q, K, Q are, respectively, the global generalized displacement, global generalized rigidity matrix, and global generalized external force. The stationary condition of variation gives K*q

=

Q,

K*

=

f(K

+ KT).

(2.38)

These are the well-known results. However it is important to note that, if displacement is continuous over all the interelement boundaries, the stress vector is also continuous in an integral sense on all the interelement boundaries, and hence the continuity conditions of bQth displacement and stress will be completely satisfied when the element size becomes infinitesimally small.

C. GENERALIZED VARIATIONAL PRINCIPLE FOR INCOMPATIBLE DISPLACEMENT ELEMENTS DERIVED FROM THE MINIMUM POTENTIAL ENERGY PRINCIPLE If the displacement element used in the computation is of an incompatible nature, then (2.23) can be considered as the constraint of variation. By means of the Lagrangian multiplier A:mm'), we may rewrite the functional II&* of Variational Principle I**A into the following functional of another generalized variational principle: (2.39) where Il&* represents (2.20), Aimm')is the Lagrangian multiplier on the interelement boundaries of elements ( m ) and (m'), and Z(mmj) represents the summation of all interelement boundaries in the body.

104

Wei-Zang Chien

By taking a variation of (2.39),we get

n&*

Hence the necessary condition for to be stationary gives not only (2.26a)-(2.26~)but also the continuity conditions (3.23),and Ajmm')

= -a,,

&/

e(m)n(m) kl j

=

e(m')

.

(m')

ukl k / nj

(2.41)

This is in fact equivalent to Eq. (2.27). Here we obtain a conclusion similar to that given in the functional of the generalized variational principle for incompatible elements by Pian and Tong (1972).From (2.41), we have

II

(rijl?!)

- u,!!f7'))Ayf"') dS =

-

/I /I

l,~ffl)u,,

e(W)n(fn)

!/lJ

a/ ;

dS

,~'"""''

,p""'l

-

M ; n f ' ) u , , c ( n f ' ) (nr')

!/XI

yi

dS.

(2.42)

p~t,,,',

Therefore, the two integrals on the right-hand side of (2.42)can be written separately into the rn th-element terms and rn'th-element terms. Equation (2.39)can be written as

(2.43) in which S y ) is the total interelement surface of element (rn). This functional can be written entirely in the same form as that given by Pian and for uv&'r)n,;''f). That is, Tong (1972)if we substitute T~'")

Incompatible Elements: Generalized Variational Principles

105

(2.44) Let us now consider 7‘:”’)as an independent variable. Taking a variation of (2.44), we obtain

+

- / \ [ ~ : ’ ~ ) 8 7 ‘ , ( ~(T,)(m’cnr, -

p

The required stationary condition 8IirG (1’1) T,“”)= uvk/ekl n,o n )

a,~~e~,?’n~m’)8u~m’] dS]. (2.45)

=

on

0 gives (2.26a)-(2.26c) and also ) :$

+ ~j’”).

(2.46)

In addition to these, there is the relation on the interelement boundaries ~ l l ’ I ~ ’ l ‘ as )

11

(lljR1)87;(’n)- u oI n ’ ” 7 ’ ~ n ’ ’ ) )

\’”””

dS

=

0.

(2.47)

I

From this equation, we fail to obtain not only the continuity condition u;”‘) = u;”“)of displacement, but also the stress vector continuity condition

q’“)

= -7’:In‘). Thus the functional (2.44) fails to fulfill the continuity conditions of displacement and stress vector when the element size becomes infinitesimally small. Another modification of the potential energy principle is due to Tong (1970) and is based on the use of separated variables for the displacement field in region 7 in general and the interelement boundary displacements. Let uimm’)be the interelement boundary displacement. Then we have

as the constraint conditions. It can be proved by means of the Lagrangian multiplier method that the functional of this generalized variational principle is

106

Wei-Zang Chien

Hence, Tong's variational principle can be written as

an** IA?

-

0.

(2.52)

This can be used for the computation of incompatible displacement elements. It may be mentioned again that T,("')and e y )in (2.49) are defined by (2.51b), (2.51~1,and (2.19). It should be noted that fifd has an additional undetermined variable, and the increasing number of undetermined quantities of generalized displacement is of course disadvantageous to the computational work. In the following, we suggest another formulation of the functional for incompatible elements, in which no additional quantities of generalized displacement need to be introduced. From (2.41), we assume that

Hence (2.39) may be rewritten as

or as

where n&*is given by (2.20). The stationary condition of Ilf& is SM,*,*,= 0, where

Incompatible Elements: Generalized Variational Principles

-

'

2 aok/(fk/ (m)n,Off) + e'ff"' nJ(fff'))(~uy + 8u(fN') )] dS.

From Eq. (2.56), we find the following additional relations: / , ( I f f ) - /p') = 0 on l p f f f ' )

f

f (Jfff)

f / k / k / "/

')If)

-

-

(,

e'ff"' ' f f f ' )

f l k l l./

r'/

on

p l f f f ' l

107

(2.56)

(2.57a) (2.57b)

This shows that all interelement continuity conditions are satisfied. Hence we have the following. Generalized Variational Principle IA3 (generalized variational principle of incompatible-displacement finite elements derived from the minimum potential energy problem): Among all the sets of ujfn)(m = 1, 2, ..., N), the one set of ulm)that makes stationary satisfies (2.26a)-(2.26~),the continuity conditions of displacement, and the stress vector (2.57) on the interelement boundaries. The functional differs from fifh given by (2.49) by the fact that no additional undetermined displacements at interelement boundaries need to be introduced. of (2.55) may be On the basis of (2.29) and (2.32), the functional expressed in matrix form as

nf&

nf&

nf&

(2.594 (2.59b)

108

Wei-Zang Chien

where S{") represents the total interelement boundaries for element (m). Now (2.59) may be assembled into a global matrix form as

fiII*AT = iqTK29 - q'Q,

(2.60)

where K2 is the assembled matrix from K\") and Kimm').It may be noted that the matrix KZhas no symmetrical property. gives The stationary condition of

nr&

Kfq = Q ,

Kf

=

B(K2

+ KT).

(2.61)

It should be noted that Jones (1964) first suggested the form of (2.39), and Greene et al. (1966) recognized the possibility of representing the by undetermined boundary reactions (2.41). Lagrangian multiplier A!mm') From this point of view, one may regard this kind of incompatible finite element as another kind of hybrid element.

D. GENERALIZED VARIATIONAL PRINCIPLE OF HYBRIDINCOMPATIBLE ELEMENTS DERIVED FROM THE M I N I M U M POTENTIAL ENERGY PRINCIPLE Let us formulate the incompatible finite-element computation based on generalized variational principle I*. The continuity condition of displacement at interelement boundaries S(llllll')

/p)- ,p')= 0 I

I

on

~ ( l l l l l r ' ~

(2.62)

may be introduced as condition of constraint, and appropriate boundary variables are used as the corresponding Lagrangian multipliers in the functional (2.17). Thus we have

(2.63) By means of the stationary condition of Hrg, we can show that (2.11, (2.2a), (2.3), (2.6a), and (2.6b) are satisfied in all the elements and (2.62) is satisfied at interelement boundaries. Furthermore, the following continuity condition of stress vectors is obtained: Aim"') = --(T(m)n(m) U J = c TI. ( mI ' ) nJ ( m ' ) . (2.64)

Incompatible Elements: Generalized Variational Principles

109

Similarly, as in (2.53), Ajmm')may be represented by Thus, (2.63) may be written as

(2.66) The stationary condition of II@j not only gives (2.1), (2.2a), (2.3), (2.6a), and (2.6b) in all the elements, but also the continuity conditions (2.62) and (2.64) of displacement and stress vector at all interelement boundaries. Thus we have the following: Generalized Variational Principle IB3 (generalized variational principle for incompatible hybrid elements derived from the minimum potential energy principle): Among all sets of ulm),up),e r ) ( m = 1, 2, ..., N ) , the one set of uf"), up),e r ) that makes rIfB%stationary satisfies (2. l ) , (2.2a), (2.3), (2.6a), and (2.6b) in all the elements, and at the same time satisfies the continuity conditions (2.62) and (2.64), or the compatible conditions of uf"', u!m)n(m) U J at all interelement boundaries. and u)"')be represented by interpoLet all the components of P:", generallation function N?' ( k = 1, 2, ..., I ) , the generalized strain a:"'), ized stress @) and generalized displacement 4:"" as follows: e(m) = N(m)&n) (1) (2.67a) a(m) = ~ ((I ) mpc,') (2.67b) ,,(m) = N(m) (2) 9( m ) (2.67~) where e(m),dm), dm), and qcm)are shown, respectively, in (2.31a), (2.31b), (2.28a), and (2.28b). ,)$: dm), pcm) are

ur',

q:),

N* N*

N* N*

I

N*

(2.68a)

110

Wei-Zang Chien

(2.68b)

dm)T = [a\m),aim),

..., aim),..., ..., ..., ..., ..., a@"], (2.68~) p(m)T= [p\"', pim),pirn)..., pi"); ..., ..., ..., ..., ..., p@], (2.68d) and =

N*(llJ)

By transforming e p ) , c$), as

"I"'),

N:"'), , .., N y ] .

(2.69)

u ! ~into ) matrix form, Eq. (2.66) may be written

in which the various K* and Q are

(2.7 1c) (2.7 Id)

where S{") is equal to the total interelement boundary surface area. Let us now assemble (2.70) into a global matrix representation. First we have

Incompatible Elements: Generalized Variational Principles

with

1 11

III

Thus the global representation of (2.70) may be written as

n,*,*,= $(Y'KT(Y - aTK;P + qTK:P

- q'QT + PTQ; (2.73) in which K: is defined in (2.72), and K&, K&, Q: and Q; are the global Q : ( m ) , and Q ; ( m ) , respectively. The condition for matrices of KT'"), IIl*B*,to be stationary yields an,*,*,= (K:(Y - K;P)' 6a + (qTK: - aTK; + Q;') SP

+ (K:P

Q*)' 6q

0,

(2.74)

KT(Y - K;P = 0,

(2.75a)

KTTq - K ; T ~+ Q; = 0,

(2.75b)

-

=

or

K:P

- Q: =

0.

(2.75~)

The solution of the above set of equations is

P = K3* - I Q-T , (Y

= K:-~K;K;-IQT,

(2.76a) (2.76b)

and q can be found from (2.75b); that is, This is the matrix equation for q.

E. GLOBALGENERALIZED VARIATIONAL DERIVED FROM THE MINIMUM PRINCIPLE COMPLEMENTARY ENERGYPRINCIPLE If the variational conditions (2.1) and (2.6a) of the minimum complementary energy principle are introduced into the generalized variational principle as constraint conditions and appropriate variables A; and p i are used as the corresponding Lagrangian multipliers, we have the functional of this generalized variational principle as

112

Wei-Zang Chien

Considering hi, pi,and uijas independent variables, the stationary condition of II; yields

an,*,= / / / [ ( & j j k / ~ k / - A;,;)~u,

+ (Cij.,j+ Fi)6AjI d~

T

- A;)6uijnjdS

-//(lTi

+ / / ( p i + Aj)8uijnjdS s,,

(2.78)

+ f i ( u i j n ; - jii)6pi dS = 0. S,,

Thus we have

+ A;.;) = 0 u - .+ F . = 0

bijk,ukl -

U,J

I

UI . - A., = 0

pi

+ A;

=

u JJ- nJ . - p.1 =

0 0

in

T,

(2.79a)

in

7,

(2.79b)

on S,,,

(2.80a)

on S,, on S,.

(2.80b) (2.81)

From (2.79), (2.80a), and (2.80b), we find A I. = u 1.

in

T,

(2.82a)

p . = -A. = -ui

on S,, + S,.

(2.82b)

Thus (2.77) may be written as

n;",= /I\[; &ijk/CTijUk/ -k r

-// s,,

U i u , p j dS -

(Cij,j

/\

- 1

+ Fi)Ui

u;(uijnj-

dT

pi) dS.

(2.83)

S,,

Generalized Variational Principle 11" (generalized variational principle for small-displacement elasticity derived from the minimum complementary energy principle): Among all sets of uijand uir the set of uijand ui that makes n,*,stationary satisfies (2.1), (2.2b), (2.3), (2.6a) and (2.6b).

F. GENERALIZED VARIATIONAL PRINCIPLE OF COMPATIBLE ELEMENTSDERIVED FROM THE MINIMUM COMPLEMENTARY ENERGYPRINCIPLE Let us study the stress-compatible elements, in which the stress vectors up)nJ!m)are continuous over all interelement boundaries; that is,

Incompatible Elements: Generalized Variational Principles +

&')u

nj( m ' )

=

on S(mm').

0

1 13

(2.84)

For this kind of stress-compatible element, we can show by making use of the stationary condition of corresponding functional that the displacement ufm)is also continuous over all the interelement boundaries, or

*y - Uj.m')

=

on

0

~ ~ l l f n f ' )

(2.85)

For this purpose, we formulate here the functional of the generalized variational principle for stress-compatible elements based on (2.83). We have

-11 .

fiiuF)ny'dS

I,

-

11 q",I

ujm)(ur'ny) - pi) d S } .

(2.86)

I,

The variation of JJ,*,; gives

If the stress vectors are compatible, then

(2.88) Hence the stationary condition aII,*,: = 0 gives the displacement-compatible condition (2.85). This proves the statement that the stationary condition of the functional of stress-compatible elements gives the compatible condition of displacement.

Generalized Variational Principle II**A (generalized variational principle for stress-compatible elements derived from the minimum complementary energy principle): Among all the sets of u,!;)and u~'"',where vim)are interelementwise compatible, the set of vim)and ulm)that makes

Wei-Zang Chien

114

::II

stationary satisfies (2.l), (2.2b), (2.3), (2.6a), and (2.6b) in all the finite elements and the continuity condition of uj’”)over all the interelement boundaries. Let us write (2.86) in matrix form. We have (2.89a) (2.89b)

where ~ ( m ) U(m), , pOIl), q011) , N(”’) ( I ) , and N$’ are defined in (2.31b), (2.28a), (2.68d), (2.28b), (2.68a), and (2.68b), respectively. Then

(2.90) in which

Ql,

Qz are defined as in (2.71b) and (2.71~1,and

ROY) =

111N~;;”TbN~I;’dr,

(2.91a)

T,,,

The global matrix representation of (2.90) is

n:

=

t/3TRhlp + qTR& - pTQziq’al.

Invoking the stationary condition on :lI

8Il,*,;= 8pT(RhiP - Qz

(2.92)

gives

+ Rlzq) + SqT(Rh2P + Q l )

=

0 , (2.93)

or (2.94a) (2.94b) Solving p from (2.9.4b) and substituting this result into (2.94a), we have

p

=

-R-’h2 Q l ,

RLq = Qz

+ RhiRh;lQi,

(2.95a) (2.95b)

and q can readily be obtained from (2.95b). Thus we find from stresscompatible elements the solution of p and q.

Incompatible Elements: Generalized Variational Principles

I I5

G. GENERALIZED VARIATIONAL PRINCIPLE FOR INCOMPATIBLEELEMENTSDERIVEDFROM THE MINIMUM COMPLEMENTARY ENERGYPRINCIPLE Let us adopt the stress-incompatible elements. Then we may use the Lagrangian multiplier hi for the condition of constraint so that the conditional variational principle transforms into a nonconditional variational principle. The corresponding functional may be written as

Variation of

n,*,&gives

+ (ar)n(m)+ a!m')nj(m'))8Ajmm')] dS.

(2.97)

1J

The stationary condition of II;f,*gives on one hand (2. I), (2.2b), (2.3), and (2.6a,b) in all the elements, and, in addition, the following continuity conditions on all interelement boundaries: a!'"'n(,'"' r J J + -~j'"m')

v(!m')n!m') v J

=

l$n)

= 0,

= uIm').

(2.98a) (2.98b)

Equations (2.98a) and (2.98b) are, respectively, the continuity conditions for stress vector and displacement. By means of (2.98b), we obtain

Thus (2.96) may be written as (2.100a)

Wei-Zang Chien

116

in which Sl”)represents the total interelement boundary surface area of the element (m).Equation (2.100a) in fact is quite similar to that used by Pian and Tong (1972) for incompatible elements. By means of (2.87), we have

-

//(crby)ni(m)

- pi)tiujm’ d ~ }

2, 11(crr)np)8ulm)+ $‘)nY’”u/’‘’)) dS.

Sp’

-

(2.100b)

(mm ) S ( m m ’ l

It can easily be seen that none of the continuity conditions can be obtained from the stationary condition an& = 0. Thus the functional (2.100) cannot be used to deal with the incompatible elements. It is shown by Pian and Tong (1972) that, in variational principle (2. loo), there are additional independent variables uim)= ujrn’)on interelement boundaries besides cf) and ujm)in T. Denote u / ’ ~=) ui( m ’ ) by u!mm’) Then (2.100) becomes

with m

The variation of IIBi; is

- [ ~ ~+ ~$~r ) Jn ~ !J m ’ ) )l a u ~ m md’ ) ~ ) .

(2.101b)

The stationary conditions of IIBXT simultaneously give the continuity conditions of the interelement displacement and the stress vector, and, therefore, the functional (2.1Ola) in fact represents the generalized variational principle for incompatible elements. To avoid the application of the additional variable u p ’ )in the generalized principle, we may take, in accordance with (2.98b),

Incompatible Elements: Generalized Variational Principles = -&(u;'")

+ u!~')).

I17

(2.102)

Hence the functional for incompatible elements can be written as

The variation of II,*,&is

The stationary condition 611;",i3= 0 not only gives (2.1), (2.2b), (2.3), (2.6a), and (2.6b) for each of the elements, but also the interelement continuity conditions & O n ( m ) + &I') (m') = 0 on S'""'), (2.105a) O J u nj $,)

= u!m')

on

S(mm').

(2.105b)

The functional (2.103) can be further simplified to the form

with nt

where I:,*

represents (2.86).

Generalized Variational Principle IIA3 (generalized variational principle for incompatible elements derived from the minimum complementary energy principle): Among all sets of a!?)and ujm)(which need not be compatible), the set of err) and ujm)that makes I I f i z 3 stationary satisfies (2.1), (2.2b), (2.3), (2.6a), and (2.6b), in every element and also the interelement continuity conditions for c$)np) and ujm). Let us now reduce (2.106) to matrix form. Using (2.89a) and (2.89b), the functional (2.106) may be written as

118

Wei-Zang Chien

with m

(2.107) in which p(m),q("), RLT), K:'"), KT('nfn'),Q?), and Qhm) are defined by (2.68a), (2.28b), (2.91a), (2.71e), (2.71f), (2.71b), and (2.71c), respectively. The expression (2.107) can be assembled in global matrix form. First, we have

with f n

Thus the global matrix form of (2.107) can be written as

ntX3 = 4PTRh*iP+ qTKf3@- pTQf + qTQT

(2.109)

where Kf3 is defined by (2.108), and R & , QT, and Q fare the assembled respectively. The first variation of global matrices of RiY), Qlm),and (2.109) is

aim),

t3n& = 8pT(R;Ip + Kb*;Tq - of)+ 8qT(K&P+ QT).

(2.110)

The stationary condition gives

K&P

RtlP

+ K;Tq

+ QT

=

0,

(2.11la)

Qf

=

0.

(2.11 1b)

-

From these two equations the solution of p and q can be abtained.

111. Generalized Variational Principle of

Incompatible Elements for the Plane Problems in Elasticity A. GENERALIZED VARIATIONAL PRINCIPLE OF

DISPLACEMENT-INCOMPATIBLE ELEMENTS POTENTIAL DERIVEDFROM THE MINIMUM ENERGY PRINCIPLE Let us now consider the plane problems of elasticity. On a unit thickness of material, there are combined actions of body force F, (LY = I , 2) and external force T, (a = 1, 2) acting on the boundary edges. The minimum potential energy principle for the plane static problem of elasticity is 6rI,p = 0,

(3.1)

Incompatible Elements: Generalized Variational Principles

I 19

where

in which the Greek indices a,p, y , and 6 range over 1 and 2 in plane problems. u, = ( U I , u2) are the components of plane displacement. The strain and stress components are e,p

cap =

(3.3a)

+ up,,),

= b(u,.p

(3.3b)

aapyseyfi

9

where aupysare elastic constants. For the isotropic materials, the stress-strain relations of plane stress problems are

where E is Young’s modulus, and v is Poisson’s ratio. On the other hand, the stress-strain relations of plane strain problems are

[I:]

=

(1

+ v)(l

-

2v)

[

I-v

v

u

1 - v

(TI2

.

.

(3.5)

1 - 2v-

*

In the medium, there are equations of equilibrium

+ F,

=

o

in A.

On the boundary sm, where external forces condition is -

a a p n p = T,

(3.6)

?, are given, the boundary

on s,.

(3.7a)

On the boundary s,,, where boundary displacements U, are given, the corresponding boundary condition is -

u, = u ,

on s,,.

(3.7b)

Also, s,

+ s,, = s = the whole boundary curve.

(3.8)

Wei-Zang Chien

120

It is easily shown that, if capand u, satisfy (3.3a), (3.3b), (3.6), and (3.7b), the condition for the functional (3.2) to be extreme gives the solution of equilibrium equation (3.6) under the boundary condition of known force (3.7a). The conditions (3.3a), (3.3b), and (3.7b) resulting from application of this variational principle can be eliminated by introducing appropriate Lagrangian multipliers. Thus we have the following. Generalized Variational Principle I*P: The solution of cap, cap, and satisfying (3.3a), (3.3b), (3.6), (3.7a), and (3.7b) makes the following functional stationary. up

(3.9) This problem is similar to the corresponding three-dimensional problem discussed in Section ILB, and the functional IT& in (3.9) is the counterpart of (2.17) for the three-dimensional problem. If we take eaPas being equal to f ( ~ , ,+~ upJ, and regard c,pnp as being identical to aapyseysnp,Variational Principle I*P can be transformed into Generalized Variational Principle I*PA with u, as the only variable. Under the combined action of body force F, in A , a given boundary force T, on s,, and a given boundary displacement ii, on s,,, the displacement u, that satisfies (3.6) and (3.7) makes functional nrPA stationary.

PA

=

I/(;

a,pyse,pey8 - Feu,) dA -

/

so

T,u, ds

A

-

I,,(u, - ii,)a,pyseysnp ds.

(3.10)

Let us now divide the plane region A into N plane elements. The elements are supposed to be compatible. Call the u, and eap in the m t h element u;? and e$’, where e$) = f(u$

+ ~6%).

(3.10a)

Then the functional (3.10) may be transformed into the following form:

Incompatible Elements: Generalized Variational Principles

I2 1

Variational Principle I**PA (generalized variational principle of displacement-compatible elements derived from the two-dimensional minimum potential energy principle): Under the combined action of body force F, in A , boundary force T, on s,, and boundary displacement 17, on s, , the plane displacement C?) ( m = 1, 2, ..., N ) of compatible elements makes the functional II,*,*,in (3.11) stationary. If these elements are incompatible, the continuity condition *;m)

- u($

=

0

on dmm')

(3.12)

on the interelement boundaries can be eliminated by means of the Lagrangian multiplier in the functional @&. Thus the principle of conditional variation can be transformed to the corresponding principle of nonconditional variation with the functional (3.13) is defined by (3.9). From the first variation of where II,*,*,

+

I

[ALmm')

Slmm'l

fi?&

we obtain

+ aupy6e~f"n~m')]6u~m" ds} (3.14)

in which we note that (see Fig. 2) dS(m)= -dS(m').

(3.15)

Thus the stationary condition of fir& not only gives (3.3a), (3.3b), (3.6), (3.7a), and (3.7b), but also the displacement continuity condition (3.12) and the stress vector continuity condition on the interelement boundaries (m') (m') A&mm') = -a,pvseys( m )np( m ) = -aapyseyS np .

(3.16)

From (3.16), it is easily shown that a$)nbm' ddm)and a$"nbm" ddm')are the resultant forces acting on the interelement boundary ddmm') (= dsCm)=

122

Wei-Zang Chien

FIG.2. The positive directions of dn and ds on interelement boundaries.

-ddm')),and they are two force vectors equal in magnitude and opposite in direction (see Fig. 2). into the same form as that of It should be noted that if we change ll,*,*, in (2.431, then the stationary condition will fail to give the continuity conditions of both interelement boundary displacements and stress vectors. Thus we ought to take the Lagrangian multiplier of form (2.53); that is, from (3.16) we have

nz,

+ drn" ys np

A?')

= -baapys(e$)nl;n'

(rn')

1,

(3.16a)

and (3.13) may be written as 1

nE3= II;",*A- 2 2

aapys(e$"nbm) + e$')nbm")(uLm)- uLm">d s

(mm')

(3.16b) or in the form

with m

in which is defined as in (3.1 I). It can be shown by the variation of not only gives the equation of (3.16~)that the stationary condition of equilibrium (3.6) and boundary conditions (3.7a) and (3.7b) in each of the elements, but also the interelement continuity of displacements and stress vectors, (3.12) and (3.16). In the functional (3.16c), there is only the single while e$) is taken to be defined as i(u:''$ + ug;). Finally we variable uLrnYm,, have the following:

n;"&

Generalized Variational Principle I**PA3 (generalized variational principle for incompatible elements derived from the minimum potential energy principle): Among all u&$ ( m = 1, 2, ..., N ) , the one uLmYm, that makes II,*,*,3 in (3.17b) stationary satisfies (3.6), (3.7a), and (3.7b), and the in-

Incompatible Elements: Generalized Variational Principles

I23

terelement continuity conditions of displacement and stress vectors (3.12) and (3.16). Denote the displacement matrix by =

@IT

(up),

(3.17a)

and denote the generalized displacement matrix (or in particular the ma(suppose that there are r nodals trix of nodal values of displacement) qCm) in an element) by q y

=

( q p ,4 p ,

..., qjm),...) 41:")).

(3.17b)

Suppose that the interpolation functions are N p ) ,Nj"), ..., N;"). We have =

NL~)~?),

(3.18)

where

Thus (3.16b) may be written as

with nr

where K E ) , KLyfn'),and

Q g )are given by

(3.23) in which nbm)is the outward normal matrix from drn).Denote the direction cosine of outward normal n, (or n l , nz); then (3.24)

124

Wei-Zang Chien

and furthermore

(3.25a)

D p =

(3.25b) (3.254 (3.25d) Equation (3.20) can be assembled into a global matrix equation. Denote

where Kpl is the global rigidity matrix. The assembled equation of (3.20) may be written as

n,*,*,,= ;tq%PIqP

-dQP.

(3.27)

Applying the stationary condition to Il;",*,,yields

Khqp -

QP

=

0,

GI= WPI+ GI)

(3.28)

from which qp can be solved.

B. GENERALIZED VARIATIONAL PRINCIPLE OF INCOMPATIBLE ELEMENTS DERIVED FROM THE

MINIMUM COMPLEMENTARY ENERGYPRINCIPLE Just as in Section II,F, the generalized variational principle of stresscompatible elements can be derived from the two-dimensional minimum complementary energy principle. The functional is similar to (2.86) and

Zncompatible Elements: Generalized Variational Principles

125

may be written as

(3.29) in which ha,,

is the two-dimensional flexibility constant, enp = bnpysvys.

(3.30)

For the incompatible elements, the functional is similar to (2.106):

with m

Transforming this into matrix representation, we have

,p

Nl;;lpPP, ubm) = Nbm)q$'", =

(3.32a) (3.32b)

in which ub"', qgm),and Nb") are defined as in (3.17a), (3.17b), and (3.19) respectively, and

126

Wei-Zang Chien

in which (3.35a)

QK) =

11

NP’TFpdA

+ lv!:,, NY’Tfipds

(3.35e)

A”,

and ~(lm)is the interelement boundary curve of element (m).Equation (3.34) may be assembled in global matrix form. In the first place, we have

with m

Thus, the global matrix form of (3.34) is

K& is defined as in (3.361, R;bl, Q;l, QZ2, respectively, are the assembled global matrix of R%\,Qg;), Q g ) .Furthermore, qp, P p , respectively, are the assembled global matrix of qy),py). The variation of II:,*,, is

From the stationary condition on the functional we obtain

from which Pp and qp can be solved.

Incompatible Elements: Generalized Variational Principles

127

IV. Generalized Variational Principle for Plate Elements of Bending A. MINIMUM POTENTIAL ENERGY PRINCIPLE OF

PLATES AND THE RELATEDGENERALIZED PRINCIPLE FOR COMPATIBLE AND VARIATIONAL INCOMPATIBLE ELEMENTS The differential equation of deflection for bending of thin plate is

V 2 V2w = FID,

(4.1)

in which D is the flexural rigidity, F is lateral loading, V 2 is the twodimensional Laplacian, and w is the lateral deflection of the plate. Let v be Poisson’s ratio. The components of bending moment and shearing force are MI1 = -D(w,ll

(4.2a)

M22 =

+ vw,22), -D(w,22 + V W J I ) ,

(4.2b)

MI2

M2l

(4.2~)

=

QI=

=

-D(1

- v)w,[~,

-D(v2w),I,

(4.2d)

Q2 = -D(V2w),2.

(4.2e)

Furthermore, the boundary moment and equivalent shearing force are M , ( w ) = - D [ v V2w + (1 H,(w) = - D [ -

a

an

- v)w,~~],

(4.3a)

[V2w + (1 -

in which p is the radius of curvature of the boundary curve, positive if convex (Chien, 1980); ( n , s ) is the coordinate system of outward normal and boundary arc length. The lateral reaction acting on a corner k of the boundary curve is Pdw)

=

-(I - v)D A A w , ~-~( 1 l p ) ~ , ~ l

(4.4)

in which Ak(w,,, - ( l / p ) ~ ,means ~ ) the increment of ( w , , , ~- ( l l p ) w s s on ) two sides of the boundary corner k in the direction of increasing s along the boundary curve. In general, we have the following boundary conditions and corner conditions:

Wei-Zang Chien

128

(1) The deflection V? and/or equivalent shearing force boundary curve

is given on the

H,

=

H

on s m I ,

(4.5a)

w

=

6

on s W I .

(4.5b)

(2) The bending moment on the boundary M or the slope of deflection along the outward normal on the boundary G,, is given by (4.6a) (4.6b) where

+

sW1 s,,,~ = sm2

(3) The corner force

+ sW2 = the whole boundary.

(4.7)

or corner deflection V?k2 is given at corner kl

or kz: Pk, = P k l

at corner k l = 1, 2 ,..., k,,

(4.8a)

wk2 = W k 2

at corner k2

1 , 2 ,..., k , .

(4.8b)

=

Let i be the total number of corners along the boundary curve; then k,

+ kM,= i .

(4.9)

The problems of the bending of thin plate have been discussed by Chien (1981) for the case of incompatible deflection elements. The results are as follows. Generalized Variafional Principle 111 (derived from the minimum POtential energy principle): Among all w(x,y ) or W ( X I ,X~),the one that makes the functional stationary satisfies the field equation (4. I), boundary conditions (4.5a), (4.5b), (4.6a), and (4.6b), and the corner conditions (4.8a) and (4.8b).

k.

(4.10)

Incompatible Elements: Generalized Variational Principles

129

in which

no =

/ID [($+ *I2

-

(--ax2 ay2 --)] d 2 ~ dA. ax ay ax ay

2(1 - v) a2w a2w - a%

dY2

A

(4.10a) Let us now divide the plate region into N finite elements and study the related variational principle. Let us further assume that besides the distributed loads F , the boundary conditions (4.5a), (4.5b), (4.6a) and (4.6b), and the corner force conditions (4.8a) and (4.8b), the plate is loaded by concentrated loads F , I at cF interior points (x,, , y, ,) and is supported by given deflection WL2 at some other c,, points ( x , ~y,?). , They are FC1 =

FcI

w , =~ W C 2

at x , , , y C I , where

L'I =

at xC2,y t 2 , where c2

=

1, 2, ..., c F ; (4.11a)

I , 2, ..., c w . (4.11b)

Furthermore, let us assume that these CF + cw points are the common corner points of the finite elements themselves. We then have the following: Generalized Variational Principle IIIA for the deflection-compatible , ( m = 1, 2, ..., N ) elements: Among all sets of the field variable d r n ) ( xy) of compatible elements, the set of w('")that makes @,A stationary gives the solution of the field equation (4.1) under the various boundary conditions (4.5a), (4.5b), (4.6a), and (4.6b), corner conditions (4.8a) and (4.8b), and supporting conditions (4.1 la) and (4. I Ib) in the interior of the plate region, where

(4.12)

(4.13a)

(F-

a2W(m) a2w(m)

-

2(1- v)

ay2

a Z w ( r n ) a2,&n)

-

-ax dy ax ay

(4.13b)

Wei-Zang Chien

(4.13~) (4.13d) the number of elements having common corner points at point k2 on plate edge, at which the plate deflection Gkg is given. the number of elements having common corner points at point c2 in the interior of the plate, at which the plate deflection GC2is given. w& j = 1,2 )...)J', j = 1, 2, ..., J", w$, the number of elements having common corner points at point k l on the plate edge, at which the concentrated load &, is given. the number of elements having common corner points at point c I in the interior of the plate, at which the concentrated load FClis given.

(4.13e)

(4.13f) (4.13g) (4.13h)

(4.13i)

(4.13j)

Generalized Variational Principle IIIA further shows that the following interelement compatible conditions are satisfied: -

W(m')

=

0

on

s(mm')

(4.14a) (4.14b)

by virtue of (4.15a)

dn(tn)= &m)

=

-ds(m').

(4.15b)

Generalized Variational Principle III**A (generalized variational principle of incompatible elements derived from the minimum potential energy principle): Among all sets of field variables w(")(x, y) (rn = 1, 2, ..., N ) of incompatible elements, the set of w ( ~that ) makes IIf;", stationary satisfies the field equation (4. I), the boundary conditions (4.5a), (4.5b), (4.6a) and (4.6b), the corner conditions (4.1la) and (4.1lb), the interelement continuity conditions (4.14a) and (4.14b) of deflection in the interior of the plate, and also the interelement continuity of bending moments and equivalent shearing forces M,( W(',)) = M,( w('"')) on s ( m m ' ) , (4.16a)

Incompatible Elements: Generalized Variational Principles H,(w(m)) = - H

(w(fn'))

on

S(mtn').

131

(4.16b)

The functional of this principle is

in which, with IILT) as defined in (4.13a),

for ni

-

1 2,ls,m,,,,, 3 [ H A W (m ,) + H , , ( W ( ~ ' ) )ds. ]~(~)

(4.17a)

all m for WI

The upper index r" in (4.17) represents the total number of elements denoted by I", 2", ..., r" with common corner points at point kl on the plate edge, and we may take any one of these elements as the 1"-element. Similarly, the upper index r"' in (4.17) represents the total number of elements denoted by iff', 2"', ..., r"' with common corner points at cI in the interior of the plate, and we may take any one of these elements as the 1"' element. The proof of Variational Principle III**A is given by Chien (1981) and is not repeated here.

B. MATRIX EQUATIONS OF DEFLECTION-INCOMPATIBLE ELEMENTS Let us now take the serendipity family of incompatible triangle elements. Suppose that each element is assigned to have either 6 or 9 degrees of the mth element may be of freedom (Fig. 3). The field function dm) written as W(m) = N ( m + + h ) . (4.18) The interpolation functions in the case of 6 degrees of freedom are

Nm)= ~ L ~ ~ ' ( L -I ~ 41, ) =

4Lj('")Ly),

i = 1, 2, 3;

(4.19a)

i=4,5,6,(i-3,j,ktake 1,2,3 in cyclic permutation). (4.19b)

132

Wei-Zang Chien

FIG.3. Incompatible triangular plate element of a serendipity family: (a) six degrees of freedom, (b) nine degrees of freedom.

In the case of 9 degrees of freedom, they are

@)

=

$ L ~ ~ ' (-L$)(L!") ~ ~ ' - $),

i = 1, 2, 3,

=

y L y ) kp ()L!"' J - f),

i = 4, 5 , 6 (i - 3, j , k take 1, 2, 3 in cyclic permutation), (4.20b)

Nm)=y

~ ( ~Lk(mm )(L) (km ) - 231,

in which

i

=

7, 8, 9 (i - 6, j , k take 1, 2, 3 in cyclic permutation), (4.20~)

N(m) = [ N I ,N 2 , ..., N$"), W(m)T

(4.20a)

= [ I , w 2 , ..., w , ] ( ~ ) ,

t = 6 or 9;

(4.2 1a)

t = 6 or 9.

(4.21b)

Lim),and Lim)are the area coordinates in the mth triangle element, and w!") is the nodal deflection of the mth element at the ith nodal point. If this kind of incompatible element is used, the deflection of the mth element and its adjacent m'th element at the common nodal points equal to each other (see Fig. 4), then the deflections of the elements at the interelement boundaries are continuous; that is, (4.14a), (4.13g), and (4.13h) are automatically satisfied. The functional II,TfAmay be simplified to

with m

Incompatible Elements: Generalized Variational Principles

+ (1 - u ) w . J m ) , v2w + ( 1 - U ) w ~ J m ’ ) ,

M,(w(”))= - D [ u v*w

M,(w(’n‘))= - D [ v

133

(4.23a) (4.23b) (4.24a) (4.24b)

evaluated with the above interpolaIn n,cl;),the terms involving H,,(w(’~)) tion functions become Hn(W(m))

=

0

H , , ( W ( ~= ) ) constant

for six degrees of freedom,

(4.25a)

for nine degrees of freedom.

(4.25b)

Thus, in the case of 6 degrees of freedom, the corresponding H n ( d m ) )is identically zero everywhere, or

I

’;!A

Hn(w(rn))(w(rn)

-

w)&On)

0

(4.26)

In this case, the boundary condition - -hI =

(4.27) 0 must be used separately in the final computation of the global matrix, in which the indicesj and ( m ) indicate thejth boundary nodal point of the mth finite element. For the purpose of calculation of a2wlan2,a2wlas2,a2wldn as, we define a as the direction angle between outward normal n and the positive x axis, as shown in Fig. 5 . The transformation between coordinate systems ( n , s) and (x, y) is J

n = x cos a

+ y sin a,

W.i

s = --x sin a

+ y cos a,

(4.28)

FIG. 4. The positions of nodal points of adjacent finite elements: (a) six degrees of freedom, (b) nine degrees of freedom.

Wei-Zang Chien

134

FIG. 5. The direction angle between outward normal n and the positive x-axis of a triangularelement.

or x

=

n cos a - s sin a,

y = n sin a

+ s cos a .

(4.29)

Thus we have

(4.30a)

cos2 a

-sin a cos a sin2 a

2 sin a cos a cos2 a - sin2 a

-2 sin a cos a

sin2 a

sin a cos cos2 a

From these relations, we find that

M A W ) = -D[l, 0 , v ]

T

= [cos2 a

=

-DT

(4.31)

+ v sin2 a , 2(1 - v) sin a cos a, sin2 a + v cos2 a ] . (4.32)

Incompatible Elements: Generalized Variational Principles

135

From (2.1), we obtain

(4.33)

where

G =

a2N2 a2N, ax ay ax ay

...

-1

a2N, ax ay

(4.34)

The result is (4.35) For t h e j t h edge of element ( m ) , we can calculate 7j in terms of aj: TT) =

[ C O Saj~ + v sin2 aj, 2(1 - v) sin aj cos aj, (4.36a) sin2 aj + v cos2 a j ] , j = 1, 2, 3;

Finite element ( m ) has three adjacent elements m' (numbered m i , m i , m; as shown in Fig. 4). The actions of the bending moment due to eledmmi),and ments m i , m i , and m; on the interelement boundaries dmmi), dmm)) are, respectively, M n , ( w ( m l ) ) = -DT(mi)G(mi)w(mi),

(4.37a)

136

Wei-Zang Chien

where

(4.39)

Thus we can evaluate the integrals in (4.22b). The results are

(4.40)

(4.41a) (4.41b) (4.41~) (4.4I d)

(4.42)

(4.42a)

(4.42b)

Similarly, we can write (4.43)

Incompatible Elements: Generalized Variutional Principles

137

(4.43a)

(4.44) (4.45) (4.45a) (4.45b) (4.45c) (4.45d) a3NI ax3

J=

d3Nt -

a3N2 ax3

a3NI a3N2 ax2 ay ax2 ay

ax3

d3N,

..

ax2 ay

a3NI a3N2 ax ay2 ax ay2

a3N, ax ay2

a3NI -

a3N2 -

a3Nt -

dy3

ay3

dY3

(4.46)

Thus we obtain (4.47a) (4.47b)

(4.48) and NiT:, fi:,), and J(m)are defined by (4.21a), (4.49, and (4.46), respectively. Therefore, (4.13a) can be written in matrix form: m ) - 1 (m)TK(m)( m )- FOn) (tn) (4.49) nil, - zw f w f W '

where K**(m) = Kh") + KF).

(4.52)

By means of similar notation, we may express (4.53) where

Pg:

= -( 1

m = [-sin

- v ) D Ak2(nG)'J), CY

cos a cosz a

-

(4.54) sin2 a sin

CY

cos a].

(4.55)

Finally, we have the assembled global matrix equation

II,*,&, bwTK**w - Qw, 2

in which

(4.56)

139

Incompatible Elements: Generalized Variational Principles

(4.58) The condition for II;",;"A, to be stationary is

sII:,*,I

=

Kw

0,

=

Q,

(4.59)

in which K

=

$(K** + K**T).

(4.60)

For simp.j supported plates, the above results can "e greatly simplified. For the incompatible triangle finite element with 6 or 9 degrees of freedom, (4.58) reduces to

(4.61b) where

C. GENERALIZED VARIATIONAL PRINCIPLE COMPLEMENTARY DERIVEDFROM THE MINIMUM ENERGYPRINCIPLE Suppose that a thin plate reaches equilibrium under the action of bending moment Map and shearing force Qa . The equations of equilibrium are Map,p =

Qa., + F where

F

=

Qa,

(4.63a)

0,

(4.63b)

is the distributed normal load. Elimination of Qa gives Map,.p

+F

= 0.

(4.64)

140

Wei-Zang Chien

In addition, there are in general the boundary conditions, the corner conditions (4.5a), (4.5b), (4.6a), (4.6b), (4.8a), and (4.8b), and also the curvature-bending moment relation

+ BapysMys where Bapysare the flexibility constants. w,ap

=

(4.65)

0,

Minimum Complementary Energy Principle IV: Among all the bending moments Map satisfying the equations of equilibrium (4.64), the boundary conditions of known edge moment and equivalent shearing forces (4.5a) and (4.6a), and corner conditions of known reaction forces (4.8a), the set of Map that minimizes IIIvalso satisfies the boundary conditions of known edge deflection (4.5b) and (4.6b), the corner conditions of known supporting deflection (4.8b), and the curvature-bending moment relations (4.65). The required functional n , is~

k.

(4.66) in which Hn =

Qn

+ Mns,.Y = Mn,n + 2MnAjs

Mn = Mapnanp,

9

(4.67a) (4.67b)

Qn

=

Map.pn,,

(4.67~)

Pk2

=

hk,Mns.

(4.67d)

Here we assume that the boundaries of the plate are composed of straight edges. We may consider the equations of equilibrium (4.63a) and (4.63b), the boundary conditions (4.5a) and (4.6a), and the corner condition (4.8a) as constraints of variation, and by means of appropriate Lagrangian multipliers A, p ( l ) ,p ( 2 ) and r p ( k l ) ,Variational Principle IV with functional (4.66) may be transformed into the Generalized Variational Principle IV* with another functional,

Incompatible Elements: Generalized Variational Principles

141

(4.69) The variations 6 M a p , 6 H , , 6 M n , 6Pkl, 6Pk2, 6A, 6 p ( , ) ,6p(2),and 8p(kl) are all independent of each other. Thus the stationary condition of II,*, gives equations of equilibrium (4.63a) and (4.63b), boundary conditions and corner conditions (4.5a), (4.6a), and (4.8a), and also (4.70a) (4.70b)

A = - p(1) A.n

=

A,n

= ~ ( 2 )

on

@,n

A

= Gk2

A

=

p(kl)

S,I

(4.704

9

on sMS2 ,

(4.70d)

on sgZ,

(4.70e)

at k2 = 1, 2,

..., k M , ,

(4.70f)

at k l

..., k,.

(4.70g)

=

1, 2,

We have A = w &I)

=

P(2) = &kl)

in A , SM.1 S , 2 ’ S,I S,? and at kl = I , 2, ..., k,, k2 = 1, 2, 9

9

..., k,,.,

(4.71)

-w

on s T l .

(4.72a)

W3n

on sT2,

(4.72b)

= WkI

at k l

=

1, 2,

..., kM,).

(4.72~)

Equations (4.70b), (4.70d), and (4.70f) are, in fact, the given boundary deflection conditions, and Eq. (4.70a) the curvature-bending moment relation. Equation (4.70a) may be written as

BapysMys

=

-w,.p

*

(4.73)

Thus, the physical nature of all Lagrangian multipliers are determined. The functional IIfv may be written as

Wei-Zang Chien

142

Generalized Variational Principle IV* (for the bending of plate derived from the minimum complementary energy principle): Among all sets of Mapand w , the set of MaPand w that makes IIfVstationary satisfies all the necessary physical conditions (i.e., equations of equilibrium, curvaturebending moment relations, and all the boundary conditions and corner conditions). It should be noted that

11

Map@pW d A =

1,

wMap.anp ds

A

=

we,, ds

-

-

1,

I\

M’,pMap,a dA

A

w,pMapnads

+

I\

M a p ~ , adA P

(4.75)

A

and also -

I,w.pMapna ds

= -

\jw.rMns + w,nMn) ds kn +k,

= k= I

wk AkM,,

+

wM,,,, ds

-

I

wqnM,ds. (4.76)

Therefore we obtain

11 A

Map,3w dA =

11

Mapw,+ d A +

\$

H f l wds

A

Substitution of (4.77) in (4.74) provides another form of II,*,, which is

This is the second form of the generalized variational principle derived from the minimum complementary energy principle.

Incompatible Elements: Generalized Variational Principles

143

Variational Principle IV*A: Among all sets of Map and w , the set of Map and w that makes III*,, stationary satisfies all the necessary physical conditions (i.e., equations of equilibrium, curvature-bending moment relations, and all boundary conditions and corner conditions). Let us now divide the plate region into N finite elements and study the related variational principles. We further assume that besides the distributed loads F , the boundary conditions (4.5a), (4.5b), (4.6a), and (4.6b), and the corner conditions (4.8a) and (4.8b), the plate is loaded by concentrated forces Fc, at cF interior points (+, ycl),and is supported by given deflection Wc2 at some other c , points ( x c 2 , yc2); that is, the plate is subjected to conditions (4.1 la) and (4.1 lb). We also require that CF + c , points be the common corner points of the finite elements concerned. Thus we have the following generalized variational principle for compatible finite elements for the bending of plate. Generalized Variational Principle IV*f: Among all sets of finite-elernent field functions M$) and won)satisfying all compatible conditions for bending moments and equivalent shearing forces

the set of and w("') that makes II&, stationary satisfies the equation of equilibrium of finite elements (4.64), the curvature-bending moment relation (4.65), boundary conditions (4.5a), (4.5b), (4.6a), and (4.6b), corner conditions (4.8a) and (4.8b), and supporting conditions in the interior of the plate (4.1 la) and (4. I lb). Furthermore, the solution of dm) thus obtained satisfies also the compatible conditions

(4.80b) Functional I&

is

144

Wei-Zang Chien

Similarly, from II,*,A,we find the generalized variational principle of deflection-compatible elements. Variational Principle IV*Af Among all sets of finite-element field functions M$’ and w(‘“)satisfying all the compatible conditions of deflecthat makes n?vAf stationary satisfies the tion, the set of M$) and w(’”) equilibrium conditions for all the finite elements (4.64), the curvaturebending moments relation (4.65), the related boundary conditions (4.5a), (4.5b), (4.6a), and (4.6b), the corner conditions (4.8a) and (4.8b), and the constraint conditions of supports in the interior of the plate (4.1la) and (4.1 Ib). At the same time, the bending moments and equivalent shearing force associated with the solution also satisfies the compatible conditions on the interelement boundaries (4.79a,b). The appropriate functional GVA~ is

(4.84) where

(4.85)

Incompatible Elements: Generalized Variational Principles

145

D. GENERALIZED VARIATIONAL PRINCIPLE OF INCOMPATIBLEELEMENTS DERIVED FROM THE MINIMUM COMPLEMENTARY ENERGY PRINCIPLE Let us consider the generalized variational principle for incompatible elements. Take the functional I&as an example. If the bending moments are not compatible, we may use the Lagrangian multipliers for the conditions (4.79a) and (4.79b) and reduce the conditional variational principle into a nonconditional principle. We call this nonconditional principle the generalized variational principle of incompatible elements. The functional is

(4.86) From the stationary conditions of this functional we can prove that

(4.87b) Thus we may change (4.86) into

(4.88)

146

Wei-Zang Chien

(4.89) for m

Hence we find the following generalized variational principle for incompatible elements of plate bending derived from the minimum complementary energy principle. Variational Principle IV**f: Among all sets of field functions of finite ),;' w('")that makes nr;*,stationary satiselements M'$, w('"),the set of M fies the equations of equilibrium of finite elements (4.64), the curvaturebending moment relation (4.65), the related boundary conditions and corner conditions (4.5a), (4.5b), (4.6a), (4.6b), (4.8a), and (4.8b), and the supporting conditions in the interior of plate (4. I la) and (4. I Ib). At the same time, the interelement continuity conditions (4.80a), (4.80b), (4.79a), and (4.79b) are also satisfied for the bending moment, equivalent shearing force, deflection, and slope of deflection. Similarly, we may establish another generalized variational principle for incompatible elements from Variational Principle IV**Af. Generalized Variational Principle IV**Af (another plate-bending variational principle for incompatible elements derived from the minimum complementary energy principle): Among all sets of field functions M'$, w(")of finite elements, the set of M'$ and w('")that makes the functional n,*,*,,stationary satisfies the equation of equilibrium (4.64), the curvaturebending moment relation (4.65), the boundary conditions (4.5a), (4.5b), (4.6a), and (4.6b), the corner conditions (4.8a) and (4.8b), and the supporting conditions (4.1la) and (4.1 lb) in the interior points of plate region A. At the same time, the interelement conditions of continuity for the bending moment, equivalent shearing force, deflection, and slope of deflection are also satisfied. The corresponding functional n,*,*,,is

Incompatible Elements: Generalized Variational PI jnciples

for nr

147

(4.91)

It is easily shown by means of (4.77) that

rI;;

(4.92)

= rIl"v*Af.

That is to say, Variational Principles IV**f and IV**Af are in fact identical.

E. THEMATRIX FORMULATION OF INCOMPATIBLE ELEMENTS FOR T H E BENDING OF T H I NPLATE Let us assume that there are t nodal points in a finite element. The nodal values of w("~) are w(;n),wy' ... w y ) ;those of M$' are M(l&), denoted by M(/) ( k = 1, 2, ..., 3t) in the order of M(,'y(',),M(,'i/l),&I!$'$',),&I!j'y('z), M(,'&, ..., , M(i& , MK:tj . We may write 9

w(m)T

= [ w p , w p

M(m)T =

..., W p ] ,

[M?,, M% 7

7

Mi?),,

M?I) MTh, Mi%I 9

9

(4.93a) 7

M W 2 ) 7 M(IT(2)

*

7

Mi%,

9

** *

9

(4.93b)

We denote by N l ( x ,y), N2(x,y ) , N3(x,y ) , ..., Nl(x,y) the interpolation functions of finite elements. Thus we have

[$:I

w(m)

= N(m)w(m) =

Ni;;)pM(m),

(4.94a) (4.94b)

where N ( m ) and "5)are defined as in (4.21a) and (3.33d). Let us consider the simplest kind of incompatible elements, that is, the serendipity family of triangle elements in Figs. 3 and 4. Thus the deflection and bending moments are continuous on the interelement boundaries. However, the

148

Wei-Zang Chien

slopes of deflection and equivalent shearing forces may not be continuous; that is, W(rn)

=

,,,(rn’)

on

S(mm‘),

(4.95)

M p

7

-M(‘n’)

on

S(JilJn’).

(4.96)

On all the common corners of adjacent elements, the corner deflections are also continuous; that is,

wl;’ = ... = HI;;‘, (2) = ... = wCI (JJ?, w;;l = W cI w p = wg) = ... = w L ~ ) , Wk2 = wg’= wL;) = ... = w&), zc2= w g

=

From definitions P c ) = AAIM!,‘:),Ply’ have

..., c,; 1, 2, ..., c F ; 1, 2, ..., k,; 1 , 2, ..., k,.

c2 = I, 2,

(4.97a)

CI =

(4.97b)

kl =

k2 =

(4.97~) (4.97d)

A(lM!‘:’, etc., we immediately

=

Thus, IIr$ may be written as

where

+ J18~~irn) - M)wv,”’ ds - J,p,H:”’)W(’~)ds

(4.100)

I

and s\“) represents the interelement boundary in the interior of the plate region for the finite element ( m ) . As for @&f, we have

Incompatible Elements: Generalized Variational Principles

149

(4.102)

aw ax aw ay aw aw w,n = -i- - - = cos a - + sin a - = N,,w dx an dy dn dX aY

(4.103)

where ax

ax

dNr

dN?

N,n = [cos a , sin a ]

. . .,

Lay. r3>”

dN,

1.

(4.104)

GJ

By means of (4.94b), we find

M,

=

I: [

Mapnanp= [cos2 a , 2 sin a cos a,sin2 a ] M I 2 = R,M,

in which

R,,= [cos2 a , 2 sin a cos a , sin2 ( Y ] N ( , ) ~ .

(4.105)

(4.106)

Similarly,

Mns = M a p a s p = [-cos a sin a , cos2 a

-

I::[

sin2 a , sin a cos a ] M 1 2

-

(4.107)

or Mns

=

RnsM,

(4.108)

where R,, is R,,v = [-cos a sin a , cos2 a

-

sin2 a , sin a cos a l N ( ~ ) ~(4.109) .

Furthermore, Mn,n =

Rn,nM,

Mns,.v = Rns,sM,

(4.110)

Wei-Zang Chien

150

and Rn,n= [cos2 a, 2 sin a cos a, sin2 (Y]N(I)P,~,

(4.1 11a) (4.1 11b)

Rns,, = [-cos a sin a, cos2 a =

-

sin2 a, sin a cos a ] N ( l ) ~ , . ~ , (4.11lc)

a

(4.11Id)

[-sin a ax, cos a

Therefore, we find

Hn

=

Mn,n

+ 2Mns,,y ==

(Rn,n + 2Rns,s)M.

(4.1 12)

The operator matrix D2 is defined as (4.113) Then we have from (4.94b) DzN(I)PM.

(4.114)

D~NwN( I )pM = w ~ ( D ~ N )1 )pM. ~N(

(4.115)

Map,,

=

Similarly, we find

Mapw,.P

=

By means of these matrix notations, (4.98) can be written as

(4.117a)

tncomputible Elements: Generalized Vuriational Principles

15 1

From (4.99) we find

in the following global matrix Substitution of (4.1 16) into (4.42) gives II,*,*, form

$:IX

=

$MTC1M + wTC2M + wTF + MTW

(4.119)

in which

rn= I

(4.119d) From the stationary condition

8n:S

=

0

(4.120)

we obtain

CIM

+ CTW + W = 0, C2M + F = 0.

(4.12 1a) (4.12 1b)

This is the matrix equation for the finite-element computation derived from rI74. Let us now consider XI,*,*,,. This gives a,*,*df") = &M(rn)TC{m)M(m) + W(m)TC!g)M(m) + W(m)TF(m) + M(m)T,jy) (4. 122) in which C\"), F(m)are defined by (4.117a) and (4.117~);the others are

Wei-Zang Chien

152

and (4.101) can be written as

Substitution of (4.122) into this equation recasts it in global matrix form:

II,T~*AA~ = hMTCIM + wTC2AM + wTF + MTW.

(4.125)

Comparison of this equation with (4. I 19) shows that except for the second term on the right-hand side of (4.123), the other terms are identical. Now Cm is defined by N

(4.126)

The stationary condition of (4.125) is (4.127)

6nr&f= 0,

from which we obtain for M and w the equations

C I M + CTAW 4- W

=

0,

C~AM +F

=

0.

(4.128)

This is the set of matrix equations for incompatible finite-element computations derived from the functional IITGAA~. The results given in this section are in complete accord with those given by Herrmann (1966, 1967) for the same problem.

V. Conclusions This article shows that all incompatible finite dements can be treated by original field variables used in corresponding compatible elements. The compatible conditions are satisfied in an integral sense on interelement boundaries and are completely satisfied in the limit of diminishing sides of finite elements. This is economical in computations with fewer degrees of freedom. We are able to reduce the unknown Lagrangian multiplier to original field variables. In this article, three problems are discussed: (1) general three-dimensional elasticity, (2) two-dimensional problems of elasticity, and (3) bending problems involving thin plates. For all cases, the generalized variational principles of incompatible elements are discussed on the bases of

Incompatible Elements: Generalized Variational Principles

153

both the minimum potential energy principle and the minimum complementary energy principle. REFERENCES Chien, W.-Z. (1980). “Variational Principles and the Finite Elements Method” (in Chinese). Science Publishers, Beijing. Chien, W.-Z. (1981). Incompatible plate elements based upon the generalized variational principles. Proc. I n t . Symp. Mixed Finite Elements, Atlunra, Georgia, April 8-10. Greene, B. E., Jones, E. E., McKay, R. W., and Strome, D. R. (1966). General variational principles in the finite element method. AIAA J . 7 , 1254-1260. Herrmann, L. R. (1966). A bending analysis for plates. Proc. 1st Conf. Matrix Methods Struct. Mech., 1965 (AFFDL-TR-66-80), 577-604. Herrmann, L. R. (1967). Finite element bending analysis for plates. J. Eng. Mech. Diu., A m . Soc. Civ. Eng. 98, 13-26. Jones, R. E. (1964). A generalization of the direct-stiffness method of stiffness analysis. AIAA J . 2 ( 5 ) . 821-826. Pian, T. H. H., and Tong, P. (1972). Finite element methods in continuum mechanics. Adu. Appl. Mech. 12, 1-59. Tong, P. (1970). New displacement hybrid finite element model for solid continua. I n t . J . Numer. Methods 2, 78-83.

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ADVANCES I N APPLIED MECHANICS, VOLUME

24

Galactic Dynamics and Gravitational Plasmas" C. C. LIN Massachusetts Institute of Technology Cambridge, Massachusetts

G. BERTIN Massachusetts Institute of Technology Cambridge, Massachusetts and Scuola Normale Superiore Pisa, Italy

...........................................

156

11. Observations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

158

1. Introduction

A. Radio Continuum Observations of M51 . . . . . . . . . . . . . . . . . . . . . . . B. Gas Motions in M81. . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Detection of Smooth, Broad Spiral Arms. . . . . . . . . . . . . . . D. The Survey by Roberts, Roberts, and Shu . . . . . . . . . . . . . . . . . . E. Single-Mode Galaxies and Coexistence . . . . . . . . . . . . . . . . . . . . . . . 111. Density Wave Theory of Spiral Structure. . . . . . . . . . . . ......... A. Some Basic Dynamic Stellar Systems . . . . . . . ......... B. Disk Galaxies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. The Lin-Shu Dispersion Relation D. Dynamic Theory of S E. Interacting Galaxies. .............. 1V. Dynamic Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Propagation of a Group of Density Waves; Conservation of Wave Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Propagation of Wave Packets C. Propagation of Sustained Wave Trains . . . . . . . . . . . . . . . . . . . . . . . V. Theory of Discrete Spiral Modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Spiral Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Methods of Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159

163 164 164 165

170 171 171 172 173 175 176 179

* This paper is reproduced from Plasma Astrophysics [(T. D. Guyenne and G. LCvy, eds.) ESA SP-161, pp. 191-205. European Space Agency, Noordwijk, The Netherlands]. Minor changes were made. For further discussion of related issues see Lin and Bertin (1984). IS5 Copyright 0 1984 by Academic Press. lnc. All rights of reproduction in any form reserved. ISBN 0-12-002024-6

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C . C . Lin and G . Bertin

C. Resonances: Their Roles in Maintenance of Spiral Modes . . . . . . . . . . D. Modes with High Quantum Numbers . . . . . . . . . . . . . . . . . . . . . . . . VI. Dynamic Approach to Classification of Galaxies . . . . . . . . . . . . . . . . . . . A. Bar Structure of Type sBB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Bar Structure of Type sB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Bar Structure of Type sAB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Normal Structure of Type sA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Dual Spiral Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VII. Concluding Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

180

182 182 183 183 184 184 185 185 187

I. Introduction A normal galaxy is primarily a collisionless, gravitationally bound stellar system, although there are other constituents such as interstellar gas, cosmic ray particles, and magnetic field. The study of the stellar system may be made on the basis of Liouville’s partial differential equation for the distribution function of point masses (the stars) in phase space and of Poisson’s equation for gravitational potential. This combined system of equations is the counterpart of the system of Vlasov and Maxwell equations in electromagnetic plasmas. Hence the term “gravitational plasma” is appropriate for the description of stellar systems. In this discussion, we mention some specific similarities and differences between these two kinds of plasmas, but our main objective is to give a report of current developments in the dynamics of stellar systems, especially in the applications to spiral galaxies. According to a scheme initiated by Hubble and refined by Sandage and de Vaucouleurs, regular galaxies are classified according to their morphology into ellipticals, normal spirals, and barred spirals. Within each category there are subdivisions. For example, the normal spirals are subdivided into Sa, Sb, and Sc galaxies in the order of decreasing disk/bulge mass ratio, increasing pitch angle, and increasing gas content. There are also transition types, such as SO and SAB. One of the challenges to theorists is to find the reasons why there are global spiral structures, especially since the bright spiral arms are actually marked out by the newly formed young stars. Another challenge is the considerable permanence of the preceding classifications, since the subdivisions are, on the one hand, correlated with a number of physical characteristics (e.g., the bulge size) that cannot change rapidly in time and, on the other hand, with the pitch angle, which appears to be more readily changeable because of differential rotation in the system. This is known as the winding dilemma.

Galactic Dynamics and Gravitational Plasmas

157

Observations, especially of the frequent existence of a regular grand design of spiral structure in differentially rotating systems,* have naturally led to the suggestion that spirals are governed by a wave phenomenon. In this view the spiral distribution of stars is a collective mode in the stellar system, which can be studied according to the principles of stellar dynamics just like the collective modes in electromagnetic plasmas. Our chapter is indeed concerned with such collective modes. After the formulation of the density waue theory and the analysis of its dynamic implications, many important observations have been made that strengthen the wave interpretation of spiral structure. According to this picture, a density wave pattern, often of relatively low amplitude, is present in the stellar component of the galactic disk, and this, in turn, induces shocks in the interstellar medium, which is much less massive than the stellar component. The shocks, in turn, trigger star formation and produce the more striking optical phenomena. Even if it is hard to decide whether a given spiral is a transient or a quasi-stationary structure, on a statistical basis the theory of quasi-stationary spiral structure is on firm ground (cf. Fig. 3 and Section 11,D). We may add that some transient spiral features (cf. Section IV,B,4) are characterized by a very small amount of relative motion between the pattern and the material. Thus their ability to trigger star formation consistent with observations (cf. Section I1,A) is still to be proved. More recently, the density wave picture has been brought to a relatively complete form, supported by a fairly coherent dynamic basis so that we may attempt to formulate a dynamic approach to the classification of spiral galaxies. We deal with the classification of galaxies into SA, SAB, and SB types, not with the subclassifications such as SAa, SAb, and SAC and their relationship to the size of the nuclear bulge, and to the gaseous content. These have often been discussed in the existing literature. We do not attempt to add to those discussions in the present article, but reserve these elaborations for a later presentation. Among the numerous dynamic issues that we do not cover are many numerical experiments on the stability of flat stellar systems, the possibility of galactic evolution due to fast internal instabilities, and the detailed behavior of shocks in the interstellar medium in the presence of bar and/ or spiral fields. On this last point, a proper discussion of the physics of the interstellar medium would be necessary.

* For a recent study of the frequency of occurrence of grand design spirals, see Elmegreen and Elmegreen (1982). A report on this work was presented at IAU Symposium 106 [see Lin and Bertin (1984)].

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In addition to the problem of spiral structure, there are, of course, many other dynamic issues that belong to the domain of galactic dynamics. For instance, the structure of the ellipticals, with their (so far) mysterious three-dimensional structure, provides an example of the richness of phenomena in collisionless stellar systems and may teach us something about the structure of the invisible halos. The dynamic role of halos around galactic disks is being studied actively, especially after the discovery of generally flat rotation curves and the suggestion by Ostriker and Peebles [(1973), see Bertin (1980)l that a global-order parameter ( t ) should govern the overall stability of axisymmetric galactic systems. One important issue is whether global parameters like t can be sufficient to characterize the dynamic properties of self-gravitating systems. The answer may be positive for those systems that possess only one length scale and is likely to be negative for those systems with many scales (cf. the discussion on dual systems in Section VI). Another large-scale phenomenon under active investigation is the warping of the outer parts of galactic disks. Warped disks have been found to be the rule rather than the exception in spiral galaxies. They appear to be related to the presence of a sizable spheroidal halo component, and their collective dynamics may prove to be an interesting probe of the outer structure of galaxies. It has been suggested that the environment in groups or in clusters of galaxies plays an important role in galactic evolution (see also Section 111,E). This would occur via galaxy-galaxy and galaxy-intergalactic medium interactions. Thus there is now revived interest for a deeper understanding of dynamic friction and related phenomena. In this regard also, one important open question is the role of large halos around galaxies in such “environmental” dynamics. All these issues indicate how powerful and complex the collective effects are in galactic systems. They would all deserve a separate and extended discussion. In the present article we have chosen to focus on the problem of spiral structure, where great progress has been made and many detailed cooperative phenomena have been investigated at length.

11. Observations Rather than list all the relevant observational material [which has been often reviewed, e.g., see Lin (1971) for an early review and Strom and Strom (1978) for a more recent review in a broader context], we would like here to emphasize only a few cases of great importance in the overall perspective of the theory of spiral structure. These offer strong encour-

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agement for the construction of a dynamic classification of galaxies. The reader is referred to the original literature for other details.

A. RADIOCONTINUUM OBSERVATIONS OF M51 Radio observations by Mathewson et al. [(1972), see Bertin (1980)l of M51 at 1415 MHz have revealed a clear delineation of a two-armed grand design spiral structure with the emission maxima lying at the inside edges of the bright optical arms in coincidence with the dust lanes. The linear polarization of the observed emission implies that it is due to synchrotron radiation. The inner radio arms are estimated to be less than 250 pc wide. These data strongly support the view that the interstellar medium undergoes a shock in the way predicted by the density wave picture. The clear correlation of the radio continuum data with many other observed properties is indicative of the collective behavior as the basis of the phenomenon of spiral structure.

B. GAS MOTIONSI N M81 Visser [(1978), see Bertin (1980)l studied the dynamics of gaseous motions in M81 (see Fig. 1). After a choice of an optimum model for the optical spiral pattern based on density wave theory, Visser calculated the gaseous response to the spiral field by using the value of the amplitude for the density wave suggested by the photometric data of Schweizer [(1975), see Bertin (1980); cf. Section II,C]. Comparison with the observed gas motions is in remarkable qualitative and quantitative agreement. Visser and Haass (1981) have been able to provide dynamic support to the whole analysis by calculating a self-excited spiral mode for a model of M81 (see Fig. 2) in agreement with the observed pattern and the theoretical pattern used by Visser.

c. DETECTION OF SMOOTH, BROADSPIRAL ARMS By means of photometric techniques, it has been possible to study the structure of the underlying old disk population in a number of spiral galaxies with a grand design [Schweizer (1975), see Bertin (1980); Elmegreen (1981), see Lin and Bertin (1984)l. The consequent detection of smooth, broad spiral arms in this massive component (M51 and M81 are within the sample studied) indicates that an underlying density wave of

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FIG. 1. Radial velocity field associated with density wave model (symbols) compared with observed velocity field superimposed on a radiograph for M81 [from Visser, H. C. D. (1978), Ph.D. Thesis, University of Groningen, Holland].

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FIG.2. Density contours for a theoretical spiral mode by Visser and Haass (1981) that provides the dynamic basis for the model of M81 by Visser.

moderate amplitude is behind the scenes and governs the other (less massive and less hot) components of galactic systems to form the observed spiral structure. The smoothness and regularity of these data indicate that a linear analysis for the dynamics of density waves is probably adequate to cover most observed situations.

D. THE SURVEY B Y ROBERTS, ROBERTS,A N D SHU Roberts et al. [(1975), see Bertin (1980)l have been able to find a statistical correlation between dynamic (theoretical) parameters and observational classifications: the morphological classification mentioned earlier and the luminosity classification of van den Bergh (see Fig. 3). This categorization of disk-shaped galaxies on the basis of the theory of density waves (which predicts a general range for pitch angles and shock strength, depending on the axisymmetric basic model of the galaxy) suggests that, at least in a statistical sense, the density wave theory of spiral structure is on solid ground. Furthermore, it adds much support to the belief that the current morphological classifications are of a persistent nature and, therefore, gives much confidence to theorists who seek a relatively simple dynamic approach to the classification of galaxies (see Section VI).

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FIG.4. A galaxy studied by V. Rubin (D100).

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FIG.5 . Effect of superposition of three spiral modes on a galaxy model [from Haass, J., Bertin, G., and Lin, C. C. (1982), Proc. Nutl. Acad. Sci. USA 79,39081. Spurs, such as the one shown in Fig. 4, could be due to the coexistence of a few modes.

E. SINGLE-MODE GALAXIES AND COEXISTENCE There is no doubt that not all spiral galaxies present a neat grand design. An interpretation is that the more regular the observed spiral structure, the smaller the number of spiral modes involved in its dynamics. Certain observed galaxies seem to demand a single dominant mode (in a sense that will be explained later; cf. Section V), and their nonbisymmetric spiral grand design rules out any tidal or nondynamic origin. However, in general, the observed structure is very often of a global twoarmed appearance with some less regular features, such as spurs (see Fig. 4). Haass’s example (Fig. 5 ) of the linear superposition of three global spiral modes (of the kind described in Section V) readily shows that the effects of coexistence are likely to play a major role in the comparison of the theory with observations. On the other hand, it would be wrong to conclude that the presence of a number of modes greater than three necessarily implies the fuzzy structures of the NGC 2841 type. In fact, a linear superposition of three regular, two-armed normal modes leads to a coherent (though, in general, evolving in time) two-armed grand design. A further discussion of coexistence is given in Section VII.

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111. Density Wave Theory of Spiral Structure

A. SOMEBASICDYNAMIC CONSIDERATIONS: NATUREOF STELLAR SYSTEMS THECOLLECTIVE Similarities between self-gravitating systems and ordinary plasmas arise from the common long-range nature of the basic forces, whereas differences arise from the opposite signs of these forces, and from the existence, in the gravitational case, of an equivalence principle that identifies the gravitational charge with the inertial mass. Classical and statistical mechanics are common fundaments of stellar dynamics and plasma physics. Systems dominated by long-range interactions are not “additive” in the sense that, in general, their behavior cannot be described in terms of the properties of their subsystems, when studied separately. Thus, if we are interested in stability properties, it is crucial to distinguish between local and global stability criteria. Furthermore, a critical role in the behavior of such systems is played by their geometry. It is appropriate to describe the kinematics of large stellar systems (in particular, galaxies) by means of a distribution function indicating the density of stars in a suitable phase space. In general, galaxies are found to be collisionless, so that the evolution of the distribution function is governed by the Vlasov equation of plasma theory. This common description, together with the common nature of the dynamic interactions, is the basis of a considerable number of analogies between stellar dynamics and the physics of high-temperature plasmas. Gravitation can be naturally balanced not only by pressure but also by rotation. In flat stellar systems rotation is the primary balancing entity. It happens that the Coriolis force in the presence of rotation has the same form as the Lorentz force for a charged particle in the presence of a magnetic field. This is another important source of analogies between stellar dynamics and plasma physics and is the reason for the close similarity of the orbital descriptions in the two cases. In this connection we also mention that rotation, in order to fit in with the gravitational field in flat systems, has to be differential and that such shear has important kinematic and dynamic consequences. Finally we notice that self-gravitating systems, like ordinary plasmas, display the interesting feature of allowing negative energy modes, that is, modes that are amplified by dissipative processes. This feature (cf. following sections) has been recognized as playing a crucial role in the problem of the origin and maintenance of spiral structure. In recent years, besides the kinetic or “stellar dynamic” approach [Chandrasekhar (1942), see Bertin (1980)], the fluid approach has had a

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revival of interest. The relevant equations are the standard Euler and continuity equations supplemented by a polytropic equation of state and Poisson's equation [see, e.g., Lin and Lau (1979)J.This approach is relatively simple and, far from the Lindblad resonances (see Section III,C for a definition), results are found to be in good agreement with the more rigorous kinetic analysis.

B. DISKGALAXIES Disk galaxies have a flat component that is an inhomogeneous system in differential rotation. In the present context the bulge-halo components of a disk galaxy are treated as fixed systems in equilibrium. In general, the disk is basically axisymmetric. In such a system most orbits are quasicircular and can be described as the superposition of a smooth circular motion with an angular velocity R(r)at radius r on an epicyclic oscillation with frequency K , defined by K~ = 4R2 ( 1 - I s ) , where s = -d In R/d In r is the shear parameter. For flat galaxies it is appropriate to introduce a (projected) surface density u. A characteristic length for such systems is In. the gaseous approach the Jeans instability provided by 6 = T G U / K ~ ~ )1, where c, is the against axisymmetric waves occurs when Qs = c , / ( K < speed of sound. In the kinetic approach the criterion is Q < 1, with Q = 0 . 9 3 5 4 ~ / ( ~where 6 ) , c is the radial velocity dispersion of the stars. If the parameter e0 = ( 6 / r )is small, an asymptotic treatment of spiral density waves is allowed. Recently, after an improved potential theory had been developed, the asymptotic approach was successfully applied to relatively open structures (with pitch angle less than 30"). The stability of spiral waves in this new regime of open patterns has been found to be dependent on a new dimensionless parameter J , in addition to Q (cf. Fig. 9). The parameter J is proportional to c0 and to the square root of the shear parameter s. Its net effect (which is the combined influence of selfgravity, shear, and tangential forces) can be approximately described in terms of an equivalent stability parameter Qe , which tends to be close to unity even when the actual value of Q departs from such value. More recently, on the basis of the dispersion relation illustrated in Fig. 9 , it has been shown that very open waves are supported when both J and Q are relatively high. The reader should consult Lin and Bertin (1984) for a detailed discussion of the regime of open waves.

C. THELIN-SHUDISPERSION RELATION The interpretation of the spiral structure as a density wave phenomenon was originally proposed by Lindblad [see Lindblad (1963);see also

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Toomre (1981)l. However, the collective behavior of the stars was first put into a satisfactory formulation and calculated by Lin and Shu. A salient point of their "density wave theory" was the hypothesis of a quasi-stationary spiral structure to meet the fundamental need for an explanation of the existence of grand design. There was also, in the beginning, the immediate necessity to provide a consistent description of the spiral phenomenon in sufficiently good agreement with the observational data. This resulted only in the derivation of a dispersion relationship that enables a spiral pattern to be calculated once the angular velocity of the pattern is specified. The problem of the excitation of spiral wave patterns was postponed, and the theory in its primitive form could not provide a method for predicting a well-defined value of the angular velocity flp of the spiral pattern from given properties of the basic axisymmetric model galaxy. To be more precise about the dispersion relationship, let A be the spacing between two successive spiral arms and v = m(R, - R)/K be the frequency parameter, where m is the number of spiral arms. Clearly, lvl is the dimensionless frequency at which the material rotating at angular velocity R encounters the gravitational field rotating at angular speed a,. The dispersion relationship calculated by Lin and Shu is then of the form lk8l = F(I.1, Q), where k = 27r/h. The specific formula defining the dispersion relationship is given in the literature cited; see, e.g., Bertin (1980). The numerical plot of the dispersion relationship is shown in Fig. 6. When IvI assumes integer values, there is a resonance condition. We note

FIG. 6 . The Lin-Shu dispersion relation for spiral waves in the case Q = 1. Group propagation is indicated by arrows. Different kinds of waves and resonances can be recognized in the diagram.

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that in a galaxy u = u ( r ) . At u = 0, the wave corotates with the particles; we call this the corotation resonance. The inner and outer Lindblad resonances are defined, respectively, by at inner Lindblad resonance, u = -1 at outer Lindblad resonance. These correspond to locations where the driving frequency is equal to the epicyclic frequency. The conditions at these resonances are more complicated than indicated by the preceding dispersion relationship. The reader is referred to the original literature for the details. Some of the behavior of these waves at these resonances is described below. In the regions between resonances, wave trains propagate in a way that is defined by the dispersion relation (cf. Fig. 6), and their amplitude is governed by the conservation of wave action. These points are discussed in Section IV. Here we just mention that a proper combination of propagating waves can give rise to stationary rotating structures, that is, to modes. u = +I

D. DYNAMIC THEORY OF SPIRAL PATTERNS 1. Equivalence of Various Mathematical Approaches

Now we turn our attention to the evolution of global wave patterns over a galactic disk. This means that we must heed the boundary conditions at the origin and at the outer edge of the disk, which is often taken to be at infinity. The analytic treatment of the evolution of a linear dynamic system or of a field can be carried out via three approaches, which are superficially different but substantially the same. These are the solution of the initial value problem by

(1) the superposition of normal modes, including possibly a set with continuous spectrum; (2) Laplace transformation or other approaches that emphasize evolution from a given state; (3) the influence function (Green’s function). To initiate the calculation, one may either set an appropriate initial condition or force the system with an external agent that has, for example, a Gaussian variation in time [cf. Toomre (1981)l. The mode approach is the simplest when the situation can be essentially represented in terms of a few (two or three) discrete modes. In such case

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the behavior of the system is usually not too sensitive to the initial conditions provided that they are “reasonable.” The dominant mode soon takes over. However, one has first to make sure that the system does not contain a set of modes with a continuous spectrum, because their role may not be clearly revealed by simple superposition. For example, a superposition of neutrally stable oscillations (part of a continuous spectrum) often leads to a solution that depends very much on the initial conditions and decays algebraically in time.

2 . Absence of Continuous Spectrum According to the general spectral theory, a linear eigenvalue problem has no continuous spectrum if the operator is regular and the domain is finite. In the case of an infinite domain, there may be a continuous spectrum, but an eigenfunction of this set corresponds to a distribution with its dominant behavior at infinity. A classical example is the Schrodinger description of the ionized state of the hydrogen atom. On the other hand, when the basic state of a system is such that the amplitude of the small perturbations dies away sufficiently rapidly at infinity, the system is effectively finite in extent. In one study of the dynamics of galaxies [Hunter (1969), see Toomre (1977)], a set of modes with continuous spectrum has been found to exist not because of the infinite extent of the disk, but because of its infinitesimal thickness. These modes should have a discrete spectrum when the disk thickness is realistically taken to be finite. The amplitude distribution of these modes also suggests that they are not of great physical significance because they are restricted to the edge regions. We must turn to the other source of continuous spectrum, that is, the possible occurrence of singularities in the interior of the galactic disk, at neither the center nor infinity. In the gas dynamic models (with or without modification of the gravitational field), one has to pay specific attention to the singularities at corotation resonance v = 0 and at Lindblad resonances v = k 1 . These singularities are not present in the stellar system. They are, so to speak, smeared out by the dispersive motions of stars. Thus, in using a gaseous model, one must be aware of the pitfalls in the improper handling of the resonances. This issue is further discussed in Section V [cf. Hunter (1969, p. 72), see Toomre (1977)l. Even in the stellar approach, one needs to be aware of another issue suggested by our experience in hydrodynamic stability [as discussed by Hunter and in the following references he cited: Case (1960, 1961), Lin (1961)l. One has to consider the self-adjustment of the system to an initial

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disturbance through the propagation of signals across the direction of flow. In the problem of hydrodynamic stability of parallel flows, there is a singularity in the comoving layer if the Reynolds number is infinite. Consequently, there is indeed an important set of eigenfunctions with a continuous spectrum. These functions have a mild singularity in the interior of the domain in question. When the Reynolds number R is large but finite, this singularity may not exist, and the corresponding eigenfunctions are wiped out. But this does not mean that the representation of the solution by the discrete modes would be convenient, because the smearing out of the singularity by the viscous forces is done by a diffusive mechanism that acts over a time scale on the order of R’”, and many higher modes are therefore required if the initial value resembles one of the singular modes of the continuous spectrum. In the present case, however, the signal propagation is governed by a wave process. The group propagation of waves across the direction of mean motion is fast. For galaxies with exponential density distribution, as indicated by observations, the time required for signal propagation across the main part of the disk is estimated to be on the order of magnitude of one period of revolution. Hence, there is no need for higher modes (with their fine structures) to be included in abundance to give an approximate description of an evolutionary process for given initial conditions that are not singular in the extreme. In any case, such singular initial conditions are unlikely to be produced naturally. 3 . Modes and Their Excitation

We therefore turn to the modal approach. A consistent dynamic theory for the excitation and maintenance of spiral structure is described in Section V. It provides the basis for our attempting to formulate a dynamic approach to the classification of spiral galaxies. Generally speaking, this implies that definite morphological types are associated with appropriate wave patterns, which are in turn determined by basic dynamic parameters in the equilibrium model. To be sure, nearly stationary wave patterns cannot be expected when several modes coexist, but they may still be in the same general range of pitch angles so that the morphological classification is not changed, even though there may be vacillations in the pitch angle. On the other hand, it is possible for a galactic system to have no unstable spiral modes. In that case an evolving spiral pattern can still be excited by external influences such as a galactic encounter (cf. Section 111,E). Theoretical investigations show that some realistic axisymmetric

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galaxy models with a disk population can support unstable spiral modes. These modes are normal spiral modes, barred spiral modes, or modes in transition between them, depending on the properties of the equilibrium model. All kinds of modes are found to be trailing, consistent with the observations.

E. INTERACTING GALAXIES We wish to emphasize that there is no contradiction between the two approaches of internal instability and external excitation. Once the role of internal dynamics is understood, external forcing is just part of the selection mechanism of spiral modes. Unless strong collisions are involved (such as those that are supposed to originate peculiar galaxies), a tidal encounter can be visualized as a mode selector. This is particularly true if the conditions of the galactic disk under consideration do not have strong spiral instabilities but only damped or weakly growing spiral modes. If the galaxy does not have unstable spiral modes, the spiral pattern excited by the close encounter would eventually die away. If there is a dominant decaying mode, the form of the spiral pattern will settle down essentially to a form that will remain unchanged for quite some time. Otherwise, the form of the pattern will evolve, but the pitch angle may still change over the same general range as determined by the dynamic parameters; so the overall form of the pattern does not change appreciably, and such cases need not disturb the statistical correlation between the morphological and dynamic classifications. One simple general comment should be made: A tidal disturbance is likely to favor bisymmetric structures (i.e., m = 2 modes). On the other hand, three-armed and multiple-armed galaxies, spurs, or asymmetric disks (of the m = l type) are naturally expected in the context of internal dynamics. Indeed, three-armed regular spirals are observed, and these are certainly not of tidal origin.* Finally, we would like to stress that, even in the context of two-armed spirals, a good modeling for a tidal encounter, which could show directly the mechanisms of mode selection and exhibit the amplitude of the induced structures, is not yet available and, indeed, offers an attractive and challenging theoretical problem. For example, a proper inclusion of three-dimensional collective phenomena is still lacking and would be especially necessary in a discussion of a tidal encounter in the presence of sizable halos surrounding galaxy disks.

* Another example in favor of this argument is provided by the observation of NGC 4254 by Iye er al. [(1982), see Lin and Bertin (1984)l.

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Moreover, orbiting satellites need not provide a highly effective mechanism for exciting spiral patterns because there is usually a mismatch of time scales. The inner part of a galaxy has a much faster “clock” than objects traveling at the outskirts of the same galaxy. In controlled numerical experiments, one could adjust the time scale of the exciting force (e.g., that in the Gaussian time variation in Toomre’s experiments). It would be interesting to examine the effect of a change in the time scale in a systematic manner to determine the range for most effective excitation and to compare such a time scale with that for the orbiting satellites and galactic encounters. Perhaps Oort [(1970), see Bertin (1980)l was correct, after all, when he made the following remarks on spiral structure at the Base1 Symposium in 1969: “But in most cases we cannot invoke interaction. Moreover, if there had been interaction in the past, it might well have contributed to disturb the regular wave pattern, but less likely to rebuild it.” Interacting galaxies have been shown [Toomre and Toomre (1972), see Toomre (1977)l to exhibit very interesting behavior responsible for the occurrence of peculiar galaxies. We do not pursue this line in the present article.

IV. Dynamic Mechanisms In this section we comment on some dynamic mechanisms, such as wave propagation and wave behavior at turning points and resonances. Only the key qualitative features are discussed. For details and more complete analysis, the reader is referred to the literature cited in the references.

A. PROPAGATION OF A GROUPOF DENSITY OF WAVEACTION WAVES;CONSERVATION The general theory of dispersive waves can be applied to the problems at hand. Toomre considered the propagation of a group of density waves and found that short trailing waves propagate away from the corotation circle, whereas long trailing waves propagate toward the corotation circle. The direction of propagation is reversed in the case of leading waves. The amplitude distribution of such waves is found to conform to the usual principle of conservation of wave action. This was first shown for the case of axisymmetric waves by Toomre, who used Shu’s calculation of the amplitude distribution of spiral density waves in a stationary state;

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FIG.7. The wave amplitude [in a dimensionless form defined by Shu (1970)l for different branches of the dispersion relation as calculated by him on the basis of the law of conservation of wave action. Note the amplitude growth as the long trailing wave (C) propagates from near resonance toward corotation.

the demonstration was completed by Shu for the case of spiral waves. The amplitude distribution predicted by this principle is shown in Fig. 7 . When tangential forces are considered, these predictions are slightly altered, but the trends indicated continue to hold. Specifically, we might note that there is a continual rise in amplitude during the outward propagation of the long trailing waves. This rise in amplitude is accompanied by a swing process; that is, the pitch angle decreases as the long trailing wave group progresses (branch C in Figs. 6 and 7). Naturally, the waves would conform to the local dispersion relationship, and it is not difficult to verify that this latter condition requires the wave packet to be sheared with the same trend (but at a lower rate) as a material concentration during the process of propagation. It has also been shown in early studies [Goldreich and Lynden-Bell (1965); Julian and Toomre (1966); see Toomre (1981)] that the leading waves have a strong tendency to swing over to the trailing form, with an accompanying transient growth, similar to the vorticity patterns in shear flows.

B. PROPAGATJON OF WAVEPACKETS A closer study of the propagation of density waves can be made in terms of either wave packets or single-frequency sustained wave trains as

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is reemphasized by Lin [see Athanassoula (1983), p. 1171. Toomre made his pioneering study by using wave packets. He made a number of numerical calculations for his local model (whose application is, however, extrapolated to the neighborhoods of Lindblad resonances v = -+1 without proper justification). The frequency distribution chosen for his group is Gaussian (and hence the time dependence is also Gaussian). He arrived at the following conclusions: (1) In regions where the group velocity vanishes, the wave propagation changes direction, with a corresponding change in the nature of the wave (e.g., from a short trailing wave to a long trailing wave). (2) Near regions of Lindblad resonance the waves show an attenuation in amplitude and eventually become absorbed there (similar to what is known as resonant absorption in plasma dynamics). (3) The system appears to be “soft” in the neighborhood of corotation where it allows a free passage of waves. From such a neighborhood, short trailing waves appear to be excited by a barlike forcing function with a Gaussian time variation. The amplitude rises considerably (almost by threefold) during the period when the driving signal is decaying as it passes through the neighborhood of corotation. (4) More recently, Toomre (1981) carried out similar calculations in the “constant velocity model” used by him and by Zang. In this case, the forcing is done by a leading wave packet propagating outward, away from inner Lindblad resonance. There is found to be a considerable piling up of energy in the nominal turning region (which happens to coincide approximately with the inner boundary of the corotation region in that example) as the wave packet is slowed down, swings to the trailing form, and grows in amplitude because of local instability. There is also considerable “leakage” of trailing waves allowed, so that (the amplified) disturbances are seen to propagate toward the outer Linblad resonance, with an appreciable rise in amplitude partly associated with the principle of conservation of wave action. Toomre refers to this as “swing amplification.” The inward propagating trailing waves are eventually absorbed at the inner Lindblad resonance and show only a moderate rise in amplitude.

C. PROPAGATION OF SUSTAINED WAVETRAINS The propagation of single-frequency sustained wave trains was studied by Mark, who used an asymptotic approach suitable for galactic disks. As one would expect, the results are more quantitative, not restricted to specific models, and, therefore, more suitable for use as building blocks

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\

short (+31

@

short (-4) Galactic Center

Outer Regions

FIG. 8. The WASER mechanism for amplification (excitation) of spiral structure at (1976), Astruphys. J . 205, 3631. corotation [from Mark, J. W.-K.

for a theory of spiral modes in a galactic disk (see Section V). Qualitatively, they are of course in agreement with Toomre’s conclusions. (1) The waves are turned back at the turning points, with exponentially decaying solutions in the forbidden zone. Specifically, as first found by Mark, a short incoming trailing wave train is refracted into a long outgoing trailing wave train. (Even though a short leading wave would propagate in the same direction, it is apparently not excited.) (2) The waves are absorbed at Lindblad resonances. In addition, one can actually calculate the rate of attenuation as the waves approach them. (3) At the corotation resonance there is the WASER mechanism (Fig. 8), which provides the basis for the excitation of spiral modes. This is similar to Toomre’s comment of “softness,” but it specifically provides a quantitative calculation of the amount of short trailing waves excited by a long trailing wave train approaching the corotation region from the inside. Specifically, it is found that the incoming wave is not only refracted back as in the case of an ordinary turning point, but there is also stimulated the emission of short waves from the corotation circle in opposite directions. The amplitude of this stimulated “radiation” depends on the parameter Q. If Q is high, there is essentially no stimulated emission. If Q is sufficiently low so that there is local Jeans instability, the stimulated wave can have a very high amplitude. It is also found that the presence of a gaseous component and the inclusion of the tangential forces ( J ) enhance the amplitude of the emitted wave so that the amplification mechanism operates even when the system is axisymmetrically Jeans stable (see Figs. 8 and 9).

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FIG. 9. Local stability diagram shows the role of the parameters Q and J [from Lau,

Y. Y.,and Bertin, G. (1978). Asrrophys. J . 226, 5081.

The process described above operates at relatively low values of J (typically up to J = 0.6). At larger values of J a new kind of overreflection occurs when an open leading wave approaches the corotation circle from inside. In the context of steady wave trains, this process has been discussed by Bertin in Athanassoula (1983, p. 119) and is illustrated more extensively in Lin and Bertin (1984). These mechanisms of refraction and stimulated emission play important roles in the maintenance of spiral wave patterns and modes, as we shall see in Section V.

V. Theory of Discrete Spiral Modes In recent years, galactic disks have been shown to be able to support discrete unstable spiral modes. These are regular spiral structures, with a grand design, that rotate rigidly around the galactic center and do not propagate in the radial direction (in contrast to what was found to be the case for density wave packets). In the linear analysis the amplitude of the mode grows exponentially in time. In many cases the growth rate is found to be compatible with the belief that these modes can reach observable amplitudes over a period of time much shorter than the Hubble time scale. Thus they are ideal as bases for the explanation of a grand design in spiral structure and for the resolution of the winding dilemma. In addition, it has been noted that the coexistence of more than one mode can be used to

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explain the existence of less regular galaxies and might even lead (when many modes are present) to situations relevant to the so-called flocculent spirals (like NGC 2841). The modes owe their excitation to the existence of differential rotation and their capacity to redistribute angular momentum in the galaxy. Mechanistically speaking, their global structure is maintained by proper superposition of different kinds of density waves: leading and trailing waves and, in each case, short, long, and possibly open waves. The transition form normal to barred spiral modes is associated with the different weights of leading and trailing components in the mode composition. Although most of the current developments deal with linear waves, the general conclusions are expected to hold for the nonlinear theory as well, at least for the weakly nonlinear cases. Even if it is quite obvious from the beginning, it should perhaps be stressed that the growth (or decay) rates of the modes depend quite sensitively on the distributions of the parameters chosen in the basic model. Local instability studies suggest that larger values of the parameters Q and E tend to favor more stability, whereas larger values of the parameter J tend to favor more instability. When results obtained by different methods are compared, the ranges of these parameters should be noted, whether we are talking about self-excited modes, externally induced decaying modes, or other processes. Otherwise, misleading conclusions could be improperly deduced. For example, in the original model of Zang [(1976), see Toomre (1981)], the value of EO = E / Qis 1/4, but in the modified model with “half-mass,” its value becomes 1/8. With Q = I .5, this still gives a relatively high value of E = 3/16, since the observed value of E in the solar vicinity is about 1/10, which is about twice as small. The following comments apply to galactic models with values of E close to those in the range of the observed value and need not apply, unmodified, to models with larger values of E and Q. A synthetic view of the different relevant regimes in a ( J , Q ) diagram is given by Lin and Bertin (1984) where we include a discussion of open waves, as is required when JQ2 > 4V%9.

A. SPIRAL MODES We present in the following a brief summary of our conclusions. A better appreciation of the theoretical aspects is provided in Sections V,B and V,C.

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1. Normal Spiral Modes

Normal spiral modes are most likely to occur when the typical surface density distribution is low at the center and the basic mass distribution includes a sizable nuclear bulge. In the simplest case, they are formed from the composition of two trailing spiral waves, a short trailing spiral wave train propagating inward and a long trailing spiral wave train propagating outward, in the region inside of the corotation circle. The inward propagating short trailing wave is refracted, at a “Q-barrier” in the general neighborhood where the bulge appears, into an outward propagating long trailing wave. At the corotation circle there is a WASER mechanism that results in the amplification (over-reflection) of an inward propagating trailing wave and the transmission of a short trailing wave propagating outward. This outward propagating wave is then absorbed at the outer Lindblad resonance or by the turbulent gaseous medium. The WASER mechanism provides for the growth of these modes, which is found to be aided by the presence of sizable tangential forces (as is the case for open structures). The eigenvalues of the discrete modes are easily determined by means of a “quantum condition” (reminiscent of many problems in classical and quantum mechanics). Some examples of the detailed properties of these modes calculated may be found in the references. In some cases, the mode may even have a noticeable “open” component.

2. Barred Spiral Modes and Spiral Modes in Transition Barred spiral modes are those that have a barlike structure near the center when m = 2. They are expected in those models in which the bulge is not present or not sizable enough to prevent waves from reaching the galactic center. Unlike the normal spiral modes, they include leading waves in addition to the trailing waves. In the simplest case, the incoming trailing wave is reflected near the center into a short feuding wave. The combined structure has a barlike character, rotating at an angular velocity lower than that of the material near the center. This is associated with a trailing spiral disturbance. Such wave patterns have been obtained by a number of authors using different methods. (The term “barlike modes” has also been extended to include similar modes for m = 1 and 3, with reflection of leading waves from the center. Indeed, for m = 1 , reflection from the center is generally possible.)

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To be sure, the incoming wave just discussed can reach the center only if the mass distribution does not have a “hole” in the middle. A study of basic mass distributions with different degrees of “missing mass” in the center shows that there is a continuous transition of spiral modes from the barlike type to the normal type. In Figure 10, we present a sequence of modes given by Haass [(1982), Ph.D. thesis, MIT]. These are the same modes for a sequence of basic states where the neighboring ones are quite close to each other. The transition from bar mode to normal mode is apparent. In the transition range, where the Q barrier is “imperfect,” the trailing wave stimulates both a long trailing wave at the Q barrier and a leading wave at the center. The amplification of the barred spiral modes is still due to an overreflection mechanism at the corotation circle helped by local Jeans instability associated with a large J parameter, that is, a large influence of tangential forces (cf. subsequent discussion of transfer of angular momentum near corotation). Not shown in the sequence of Figure 10 is a kind of mode that tends to occur in the regime of high Q and high J . Such open bar modes have a

FIG. 10. The same spiral mode is followed in a one-parameter sequence of equilibrium models, which are characterized by different properties of the nuclear bulge. Correspondingly, the mode is found to undergo a transition from a barred (sB) to a normal (sA) spiral mode [from Haass (1982). Ph.D. thesis, MIT].

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two-lump appearance and are maintained by a superposition of very open leading waves and very open trailing waves. In the regime just described the long wave branch of the dispersion relation tends to be suppressed. The excitation mechanism can be seen as a new type of overreflection [cf. Athanassoula (1983, p. 119) and Lin and Bertin (1984)] operating at corotation. To summarize, a spiral mode consists, in general, of a short trailing wave train, a long trailing wave train, and a leading wave component near the center at different levels of prominence. The spectrum of a Fourier analysis of such a spiral mode should tell its precise nature most clearly.

B. METHODSOF ANALYSIS The preceding results were obtained by using the method of asymptotic analysis, combined with a more general fluid dynamic theory. A number of other methods of approach have also been adopted. Among these we should mention the exact numerical integration of stellar models with a high degree of symmetry [e.g., Zang (1976); see Toomre (1981)] and numerical integration of fluid models (e.g., Bardeen, Erickson, Aoki, Iye, and others). Some of these calculations, though important, are somewhat restricted: For example, the distribution of surface density in Zang’s elegant model varies inversely with the radial distance, which implies a much larger scale than that of the exponential disk. On the other hand, asymptotic methods have enabled us to develop a flexible and simple theory for calculating spiral modes in a wide class of realistic galaxy models. This has been done both in the simple fluid model [cf. Lin and Lau (1979) and the references therein] and in the more appropriate stellar dynamic theory [cf. Bertin (1980) and the references therein]. In addition, a more recent and general fluid theory, which includes the potential theory in its exact integral form, has enabled us to reach very open configurations and even bar modes (cf. Section V,A). This approach has clearly identified a few parameters (such as c 0 , J , in addition to Q ) that contain much of the physics of the phenomenon and are bound to play important roles in the problem of the dynamic classification of galaxies. It has also proved to be of great help to theorists because of the convenience in including a correct modeling of the complex structures of a galaxy; for example, the mass in the bulge-halo component and the distribution of mass and velocity dispersion in the disk component. In particular, it has been found to be quite easy to construct specific models for comparison with observations (e.g., Milky Way, M81). The physics of important features or effects, such as the thickness of the galactic disk, the presence of a linearly responsive gas

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component, or the existence of an active halo, can be incorporated into an asymptotic theory. All these go far beyond the infinitesimally thin disk model. In other cases the physics of certain kinetic effects, such as resonance, is included, even in the fluid model, either by modifying the relevant terms according to the results of the stellar dynamic treatment (at corotation) or by a proper choice of boundary conditions (for the Lindblad resonances). These objectives cannot be so easily accomplished with some other approaches.

C. RESONANCES:THEIRROLESI N MAINTENANCE OF SPIRAL MODES Transfer of angular momentum and energy between the wave system and the stars can take place where the stellar response is resonant, that is, where the wave resonates either with the bulk motion of the stars (corotation resonance, u = 0) or with the epicyclic motions (Lindblad resonances, v = k l ) . The inner Lindblad resonance (ILR) ( u = -1) can be absent, depending on the properties of the rotation curve and the value of the pattern frequency of the spiral wave. In general, one-armed waves do not have ILR. In addition, a Q profile rising in the central region (to simulate the presence of a spheroidal bulge) can prevent the wave from reaching the ILR and limit its influence on the dynamics of spiral structure. As a result of the “finite temperature” of the galactic disk, the resonant regions in the galaxy are not circles but annuli with a width of about two epicyclic radii. In analogy with the Landau (1946) discussion of wave-particle interaction in ordinary plasma physics, one finds that resonances are not singular regions but act as sources (or sinks) for wave action (i.e., for wave energy and angular momentum). In general, stars are found to absorb incoming waves at the Lindblad resonances. The presence of the outer Lindblad resonance helps excite the mode because it is consistent with an outer boundary condition of angular momentum radiation. The coupling of the region inside corotation ( u < O ) , where the wave energy density is negative, with the region outside corotation ( u > O ) , where the wave energy density is positive, through angular momentum transfer is found to be a powerful mechanism (WASER) for mode excitation (cf. previous discussions in Section IV). On the other hand, the absorption of waves at an ILR can interrupt the relevant wave cycle by preventing waves from reflecting back from the central to the outer regions. This is found to damp the spiral mode [cf. also numerical results by Zang (1976); see Toomre (1981)l. Thus the welcomed role of ILR is that of considerably reducing the number of unstable modes that can be sup-

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ported by a stellar disk (cf. Section V,D). Corotating stars, in general, help excite the mode, but their effect is less dramatic than previously expected. In general terms, one can summarize the role of resonances by saying that they have no singular properties but act like dissipation. One can thus expect neutral modes to be usually discouraged by their presence. When the dynamics of galactic disks is studied in the fluid model, singularities appear at the corotation resonance ( v = 0) and at the Lindblad resonances (v = k l ) . Thus there is tremendous peculiarity in the behavior of the fluid model at these resonances, especially near the Lindblad resonances, where the amplitude of the disturbance can become very large. Indeed, in the case of neutral oscillations, it becomes infinite. In the case of growing modes, this tendency is somewhat tempered by the growth rate (a detuning process). If the fluid model has no pressure, the tendency toward large or infinite amplitudes is even further enhanced. A specific example of this disparity is the neutral “C modes” calculated by Erickson [(1974), see Toomre (1981)l. In his model pressure is absent, but the gravitational potential is softened. In the axially symmetric case, discrete rings are used. From his results the existence of a continuous spectrum has been suggested. The discrepancy with the stellar system is most easily noticed through an examination of the amplitude distribution of his modes. [See Erickson (1974), Fig. 17, p. 891. The essential part of his C mode is located in the vicinity of the Lindblad resonance, with large amplitudes limited to a small range. But this is a reflection of the behavior of the fluid model and does not properly describe that of a stellar system. In the latter system the Landau damping mechanism should be active, and we would expect discrete damped modes, with very small amplitudes at the circle of Lindblad resonance and outward from there. In other situations, especially when there are strong gradients of the various parameters in the basic distribution, it may be necessary to use the stellar model to avoid the exaggerated strength of the instabilities that could arise in the fluid model. This issue is especially important in connection with the “edge modes” discussed by Toomre [cf. Toomre (1981, Fig. 12, the D mode)]. In conclusion, in studying the large-scale dynamics of galaxies it is of crucial importance to have a proper inclusion of stellar dynamic effects at resonances, because a kinetic treatment (as is appropriate for collisionless systems) incorporates “dissipative” phenomena of the Landau type, which remove the singularities (and hence the related continuous spectrum) that are present in the fluid model. This is a necessary step to eliminate spurious properties of the singular fluid model, which might be mistakenly attributed to the stability of stellar systems, and to reduce the

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exaggerated strength of instabilities that could arise in the fluid model, especially when there are strong gradients of the various parameters involved.

D. MODESWITH HIGHQUANTUM NUMBERS In the study of a fluid model of a galaxy disk, one usually finds for a given m a large number of low-frequency or “high-n,” modes. In general, these tend to have lower growth rates with respect to the higher frequency modes (such as those displayed in Fig. 10). Nonetheless, they may give one the impression that too many modes are bound to coexist in a galactic disk, a fact that is hard to reconcile with the appearance of spirals with grand design. Similarly, the investigation of fluid models shows that “high-m” modes, being usually quite open, have in general larger growth rates than m = 2 modes. This again leaves one with the impression that the theory is not correct, because it would predict many multiple armed spirals (m 2 3), which are not so common in the sky. The answer to both paradoxes lies in the stellar dynamic effects at ILR. The presence of a modest concentration of nuclear mass in the center can easily damp, via ILR, the higher m modes and, at the same time, the low-frequency higher n modes. Indeed, the very fact that bisymmetric spirals are quite common indicates that ILR plays an important role in mode selection. Conclusions of this kind were reached in our earlier experience with pure trailing spirals [Bertin et al. (1977), see Bertin (1980)l.

VI. Dynamic Approach to Classification of Galaxies Our theoretical results, such as those already cited, indicate a simple, promising, working program-partially completed-aimed at the dynamic classification of spiral galaxies. In broad outline, our dynamic approach is as follows. Normal spiral modes and barred spiral modes correspond to normal spiral galaxies (SA) and some barred spirals (SB). In the present perspective, the prominent types of modes that occur depend primarily on the mass distribution in the galaxy, but also on other properties such as velocity dispersion. Barred galaxies, characterized by massive bars, should be treated as a separate category, since the basic mass distribution is not axisymmetrical. Modes with m = 2 are the most commonly observed, because of the nature of the curve il - ~ 1 2a, s already noted in the early phase of the development of theory (cf. Section V,D).

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Those with m = 1 should also occur often. One might even speculate that they always coexist with modes with m = 2. Certainly in M3 1, the recent data on the asymmetry of the gaseous distributions and motions suggest the coexistence of such a mode. We now describe these speculative conclusions in greater detail in the following.

A. BARSTRUCTURE OF TYPEsBB If the basic mass distribution has a large rotating ellipsoidal (nonaxisymmetric) component and a surrounding disk component with differential rotation, there is naturally to be expected a spiral structure in the disk driven by the “bar.” Indeed, it has been shown (by Sanders, Huntley, Roberts, Liebovitch, and others) that even a rotating broad oval distribution of matter with rather small eccentricity can drive a disk of gas into barlike features with rather narrow bars. In such cases the optical appearance does not coincide with the distribution of mass. The spiral feature is caused by the tendency toward epicyclic motion of the material in a rotating system when it is driven by a rotating bar. Stars with low dispersive speeds would respond to such driving just like the gas. This redistribution of matter would produce a spiral gravitational field, which enhances the spiral feature further. Stars with higher dispersive speeds would respond more weakly. But because of their higher total mass, they are also expected to contribute to the spiral gravitational field. In such cases, however, the spiral waves do not contribute very much to the barlike feature. The oval distortion is perhaps governed by processes similar to those of rotating ellipsoids [cf. Chandrasekhar (1969)], see Bertin (1980)l.

B. BARSTRUCTURE OF TYPEs B When a galaxy is entirely composed of a differentially rotating galactic disk (e.g., like that in the Kuzmin-Toomre sequence) without a nuclear bulge or halo, there is a strong tendency for a barlike instability to develop near the center. This may be expected by analogy with the classical solutions for rotating ellipsoids, since the system has been flattened by rotation. It is expected that the Ostriker-Peebles criterion would apply. In contrast, there may be bar modes where the spiral density wave pattern is composed essentially of a leading wave and a stronger trailing wave.

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Recent calculations made by Toomre (1981) and Haass [(1982), see Lin and Bertin (1984)J have shown that, near the center, the mode may be described as composed of an outgoing leading wave and an incoming trailing wave, if there is a free passage of waves near the center. (Still, keep in mind that the origin is a node of the wave pattern.) There is then a barlike structure near the center with a trailing spiral pattern outside the corotation circle, if the wave pattern can propagate through that neighborhood. We refer to such wave patterns by the designation sB and assume that it essentially falls within the designation SB used by Hubble, Sandage, and de Vaucouleurs on the basis of observational data. A plausible example of such a galaxy could even be NGC 1300. This galaxy also has a small disklike structure near the center. It may be associated with a very small spherical nucleus, which does not effectively interfere with the primary mechanism of the formation of the bar mode. However, many of the galaxies classified as SB according to Sandage and de Vaucouleurs may belong to the sBB category in our dynamic classification. Galaxies with a mildly ellipsoidal, basic Population I1 mass distribution may be expected to “force” a prominent barlike sB mode if there is already an instability of this type. The phase angle of the modes is then expected to be correlated with the directions of the principal axes of the ellipsoidal distribution. Thus, the observed spiral features in a galaxy can be used to infer the details of the basic Population I1 mass distribution.

C. BARSTRUCTURE OF TYPEsAB If the mass distribution is that of a pure disk, but the velocity dispersion is gradually increased in the central regions from the situation just considered (so that the Q parameter becomes enhanced there), the incoming short waves will not be returned as a pure leading wave from the center. There is also refracted a long trailing wave component propagating outward. As this component becomes more important, the bar structure becomes less prominent, and the spiral pattern gradually takes on the features of a normal spiral. This is then a spiral pattern of a transition type similar to the concept of the S/SB classification or the SAB classification introduced by de Vaucouleurs. We therefore refer to such spiral patterns by the designation sAB.

D. NORMAL STRUCTURE OF TYPEsA These are galaxies where the waves propagating from the center outward are essentially long waves of the trailing type, as fully discussed in

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Section V,A, 1. They are designated as spiral patterns of the type sA to correspond to the classification SA of de Vaucouleurs. Galaxy models with a substantial nuclear bulge or halo tend to have spiral patterns of this type. In such galaxies, very little mobile mass can be assigned to the central part of the disk component. Thus, the Q parameter can be high even when the dispersion velocity is moderate. Therefore the tendency toward distortions typical of rotating ellipsoids is not so prominent. Indeed, the bulge-halo population would also tend to discourage barlike formation [see Berman and Mark (1979)l.

E. DUALSPIRAL STRUCTURES A number of galaxies (such as NGC 5364, NGC 210, and NGC 4314) show dual spiral structures. Another such spiral (NGC 1512) has been studied by Lindblad and Jorsater. These structures can be explained in terms of the density wave concept as follows. Consider, for example, the galaxy NGC 5364. In “The Hubble Atlas of Galaxies” the following description is given: NGC 5364 is one of the most regular galaxies in the sky. The thin, bright, internal ring appears to be completely closed. There is spiral structure internal to this ring, but the inner arms starting from the nucleus are not well defined. The regular outer arms begin tangent to the bright ring. Two external arms are present. Their junction on the ring is difficult to find. [Sandage (1961), see Lin and Lau (1979).] In our view, the outer spiral structure is a regular normal spiral pattern. There is a nuclear bulge and a halo in this galaxy, and hence the ILR ring. This nuclear bulge of the older stars (Population 11) is on the scale of the size of the ring (or larger), much larger than the size of the tiny nucleus shown in the photograph. The disk component directly under the nuclear bulge has sufficient mass to develop into an entirely separate spiral structure.

VII. Concluding Remarks In applying our theory, we must keep an open mind with regard to other physical processes that produce spiral features. Although the spiral features in neat spirals of the M81 type (which has a regular grand design) and fuzzy spirals such as NGC 2841 (which has no apparent grand design)

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could both be due to density waves, they may well have been produced by different physical processes. In galaxies of the NGC 2841 type, there appears to be almost complete rotational symmetry. Either there is a superposition of many modes, or other mechanisms (e.g., induced star formation) may be present. An examination of the physical conditions in the interstellar medium is strongly urged for such galaxies in order to decide whether the process of induced star formation can actually continue over long distances. (Induced star formation is probably present in irregular galaxies, but this is outside the scope of our present discussion.) Observations of large spiral galaxies (with coherent spiral structure up to 100 kpc from the galactic center, as suggested for UGC 2885) offer another dynamic puzzle. In such galaxies the “clock” is running very slowly in the outer parts and there the winding dilemma is not serious; still the connection of the spiral structure with the central regions seems to demand once again a wave interpretation of the observed structures. In addressing these and other questions arising from observations it is very important to keep in mind the motivation for the original hypothesis of quasi-stationary spiral structure. The hypothesis is specifically directed at the central issue posed by Oort: A pattern with a grand design exists in many galaxies, even when the spiral structure is “hopelessly irregular and broken up.” It was also stated in the early papers that the superposition of a few modes would be suitable for the representation of patterns that are not permanent but comparatively transitory [cf. Lin (1965, 1970) cited in Lin (1971) for emphasis on coexistence, including the possible presence of material clumps, waves, and wave patterns]. From the original hypothesis of quasi-stationary spiral structure the theory has recently developed into a more complete dynamic form as a theory of superposition of spiral modes. In the present article we have stressed the point that the modal approach is best justified in the absence of a continuous spectrum of oscillations in stellar disks. Indeed there is as yet no evidence of a continuous spectrum. In support of this view, we have discussed the role of dissipative mechanisms in providing discrete modes. In galactic systems, dissipation can be induced either by gas turbulence or by the collisionless mechanism of Landau damping. In the present article, on the basis of a number of observational and theoretical results that have accrued during recm: years, we have arrived at the following proposal. Starting from the existence of discrete spiral modes, stable or unstable, we have attempted to formulate a dynamic classification scheme for spiral galaxies into categories sA, sAB, sB, and sBB. This proposal is an appealingly simple program, which may require improvements and modifications after more complete observational and dynamic studies are made on the properties of spiral galaxies. An optimis-

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tic expectation for the success of such a program will provide the guiding spirit for future investigations. ACKNOWLEDGMENTS This work was partly supported by Consiglio Nazionale delle Ricerche, under the United States-Italy Cooperative Science Program, and partly by the National Science Foundation. We wish to thank Jon Haass for his help in the preparation of the paper and for his permission to quote some of his results before they are published.

REFERENCES Because of space limitations and the wide range of the literature involved, we are unable to list here all the relevant references. This will be done in a more detailed forthcoming publication. The interested reader may find most of the references needed in the following publications. Athanassoula, E., ed. (1983). “Internal Kinematics and Dynamics of Galaxies,” Proc. IAU Symp. 100, Besanqon, France, August, 1982. Reidel, Dordrecht, The Netherlands. Bertin, G. (1980). On the density wave theory for normal spiral galaxies. Phys. Rep. 61 ( I ) , 1-69 (217 references). Lin, C. C. (1971). Theory of spiral structure. Highlights Astron. 2 , 88-121 (125 references). Lin, C. C., and Bertin, G. (1984). Formation and maintenance of spiral structures in galaxies. I n “The Milky Way Galaxy” (H. van Woerden, ed.), Proc. IAU Symp. 106, Groningen, The Netherlands, May, 1983. Reidel, Dordrecht, The Netherlands. In press (39 references). Lin, C. C., and Lau, Y. Y. (1979). Density wave theory of spiral structure of galaxies. Studies in Appl. Math. 60,97-163 ( I 10 references). Strom, S. E., and Strom, K. M. (1978). The evolution of disk galaxies. I n “Structure and Properties of Nearby Galaxies” (Berkhuisen and Wielebinski, eds.), Proc. IA U Symp. 77, pp. 69-95. Reidel, Dordrecht, The Netherlands (52 references). Toomre, A . (1977). Theories of spiral structure. Ann. Rev. Astron. Astrophys. 15, 437-478 (145 references). Toomre, A. (1981). What amplifies the spirals? I n “The Structure and Evolution of Normal Galaxies” (S. M. Fall and D. Lynden-Bell, eds.), pp. I 11-136. Cambridge Univ. Press, London and New York (18 references).

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ADVANCES IN APPLIED MECHANICS, VOLUME

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Strange Attractors in Fluid Dynamics* JOHN MILES Institute of Geophysics and Planetary Physics Scripps Institution of Oceanography University of California, San Diego La Jolla, California

I. Introduction . . . . . . . . .............................. 11. The Spherical Pendulum. . . . . . . . . . . . . . . . . . . . . ............ 111. Lorenz's Convection Model . . ....................... IV. The Howard-Malkus-Welander Convection Model . . . . . . . . . . ..... V. Mathematical Routes to Turbulence . . . . . . . . .. A. Bifurcation Sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Landau's Model. . . . . . . . . . . . ... ..... C. The Ruelle-Takens Scenario . . . ........................ D. The Period-Doubling Scenario. . . . . . . . . ........ E. The Intermittent Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V1. Conclusions . . . . . . . . . . . . . . . . . . . . . . .

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189 193 199 202 204 204 205 205 208 21 I 212 212

I. Introduction Given for one instant an intelligence which could comprehend all the forces by which nutiire is animated and the respective situation of the beings who compose i t - u n intelligence suficiently vast to siibmit these data to analysis-it would embrace in the same ,formula the movements of the greutest bodies q f the universe and those of the lightest atom;fr,r it. nothing would be uncertain and the fiitiire, as the past, worcld he present t o its eyes. PIERRE S I M O N , MARQUISDE LAPLACE (C. 1795)

This powerful expression of eighteenth-century optimism is an epitome of the goal of classical physics, to predict. There is, we now know, a central impediment to the achievement of that goal, even within the domain of

* Otto Laporte Memorial Lecture, Division of Fluid Dynamics, American Physical Society, University of Houston, Houston, Texas, November 20, 1983. 189 Copyright 0 1984 by Academic Press. Inc. All rights of reproduction in any form reserved. ISBN @12-002024-6

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classical physics: the macroscopic sensitivity of some phenomena to small changes in initial conditions, in consequence of which a deterministic system may exhibit chaotic motion. The explicit discovery of this impediment stems from PoincarC’s (1892) studies of the three-body problem, although, as Berry (1978) points out, it was adumbrated by Maxwell, who remarked that (Campbell and Garnett, 1882)

It is a metaphysical doctrine that from the same antecedents follow the same consequents.. .. But it is not of much use in a world like this, in which the same antecedents never again concur, and nothing ever happens twice.. .. The physical axiom which has a somewhat similar aspect is ‘that from like antecedents follow like consequents.’ But here we have passed from sameness to likeness, from absolute accuracy to a more or less rough approximation. There are certain classes of phenomena ... in which a small error in the data only introduces a small error in the result .... The course of events in these cases is stable. There are other classes of phenomena which are more complicated, and in which cases of instability may occur, the number of such cases increasing, in an extremely rapid manner, as the number of variables increases.. .. If, therefore, [the physicist is] led in pursuit of the arcana of science to the study of the singularities and instabilities, rather than the continuities and stabilities of things, the promotion of natural knowledge may tend to remove that prejudice in favor of determinism which seems to arise from assuming that the physical science of the future is a mere magnified image of that of the past. PoincarC and his successors [notably Birkhoff (1927), Kolmogorov (1954), Arnol’d (1963, 1979), and Moser (1962); see Lichtenberg and Lieberman (1983) for references and discussion] dealt with conservative (Hamiltonian) systems, and it appears to have been tacitly assumed by many physicists that dissipation eliminates the transition to chaos. In fact, dissipative systems also admit chaotic solutions, of which the quintessence is the strange attractor. Chaotic solutions of dissipative systems with regular forcing were discovered by Lorenz (1963)+, who, following Saltzman (1962), modelled thermal (Rayleigh-BCnard) convection by a set of three first-order, nonlinear differential equations (see Section 111) and obtained solutions like Such solutions certainly had been observed by others (including Saltzman) in various contexts, but it seems fair to say that Lorenz was the first to realize the far-reaching implications and to introduce the concept of a strange attractor (even though he did not use that term). +

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FIG.1. Chaotic trajectory on the Lorenz attractor for r = 28, IT = 10, and b = 8/3 [calculated by Lanford (1977)l. The trajectory starts from (a point close to) the saddle point at X = Y = 2 = 0 and winds about the saddle points at X = Y = ?[b(r - 1)]1’2, Z = r - I . From Lichtenberg and Lieberman (1983).

that in Fig. I . These solutions are distinguished from regular solutions by at least three features: (i) they are exponentially sensitive to small changes in the initial conditions; (ii) their power spectra are irregular and comprise broadband components; (iii) the surfaces on which they lie in the space of the dependent variables ( X , Y , 2 in the present case) are of fractional dimension [for which reason Mandelbrot (1980, 1982) has suggested that fractal attractor is preferable to strange attractor]. The term strange attractor for such solutions (or, more precisely, the manifold of all such solutions for a particular set of parameters) was introduced by Ruelle and Takens (1971), who discovered the phenomenon independently of Lorenz and suggested its relevance to turbulence. Their paper and the associated rediscovery of Lorenz’s work led to a flood of publications that is still growing and to which scientists from a wide variety of fields have contributed. Some of these writers appear to suggest that the study of turbulence dates from the discovery of strange attractors-e.g., Manneville and Pomeau (1980) declare that “Since the publication of the paper by Ruelle and Takens (1971) the onset of turbulence has become a major source of studies both experimental and theoretical”-and references to Reynolds,* Om. Sommerfeli , Heisenberg, Tollmien, It is a happy thought that the present lecture was delivered during the centennial of the classic experiments of Osborne Reynolds (1883).

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Schlichting, Schubauer and Skramstad, Lin, and other pioneers in hydrodynamic stability are rare in this new literature. Such enthusiasm may have put off some members of the fluid-mechanics community, who are prone to emphasize (cf. Monin, 1978) that true turbulence is both spatially and temporally chaotic and that the energy in fully developed turbulence is spread over many decades of wave number, in contrast to the octave or two that typically characterizes chaotic motion on a strange attractor. It is well to remember, on the other hand, that much of our progress in understanding turbulence has come from rather special and highly idealized problems. Early attempts to solve the Orr-Sommerfeld equation for the disturbances associated with uniform, relative motion of two parallel plates (plane Couette flow) led to the prediction of stability for all Reynolds numbers, in apparent disagreement with experiment (it is now known that the theoretical prediction is correct for infinitesimal disturbances). Taylor (1923), referring to this work and recalling Orr’s (1907) opinion that “It would seem improbable that any sharp criterion for stability of fluid motion will ever be arrived at mathematically,” remarked that: This problem has been chosen because it seemed probable that the mathematical analysis might prove comparatively simple; but even when the discussion is limited to two-dimensional motion it has actually proved very complicated and difficult. On the other hand it would be extremely difficult to verify experimentally any conclusions which might be arrived at in this case, because of the difficulty of designing apparatus in which the required boundary conditions are approximately satisfied. It is very much easier to design apparatus for studying the flow of fluid under pressure through a tube, or the flow between two concentric rotating cylinders. The experiments of Reynolds and others suggest that in the case of flow through a circular tube, infinitely small disturbances are stable, while larger disturbances increase, provided the speed of flow is greater than a certain amount. The study of the fluid stability when the disturbances are not considered as infinitely small is extremely difficult. It seems more promising therefore to examine the stability of liquid contained between concentric rotating cylinders [circular Couette flow]. If instability is found for infinitesimal disturbances in this case it will be possible to examine the matter experimentally. Taylor then proceeded to carry out both the stability calculations and the experiments for circular Couette flow and concluded with the characteristic statement that “The accuracy with which the observed and calculated sets of points ... fall on the curves [see Fig. 2 herein] is remarkable when it is remembered how complicated was the analysis employed in obtaining

Strange Attractors in Fluid Dynamics

193

FIG.2. Taylor’s (1923) comparison between observed ( 0 ) and calculated ( 0 ) speeds for the transition to instability in circular Couette flow = angular velocity of outerhnner cylinder, v = kinematic viscosity).

them.” Against this background, it seems fitting that the problem of circular Couette flow is one for which strange attractors clearly have some relevance (see Section V). The Rayleigh-BCnard convection problem studied by Lorenz (described in more detail in Section 111) requires two nonlinear, partial differential equations for its description, even after the assumptions of incompressibility and two-dimensional flow and the Boussinesq approximation, and Lorenz’s reduction of these equations to three first-order, nonlinear, ordinary differential equations raises difficult questions (see below) about the relevance of strange attractors for fluid dynamics. I therefore have chosen to consider first a physical problem that is described directly by ordinary differential equations and that is readily visualized, the resonant response of a spherical pendulum to a planar oscillation of its point of support. The results for this two-degree-of-freedom system illustrate many of the qualitative features of a multi-degree-of-freedom system and suggest the complications to be expected for a fluid with its infinity of degrees of freedom.

11. The Spherical Pendulum The spherical pendulum is perhaps the simplest example of a twodegree-of-freedom oscillator with identical natural frequencies, and its

John Miles

194

forced motion is a prototype of internal resonance. I first examined this problem more than 20 years ago (Miles, 1962) in connection with the analogous problem of fluid sloshing in a moving circular tank (Hutton, 1963) and recently revisited it (Miles, 1984a) in order to investigate the relevance of strange attractors to the aperiodic solutions that had been observed (but not pursued) in my earlier, analog computer studies. Let 1 be the length of the pendulum, 00 =

(2.1)

(g/1)”2

its natural frequency, and S its damping ratio (such that its free motion for small displacements is oscillatory for S < 1 and nonoscillatory for S > 1). We suppose that its point of support is subjected to the horizontal displacement (xo, yo) = El(C0S w t , O ) ,

0 < E << 1

(2.2)

and seek the horizontal displacement (x, y) of the pendulum bob. The restriction E << 1 suggests that all nonlinear terms be neglected in the equations of motion, which then take the form

+ 2sw0 -&d + w;) The steady state (Swot

-

w)

(x, y ) = Ew;l(COS w t , 0).

(2.3)

solution of Eq. (2.3) has the form

x = A cos wt

+ B sin w t ,

y = 0.

(2.4a) (2.4b)

The amplitudes A and B are proportional to E ; nevertheless, they may be large if 6 is sufficiently small and w approximates wg (resonance), and the neglect of nonlinear terms then is manifestly inappropriate. The dominant nonlinear terms are cubical and of the form x3, xy’, x2y and y3. Therefore, energy may be transferred between the two degrees of freedom, and the solution(s) for which y = 0, although still admissible, may not be stable. We begin the analysis of the nonlinear problem by remarking that, in the first approximation, (d2/dtZ)(x,y) = -w2(x, y) and cancels the term w;(x, y ) at resonance (w = wo); accordingly, if S is sufficiently small, the forcing term of amplitude E must be balanced by the dominant cubical ) . last concluterm, from which it follows that x and y must be O ( E ” ~This ), suggests the sion continues to hold if w2 - w; and S are O ( E ~ ” which introduction of the damping and tuning parameters

2s a = -$13 ’

(2.5a)

Strange Attractors in Fluid Dynamics

195 (2.5b)

[the factor of 2 in Eq. (2.5a) simplifies the subsequent development]; only that parametric neighborhood in which both a and v are not large is of interest, since the linear approximation remains valid if either a or v is large. The time dependence for the nonlinear problem no longer can be expected to be simple harmonic, but it is plausible to assume that it is approximately so and to pose the solution in the form

+ q l ( T ) sin at], = El131[p2(7) cos wt + q2(7) sin wt],

x = EI/~/[P,(T) cos wt

(2.6a)

y

(2.6b)

(2.7) is a dimensionless slow time (it follows from the preceding arguments that the appropriate unit of this slow time is I / E ~ / ~ W )The . substitution of Eq. (2.6) into the cubical:y augmented form of Eq. (2.3) then yields, after the neglect of higher order terms as E + 0, the evolution equations (dldT +

PI 41, ~

2

42) ,

+ ( P I QI, P2, Q2)

=

(0, 1, 0 , 01, (2.8)

where PI,... are cubic functions of pI,.... The divergence of ( P I ,QI, P2, Q2)vanishes; accordingly, the divergence of the system (2.8) in the fourdimensional phase space is

from which it follows that an element of volume in the phase space contracts like exp(-4a~), and any trajectory ultimately must be confined to a limiting subspace of dimension smaller than four. The simplest solutions of Eq. (2.8) areJixedpoints, at which Pi = 4; = 0 (i = 1, 2) and the motion of the pendulum is simple harmonic. These solutions may be either planar (p2 = q2 = 0) or nonplanar and either stable or unstable (a solution that is slightly displaced from a stablehnstable fixed point tends to move toward/away from such a point), depending on a and v. The locus of fixed points in a v , E-plane with a fixed, where E

=

p:

+ q! + p $ + q$

(2.10)

is a measure of the energy of the pendulum, is a resonance curue, an example of which is shown in Fig. 3 (Miles,1984a). The solid lines comprise stable points, while the dashed lines comprise unstable points.

I96

John Miles

Branches 1 (which extends to u = -@J)and HI3 (which extends to u = m) represent stable, planar solutions; in addition, there is a very small branch III,, at the top of the resonance curve between branches I1 and 1112, that is stable but difficult to realize. Branches I1 and 1112represent unstable, planar solutions. Branch IV3 represents stable, nonplanar solutions; in addition, there is a small branch IV, , between IV2 and the junction of branches I1 and 111,that is stable but, like 1111, difficult to realize. Finally, branches IV2 and V represent unstable, nonplanar solutions. The junction between two or more branches is a bifurcation point, the number of which decreases from seven for 0 < a < 0.433 to zero for a > 0.625. Of the seven for a < 0.433, five are of PoincarC’s type, at which one of the eigenvalues for small perturbations vanishes and then becomes positive, and two are of Hopf’s type, at which the real part of a pair of complexconjugate eigenvalues changes sign. The configuration shown in Fig. 3 for a = 1/4 remains qualitatively valid for 0 < a < 0.433. Various simplifications take place as a is increased from 0.433 to 0.625; there are no nonplanar fixed points, and the resonance curve represents only stable planar solutions, for a > 0.625. Numerical experiments reveal that if a single stable fixed point exists for a particular value of u the solution ultimately will tend to that point; if either two or three stable fixed points exist (it is impossible to have more than three in the present problem) for a particular u , which point is attained depends on the initial conditions. There is some range of u in which there are no stable fixed points if a < 0.441; if a < 0.433 this range is u3 < u < u 2 + ,where v3 is the bifurcation point between the planar branches I and I1 and uz* is the Hopf bifurcation point that separates the nonplanar branches IV2 and IV3. Numerical experiments for a = 1/4 and u3 < u < u2* yield the p1-p2 projections and the power spectra shown in Fig. 4 (Miles, 1984a). The trajectories spiral in to fixed points if u > u2* = 0.072. The asymptotic (7m) trajectories for 0.072 > u > -0.04, e.g., u = 0.05 in Fig. 4, are periodic limit cycles that project as ovals in the p I - p 2 plane; the corresponding spectra comprise the fundamental frequency fi and its harmonics, with fi decreasing from& = 0.160, the (analytically) calculated frequency of the nascent limit cycle at the Hopf-bifurcation point, with decreasing u. Very weak period doubling is observed at u = -0.04, and the half-order subharmonics increase in strength as u is decreased from -0.04 to -0.144; see u = -0.10 in Fig. 4. Period quadrupling (i.e., a second period doubling) first occurs (with decreasing u ) in the interval (-0.13, -0.14), and the quarter-order subharmonics increase in strength down to -0.144; see v = -0.142 in Fig. 4. Period octupling (a third period doubling) is observed at u = -0.144; see Fig. 4. The metamorphosis of the trajectories from almost elliptical ovals through increasing convolutions at each period doubling is

Strange Attractors in Fluid Dynamics

197

4.0

3.0

,/E

2.0

1 .o

9

-t

Fic. 3. Resonance curves for spherical pendulum for a = 1/4 (Miles, 1984a). Branches 1-111 correspond to planar and branches IV and V to nonplanar motion. The solid and dashed portions of the curves correspond to stable and unstable motion, respectively.

illustrated by the results for v = 0.05, -0.10, -0.142, -0.144, and -0.145 in Fig. 4 after allowing for the fact that the initial conditions for v = -0.144 were altered to obtain the complement of the trajectory that had been obtained for initial conditions identical with those for v = 0.05, -0.10 and -0.142. [The phase-space equations (2.8) are invariant under the reflection ( p 2 , q2)+ - ( p 2 , q 2 )and therefore admit asymptotic solutions in complementary pairs; which member of the pair is attained depends on the initial conditions. This nonuniqueness of the asymptotic solution has its counterparts, typically with higher multiplicity, in Rayleigh-Bbnard and circular Couette flows.] Chaotic trajectories occur throughout the v interval 1-0.145, -0.201. The typical pl-p2 trajectory within this interval, e.g., v = -0.15 (see also v = -0.5) in Fig. 4, appears to be symmetric with respect to p2 = 0 after a sufficiently long time, although it may spend substantial intervals in either

John Miles

198

the top or bottom half of this pattern, transferring from one to the other half at aperiodically spaced times that are quite sensitive to the initial conditions. The available evidence (broad-banded spectra, sensitivity to initial conditions, and intermittent transfer between two complementary domains of the phase space) strongly suggests that these trajectories lie on a strange attractor of Lorenz’s type. The trajectories for v = -0.145 and -0.20 (see Fig. 4) are exceptional in being confined to one of the two (upper and lower) domains of the putative attractor, which then appears to be of Rossler’s type [see Lichtenberg and Lieberman (1983), p. 3871. The complement of the trajectory shown in Fig. 4 for v = -0.145 was obtained by altering the initial conditions from ( p I,41, p 2 , 42) = (0, 0, 0, 1)

F v=

0.05

Y =

-0.15

-0.1

-0.142

-0.144

-0.145

-0.2

-0.35

-0.45

-0.5

F

FIG. 4. Phase-plane projections and power spectra for spherical pendulum (Miles, 1984a);pl,zare defined by Eq. (2.6), L = logloP,where P is the power spectrum of E [Eq. (2. lo)], and F is the dimensionless frequency.

Strange Attmctors in Fluid Dynamics

199

to (0, 0, 1, 0); on the other hand, essentially the same trajectory as that shown for v = -0.20 was obtained with each of the three sets of initial conditions (0, 0, 0, l ) , (0, 0, 1, 0), and (0, 1, 4,1). The trajectories for v = -0.21 and -0.22 are limit cycles, with period doubling as Y increases from -0.22 to -0.21. Those for v = -0.23 and -0.24 are chaotic and appear to lie on strange attractors of Lorenz’s type. Those for -Y = 0.25 (.01) 0.30 (.05) 0.45 are periodic, with period doubling at -0.25 as v increases to -0.24 and at -0.40 and -0.45 as v decreases to -0.50, where a strange attractor (presumably) of Lorenz’s type again occurs; see v = -0.35, -0.45, and -0.50 in Fig. 4. It seems likely that period quadrupling and octupling could be observed through finer sampling over the spectral intervals that separate those of chaotic trajectories, just as in the v interval (-0.13,0.145). Moreover, the appearance of strange attractors within the narrow interval (-0.22, -0.25) suggests that there could be additional such subintervals within the interval (-0.30, -0.50). The v interval (-0.50, -0.92) also contains subintervals of both periodic and chaotic trajectories. A mathematically similar problem is posed by a two-mode description of the resonant motion of a weakly damped, stretched string that is driven by a simple harmonic, planar force (the appropriately normalized phasespace equations differ only in the numerical value of a single coefficient). The planar, harmonic response of the string is unstable in a certain neighborhood of resonance, just as with the pendulum; on the other hand, there is no spectral neighborhood in which both the planar and the nonplanar, harmonic response of the string are unstable, by virtue of which the asymptotic (wt -+ m) motion appears to be regular for all frequencies (Miles, 1984b). A second, mathematically analogous problem, already mentioned, is posed by a two-mode description of fluid sloshing in a circular tank that is subjected to a horizontal, planar oscillation. If the depth exceeds a certain critical value the solution resembles that for the spherical pendulum and may be chaotic in some neighborhood of resonance. If the depth is inferior to this critical value the solution resembles that for the string and may be nonplanar but not chaotic.

111. Lorenz’s Convection Model Lorenz (1963),following Saltzman (1962), considered two-dimensional, Rayleigh-BCnard convection of an incompressible fluid to which heat is supplied at the lower of two parallel boundaries. The original problem

John Miles

200

(Rayleigh, 1916) is governed by four partial differential equations: the two components of the Navier-Stokes equation in the Boussinesq approximation, with the buoyancy force included in the vertical component, the equation of continuity, and the heat-conduction equation; these may be reduced to two partial differential equations through the introduction of a stream function and the elimination of the pressure (which leads to a partial differential equation for the vorticity). Saltzman (1962) had attacked this problem by truncating Fourier-series representations of the stream function and temperature (initially at 52, and ultimately at 7, terms). In the published version of his work, Saltzman reported that “except for very large Rayleigh numbers (e.g., [r] > 20) [the truncated expansions] approach a steady cellular form” and presented results only for r 5 10, where r is the ratio of the Rayleigh number to its critical value. In a private communication to Lorenz, he also observed that (as reported by Lorenz) “In certain [of Saltzman’s] cases all except three of the dependent variables eventually tended to zero, and these three variables underwent irregular, apparently nonperiodic fluctuations .” Saltzman’s results led Lorenz to approximate the appropriately normalized stream function and temperature by the truncations $

=

d2X(t) sin(raxlh) sin(rzlh),

T

=

d 2 Y ( f ) cos(naxlh) sin(rz1h)

(3. la) -

Z ( t ) sin(2rzlh),

(3.lb)

where X , Y , and 2 are dimensionless Fourier amplitudes, t is a dimensionless time, x and z are the horizontal and vertical coordinates, h is the vertical separation of the boundaries, and a is the aspect ratio. Note that Eqs. (3.la) and (3.lb) correspond to stress-free boundaries at z = 0, h and that stress-free, insulated boundaries could be placed at x = 0, hla. The substitution of Eqs. (3.la) and (3.lb) into the vorticity and heat-conduction equations yields the three first-order, nonlinear differential equations

+ PY,

x

=

-PX

Y

=

rX - Y

Z

=

-bZ

-

XZ,

+ XY,

(3.2a) (3.2b) (3.2~)

where P = Y / K is the Prandtl number, Y is the kinematic viscosity, K is the diffusivity , r =

R

R,

=

RIR,,

(3.3a)

(gah3 AT)/vK,

(3.3b)

+

(3.3c)

= ~ ~ 4 ( 1a2)3lia2,

Strange Attractors in Fluid Dynamics

20 1

R is the Rayleigh number, R , is its critical value (for the transition from conduction to convection), a is the coefficient of thermal expansion, AT is the imposed temperature difference between the lower and upper boundaries, and b = 4/(1

+ a2).

(3.4)

These Lorenz equations, as Eqs. (3.2a)-(3.2c) are now called, admit the null solution X = Y = 2 = 0, which corresponds to heat transfer through pure conduction (for which the temperature gradient is uniform), and this null solution is stable/unstable for r 5 1, exactly as Rayleigh had inferred from the full partial differential equations (not surprisingly, since Lorenz’s minimum truncation was informed by Rayleigh’s analysis). Lorenz, following Saltzman, chose b = 8/3 which corresponds to a = 1/ d 2 ) and R, = 27r4/4, as obtained by Rayleigh at the onset of convection,+ and P = 10 (P = 30, 6.8, 0.71, and 0.03 for silicone oil, water, air, and mercury, respectively). Then, for 1 < r < 24.06, Eqs. (3.2a)-(3.2c) admit a stable, fixed, nonzero solution that represents steady convection. Steady convection also is possible for 24.06 < r < 24.74, depending on the initial conditions, but for 24.74 < r < 99.52 the solution is irregular and exponentially sensitive to initial conditions (a small change in the initial conditions implies an initially exponential separation of the initially contiguous solutions); see Fig. 1. The interval 99.52 < r < 313 contains windows of stable periodic motion, at least some of which are separated from chaotic motion by period-doubling cascades (these windows are similar to those described above for the pendulum). Only stable periodic motions are obtained for r > 313. See Robbins (1979) and Sparrow (1982) for details. Subsequent calculations by McLaughlin and Martin (1975) and Curry (1978) have shown that higher-order truncations of, or changes of the Prandtl number in, the Rayleigh-BCnard problem, alter the bifurcation sequence that precedes chaos. Moreover, Moore and Weiss’s (1973) nulo-’, merical solutions of the full two-dimensional equations for P = 1, 6.8, 16, lo2, and are regular for r 5 lo3 (their highest value), which implies that chaotic solutions of the two-dimensional Rayleigh-BCnard problem are an artefact of truncation. [Moore and Weiss (1973) do not refer to Lorenz or to the possibility of chaotic solutions, and this last inference is due to Toomre et al. (1982). See also Lichtenberg and Lieberman (1983), p. 446, and Curry et ul. (1984).] Similarly, Marcus (1981) has explored the effects of truncation for spherical convection in considerable detail and concludes that the bifurcation sequence may be sensitive to the The minimum value of R, for rigid boundaries is 1707 at u = 0.992.

202

John Miles

order of truncation and that the number of modes required for an accurate representation of the original convection problem tends to increase with r. It should be emphasized, however, that both laboratory experiments (see below) and numerical solutions of the full three-dimensional equations yield chaotic convection in rather broad parametric regimes, and the proper choice of modes (in particular, three- versus two-dimensional) is at least as important as the order of truncation. It also is worth remarking that Lorenz’s sequence of states requires P > b + 1, where 0 < b < 4; if P < b + 1 the basic modes must incorporate a transverse waviness of the convection rolls. The preceding summary only begins to describe the rich structure of the solutions of the Lorenz equations. A much more complete description has been given by Sparrow (1982), but it seems likely that much remains to be learned. DaCosta et al. (1981) have extended Lorenz’s model to doublediffusive convection, using Veronis’s (1965) five-mode truncation of the full partial differential equations. A numerical treatment of the full twodimensional equations for this problem has been given by Moore et al. (19831, who report the bifurcation sequence: periodic oscillations + period doubling cascade + chaos.

IV. The Howard-Malkus-Welander Convection Model The severe truncations required for the description of Rayleigh-BCnard convection by the Lorenz equations naturally raises the question of whether these equations have any direct physical significance. This question has been answered affirmatively by Malkus and Howard (Malkus, 19721, who considered a modified version of Welander’s (1967) convection loop (for which Welander had obtained numerical solutions with alternating bursts of periodic and aperiodic flow).” Consider (see Fig. 5 ) a vertically oriented, toroidal tube of crosssectional area A and major radius a ( A << n u 2 ) that is filled with an incompressible fluid, the density of which decreases linearly with the temperature, and is subjected to a prescribed temperature at its outer wall. The equation of motion, which incorporates pressure gradient, friction, and buoyancy, is

+

See Coullet (1979) for a mechanical model that is described by the Lorenz equations.

Strange Attractors in Fluid Dynamics

203

I

FIG. 5 . The Howard-Malkus-Welander convection loop.

where q , p , and Tare the circumferential velocity, pressure, and temperature averaged over the cross section, v is an effective friction coefficient (v = 8 ~ p / p o for A Poiseuille flow of a fluid of viscosity p ) , po is the density at T = To, Q is the thermal expansion coefficient, and -g sin 8 is the circumferential component of the gravitational acceleration. The variation of the density is neglected except in the buoyancy term (the Boussinesq approximation). The corresponding equation of continuity is

(4.2)

aq/ae = 0,

which implies q = q ( t ) . The heat-transfer equation, on the assumptions that the wall of the tube has the prescribed temperature TW(@and that thermal conduction is negligible compared with thermal convection in the fluid, is aT q a T + -a -ae = k(T, - T), (4.3) at where k is the heat-transfer coefficient for the wall. Now suppose, for simplicity, that T,(o)

=

T, cos

e

(4.4)

(the results may be generalized for an arbitrary, Fourier-series representation of Tw).Substituting Eq. (4.4), together with r

T(0,t) = Co(r) + x [ C , ( t )cos n0 n=

I

+ S , ( t ) sin n e ] ,

(4.5)

into Eqs. (4.1) and (4.3) and averaging Eq. (4.1) over 8 (which eliminates

John Miles

204 the pressure), we obtain

+ IagS), = --a-InqS, + k(61,T,

(4.6a)

q = -vq

C,

- CJ,

(4.6b)n

S, = a-InqC, - kS,,

(4.6~1,

where the overdots signify differentiation with respect to t , and 61, is the Kronecker delta. It is a crucial virtue of the model that, although q enters each of Eqs. (4.6a), (4.6b),, and (4.6c),, Eqs. (4.6a), (4.6b)1,and (4.6~)) are independent of the remaining equations-i.e., Eqs. (4.6a)-(4.6c) satisfy what Sommerfeld calls the “requirement of finality.” Introducing X, Y , Z , and 7 through q = akX,

S1 = r-’T,Y,

C , = T,(I

in (4.6a), (4.6b)I and ( 4 . 6 ~ and ) ~ ~choosing b

P Y

=

-

=

r-lz),

kt = 7

(4.7)

1 and

ulk,

(4.8a) (4.8b)

= f(akv)-’gaT,,

as the counterparts of the Prandtl number and the normalized Rayleigh number, we obtain the Lorenz equations [Eqs. (3.2a)-(3.2~)].It follows that the Lorenz equations have a direct correspondence to a laboratoryreproducible fluid system without the uncertainties implied by truncation in Lorenz’s original model. Given the solution of Eqs. (4.6a), (4.6b)), and (4.6~))for q ( t ) , Eqs. (4.6b), and ( 4 . 6 ~for ) ~n = 0, 2, 3, ... may be solved in independent pairs. It suffices to note here that

c,’+ s,’ = (c,’+ sX),=~

e-2k‘,

n = 0, 2, 3 ,...,

(4.9)

and hence that the asymptotic solution of the problem is completely described by the solution of the Lorenz equations for q , CI, and SI.

V. Mathematical Routes to Turbulence

A. BIFURCATION SEQUENCES At least four different bifurcation sequences from laminar to turbulent flow have been proposed. The topological elements of these sequences and their physical manifestations for Rayleigh-Benard convection are: (a) fixed points (steady conduction for R < R , or steady convection for R > &),

Strange Attractors in Fluid Dynamics

205

(b) stable limit cycles (periodic convection), (c) doubly (or multiply) periodic limit cycles (doubly or multiply periodic convection), (d) chaotic motion on strange attractors (weakly turbulent convection).

B . LANDAU’S MODEL Landau ((1944); see Landau and Lifshitz (1959)l posited a model in which a sequence of periodic, finite-amplitude disturbances with incommensurate frequencies is excited as some control parameter, for example the Reynolds number, increases through a corresponding sequence of Hopf-bifurcation values (see Table I). The flow then would become increasingly complicated and ultimately (the argument runs) would appear to be chaotic. Among the several objections that have been raised against this model, perhaps the most compelling is that observed power spectra do not show the necessary sequence of lines and always contain broadband components after either the appearance of two or three (or at most four) incommensurate (as opposed to harmonically or difference-related) frequencies or a period-doubling cascade (as in the example of the pendulum). It probably is fair to say that the Landau model now is primarily of historical interest, but it should not be overlooked that it provided the basis for most of the early (1958-1978) analytical work in nonlinear hydrodynamic stability.

C. THERUELLE-TAKENS SCENARIO Ruelle and Takens (1971) conjectured that a strange attractor would appear at the third bifurcation in the Landau sequence, so that at most two incommensurate frequencies could appear prior to the transition to chaotic flow (see Table I). [The conjecture that a triply periodic motion is “generically” unstable has been proved by Newhouse et al. (1978) for a special class of attractors.] This scenario, with some variation in the number of incommensurate frequencies that appear prior to chaos, has been observed in both Rayleigh-BCnard convection and circular Couette flow. Figure 6 (Gollub and Benson, 1980) shows five different types of spectra for Rayleigh-BCnard convection for P = 5. In the top row ( R / R , = 31.0), the flow is singly periodic (only& (sic) and its second harmonic appear above the instrumental noise); in the second row (RIR, = 35.0) the flow is doubly periodic (two incommensurate frequencies, fi and fi , their

TABLE I BIFURCATION SEQUENCES” Model Landau

Stationary point

RuelleTakensNewhouse

Stationary point

Feigenbaum

Stationary point

PomeauManneville

a

-

Bifurcation Sequence

Stationary point

(Hopf)

Singly periodic orbit

(Hopf)

Singly periodic orbit

(Hopf)

(Hopf)

Singly periodic orbit (period T ) Singly periodic orbit

After Lichtenberg and Lieberman (1983), p. 448.

(Hopf)

Doubly periodic orbit

(Hopf)

Doubly periodic orbit

-

Singly (Periodperiodic doubling) orbit (period 2T) (Reverse tangent)

Intermittent chaotic motion

(Hopf)

-

(Perioddoubling)

Triply periodic orbit Strange attractor Singly periodic orbit (period 4n

- --

Turbulent motion

--

Strange attrac tor

..

(Hopf)

...

(Perioddoubling)

Strange Attractors in Fluid Dynamics 0.2

0 -0.02

207 10-1

-p

10-3

-

10-5 0.02

lo-’

0

10-3 -0.02

-;

-

10-5

-L

c

lo-’

0

L

I

N

-0.02

10-3

-0.04

N

b

10-5

E L)

--

I

0

lo-’

x.

Q

-0.02 10-3

-a04

10-5 0.24

10-7

0.16 10-3

0.08

10-5

0

0.1

a2

lo-’ 0.3

f(Hz)

FIG.6. Velocity records and corresponding power spectra for Rayleigh-Bknard convection with P = 5 and five different heat inputs (Gollub and Benson, 1980).

harmonics, and various differences appear above the instrumental noise); in the third row (RIR, = 45.2), phase locking, withj$& = 9I4, has occurred, and the basic frequency fL = fit9 = fit4 and its harmonics appear; in the fourth row (RIR, = 46.8), broadband components (well above the instrumental noise) appear along withfi andfi and a few of their descendents; in the fifth row (RIR, = 65.4), the flow is strongly chaotic. Figure 7 (Gollub and Benson, 1980) illustrates the presence of three incommensurate frequencies, f i ,fi,and (the frequencies of the remaining peaks are linear combinations offi ,fi, andf3); see also Fauve and Libchaber (1981). As many as four independent frequencies have been observed in some experiments (Gollub and Benson, 1980; Gorman et al., 1980).

John Miles

208

McLaughlin and Orszag (1982) have carried out three-dimensional numerical integrations of the Boussinesq equations (which govern RayleighBCnard convection) using a 16 x 16 x 17 modal expansion with no-slip boundary conditions and P = 0.71 (as for air) and obtained results that support the Ruelle-Takens scenario and are qualitatively consistent with the experimental observations of Gollub and Benson after allowing for the larger Prandtl numbers (2.5 and 5.0) in these experiments. Their spectra are presented in Fig. 8. A single frequency,fi , dominates the spectrum for RIR, = 3.81 (I have divided McLaughlin and Orszag's values of R by R, = 1707). A second frequency,fi, appears at slightly larger values of RIR,, and the spectra are dominated byfi andfi and their harmonics and heterodynes up to about RIR, = 8.8, above which the spectra are broadbanded. A third frequency, f3, appears at RIR, = 5.27 and very weakly at RIR, = 7.03 but is much less prominent than that in Fig. 7.

D. THEPERIOD-DOUBLING SCENARIO Feigenbaum (1978) proposed that the observed bifurcation sequence for the solution of a set of nonlinear differential equations should resemble that of certain one-dimensional mappings, in which the period doubles at each bifurcation until an accumulation point is reached at a finite value of the control parameter, a small increase of which then leads to chaos. The first step then is a Hopf bifurcation from a fixed point to a singly periodic

loo

I

f

IHz)

FIG. 7. Power spectrum containing three incommensurate frequencies in RayleighBCnard convection with P = 5 and RIR, = 42.3 (Gollub and Benson, 1980). The frequencies of all peaks are at linear combinations of fi ,f 2 , and h .

Strange Attractors in Fluid Dynamics

209

10

t

fl

f

0

(a)

10

266 0 ‘d

f

26.6

(b)

‘d

f

39.9 -

Lfl-fz

10

10

S %

2

-

M

0

0

f

(el

26.60 rd

cr)

‘4

FIG.8. Calculated power spectra for three-dimensional Rayleigh-Benard convection with no-slip (rigid) boundary conditions, P = 0.71, and R/R, = (a) 3.81, (b) 5.27, (c) 5.85, (d) 7.03, (e) 8.78. (f) 14.64 (McLaughlin and Orszag, 1982).

John Miles

210 100 10-2 10-4

7 100110-6 I

? ‘w

10-2

--

10-4

E

k

4

1Ool10-6 10-2 e-4

AI I>

10-6

0.0

0.1

0.2

0.310.0 f

0.1

0.2

0.3

IHr)

FIG.9. Period-doubling sequences in Rayleigh-Btnard convection with P = 2.5 [after Gollub e t a ! . (1980)].

orbit, as in both the Landau and Ruelle-Takens scenarios. The subsequent bifurcations are usually labelled pitchfork. The spectrum after the first bifurcation comprises a single frequency, say fi , and its harmonics, 2fi, 35, ...; ifi,jfi, ... appear after the second bifurcation; if,,2fi ,jfi, ... appear after the third bifurcation, etc. This scenario is followed by the spherical pendulum (Fig. 4, v = .05, -0.10, -0.142, -0.144). It also has been observed in Rayleigh-BCnard convection; see Fig. 9 (Gollub et al., 1980). Feigenbaum’s model predicts that, after several bifurcations, the amplitudes of the subharmonics generated at each bifurcation should be 8.2 dB down from those of the preceding bifurcation. This prediction is tested in Fig. 9d, where the line drawn through the peaks at ifi, #fi, ... is found to be approximately 8.2 dB below the line drawn through the peaks at if,,

4h, .... The preceding patterns typically were reproduced on decreasing RIR, , but the appearance of the incommensurate (with ft) frequency f. at

Strange Attractors in Fluid Dynamics t

I

I

- -0.04 1 0

-0.04

I

RIR, = 100.4

0.12

'(0

21 1

1 0

400

,

I

,

800

1200

1600

,

I

,

400

800

1200

t

2000

,I 1600

2000

(51

FIG. 10. Velocity records in Rayleigh-BCnard flow with P = 5 (Gollub and Benson, 1980): (a) doubly periodic flow at RIR, = 100.4; intermittent (doubly periodidchaotic) flow at RIR, = 102.8. The flow becomes completely chaotic for somewhat larger RIR,.

RIR, = 27 (Fig. 9f)was a significant exception. A further decrease of R1 R, to 26 reproduced Fig. 9b.

E. THEINTERMITTENTSCENARIO Manneville and Pomeau (1980) have discussed an intermittent transition for the Lorenz model in which the solution alternates between a singly periodic limit cycle and a strange attractor [see also Shimada and Nagashima (1978)l. They have suggested, through a mapping example, how this might occur [see also Eckmann (1981)], and it has been observed in Rayleigh-BCnard convection (Libchaber and Maurer, 1979; Gollub and Benson, 1980); see Fig. 10. It should perhaps be emphasized that the intermittency in this model has no direct connection with the observed intermittency in turbulent boundary layers (Hinze, 1959). The latter phenomenon, which is typical of turbulence at free boundaries (e.g., the mixing zone at the interface between a jet and a surrounding fluid), is basically spatial in character, whereas the Manneville-Pomeau intermittency is basically temporal; moreover, the intermittency in the boundary-

212

John Miles

layer is between slow fluctuations of low intensity and rapid fluctuations of high intensity, rather than between periodic and aperiodic flow. The experimental and theoretical evidence just cited strongly suggests the existence of strange attractors in some parametric regimes for Rayleigh-BCnard convection. Recent measurements and calculations by Brandstater et al. (1983) provide even more compelling evidence for circular Couette flow. They calculated (from their experimental data) the largest Lyapunov exponent and the metric entropy, each of which is positive for, and measures the exponential separation of different trajectories on, a strange attractor. They also calculated the fractal dimension of the attractor and found it to be less than 5 for Reynolds numbers up to 30% over the critical value at the onset of chaos, whereas the number of degrees of freedom of the fluid is extremely large.

VI. Conclusions Summing up, it seems fair to say that strange attractors are definitely relevant to the transition from regular to chaotic flow in some parametric regimes, whether or not the chaotic flow is called turbulent, weakly turbulent, or simply aperiodic. Whether strange attractors are relevant to shear flows at those high Reynolds numbers that are the hallmark of fully developed turbulence is another matter. Moreover, even though there are impressive correlations between the theoretical models and the experimental observations, the models do not yet provide a priori predictions and are, in that crucial aspect, deficient. As Hoyle (1957) has Alexandrov say in The Black Cloud, “Correlations after experiments done is bloody bad. Only prediction is science.” ACKNOWLEDGMENTS I am indebted to H. D. Abarbanel, S. H. Davis, R. J. Donnelly, J . P. Gollub, 0. Lanford, W. Malkus, S. A. Orszag, B. Saltzman, H. L. Swinney, M. Tabor, and J. E. Weiss for aid in the preparation of the lecture on which the foregoing is based. This work was supported in part by the Physical Oceanography Division, National Science Foundation, NSF Grant OCE-81-17539, and by a contract with the Office of Naval Research.

REFERENCES Berry, M. V . (1978). Regular and irregular motion. Top. Nonlineur Dyn. AIP Conf. Proc., NO. 46, 111-112. Brandstater, A., Swift, J.. Swinney, H. L., Wolf, A., Farmer, J . D., Jen, E., and Crutchfield, J. P. (1983). Low-dimensional chaos in a system with Avogadro’s number of degrees of freedom. Submitted to Phys. Reu. Lett.

Strange Attractors in Fluid Dynamics

213

Campbell, L., and Garnett, W. (1882). “The Life of James Clerk Maxwell.” Macmillan, London. Coullet, P., Tresser, C., and Arneodo, A. (1979). Transition to stochasticity for a class of forced oscillators. Phys. Lett. A 72, 268-270. Curry, J . H. (1978). A generalized Lorenz system. Commun. Math. Phys. 60,193-204. Curry, J. S., Herring, J . R.. Lonearic. J . , and Orszag, S. A. (1984). Order and disorder in two-and three-dimensional Benard convection. J . Nuid Mech. (in press). DaCosta, L. N., Knobloch, E., and Weiss, N. 0. (1981). Oscillations in double-diffusive convection. J. Fluid Mech. 109, 25-43. Eckmann, J.-P. (1981). Roads to turbulence in dissipative dynamical systems. Rev. Mod. Phys. 53, 643-654. Fauve, S . , and Libchaber, A. (1981). Rayleigh-Btnard experiment in a low Prandtl number fluid, mercury. In “Chaos and Order in Nature” (H. Hacken, ed.), pp. 25-44. SpringerVerlag, New York. Feigenbaum, M. J., (1978). Quantitative universality for a class of nonlinear transformations. J. S f a t . Phys. 19, 25-52. Gollub, J. P., and Benson, S. V. (1980). Many routes to turbulent convection. J . FIuid Mech. 100, 449-470. Gollub, J. P., Benson, S. V., and Steinman, J. (1980). A subharmonic route to turbulent convection. Ann. N Y Acad. Sci. 357, 22-27. Gorman, M., Reith, L . A., and Swinney, H. L. (1980). Modulation patterns, multiple frequencies, and other phenomena in circular Couette flow. Ann. N Y Acad. Sci. 357, 10-27. Hinze, J. 0. (1959). “Turbulence,” pp. 12, 289, 402ff. McGraw-Hill, New York. Hoyle, Fred (1957). “The Black Cloud.” Harper, New York. Hutton, R. E. (1963). An investigation of resonant, nonlinear, nonplanar, free surface oscillations of a fluid. N A S A Tech. Note D-1870. Landau, L. D., and Lifshitz, E. M. (1959). “Fluid Mechanics,” Section 27. Pergamon, London. Lanford, 0. (1977). In “Turbulence Seminar” (P. Benard and T. Ratiu, eds.). Lecture Notes in M n t h . 615, 114. Laplace, Pierre Simon (Marquis de) (c. 1795). “A Philosophical Essay on Probabilities” (translated from the Sixth French Edition by F. W. Truscott and F. L. Emory), 1951, p. 4. Dover, New York. Libchaber, A., and Maurer, J. (1980). Une experience de Rayleigh-Btnard de geometric rtduite: multiplication, accrochage et demultiplication de frequencies. J . Phys. Colloy. 41, 51-56. Lichtenberg, A. J., and Lieberman, M. A. (1983). “Regular and Stochastic Motion.” Springer-Verlag. New York. Lorenz, E. N. (1963). Deterministic nonperiodic flow. J. A f m o s . Sci. 20, 130-141. Malkus, W. V. R. (1972). Non-periodic convection at high and low Prandtl number. M e m . Soc. R. Sci. Liege (6)4, 125-128. Mandelbrot, B. (1980). Fractal aspects of the iteration O f 7 4 Az(1 - z)for complex A and z. Ann. N Y A c a d . Sci. 357, 258. Mandelbrot, B. (1982). “The Fractal Geometry of Nature,” p. 195. Freeman, San Francisco. Manneville, P., and Pomeau, Y. (1980). Different ways to turbulence in dissipative dynamical systems. Physica D 1, 219-226. Marcus, P. S. (1981). Effects of truncation in modal representations of thermal convection. J. Fluid Mech. 103, 241-255.

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McLaughlin, J. B., and Martin, P. C. (1975). Transition to turbulence in a statically stressed fluid system. Phys. Rev. A U ,186-203. McLaughlin, J. B., and Orszag, S. A. (1982). Transition from periodic to chaotic thermal convection. J . Fluid Mech. 122, 123-142. Miles, J. W. (1962). Stability of forced oscillations of a spherical pendulum. Quart. Appl. Math. 20, 21-32. Miles, J. W. (1984a). Resonant motion of spherical pendulum. Physica D . In press. Miles, J. W. (1984b). Resonant, non-planar motion of a stretched string. J . Acoust. Soc. A m . (in press). Monin, A. S . (1978). On the nature of turbulence. Sou. Phys. Usp. 21, 429-442. Moore, D. R., and Weiss, N. 0. (1973). Two-dimensional Rayleigh-BCnard convection. J . Fluid Mech. 58, 289-312. Moore, D. R., Toomre, J., Knobloch, E., and Weiss, N. 0. (1983). Period doubling and chaos in partial differential equations for thermosolutal convection. Nature (London) 303, 663-667. Newhouse, S., Ruelle, D., and Takens, F. (1978). Occurrence of strange axiom A attractors near quasi-periodic flows on T “ , m 2 3. Comm. Muth. Phys. 64, 35-40. Orr, W. McFarland (1907). The stability or instability of the steady motion of a perfect liquid and of a viscous liquid. Proc. R . Irish Acad. A 27, 9-68, 69-138. Poincark, H. (1892). “Les Mtthodes Nouvelles de la MCcanique CCleste.” Gauthier-Villars, Paris. Rayleigh, Lord (1916). On convection currents in a horizontal layer of fluid, when the higher temperature is on the underside. Philos. Mag. 32, 529-546; Scientific Papers 6, 432-446. Reynolds, Osborne (1883). An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels. Phil. Trans. R . Soc. London 174, 935-982; Scientific Papers 2, 51-105.

Robbins, K. A. (1979). Periodic solutions and bifurcation structure at high R in the Lorenz model. SIAM J . Appl. Math. 36, 457-472. Ruelle, D., and Takens, F. (1971). On the nature of turbulence. Comm. Math. Phys. 20, 167192; 23, 343-344. Saltzman, B. (1962). Finite amplitude free convection as an initial value problem-I. J . Atmos. Sci. 19, 329-341. Shimada, I . , and Nagashima, T. (1978). The iterative transition phenomenon between periodic and turbulent states in a dissipative dynamical system. Prog. Theor. Phys. 59, 1033-1036. Sparrow, Colin (1982). “The Lorenz Equations: Bifurcations, Chaos and Strange Attractors.” Springer, New York. Taylor, G. I. (1923). Stability of a viscous liquid contained between two rotating cylinders. Phil. Truns. R . Soc. London, Ser. A 223, 289-343; “The Scientific Papers of Sir Geoffrey Ingram Taylor” (G. K. Batchelor, ed.), Vol. 1V. Cambridge Univ. Press, London and New York. Toomre, J., Gough, D. O., and Spiegel, E. A. (1982). Time-dependent solutions of multimode convection equations. J . Fluid Mech. 125, 99-122. Veronis, G. (1965). On finite-amplitude instability in thermohaline convection. J . Mur. Res. 23, 1-17. Welander, P. (1967). On the oscillatory instability of a differentially heated fluid loop. J . Fluid Mech. 29, 17-30.

ADVANCES I N APPLIED MECHANICS, VOLUME

24

Nonlinear Ship Waves HIDEAKI MIYATA Department of Naval Architecture University of Tokyo Tokyo, Japan

TAKA0 INUI Department of Mechanical Engineering Tamagawa University Tokyo, Japan

I. Introduction . . . . . . ...................... 11. Nonlinear Waves Generated by Ships. . . . . . . . . . . . . . . . . . . . . . . . . . . A. Ship Wave Formation ................ B. Discontinuity an .... 111. Characteristics of Waves around Wedge Models. . ........

A. Configuration of B. Velocity and Pressure Distribution in Waves. .

...............

IV. Modified Marker-and-Cell Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Computational Procedure.

215 218 219 228 242 251 264

....

B. Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

271

V. Computed Waves around Wedge Models. . . VI. Concluding Remarks . . ......................

287

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

I. Introduction The waves generated by ships advancing steadily in deep water have been considered as typical linear dispersive ones. Many wave-makingresistance theories have been proposed by mathematicians and naval architects, most of which are based on this linear postulation. However, in spite of the efforts devoted to the improvement of wave-making-resislance theories, their effectiveness is very limited. They only succeed in a 215 Copyright 0 1984 by Academic Press, Inc. All rights of reproduction in any form reserved. ISBN 0-12-002024-6

216

Hideaki Miyata and Takao Inui

qualitative evaluation of the wave resistance of very fine ships, such as container carriers. As new theories have been advanced, they have received either partial or no acceptance by ship designers. In two cases linear wave-making theories have shown their practical effectiveness and have made a remarkable contribution to the design of ship forms having minimum wave resistance. One is the development of bow bulbs by Inui (1962) and the other is the wave-analytical procedure for minimizing wave resistance by Tsutsumi (1978) and Matsui et al. (1980). The former has been derived intuitively by observation of waves, and the latter by careful analysis of measured wave records. It is very rare that a theory by itself contributes to the minimization of wave resistance. The design of ship forms with minimum resistance still rely to a great extent on towing tests using a series of ship models and the experiences of the designer. The inadequacy of the theories so far developed is revealed in experimental results obtained by using models. The wave resistance derived from a measured wave record (RwP)is usually smaller than that derived except in the case of very fine from the towing force measurement (Rw), ship models. The extreme case is seen in the paper by Inui et al. (1979), in which the curve of Rw versus Froude number shows not only quantitative disagreement but also very poor qualitative resemblance. The cause of this discrepancy has been shown to consist of the complicated nonlinear phenomena in the near field of ships. Some ship waves of full-scale ships are shown in Figs. 1 and 2. Nonlinear fluid motions involving white bubbling are evident around the ships, while Kelvin’s linear dispersive wave system is present in the far field. It may also be noted that the nonlinear wave formation depends on the configuration of the ship forms. Baba (1969) noticed the singular free surface phenomena around the bow of full hull forms, and he interpreted them as the breakdown of waves. The resistance caused by the phenomena was classified as a new component of viscous resistance, because it was measured as a momentum loss far behind ship models. In the same year, Taneda and Amamoto (1969) displayed the experimental results of flow visualization of free surface flows around circular cylinders and a full hull form. They showed that intense vortical flows are generated from the bow and stern of a full hull form and called them necklace vortices or stern jump vortices. These two pioneering works were followed by those of Takekuma (1972), Taneda (1974), Baba and Takekuma (1975), Baba (1975, 1976), Eggers (1981), and Kayo and Takekuma (1981), who, however, could arrive at no consistent explanation. Miyata and co-workers supposed that the complicated nonlinear free surface phenomena called breakdown of waves or necklace vortices were

Nonlinear Ship Waves

217

FIG. 1 . Waves generated by a bulk carrier photographed from an aircraft. (Reprinted courtesy of the Japan Foundation for Shipbuilding Advancement.)

FIG. 2. Waves generated by a passenger boat photographed from an aircraft. (From Ships of the World.)

Hideaki Miyata and Takao Inui

218

consequent results of nonlinear wave making in the near field, and over a period of five years they carried out experimental investigations into the characteristics and structure of nonlinear waves, as well as numerical analyses of nonlinear wave formation by a finite-difference method. This chapter deals with the studies they made at the Experimental Tank of the University of Tokyo. In Section I1 characteristics of nonlinear waves around ship models are experimentally studied, and in Section I11 more detailed characteristics and structure are clarified with simple wedge models. A numerical method of simulating nonlinear wave formation around the bow is described in Section IV, and the computed results are discussed and compared with experimental results in Section V.

11. Nonlinear Waves Generated by Ships Results of the experiments aimed at clarifying the characteristics of waves around ship models are presented and discussed in this section. The model ships whose particulars are listed in Table I were towed at a TABLE I PRINCIPAL PARTICULARS OF S H I P

MODELS

Variables Length

Lw

Breadth B

Model

(m)

(m)

Draft d (m)

M40 M4 1 M42 M43E1

2.000 2.000 2.000 2.100

0.308 0.364 0.444 0.364

0.105 0.105 0.105 0.105

0.543 0.543 0.543

M43E2 M43E3 M43B1 M4382 WMI-B

1.980 1.860 1.860 1.860 2.400

0.364 0.364 0.364 0.364 0.240

-

WMI-C

2.400

0.240

0.105 0.105 0.105 0.105 0.050 0.060 0.060

WM2

2.400

0.480

0.060 0.100 0.150

Fineness

Cb

-

0.680 0.667 0.667

Remarks Fine ship Wide-beam fine ship Entrance angle a = 5" a = 10" a = 15"

Attached with a flat bulb Attached with a cylindrical bulb Parabolic waterlined, wall-sided model with a blunt bow Parabolic waterlined, wall-sided model Parabolic waterlined, wall-sided model

Nonlinear Ship Waves

219

constant advance speed in the experimental tank whose length, breadth, and depth are 86 m, 3.5 m, and 2.5 m, respectively. Wave patterns were photographed and the velocity components of flows and momentum loss far behind the ship models were measured. The x axis is parallel to the centerlines of the ship models and positive aftward, the y axis is oriented laterally, and the z axis is oriented upward. The origin of the coordinate system is located at the center of the ship model, that is, a half length behind the fore end. The reference lengths for Froude numbers Fn and Fd are the length (Lpp)and draft ( d )of the ship models, respectively.

A. SHIPWAVE FORMATION The technique of photographing waves with an aluminum powder film on the water surface is often used at the Experimental Tank of the University of Tokyo. Very fine aluminum powder is spread over the water surface, and it forms a very thin film that can be split by a violent disturbance. The photographs obtained by this technique are suitable for a qualitative grasp of wave formation. A typical wave pattern image obtained by this technique is shown in Fig. 3 for a wide-beam fine ship model. The wave system is not like that of Kelvin’s waves in the near field of the ship model. Two remarkable waves are originated from the forepart of the model, and their crest lines are nearly parabolic or straight. The appearance of the water surface is quite different in front of and behind the wave-crest lines. Behind the wave crest the water surface seems to be turbulent and the aluminum powder film is split, showing the presence of discontinuous flow on the free surface. A linear dispersive wave system is observed at the left corners of the picture, and it is deformed by a continuation of the crest line of the second singular wave. The black region behind the ship model, where the aluminum powder is displaced laterally and bare water surface is revealed, is supposed to be an effect of boundary layer separation. Figures 4-6 are cases of simple models WM1-C and WM 1-B. The sidewalls of both models are vertical, and the waterline of WMI-C is a parabolic curve. WMI-B is a modification of WM1-C, with a blunt head form attached. The bow waves of the sharp-ended model WMI-C magnify their strength and the slope of the forward face with an increase in Froude number. The bow waves of the round-ended model WMI-B evidently show intense nonlinearity in a wide range of advance speeds. Their appearance is very similar to a turbulent bore or a hydraulic jump. The two wave fronts are jumps of water surface that involve violent turbulent

220

Hideaki Miyata and Takao Inui

FIG.3. Waves around wide-beam fine ship model M42 advancing at Fn = 0.267. The water surface, covered with an aluminum powder film, is split near and behind the model.

motion. The nonlinear wave motion, which is similar to that in shallow water, can be generated in the vicinity of ship bows in deep water. Photographs taken from above are seen in Fig. 6. Two nonlinear waves are observed, and they magnify their intensity according to the increase of advance speed. The discontinuous wave motion enlarges the black region into which aluminum powder cannot be transferred. The foremost nonlinear wave keeps a circular shape, whereas the second is circular at Fn = 0.16 but is transformed into an oblique shape at higher Froude numbers. The first nonlinear wave is analogous to a detached shock wave in supersonic flow, and the second, with a nearly straight wave-crest line at a high Froude number, is analogous to an attached shock wave. Water flows on the free surface are visualized with paper tips, 5 mm in diameter, used as tracers, as seen in Fig. 7 . The two nonlinear waves produce lines of discontinuity, across which water flows undergo abrupt change. Wave-pattern pictures of a simple-parabolic waterlined model WM2 are present in Figs. 8-10 for three different draft conditions. It is shown that the angle of the wave-crest line to the centerline ( p ) is decreased by the increase of advance speed, and that the increasing draft increases this

FIG.4. Bow waves of sharp-ended model WMI-Cat five advance speeds: Fn 0.16, 0.20, 0.24, and 0.28.

=

0.12,

FIG. 5 . Bow waves of round-ended model WMI-B at five advance speeds: Fn = 0.12, 0.16, 0.20, 0.24, and 0.28. The draft of the model is 0.06 m.

N W N

FIG.6. Bow-wave formation of WMl-B under the same conditions as Fig. 5. The aluminum powder film is violently split, which illustrates the discontinuity of surface flow. The draft of the model is 0.06 m.

FIG.7. Free surface flows around the bow of round-ended model WMI-B, photographed using a stroboscope and with circular water tips as tracers. The Froude numbers based on draft are 0.8, 1.0, and 1.2. The draft of the model is 0.05 m. 224

FIG.8. Wave-patterns of simple-parabolic-waterlined ship model WM2 at three advance speeds: Fn = 0.22, 0.26, and 0.30. The draft of the model is 0.06 m .

FIG.9. Wave-patterns of simple-parabolic-waterlined ship model WM2 at three advance speeds: Fn = 0.22, 0.26, and 0.30. The draft of the model is 0.10 m.

FIG. 10. Wave-patterns of simple-parabolic-waterlined ship model WM2 at three advance speeds: Fn = 0.22, 0.26, and 0.30. The draft of the model is 0.15 m. 227

228

Hideaki Miyata and Taka0 Inui

angle. The second nonlinear wave, which is present under a shallower draft condition (Fig. 8) disappears under a deeper draft condition (Fig. lo), in which the foremost nonlinear wave is huge. The change of the foremost nonlinear wave with the increase of advance speed is clearly observed in Fig. 1 1 , where the ship model is M43E3 whose bow form is like a wedge of 30". The round wave formation at Fn = 0.1667 is transformed into an oblique straight-lined formation at Fn = 0.2041, and henceforth the angle p is gradually decreased. The effect of the angle of the fore end (entrance angle) of the ship models is seen in Figs. 12 and 13. The three models have wedgelike bows, and the entrance angles, which are one-half the wedge angles on the undisturbed waterline, are 5", lo", and 15". The angle of the wave-crest line p is in close relation to the entrance angle. The generation and formation of nonlinear waves around a bow depend on the configuration of the bow, the draft of the models, and the Froude number. The linear dispersive wave system does not share these properties. Almost all ocean-going ships are equipped with bow bulbs, large or small. The effect of bow bulbs in reducing wave resistance consists at least partially in weakening the nonlinear bow wave, as seen in Fig. 14. By attaching two kinds of bulbs, B1 and B2, the angle of the wave crest p is decreased and the black region disappears. In most cases the decrease of p is accompanied by the decrease in the wave height of the nonlinear wave, and consequently the wave resistance is reduced. The angle p is a convenient measure of the strength of nonlinear waves.

B. DISCONTINUITY AND ENERGY DEFICIT A wave-contour map of M42 under the same conditions as in Fig. 3 is shown in Fig. 15. The contour lines in the near field are distorted, especially about the second nonlinear wave. In order to know the velocity change across the two wave fronts, the flow velocities were measured by a five-hole pitot tube with a sphere-head diameter of 6 mm. The results are shown in Fig. 16, in which the velocities are nondimensionalized with respect to the uniform stream velocity. The total velocity drops rather suddenly at the wave front, and the components normal to the wave-front line are considerably decreased, whereas the tangential components are almost conserved. A more detailed measurement was conducted with WM2 under the same conditions as for the top of Fig. 8, that is, d = 0.06 m and Fn = 0.22. The wave-contour map is present in Fig. 17 for the region in which velocity measurements were conducted. The wave slope is very steep at the

FIG.1 I . Bow waves of ship model M43E3 at four advance speeds, illustrating their intensificationand deformation with theincrease of advance speed: (a) Fn = 0.1667; (b) Fn = 0.2041; (c) Fn = 0.2500; (d) Fn = 0.3015.

FIG. 12. Bow waves of a series of ship models at Fn = 0.22 with entranc? angles 5", lo", and 15". The angle of the wave crest is increased with the increase of the entrance angle. 230

FIG.13. Bow waves of a series of ship models at Fn = 0.2774, with entrance angles So, lo", and 15". The angle of the wave crest is increased with the increase of the entrance angle. 23 I

FIG. 14. Bow waves of bow-bulb-series ship models illustrating the effect of bulbs on nonlinear bow-wave formation at Fn = 0.2673: (a) M43E3 without a bulb; (b) M43B1 with a flat bulb; (c) M43B2 with a cylindrical bulb.

Nonlinear Ship Waves

233

FIG.15. Wave-height contours of wide-beam fine ship model M42 advancing at Fn = 0.267. The wave height is indicated in millimeters.

wave front. Three kinds of distributions of measured disturbance velocities are shown in Figs. 18-20, distributions on horizontal planes, vertical planes, and lateral lines, respectively. The indications of these figures are that the disturbance velocities behind the wave crest are large, that they are nearly normal to the wave crest line, and that the region where water flow undergoes an abrupt change in velocity is limited to within the thin layer near the free surface. The change in the measured velocity vector, including uniform flow velocity, was traced along the streamlines of the calculated double model

FIG.16. Velocity change across wave fronts A and B of M42 at Fn = 0.267. The dashed vectors are velocity components normal and tangential to the wave fronts. The tangential components are almost conserved.

234

Hideaki Miyata and Taka0 Inui

E

FIG.17. Detailed wave height of WM2. The notations ss 8 and ss 7 indicate the longitudinal positions of 20 and 30% of the ship length behind the bow, respectively. Open circles indicate points at which measurements of velocity components are undertaken. The wave height is in millimeters.

Nonlinear Ship Waves

235

c Y

FIG.18. Distribution of disturbance velocity vectors on a horizontal plane 10 rnm below the disturbed free surface. (Z' =

I

5 - 1 cm).

-0.2 u

. - . - - A /

ss7 Yl2

-

-

v.'200

,

-

+

*

'

'

q

p

+ *

.

-

-

t

#

* .

.

=

-

-

'

* I

300

t

,

f ' , , * . . I

400

FIG. 19. Distribution of disturbance velocity vectors on vertical planes.

Hideaki Miyata and Takao Inui

236 ( a

1

UV.W/U

0.2

0.I

0

-0.1

-0.2 -60

I

I .

0.1

0-

-0.1

.

-0.2

-

FIG.20. Lateral distribution of disturbance velocity components on curved lines (a) 10 and 30 mm below the free surface; (b) 50 and 70 mm below the free surface. The longitudinal location of measurements is at ss 7A (25.8% of ship length behind the bow), and w is positive downward.

flow shown in Fig. 21. The results are presented in Fig. 22 for six streamlines. The origin of the vectors is concentrated, and then the heads of the vectors are almost on the dotted lines that are drawn normal to the wavefront lines. The change in velocity is mostly in the direction normal to the wave-front line.

Nonlinear Ship Waves

237

FIG. 21. Streamlines of double model flow on which velocity vectors are traced in Fig. 22.

The apparent resemblance of the nonlinear waves to nonlinear shallow water waves, the already-mentioned jumplike velocity change, and other singular properties suggest the dissipative property of the waves. The energy of the nonlinear waves may be partly dissipated into momentum loss far behind the ship models. In order to recognize the consequence of

FIG.22. Change of velocity vectors on the streamlines in Fig. 21; a-g are lateral locations defined in Fig. 21.

238

Hideaki Miyata and Takao Inui

dissipation, the loss of velocity head was measured on the vertical planes about half a model length behind several ship models, and the results are shown in Figs. 23-26, which are contour maps of head loss in absolute value of water head. Loss of velocity head that cannot be considered due to the viscous effect is recorded near the free surface in every figure. The magnitude of head loss laterally extended seems to be closely related to the magnitude of nonlinear waves observed in wave-pattern pictures. The increase of beam length magnifies the head loss due to dissipation of wave energy, as seen in Fig. 23. The shallower the draft of the model ship, the wider the contour lines extend laterally, as seen in Fig. 24. The effects of entrance angles and bow bulbs, qualitatively understood in Figs. 12-14, can be also realized in Figs. 25 and 26.

200

J

4co

300 \

/

05

200

300

4 00

100 150

200

FIG.23. Contours of loss of velocity head measured at 0.85 m behind the aft end of the ship models. (a) M40, (b) M41, and (c) M42 at Fn = 0.267. Numbers on the contour lines indicate head loss ( H o - H ) in mmAq.

Nonlinear Ship Waves

239

The resistance components of bow-bulb-series ship models are illustrated in Fig. 27. Because of the small size of the ship model, the viscous resistance component occupies a large area. The resistance component RML(marked c) is obtained by integrating the contour maps in Fig. 26.

c

FIG. 24. Contours of loss of velocity head, measured at 1.20 m behind the aft end of model WM2 under three draft conditions. (a) d = 100 rnm, (b) d = 150 mm, and (c) d = 200 mm, at Fn = 0.22.

240

Hideaki Miyata and Takao Inui

E 0 60 100 150

200 { (mm) E 2

0 60 100 150

200 f

e 400

300 /

500

(mm) Y 6 00

/

( c )

FIG.25. Contours of loss of velocity head measured at 0.85 m behind the aft ends of entrance angle-series ship models at Fn = 0.267. (a) M43EI. a = 5”; (b) M43E2, a = 10”; (c) M43E3, a = 15”.

This component is greater than the viscous resistance component (marked a). The wave resistance derived from the towing test is partly grasped as the linear wave-pattern resistance RWPin the far field and is partly dissipated into momentum loss far behind the ship models. From the experimental results described, the typical characteristics of nonlinear waves in the near field are summarized. They are (1) steepness of the wave slope, (2) formation of lines of discontinuity, (3) turbulence on the free surface on and behind the wave fronts, (4) systematic change of the wave-crest-line angle, depending on the F’roude number and the ship-

24 1

Nonlinear Ship Waves L

0

i

100

(mm) Y

200

300

400

600

5.30

60 100

150

200

(mm) Y 6 00

5 00

FIG. 26. Contours of loss of velocity head measured at 0.96 m behind the aft ends of bow-bulb-series models at Fn = 0.289. (a) M43E3, without bulb; (b) M43B1, with a flat bulb; (c) M43B2, with a cylindrical bulb.

a

M43E3

I

C

a

M 4 3 81

I

b

C

0

100

200

300 (g)

FIG.27. Resistance components of bow-bulb-series ship models: a, viscous resistance; b, wave resistance derived from towing test; c, resistance derived from head-loss contours in Fig. 26; hatched area, wave-pattern resistance derived from wave analysis in the far field.

242

Hideaki Miyata and Takao Inui

model configurations, ( 5 ) satisfaction of a kind of shock relation across the wave fronts, and (6) dissipation of wave energy into momentum loss far behind the ship models. From these characteristics, analogous to shock waves in supersonic flow and nonlinear shallow water waves, nonlinear waves in the near field of ships in deep water are called free surface shock waves (FSSWs). More detailed experimental results are presented by Inui et al. (1979a, 1979b), Kawamura et al. (1980), Miyata (1980), and Miyata et a f . (1980).

111. Characteristics of Waves around Wedge Models Further detailed investigations were conducted with the simplest threedimensional models, that is, wedge models, to clarify the characteristics and structure of nonlinear waves (Takahashi et al., 1980; Miyata et al., 1982a). The configurations of typical wedge models are shown in Fig. 28. Four wedge models were produced, that is, two with entrance angle a (one-half the wedge angle) of lo", one with 5", and one with 20". They can be combined into wedge models with a , of 15", 25", 30", 35", 40", and 45". The origin of the x coordinate was located at the fore ends (FP) of the wedge models, and the locations and measured flow variables were nondimensionalized as much as possible. The reference length for the Froude number was the draft of the models, and the range of advance speed was decided so as to cover the Froude number based on the draft (Fd) of practical hull forms; that is, Fd was varied from 0.5 to 1.5, as far as possible. The models were rigidly connected to the towing carriage. Wave pictures were photographed by a 35-mm camera, which was placed about 1.5 m above the water surface. The wave profiles, from which wave-contour maps were drawn, were measured by a wave-height recorder of contact servo type. The velocity components and pressure were measured by a five-hole pitot tube of NPL type, whose outer and inner diameters were 2.1 and 0.4 mm, respectively.

-996

1ry-=J

FIG.28. Configurations of two typical wedge models.

Nonlinear Ship Waves

243

A. CONFIGURATION OF WAVESAROUND Bows Wave formation around the bows of wedge models is ruled by three entrance angle ( a ) ,and draft parameters, namely, speed of advance (U), ( d ) of the models. When these parameters are changed one by one, the bow waves vary considerably, as seen in Fig. 29. The Froude number must be a governing parameter for gravity waves, and for the present case the reference length should be the draft of the models. To know whether Froude's law of similarity is fulfilled or not, wave formation at constant Froude numbers based on draft (Fd) was compared on the basis of two or three draft conditions. Figures 30-33 present comparisons of photographed waves for two wedge models at two Froude numbers for each. The pictures in each figure are printed so that the length of draft, which is actually different, is constant; in other words, the wave formation is compared in a nondimensional form with respect to draft length. However, the position of wave fronts relative to the shape of wedge models is not the same in these pictures, since the location of the camera is fixed, whereas the height of the models above the water surface and the absolute distance of waves from the models vary according to the change of draft. Taking into account the virtual shift of relative location, we recognize that the formation of wave patterns is ruled by Fd except for the turbulent appearance. The formation of nonlinear waves, called FSSWs, around the bows demonstrates visually the fulfillment of the Froude law of similarity. Free surface shock waves are classified into two types for convenience, namely, normal and oblique. A normal FSSW, which is analogous to a detached shock wave in supersonic flow and is, in most cases, detached from the bow, shows a round-shaped formation, as seen in Figs. 30, 32, and 33 and its front line makes an angle of n/2 to the centerline of the wedge models on the centerline. On the other hand, an oblique FSSW, which is analogous to an attached shock wave, shows nearly straight wave-front lines attached to the body surface, as seen in Fig. 31. In the case of a wedge model of a = 20", the foremost FSSW is a normal one until Fd = 0.95, and then it is somewhat suddenly transformed into an oblique one. Thereafter, it remains oblique with its wave-front angle p being decreased with increasing Fd. In contrast to the case of a = 20°, the wedge model of a = 45" generates only normal FSSWs, and they are never transformed within the tested range of Fd. The appearance of the free surface is unsteady and turbulent on and behind the wave fronts. The turbulence is intensified with the increase of advance speed. Especially, the normal FSSWs generated by wedges of large entrance angle involve violent unsteady motion and air entrainment on the free surface.

FIG.29. Photographs of bow waves illustrating variations due to changes of conditions: (a) a = 5", d m, V = 1.5 d s ; (c) (I = Y, d = 100 mm, V = 1.0 d s ; (d) (I = 20", d = 100 mm, V = 1.5 d s .

= 10 mm, V = 1 .O

m/s; (b) (I = Y, d = 100

Nonlinear Ship Waves

245

FIG.30. Wave patterns of a wedge model of a = 20" at Fd = 0.8: (a) d = 0.05 m; (b) d = 0.10 m;(c) d = 0.15 m.

FIG.31. Wave patterns of a wedge model of cy 0.10 m ;( c ) d = 0.15 m. 246

=

20" at Fd

=

1.7: (a) d = 0.05 m; (b) d =

FIG.32. Wave patterns of a wedge model of a! = 45”at Fd 0.15 m.

247

=

0.8: (a) d = 0.10 m; (b) d =

248

Hideaki Miyata and Takao Inui

FIG.33. Wave patterns of a wedge model of a 0.15 m.

=

45" at Fd

=

1.4: (a) d

=

0.10 m; (b) d

=

Nonlinear Ship Waves

249

0.20 -0.1

FP

0.1

0.4

FIG.34. Longitudinal wave profiles on the lines parallel to the centerline of a wedge model of (Y = 20" at d = 0.10 rn and Fd = 0.8.

Some typical examples of measured longitudinal wave profiles are shown in Figs. 34-37. Wedge models are towed from right to left. The coordinates and wave heights are shown in dimensional form, and the slope of 30" is illustrated for reference. Figures 35 and 36 have the same conditions, except for the absolute value of the draft, in which the similarity law of Fd is satisfied. The wave slope at the forward face is so steep as to exceed 30" near the centerline. Actually the maximum slope of 65" is recorded in the wave profiles near the centerline of the wedge model of (Y = 20"in Figs. 35 and 36, and the unsteady fluctuation of the free surface is scarcely recorded there, whereas wave profiles at a distance from the centerline show relatively small slopes of around 30", and the unsteady fluctuation is clearly recorded on and behind the damped wave profiles. The unsteady fluctuation, which sometimes gives arbitrariness to the wave profiles, is supposed to ease the very steep slope of FSSW. It seems to occur in the process of dissipating the energy concentrated on the very steep waves. Some wave profiles, which are nondimensionalized and as a result smoothed, neglecting the fluctuation, are compared in Figs. 38 and 39. The wave height is nondimensionalized with respect to the water head of uniform stream H ( U 2 / 2 g )and the coordinate with respect to the draft.

0.5

250

Hideaki Miyata and Takao Inui

e 0.02 0.04

0.06

-

E >

0.08 0.10 0.12 0.14 0.16

0.18 0.20

FIG. 35. Longitudinal wave profiles on the lines parallel to the centerline of a wedge model of a = 20" at d = 0.10 rn and Fd = 1 . 1 .

The accord between the two draft conditions is, on the whole, good, and we can be convinced of the fulfillment of the law of similarity. However, there exists some scale effect; that is, the height of the crest is attenuated when the absolute speed of advance is small, and the profiles behind the crest show slight differences. The viscosity is supposed to play a role in the mechanism of dissipation involving unsteady fluctuation. The experimental results indicate that the nonlinear waves are ruled by the Froude number based on draft. However, this law of similarity might be violated when the draft of the model is extremely large or small in comparison with beam length. Notwithstanding the complexity of the law of similarity, we can admit the validity of the similarity of Fd as far as the usual dimensions of practical hull forms are concerned. For a full-scale ship 300 m in length and 20 m in draft, Fd = 0.8 means that the Froude number based on length (Fn) is 0.21, and when the ship's draft is decreased to 10 m, the Fn is 0.146.

Nonlinear Ship Waves

25 1

FIG. 36. Longitudinal wave profiles on the lines parallel to the centerline of a wedge model of a = 20"at d = 0.15 m and Fd = 1.1.

The wave-front-line angle /3 can be determined as a function of Fd and entrance angle a . On the basis of numerous wave-pattern pictures this relation is illustrated in Fig. 40. The variation of /3 with an increase or decrease of Fd is relatively small when a is small.

B. VELOCITY AND PRESSURE DISTRIBUTION IN WAVES

Two typical conditions were chosen for the measurement of disturbance velocities and static pressures of wave motion in the neighborhood of the foremost nonlinear waves around wedge models; that is, Fd = 1.1 for the wedge model of a = 20" and Fd = 1.0 for the wedge model of a = 45". Measurements were conducted under two draft conditions, d = 0.10 and 0.15 m, for both wedge models. Two vertical x-z planes parallel to the

252

Hideaki Miyata and Takao Inui

0.30

0.40

I -0.3

-0.2

-0.1

0.1

FP

0.2

0.3

0.4

Xlmt

FIG. 37. Longitudinal wave profiles on the lines parallel to the centerline of a wedge model of a = 45" at d = 0.10 rn and Fd = 0.8.

centerline at yld = 1.0 and 2.0 were chosen for the case of a = 20°, and one at yld = 1.O for the case of a = 45". Wave height and vertical location of measurement were nondimensionalized with reference to H ( U 2 / 2 g ) . Distributions of measured velocity components, nondimensionalized with reference to the speed of uniform stream U,are presented in Figs. 41-46 for the wedge model of a = 20". The difference in velocity components is very small between the two draft conditions, most of which is attributed to the small difference in wave height. The law of similarity of Fd is demonstrated by the actual flow velocities. Disturbance velocities u and u steeply reach great values near the wave crest and consequently a line of discontinuity is formed, across which the water flow undergoes an abrupt change in velocity. Large forward and outward disturbance velocities suddenly appear at the wave front, and the vertical slope of velocity becomes steep in the thin layer near the free surface. Disturbance velocity vectors on horizontal and vertical planes are illustrated in Figs. 47 and 48. The measured velocities on the two curved lines

Nonlinear Ship Waves

253

FIG. 38. Nondimensional wave profiles on two longitudinal lines separated from the centerline by (a) the draft length and (b) twice the draft length for a wedge model of a = 20" at Fd = 1 . 1 : (-) d = 0.10 m; (---) d = 0.15 m.

parallel to the centerline and the disturbed free surface are used. On the more deeply curved line 1.lH below the free surface, the disturbance velocity vectors are gradually weakened with increase of distance from the wedge surface (to the left in the figures), while on the line very close to the free surface (0.lHbelow the free surface) they gradually increase and reach a great value, more than half of the uniform stream velocity, at the wave front and then suddenly decrease before the wave front. The measured velocity distributions of the wedge model of a! = 45" are presented in Figs. 49-5 1. The wave profile is not pulselike but similar to that of a bore, showing constant high elevation of the free surface behind the wave front. The case of higher absolute speed of advance shows higher wave elevation. The difference between the two draft conditions is presented behind the wave front where the wave elevation indicates some scale effect; however, the behavior of the velocity components is almost the same. The u and u components of the disturbance velocity are suddenly increased at the wave front, and, contrary to the case of a = 20", the region where the vertical gradient of velocity is large is widely extended behind the wave front. The vertical variations tend to be less steep very close to the free surface in the rear region where the free surface elevation remains

254

Hideaki Miyatu and Takao Inui 3/ H

0.8

0 X/d

0

1

r0.6

(bl

10

t

-1

x/a

0

1

2

FIG. 39. Nondimensional wave profiles on two longitudinal lines separated from the centerline by (a) the draft length and (b) twice the draft length for a wedge model of a! = 45" at Fd = 0.8: (-, ---), same as in Fig. 38. Froude's law of similarity is almost satisfied.

0.5

1.5

1.0

2.0

Fd

FIG. 40. Variation of the wave-front-line angle p, which systematically depends on the entrance angle a! and the Froude number based on draft Fd.

255

Nonlinear Ship Waves

-1.5L

FIG.41. Distribution of velocity component 1 + u / U on a vertical plane parallel to and a draft length away from the centerline for a wedge model of a = 20" at Fd = 1.1: (x) d = 0.1 m, U = 1.089 m/s; (0)d = 0.15 m, U = 1.334 m/s.

A

05-

O.O

-1.5

t

I

j4

00

04

FIG.42. Distribution of velocity component u/U on a vertical plane parallel to and a draft length away from the centerline for a wedge model of a = 20" at Fd = 1.1: (x, 0)same as in Fig. 41.

Hideaki Miyata and Takao Inui

256

-1.5

1

FIG. 43. Distribution of velocity component w/U on a vertical plane parallel to and a draft length away from the centerline for a wedge model of a = 20" at Fd = I . I : (x, 0)same as in Fig. 41.

T

0.5

FIG.44. Distribution of velocity component 1 + uiU on a vertical plane parallel to and twice the draft length away from the centerline for a wedge model of a = 20" at Fd = I. 1: (x, 0) same as in Fig. 41.

Nonlinear Ship Waves

257

I

0.5

-I 5'

FIG.45. Distribution of velocity component u/U on a vertical plane parallel to and twice the draft length away from the centerline for a wedge model of a! = 20" at Fd = 1. I: (x, 0) same as in Fig. 41.

T

05

FIG. 46. Distribution of velocity component w/U on a vertical plane parallel to and twice the draft length away from the centerline for a wedge model of a! = 20" at Fd = I . 1: (x, 0) same as in Fig. 41.

258

Hideaki Miyata and Taka0 Inui

FIG.47. (a) Horizontal and (b) vertical disturbance velocity vectors on the line parallel to the centerline and the disturbed free surface for the case of (Y = 20". Fd = I . I , and y l d = 1 .O: (-) 0. I H below and (---) I . 1H below the disturbed free surface.

flat. The water flow seems to travel from the wave front to the aft region so that the shear at the free surface, which is produced by the nonlinear wave motion, is released, involving turbulence on the free surface. Disturbance velocity vectors on horizontal and vertical planes are shown on three curved lines in Fig. 52. The horizontal disturbance velocity, which is gradually attenuated away from the wedge surface in the deeper region, as can be seen in Fig. 52c, is abruptly decreased at the wave front in the shallower region adjacent to the free surface, as seen in Fig. 52a.

.

J"

0.2u

FIG.48. (a) Horizontal and (b) vertical disturbance velocity vectors on the line parallel to the centerline and the disturbed free surface for the case of a = 20". Fd = I.I , and y / d = 2.0: (-) 0.1H below and (---I I.IH below the disturbed free surface.

Nonlinear Ship Waves

259

z'H E

0

'id

0.5

-QE

-1.c

FIG.49. Distribution of velocity component I + u / U on a vertical plane parallel to and a draft length away from the centerline for a wedge model of 01 = 45" at Fd = 1 .O: (x, 0)same as in Fig. 41.

The velocity head of uniform upstream is decomposed into a pressure head and a velocity head by the wave motion around the wedges, and the total head is conserved on the same streamline unless it is lost by discontinuous phenomena. In most cases, FSSW simultaneously involves discontinuous phenomena and energy deficit at the wave. front. The energy deficit, which is evaluated with measured velocity components and static pressure as a difference of the sum of pressure and velocity heads from

05

0

-05

FIG.50. Distribution of velocity component ulU on a vertical plane parallel to and a draft length away from the centerline for a wedge model of CY = 45" at Fd = 1.0: (x, 0)same as in Fig. 41.

260

Hideaki Miyata and Tukao Inui

-ao m -az o

-O

FIG. 51. Distribution of velocity component wlU on a vertical plane parallel to and a draft length away from the centerline for a wedge model of a = 45" at Fd = 1 .O.

w M A L PLANE

( a

1

Z-5-QIH

0

04

VERTICAL PLANE

FIG.52. Horizontal and vertical disturbance velocity vectors on three lines parallel to the centerline and the disturbed free surface (a) 0.1H below the free surface, (b) 0.15H below, and ( c )0.25H below for a wedge model of a = 45", d = 0.15 m, Fd = 1.0, and yld = 1 .O.

Nonlinear Ship Waves

26 1

the velocity head of uniform upstream, is shown in Fig. 53 for a wedge model of a = 20". The measured values on the curved line 0.1H below the disturbed free surface are used. Energy deficit occurs at the wave front and is reduced behind the wave front probably as a result of diffusive effect. The damped round wave profile at yld = 2.0 (Fig. 53b) is supposed to be a consequence of easing the steep slope by dissipation, and therefore, the amount of energy deficit is larger in comparison with the case at y l d = 1.0 (Fig. 53a). This damping phenomenon may be essentially the same as the wave breaking phenomenon.

0.4 0.2

-

0 .

Y-M

2

1

0.3

o.2L-

Q 9

0.1

M.

0

@

3

Q

X

S

0

( b )

FIG.53. Wave profiles and head loss at the crest of bow waves of a wedge model of a 20" at Fd = 1 . 1 , on the line of (a) y / d = I .O and (b) yld = 2.0.

=

262

Hideaki Miyata and Taka0 Inui

0

L o G O 5- O

-0.5

PI

I / ?p

u?

I

1

-

, a5 0 oi 05

o

y

o

10

nonFIG.54. Distribution of static pressure (--x--) and loss of velocity head (-G) dimensionalized by the velocity head of uniform stream for a wedge model of (I = 45", d = 0.15 m, Fd = 1.0, and y / d = 1.0 (under same conditions as Figs. 49-51).

The loss of velocity head defined as 1 - ( U + u2 + u2 + w 2 ) / U 2is calculated with measured velocity components u , u , and w ,and compared with the measured static pressure P, nondimensionalized by dpU2 in Fig. 54 for the case of a = 45". The hydrostatic pressure component pgz is not included in P , . The loss of velocity head (solid line with circle) should be compensated by the increase of static pressure (dotted line with an ex) except when discontinuous phenomena that cause energy deficit occur. The agreement of the two values is good before the wave front, and the difference suddenly becomes maximum on the forward face of the nonlinear wave. The difference implies that the nonlinear wave involves energy deficit near the free surface. Hence the energy deficit is gradually diffused and spread into the deeper region. It is noted that the depthwise distribution of static pressure is nearly vertical, which means that it is almost hydrostatic in the neighborhood of the free surface and is similar to the property of nonlinear shallow water waves. The detailed depthwise variation of measured static pressure is shown in Fig. 55. The measured values marked by circles are compared with an exponentially decreasing variation (solid curve) for which the wave number KO is assumed to be g / U 2 and the amplitude A is determined so that it agrees with the measured value on the free surface. The depthwise variation is similar to a hydrostatic one. This tendency is more evident in the case in which an intense circular (normal) nonlinear wave occurs as when a = 45". A free surface shock wave is supposed to have four developmental stages, namely, (1) formation of very steep nonlinear waves, (2) breaking or damping of the wave crest and occurrence of energy deficit, (3) diffusion of energy deficit with turbulence and sometimes air entrainment on

263

Nonlinear Ship Waves L'3'H

z-l/H

FREE WRFACE

0.4

0.2

OL

FREE SURFACE

0.2

0.3

05

0.6

e/',2PU' ( a )

( b l

FIG.55. Vertical distribution of static pressure: (a) a = 20", d = 0.15 m, Fd = 1.1, and yld = 1.0; (b) a = 45", d = 0.15 m, Fd = 1.0, and yld = 1.0: (-1 A e x p ( K o z ) 3 )(---) ; hydrostatic.

the free surface, and (4)formation of a momentum-deficient wake far behind. The most substantial feature of FSSWs is the generation of nonlinear steep waves in the near field of an advancing floating body. The wave height is often larger than if2 ( U 2 / 2 g ) ,and the maximum slope of the forward face sometimes exceeds 60". As a natural consequence, the flow velocity is considerably decelerated and the disturbance velocity due to the nonlinear wave motion is of the same order of magnitude as the speed of uniform stream. This wave formation is mostly governed by the Froude number based on draft and ship configuration, and the variation of the wave-front-line angle and the transition of a normal circular wave pattern to an oblique attached one are common to nonlinear shallow water waves and analogous with supersonic shock waves. A part of the wave energy of the nonlinear steep wave is dissipated at the wave front and transformed into momentum loss far behind the ship; on the other hand another part of the wave energy is likely to be supplied to the dispersive linear wave system that propagates to the far field. The nonlinear steepening of the nonlinear wave is partly compensated by dispersive spread and is partly eased by dissipation. Therefore, the waves of ships possess both dispersive and dissipative properties. The dissipation becomes dominant when the free surface is turbulent and breaking or damping of the wave crest is remarkably strong. For shallow-drafted full hull forms, such as oil tankers in ballast condition, the dispersion is almost negligible, but the wave energy in the near field is entirely dissipated. However, for ships with very fine hull forms, such as

264

Hideaki Miyata and Taka0 Inui

high-speed container carriers, the wave energy is mostly spread by dispersion. It is supposed that a kind of viscous effect plays a certain role in the process of wave damping and energy dissipation. The turbulence on the free surface is concerned with this dissipative behavior. The scale effect in the wave profile is attributed to the role of viscosity in dissipation. A kind of viscosity involving turbulence also diffuses the energy deficit produced at the wave front in the rear region.

IV. Modified Marker-and-Cell Method The characteristics of the nonlinear waves called free surface shock waves, as described in the preceding sections, are so complicated that it seems very difficult to explain them theoretically. Simplification of the governing equations is not acceptable, and the direct numerical method must be employed for the present problem. The modified marker-and-cell (MAC) method developed by Welch et al. (1966) is one of the most suitable solution methods for nonlinear free surface problems. It is a finitedifference technique for solving the time-dependent Eulerian or NavierStokes equations of incompressible hydrodynamics by special treatment of the free surface. Chan and Street (1970a,b) improved the MAC method and called the improved version the Stanford University modified markerand-cell (SUMMAC) method. The SUMMAC method was further modified, so as to make the analysis of nonlinear waves around wedge models possible by Masuko et al. (1982), Miyata et al. (1981), and Suzuki et al. (1981). The method is called the Tokyo University modified marker-andcell (TUMMAC) method. It is a three-dimensional version and can analyze water flows with free surface caused by advancing floating bodies in open water areas of infinite depth. For the time being the configuration of the floating body is restricted to wedge shapes, and the domain of computation is limited to the vicinity of the bow of wedge models. Thus the computations by the TUMMAC method are related to the phenomena described in the previous section.

A. COMPUTATIONAL PROCEDURE The algorithm of the MAC method, which is common to the TUMMAC method, is briefly described for convenience (see Fig. 56). The domain of computation is divided into rectangular cells whose length and width in x,

265

Nonlinear Ship Waves

I

PRESSURE CAL.

& NEW VELOCITY CAL.

J

I S H I P SURFACE CONDITION (WITH NEW VELOCITY) 1 i

* OUTER BOUNDARY CONDITION

4

1 MVEMENT OF MARKER PARTICLES

(KINEMATIC CONDITION)

1

D E F I N E FREE SURFACE SHAPE

FIG.56. Computational procedure of TUMMAC-I.

y , and z directions are DX, DY, and D Z , respectively, as seen in Figs. 57 and 58. The time is advanced by time increment DT. The primary dependent variables are the pressure and velocity components of the fluid. The pressure is specified at the center of each Eulerian cell, and a staggered mesh is used in the velocity component placement, which specifies the normal velocities at the Eulerian cell boundaries (see Fig. 57). The Navier-Stokes equations in conservative form are

-auat+ -

d(u2)

ax

+-d(uv) + -d(uw) --- aY

dZ

a+ ax

d(v2) a(uw) - _ _ a+ -avat+ - d(uu) ++-aY dZ ax

aw -+at

a(uw)

ax

a(w2) = +-a(vw) aY az +

a+

in which u , v , and w are velocity components in x , y , and z directions,

266

Hideaki Miyata and Takao Znui

I

x

FIG.57. Velocity points for the donor-cell method ( u component).

respectively; $ is a pressure divided by the density of fluid; and I, is a kinematic viscosity. By forward differencing in time and centered differencing in space for all terms except the convective terms, Eqs. (4.1) become

In the preceding equations subscripts are used for the cell location and superscripts for the time level. Variables with superscript (n + 1) are related to the ( n + 1)st time step, and those lacking a superscript are

Nonlinear Ship Waves

267

evaluated at the nth step. The convection terms are denoted by UC, V C , and WC whose expression is described later on. The expression for the velocity components at the ( n + 1)st time step is written in a compact form by combining all terms except the pressure gradient term and denoting them by 6, q , and &' as follows.

(4.3)

The divergence D at the ( n + 1)st time step is obtained from Eqs. (4.3) as

Hideaki Miyata and Takao Inui

268

Setting D = 0 is required to rigorously conserve mass, and it is aimed at the ( n + 1)st time step; that is, D$l in Eq. ( 4 . 4 ) is set at zero. Thus the equation for the pressure is derived as $..U k

=

1 2 [ ( 1 / D X 2 )+ ( 1 / D Y 2 )+ ( 1 / D Z ) 2 ] $i+ljh

(

+

$i-Ijk

DX2

+ $ i j + l k D+Y 2$ i j - l k

Since Rijk is determined when the velocity field is given, Eq. (4.5) is a Poisson equation for pressure. The momentum equations (4.3) and the Poisson equation (4.5) are the principal equations to be solved. Equations (4.3) are hyperbolic equations, which are solved as an initial value problem, and Eq. (4.5) is an elliptic equation, which is solved as a boundary value problem. The solution is advanced in time by a series of repeated steps. First the Poisson equation (4.5) is iteratively solved under given initial boundary conditions, and then new velocity components are derived from Eqs. (4.3). A new source term Rijk for the Poisson equation is calculated from the new velocity field, and the cycle is repeated. Marker particles are used to tell the new location of free surface. This solution algorithm is suitable for transient problems, although in this paper it is applied to a steady wave-making problem by letting a transient solution approach to a steady state. Two finite-difference representations of the convective terms of Navier-Stokes equations [UC,VC, and WC in Eq. ( 4 . 2 ) ] ,that is, centered differencing and second-order upstream differencing, are employed. The descriptions of UC by the two differencing methods are as follows, and the descriptions of VC and WC are abbreviated here.

Nonlinear Ship Waves

269

By centered differencing,

The second-order upstream differencing depends on the flow direction, and the velocity component u for the x-directional gradient, for example, is estimated at the midpoints of the velocity points (see Fig. 57). This differencing method is called the donor-cell method and is written as

where

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Hideaki Miyata and Taka0 Inui

The numerical stability conditions for the finite-difference equations are examined following Neumann's method [Roache (1976) and Fromm (1964)l.This method is valid for linear equations, and the difference equations are linearized; moreover, the pressure-gradient and the gravitational terms are ignored. Therefore, this stability analysis gives only approximate conditions. The derived stability condition for the case of centered differencing of the convective terms is 8 [ ( 1 / D X 2 )+ ( I / D Y 2 )+ ( 1 / D Z 2 ) ] Rn D T N D [ ( l / D X )+ ( I / D Y ) + ( 1 / D 2 ) ] 2' 2 2 D T N D . d 2 1( m + ~ 1 y z

(4.8)

where D T N D = DT U l d and R, = U * d / v . This gives lower and upper limits of the Reynolds number. The Courant condition, which implies that the fluid is not permitted to cross more than one cell in one time step, requires simultaneously the following condition that gives an upper limit to the nondimensional time increment: 7

DTND

L

5

(4.9)

[ ( l / D X )+ ( l / D Y ) + ( I / D Z ) ] d '

For the case of the first-order upstream differencing the stability condition requires the following limitations: (4.10) 1 2uDT(m

1 + ~ 1 y +2m) 5 1 - DT - + - + - . (d"X D Y DZ

"1

(4.11)

The former is the Courant condition and the latter gives an upper limit of kinematic viscosity. The model equation for the donor-cell method, which is used for the numerical stability analysis, is almost the same as that for the first-order upstream differencing, and, therefore, the approximate stability conditions for the donor-cell method is supposed to be nearly equivalent to Eqs. (4.10) and (4.11). The Poisson equation (4.5)is iteratively solved by the following equation: rJk

=

$5 + w(Jl;&

-

+%).

(4.12)

Nonlinear Ship Waves

27 1

The superscripts rn and (rn + 1) denote the iteration number, and w is a relaxation factor; qCa1 is the value calculated by Eq. (4.5). The iteration is continued until the difference in pressure between the (rn + 1)st and rnth steps converges within an allowable error. The successive overrelaxation (SOR) method is employed to solve the Poisson equation for pressure [Eq. (4.5)]. The new pressures successively obtained at the (rn + 1)st step are used, and the relaxation factor w is greater than unity. The TUMMAC method for water flows around wedge models in deep water is called TUMMAC-I so that it is distinguished from other versions of TUMMAC under development. The solution of the TUMMAC-I is remarkably influenced by the different differencing schemes for the convective terms. The TUMMAC-I that employs centered differencing for the convective terms is designated as TUMMAC-IB and the one that employs donor-cell differencing as TUMMAC-IC. Both employ the SOR method for the solution of the Poisson equation.

B. BOUNDARY CONDITIONS The kinematic free surface condition is satisfied by the movement of marker particles. The particle velocities are obtained from the u , u , and w fields by interpolation or extrapolation. For example, the velocity ux in Fig. 59b is calculated by the following interpolation formula:

- u1) + c / 2 * (u4 - u3) + f[2a2(u2+ u1 - 2u0) + 2c2(u4 + u3 - 2u0) + ac(us - u7 - & + us)].

ux = uo

+ a12

(u2

(4.13)

where a = h/DX, and c = 11DZ. The wave profile is defined by the new positions of the markers in the same manner as with the original MAC method, but the starting points of the marker particles are iteratively determined so that the new positions of marker particles are located straight above the center of the cells, as shown on the right in Fig. 59a. The dynamic free surface condition is satisfied by letting P = PO= 0 at the exact location of the free surface. “Irregular stars” of the SUMMAC method are employed and extended to the three-dimensional case. In the two-dimensional case the pressure Pik in Fig. 59c is obtained by the following equation:

272

Hideaki Miyata and Takao Inui

-DX

t DZ

+

ul*

UL 0

Po

U8 0

ux / - 4

v

4 u5 0

uo

4

u3 0

U P

*

U8 0

( b)

(C)

FIG.59. Free surface condition: (a) movement of markers; (b) velocity interpolation; (c) irregular star.

for the three-dimensional case it becomes

(4.15)

In Eq. (4.15) P on the right-hand side is set at zero for the cells on the free surface. The viscous contribution to the free surface condition is omitted. On the surface of the floating body a free slip condition is imposed, and the viscous effect on the body surface is ignored. The length and width of the cells, DX and DY,are determined so that the waterlines of the body bisect the horizontal sides of the cells, as shown in Fig. 60. In order to satisfy the body surface condition, as well as the zero-divergence condition for the boundary cells, the unknown velocity components 143 and u4 in Fig. 60 are determined by the following equations.

u3 =

M I cos

2a

+ u I sin 2a,

u4 = (u2 cos a

+ u2 sin a ) cos

a. (4.16)

The depth of the computational domain is two or three times the draft of the floating body, and at the bottom boundary the previously calculated

Nonlinear Ship Waves

273

7 FIG.60. Body surface condition: (0) u known; (A)u known: ( 0 )U unknown.

velocities of the double model flow is given so that they are smoothly connected to the computed velocity field. It is well known that the open-boundary condition can very sensitively affect the solution, and, therefore, the reflection of waves from open boundaries must be avoided. A number of open-boundary conditions were tested and finally the condition illustrated in Fig. 61 was chosen. The velocity gradient along the local stream line at the open boundary is set at zero. The unknown u on the boundary ( 0 )is obtained by linearly interpowhich is set at the same value with the upstream u (0). lating the u (0) For the v field the same procedure is followed. The initial condition of computation is at rest; that is, the flow velocities are set at zero, and the hydrostatic pressures are given at the centers of the cells. Then the inflow velocity is gradually increased at constant acceleration until it coincides with the assumed speed of advance of floating bodies. About 100 time steps are required to accelerate the flow velocities, and after the acceleration stage the computation is continued until the wave formation reaches the steady state.

FIG. 61. Open-boundary condition: (---) local stream: (0) u known: ( A ) u known: (0) u extrapolated; (0)u unknown.

s

2

6

a

ffideakiMiyata and Taka0 Inui

274

V. Computed Waves around Wedge Models Two versions of TUMMAC-I are used for the computation of waves around wedge models, TUMMAC-IB and TUMMAC-IC. The difference between the two is that the former employs centered differencing for the convection terms and the latter donor-cell differencing. Two wedge models whose entrance angles a are 20" and 45" are chosen for the computation. The length and depth of the wedge model of a = 20" are 720 and 100 mm, respectively, and the cell dimensions D X , DY, and DZ are 36, 13, and 25 mm, respectively. The number of cells is 32 x 40 X 12. The values for the wedge model of a = 45" are 400 and 100 mm; DX, DY, and DZ = 25, 25, and 25 mm; and 37 x 27 x 14. For each model computations are conducted at three Froude numbers based on draft Fd. The parameters used for the computations by the TUMMAC-IC method are listed in Table 11. The computation by the TUMMAC-IB method is conducted under one condition of a = 20" and Fd = 1.1. The TUMMACIB method necessitates the introduction of artificial viscosity to stabilize the solution. The Reynolds number based on the draft of the wedge is 10, and this value is close to the lower limit of the limitation on the Reynolds number given by Eq. (4.8). The computed results by the TUMMAC-IC method for the wedge of a! = 20" are shown in Figs. 62-64. The computed waves do not reach the steady state at the two-hundredth time step, while the acceleration is stopped at the one-hundredth time step. Then, at the three-hundredth time step the waves reach the steady state. The variation of wave-height contours and perspective views at three speeds of advance are presented in Figs. 63 and 64.The contours are drawn at an interval of 10% of H ( U2/ 2g). The experimental results show that the foremost wave is roundshaped at low speeds of advance and that it is transformed into straightTABLE I1 CONDITIONS OF COMPUTATIONS

U

DT

a

Fd

(rnlsec)

(set)

20"

0.8

0.792 1.089 1.386

0.00631 0.00459 0.00361

0.594 0.792 0,990

0.00421 0.00631 0.00505

1.1 1.4

45"

0.6 0.8

1 .o

DTND 0.05

V

w

0

I .5

0

I .5

0.025 0.05

FIG.62. Time sequence of the wave pattern around a wedge model of (Y = 20", d = 0.10 m. at Fd = 1 . 1 , computed by TUMMAC-IC. The upstream velocity is accelerated up to the one-hundredth step and the waves reach a steady state at the three-hundredth step: (a) 200 steps, 0.72 sec; (b) 300 steps, 1.08 sec; (c) 400 steps, 1.44 sec.

-4

-3

-2

-1

0

1

2

3

4

5

6

7

-4

-3

-2

-I

0

I

2

3

4

5

6

7

-4

-3

-2

-I

0

I

2

3

4

5

6

7

FIG.63. Wave-height contours computed by TUMMAC-IC for a wedge model of a = 20" at three speeds of advance: (a) Fd = 0.8; (b) Fd = 1 . 1 : (c) Fd = 1.4. Wave height and coordinates are nondimensionalized with respect to H and d . 276

FIG.64. Perspective views of waves around a wedge model of (Y = 20" computed by TUMMAC-IC at three speeds of advance: (a) Fd = 0.8; (b) Fd = 1.1 ; (c) Fd = 1.4. Wave height is nondimensionalized by H . 277

278

Hideaki Miyata and Takao Inui

lined ones at the Froude number (Fd) greater than 0.95. Hence the angle of the foremost wave-crest line to the centerline is decreased with the increase of Fd. The computed waves also illustrate this qualitative variation. Comparing these contour maps with experimental ones, we note that the computed wave height is about 70% of the measured wave height, and that the phase of the waves is shifted slightly backward. These are presumably due to numerical dissipation and phase error caused by the numerical scheme. The computed results by the TUMMAC-IC method for a wedge of (Y = 45" are presented in Figs. 65-67. In this case it is hard to have a very steady solution, and the waves are fluctuating even at the four-hundredth time step as can be seen from Fig. 65, which is quite different from the case of (Y = 20". The foremost wave continues to be round-shaped in a wide range of advance speeds and is not transformed into straight-lined waves; that is, the foremost nonlinear wave is almost always a normal FSSW, as illustrated in Figs. 66 and 67, which show nearly steady states of the computed solution. It is indicated that the normal FSSW around the bow is enlarged and the wave slope on the forward face becomes steep with the increase of Fd, which may cause breaking of the wave crest and unsteady fluctuation of the free surface at high speeds of advance. The wave height in these figures is nondimensionalized by the reference length of H , and, therefore, the computed wave height is nearly invariant; that is, the maximum wave height is from 70 to 90% of H . In Fig. 68 the wave profiles computed by the TUMMAC-IB and -1C methods are compared with the measured wave profiles. The maximum wave height by the TUMMAC-IB method agrees well with the measured, whereas that by the TUMMAC-IC method is only about 70% of the measured. The difference between the computed and the measured wave heights is large at a distance from the body and in the region behind the wave crest, where the physical dissipation plays a certain role involving turbulence on the free surface. The same comparison is present in Fig. 69 in the form of a contour map, in which the wave height is indicated at the interval of O.1H. The difference of the wave phase among the three is most evident on the wedge surface. The pressures at 180 static pressure holes on the surface of the wedge model of (Y = 20" were measured at three speeds of advance, Fd = 0.8, 1.1, and 1.4, and the results are illustrated in the form of contour maps of the pressure coefficient nondimensionalized with respect to apU2 in Fig. 70. At the forward end the pressure reaches the stagnation pressure on the vertical line below the undisturbed water line ( z = 0), but not above the line of z = 0. The water flow in the vicinity of the forward end above the undisturbed free surface is supposed to have complicated properties such

FIG. 65. Time sequence of the wave pattern around a wedge model of a = 45", d = 0.10 m, at Fd = 0.8, computed by TUMMAC-IC. The upstream velocity is accelerated up to the one-hundredth step and the waves continue to oscillate at the three-hundredth and fourhundredth steps: (a) 200 steps, 1.01 sec; (b) 300 steps, 1.52 sec; (c) 400 steps 2.02 sec.

-5

-4

-3

-2

-1

0

1

2

3

-5

-4

.3

-2

-1

0

1

2

3

FIG.66. Wave-height contours computed by TUMMAC-IC for a wedge model of a = 45" at three speeds of advance: (a) Fd = 0.6; (b) Fd = 0.8; (c) Fd = 1.0. Wave height and coordinates are nondimensionalized with respect to H and d . 280

FIG.67. Perspective views of waves around a wedge model of a = 45” computed by TUMMAC-IC at three speeds of advance: (a) Fd = 0.6; (b) Fd = 0.8; (c) Fd = 1.0. Wave height is nondimensionalized by H . 28 1

282

Hideaki Miyata and Takao Inui

FIG.68. Comparison of wave profiles on two longitudinal lines, y l d wedge model of a = 20"and d = 0.10 m at Fd = I . I : (-) measured; (---) (---) TUMMAC-IC.

=

I .O and 2.0. A TUMMAC-IB;

as that of a vortical flow. The increasing speed of advance lengthens the wavelength on the wedge surface, and, as a consequence, the region of high pressure is extended behind. The contour maps of pressure distribution on the wedge surface computed by the TUMMAC-IC method are shown in Fig. 71 under the same condition as for Fig. 70. The steep variation of pressure at the fore end of the wedge indicated by the intimate contour lines in Fig. 70 cannot be shown in Fig. 71 because of the coarseness of the cell division. As already mentioned, the solution of the TUMMAC-IC method suffers from considerable numerical dissipation, and the computed wave height is, on the whole, smaller than the measured. This consequently leads to smaller pressure coefficients computed on the wedge surface, as seen in Fig. 71, although the qualitative change of pressure distribution due to the increase in Froude number is explained by the computed results by the TUMMAC-IC method. At Fd = 1 . 1 of the wedge model of a = 20°, the pressure distributions are compared between two computed results and the measured in Fig. 72. As the computation by the TUMMAC-IB method scarcely involves numerical dissipation, the computed pressure due to wave motion rises close to the measured value. However, the rather large difference of phase somewhat invalidates the usefulness of the TUMMAC-IB method. The computation of water flows around steadily advancing wedges by the TUMMAC-I method provides a successful qualitative explanation of

283

Nonlinear Ship Waves

- 2.0

-1.0

0

1.0

20

3.0

4.0

X/d

Yf d

FIG.69. Comparison of wave-height contours of a wedge model of (Y = 20" at Fd = 1 . 1 . (a) Measured; (b) computed by TUMMAC-IB; (c) computed by TUMMAC-IC.

the steep wave formation around bows. However, further efforts are needed to diminish numerical dissipation and phase error. Only after this kind of improvement of the present numerical scheme is achieved, can we hope that the foremost wave formation will be thoroughly explained. Nevertheless, the present numerical analysis can be employed only to explain the generation of steep nonlinear waves, namely, the first stage of

Hideaki Miyata and Takao Inui

284

(b)

-

- 0.1

0

0

I

I

0.1

0.2

0.3

0.1

0.2

0.3

I

x(m)

0.4

X(m)

0.4

(EP)

0-

7 fm\

- 0.1

I

I

0 (F. I?)

FIG.70. Measured pressure distributions on the surface of a wedge of a = 20" at three speeds of advance: (a) Fd = 0.8; (b) Fd = 1 . 1 ; (c) Fd = 1.4. Numbers on the contour lines are pressure coefficients.

Nonlinear Ship Waves

285

(a)

0

Z (m)

I

-0.1

0

0.1

0

0.1

(EP)

0.2

X(ml

0.3

0.4

(b)

0-

Z (ml

-0.1

0.4

".l

0 (El?)

0.1

0.2

X(ml

0.3

0.4

FIG. 71. Pressure distribution on the surface of a wedge of a = 20" computed by TUMMAC-IC: (a) Fd = 0.8; (b) Fd = 1 . 1 ; (c) Fd = 1.4.

286

Hideaki Miyata and Taka0 Inui

-0.1

0

'

0.1

0.2

X(m)

0.3

0

0.1

0.2

X(m)

0.3

0

0.1

0.2

x(m)

0.3

I

1

I

(F.f?)

FIG. 72. Comparison of pressure distribution on the surface of a wedge of a = 20" at Fd = I . I . (a) Measured: (b) computed by TUMMAC-IB; (c) computed by TUMMAC-IC.

the complicated time developing process of nonlinear wave motion described in Section 111. The very complicated phenomena of the occurrence of energy deficit and the following stages, that is, the processes of dissipation and diffusion with turbulence, cannot be solved by the present method or similar ones. Some great intuition will be necessary.

Nonlinear Ship Waves

287

VI. Concluding Remarks The investigations concerned with the nonlinear waves called free surface shock waves have been extended to three categories: (1) experimental investigations into the hydrodynamic structure, (2) numerical analysis by direct computation of the Navier-Stokes equations, and (3) applications to hull form design. The most substantial is the first approach, and it provides a basis for the other two. The second is a straightforward approach to the theoretical explanation and at present is under development, having succeeded in qualitative explanation. The studies that belong to the third category by Nito et al. (1981) and Miyata et al. (1982b) are not described in this article, although they are most important to naval architects. The methods of hull form improvement are now empirical, and their availability is limited by their own assumptions. In the future the numerical method will be improved and can be applied to hull form design. For hull form design or the optimization of hull forms, quantitative accordance with experimental results is not always necessary and qualitative accuracy can be very useful. ACKNOWLEDGMENTS The authors are indebted to Professor T. Y. Wu of the California Institute of Technology who has kindly advised and encouraged them to write this article. They also wish to express special gratitude t o the colleagues and graduate students who worked hard on the nonlinear ship wave problem and to N. Takiura who has carefully typewritten the manuscript.

REFERENCES Baba, E. (1969). A new component of viscous resistance of ships. J . Soc. Nau. Archit. J p n . 125,9-34. Baba, E. (1975). Blunt bow forms and wave breaking. STAR-Alpha Syrnp. SOC.Nav. Archit. Mar. Eng., New York. Baba, E. (1976). Wave breaking resistance of ships. Proc. I n f . Sernin. W a w Resistance. pp. 75-92. Baba, E., and Takekuma, K. (1975). A study on free surface flow around bow of slowly moving full forms. J . Soc. Nau. Archif. J p n . 137, 1-10. Chan, R. K.-C., and Street, R. L. (1970a). SUMMAC-a numerical model for water waves, Technical Report 135. Dept. Civil Eng., Stanford Univ., Stanford, California. Chan, R. K.-C., and Street, R. L. (1970b). A computer study of finite amplitude water waves. J. Compu. Phy. 6, 68-94. Eggers, K. (1981). Non-Kelvin dispersive waves around non-slender ships. Schiffsfechnik 28, 223-252. Fromm, J. E. (1964). Time dependent flow of an incompressible viscous flow. Mefh. Compu. Phy. 3, 345-386.

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Hideaki Miyata and Takao Inui

Inui, T . (1962). Wave-making resistance of ships. So(,. N u u . Archit. Mur. En#. Truns. 70, 283-353. Inui. T.. Kajitani, H., and Miyata, H. (1979a). Experimental investigations on the wave making in the near-field of ships. Kunsui Soc. Nuu. Archif. J p n . 173, 95-107. h i , T., Kajitani, H., Miyata, H., Tsuruoka, M., Suzuki, A.. and Ushio, T. (1979b). Nonlinear properties of wave making resistance of wide-beam ships. J. Soc. Ncru. Archit. J p n . 146, 19-27. Kawamura, H., Kajitani. H., Miyata, H., and Tsuchiya, Y. (1980). Experimental investigation on the resistance component due to free surface shock waves on series ship models. Kansai Soc. Nau. Archit. J p n . 179, 45-55. Kayo, Y., and Takekuma, K. (1981). On the free surface shear flow related to bow wavebreaking of full ship models. J. Soc. Nau. Archit. J p n . 149, 11-20. Masuko, A., Miyata, H., and Kajitani, H. (1982). Numerical analysis of free surface shock waves around bow by modified MAC-method (2nd report). J . Soc. Nuu. Archit. J p n . 152, 1-12. Matsui, M., Tsuda, T., Ohkubo, K., and Asano, S. (1980). A method for optimization of ship hull forms based on wave-pattern analysis data. J . Soc. Nuu. Archit. J p n . 147, 10-19.. Miyata. H. (1980). Characteristics of nonlinear waves in the near-field of ships and their effects on resistance. Proc. 13th Symp. Nauul Hydrodynumics, 335-351. Shipbuilding Research Association of Japan. Miyata, H., Inui, T., and Kajitani, H. (1980). Free surface shock waves around ships and their effects on ship resistance. J. Sor. N a v . Archif. J p n . 147, 1-9. Miyata, H., Suzuki, A., and Kajitani, H. (1981). Numerical explanation of nonlinear nondispersive waves around bow. Proc. 3rd Int. Conf. Numericnl Ship Hydrodynumics, pp. 37-52. Miyata, H., Masuko, A., Kajitani, H., and Aoki. K. (1982a). Characteristics of free surface shock waves around wedge models (2nd report). J. Soe. Nuu. Archif. J p n . 151, 1-14. Miyata, H., Kajitani, H., Nito, M.. Aoki, K., Nagahama. M., and Tsuchiya, Y. (1982b). Free surface shock waves and methods for hull form improvement (2nd report). J. Soc. Nuu. Archit. Jpn. 152, 13-21. Nito, M., Kajitani, H., Miyata, H., and Tsuchiya, Y . (1981). Free surface shock waves and methods for hull form improvement (1st report). J. Soc. Nuu. Archif. Jpn. 150, 19-29. Roache, P. J. (1976). "Computational Fluid Dynamics." Hermosa. Albuquerque, New Mexico. Suzuki, A., Miyata, H., Kajitani, H., and Kanai, M. (1981). Numerical analysis of free surface shock waves around bow by modified MAC-method (1st report). J . Soc. Nuu. Archit. Jpn. 150, 1-8. Takahashi, M., Kajitani, H., Miyata, H., and Kanai, M. (1980). Characteristics of free surface shock waves around wedge models. J. Soc. Nou. Archit. Jpn. 148, 1-9. Takekuma, K. (1972). Study on the non-linear free surface problem around bow. J. Soc. N a v . Archit. J p n . 132, 1-9. Taneda, S. (1974). Necklace vortices. J. Phvs. Soc. Jpn. 36-1, 288-303. Taneda, S., and Amamoto, H. (1969). Necklace vortex around bow. Bulletin No. 31 Research Inst. Appl. Mech., Kyushu Univ., Japan pp. 17-28. Tsutsumi, T . (1978). An application of wave resistance theory to hull form design. J. Soc. Nau. Archit. J p n . 144, 1-10. Welch, J. E., Harlow, F. H., Shannon, J. P., and Daly, B. J. (1966). The MAC method. Los Alamos Scientific Lab. Report, LA-3425. Univ. California, Los Alamos. New Mexico.

ADVANCES I N APPLIED MECHANICS, VOLUME

24

The Mechanics of Rapid Granular Flows STUART B. SAVAGE Department of Civil Engineering and Applied Mechanics McGill University Montreal, Quebec

And you may scoop up poppy seed as easily As water, which will also, if you spill it, Glide away with as ready a downward flow. -Lucretius, De Rerum Natura [R. C. Trevelyan, tr. (1937)]

I. Introduction. . . . . . 11.

. .. . . . . . ... .... .. . . . .. . . . ... . . .. . . . .. . Plan of This Article. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preliminary Discussion of Some Granular Flow Regimes . . . . . . . . . . . . . .

A. Review of Bagnold’s Papers on Fluid-Solid Mixtures B. Modes of Flow for Dry Cohesionless Gra C. Summary of Limiting Flow Regimes . . . 111. Flows in Vertical Channels and Inclined Chutes . . . . . . . . . . . . . . . . . . . . A. Vertical Pipes and Channels. . . .................. B. Experimental Observations of C. Flows around Obstacles. . . . D. Granular Jumps . . . . , . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . ....... IV. Rheological Test Devices and Experiments A. Quasi-Static Shear Devices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... B. Dense Suspensions . . . . . . . . . . . . . . . . . . . . C. High Shear-Rate Devices for Dry Materials. . . . . . . . . . . . . . . . . . D. Some Remarks Concerning High Shear-Rate Viscometric Experiments . .

V. Theories for Rapid Granular Flows. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... ... A. Continuum Theories . . . . . . B. Microstructural Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Numerical Modeling . . . . . . . . . . . . . . . . V1. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References. . .. . . ... .. ..... .............. I

290 292 292 293 296 302 302 304 308 320 320 321 322 324 326 333 335 336 343 356 358 359

289 Copyright 0 1984 by Academic Press, Lnc. All rights of reproduction in any form reserved. ISBN 0-12-002024-6

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I. Introduction A “bulk solid” or “granular fluid” may be defined as an assembly of discrete solid components dispersed in a fluid such that the solid constituents are in contact or near contact with their neighbors. Bulk solids comprise one member of a larger class of two-phase disperse systems made up of solids and fluids; dilute suspensions form another related member, which is perhaps more familiar to fluid mechanicists (Cox and Mason, 1971; Batchelor, 1974; Herczynski and Pienkowska, 1980; Leal, 1980; Russel, 1981). Although the fluid phase plays a major role in determining the dynamics of dilute suspensions, it has relatively less influence on bulk solids behavior. For flowing bulk solids, the solids fraction v (volume of solids per unit bulk volume) typically is between 0.3 and 0.6, and thus it can approach the value corresponding to the densest-possible packing of the particles. Direct interactions between the individual solid constituents are frequent; the bulk behavior is governed largely by interparticle forces, friction, and collisions. In some instances the effects of the interactions between the fluid and solid components may ba small because the interstitial fluid has relatively small density and viscosity, as in the case of a gas. In general, the material behavior is very complex, and its understanding requires the melding of ideas from traditional fluid mechanics, plasticity theory, soil mechanics, rheology, and kinetic gas theory. Some examples of bulk solids are mineral concentrate, ore, coal, sand, crushed oil shale, grains, cereals, animal feed, granular snow, pack ice, powders, and pharmaceutical pills. Information about the mechanics of the flow of these kinds of materials is essential for the understanding and solution of a wide range of technological and scientific problems related to materials-handling engineering, pneumatic transport, flows in slurry pipelines, mineral and powder processing, fluidized bed combustion of coal and wastes, stability of tailings dumps, flows in pebble bed nuclear reactors, rock falls, debris flows, subaqueous grain flows, snow avalanches, ice jams and drift of pack ice, sediment transport in rivers, dynamics of planetary rings, etc. Probably the earliest application of granular flows was the hourglass or sand clock. These devices were in common use by the end of the thirteenth century for the measurement of the speed of ships and were used during the Middle Ages by scholars to regulate the routine of their studies and by the clergy to time their sermons (Balmer, 1978). Although some early investigations of granular materials were conducted by Hagen (1852), who studied granular flow through apertures (as in the hourglass), and by Reynolds (1885), who formulated the idea of dilatancy (the expan-

The Mechanics of Rapid Granular Flows

29 1

sion of a closely packed assemblage of particles when the bulk is deformed), little further work was done until more recent times. Extensive research has been performed on very slow granular flows in the context of soil mechanics, in which the inertia effects associated with both the individual grain interactions and the bulk deformation are negligible. Similarly, two-phase solid-fluid flows in which the fluid effects are significant-for example, as in fluidized beds, dilute suspensions, and sediment transport in rivers-have received wide attention. The present review deals with work that falls somewhere between these two kinds of flow regimes, the rate-independent and the fluid-dominated. It will focus on detailed studies of the mechanics of bulk solids undergoing deformations rapid enough for fluidlike behavior to be exhibited, but in which the interstitial fluid component plays a subsidiary role in the dynamics. One of the central problems is the determination of the constitutive equations to describe the fluxes of mass, momentum, and energy. Information about the relationship between stresses and strain rates, the detailed mechanisms that control the development of stresses, diffusion, heat transfer, etc., and formulations of these mechanisms in continuum terms would enable one to calculate the bulk behavior in various flows. These kinds of things are the main concern of the present article. Granular flows have been the subject of a number of reviews. Mroz (1980) and Spencer (1981) have dealt with relatively low strain rates where grain inertia effects are negligible. Spencer (1981) concentrated on the “double-shearing’’ plasticity theory models of deformation that admit simultaneous shearing along both families of the stress characteristics (Mandel, 1966; de Josselin de Jong, 1959, 1971, 1977; Spencer, 1964; Zagaynov, 1967). Mroz (1980) attempted to provide a unified view of various analyses of both the continuum and particulate kind by showing how they could be encompassed by plasticity theory, which accounts for hardening and softening behavior. Brown and Richards (1970) deal primarily with material properties, statics, and materials-handling applications of flow in bins and hoppers. Weighardt (1975) discussed the drag forces on bodies that are moved through granular materials and the discharge of particulate materials from bins. Nedderman et al. (1982) and Tuzun et al. (1982) have provided extensive reviews of both theoretical and experimental work on granular flows in bins and hoppers, dealing with flow rates, velocity distributions, and techniques for measurement of the velocity and bulk density. Jenkins and Cowin (1979) have described the various continuum theories for rapidly flowing granular materials. The subject matter and orientation of this last review are closest to that of the present paper. A discussion of granular flows in a geophysical context is beyond the

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scope of the present article. Nevertheless, the subject is a fascinating one, and the reader is directed to recent reviews and papers dealing with the following subjects: snow avalanches (Mellor, 1978; Hopfinger, 1983), rockfalls (Hsu, 1975; Erismann, 1979), subaerial debris flows (Takahasi, 1981), submarine debris flows (Middleton and Hampton, 1976), motion of pack ice (Rothrock, 1975; Sodhi, 1977), and sediment transport (Komar, 1976; Raudkivi, 1976). Most of these works point out the importance of the fundamental studies of Bagnold (1954, 1956, 1966). Even a brief perusal of the reviews just mentioned will suggest numerous ways in which the work described in the present review can be applied toward a better understanding of these geophysical flows.

PLANOF THISARTICLE We begin Section I1 with a review of the classical papers of Bagnold and a discussion of the various modes and regimes of granular flow. A dimensional analysis provides a physical background to the detailed review of experimental and theoretical work that follows. Section 111 reviews flows in vertical channels and inclined chutes; it deals primarily with experimental observations of stress, velocity, and bulk-density fields. Laboratory devices and viscometric type experiments designed to determine the stress-strain-rate behavior are discussed in Section IV. Quasi-static testers, suspension viscometers, and high-shear-rate devices are described. Section V reviews theories for high-shear-rate granular flows. It includes continuum models as well as analytical and numerical microstructural models that consider the details of collisions between particles.

11. Preliminary Discussion of Some Granular Flow Regimes Granular flows can be further classified in terms of different flow regimes, each having distinct characteristics. Although the main emphasis of this review is on the rapid flow regime, in which the effects of particle grain inertia are dominant, it is important for several reasons to point out the kinds of flow behavior that exist in other flow regimes. For example, in some cases of granular flow, portions of the granular material may be “locked” together and nearly rigid; the mechanics of deformation of these portions is different from those where grain inertia is important. In other cases, interstitial fluid viscosity and density may be important.

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Some appreciation of these effects is necessary to understand better the experimental results and t o recognize the limitations of the theoretical treatments in the papers reviewed subsequently. This section first reviews Bagnold’s pioneering work on granular flow and in particular examines his classifications for suspensions of solids in fluids. Further classifications of the flow of dry granular materials are then discussed.

A. REVIEWOF BAGNOLD’S PAPERSON FLUID-SOLID MIXTURES The present level of understanding of granular flows, which are influenced by grain inertia effects, owes much to the now-classical papers of Bagnold (1954, 1956, 1966). His work was motivated by an interest in the mechanics of phenomena such as the bed load transport of sediment in rivers. Bagnold (1954) performed experiments on neutrally buoyant, uniformly sized spherical wax beads suspended in Newtonian fluids (water and glycerine-water-alcohol mixture) and sheared in a coaxial rotating cylinder apparatus. By using flexible rubber to form his inner cylinder wall he could measure both the torque and the normal stress in the radial direction when various concentrations of grains were sheared by rotating the rigid outer cylinder. Bagnold distinguished three different regimes of flow behavior, which he termed macroviscous, transitional, and grain inertia. Classification of a particular shear flow, u,(x2)(Fig. 1) depends on the value of a dimensionless shear group [subsequently named by Hill (1966) as the “Bagnold number”] N

=

A”’pf(~’(~1,2)/p,

(1)

where pf and p are the mass density and viscosity of the interstitial fluid,

t

I 00-03

1 FIG.I .

000 ik-s

;I

Definition sketch for Bagnold’s (1954) analysis of sheared granular material.

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a is the particle diameter, u1,2is the velocity gradient, and A is the “linear concentration” of particles. If the mean distance between particle centers is defined as ba and the resulting free distance is s (Fig. I ) then

b

=

1

+ s/a = 1 +

l/A,

(2) defining A as equal to a / s . Note that the solids fraction v may be expressed in terms of A as v

=

v,/b3

=

d ( 1

+

l/A)3,

(3)

where vm is the maximum possible solids concentration when A = ~0 [for uniform spheres vw = r / ( 3 ~=)0.74051. Bagnold derived simple analyses to explain the rheological behavior in the two limiting regimes, the macroviscous and the grain-inertia regimes. In his macroviscous regime, corresponding to small N , viscosity is dominant and the shear and normal stresses are linear functions of the velocity gradient ~ 1 . 2 Bagnold . attributed the presence of the normal stress in the radial direction (which he called a “dispersive” pressure) to a statistically preferred anisotropy in the spatial particle distributions, as sketched in Fig. 1. More pertinent to the present review is his grain-inertia regime, corresponding to large values of N , in which the interstitial fluid plays a minor role and the major effects are due to particle-particle interactions. Bagnold argued that the main mechanism for momentum transfer is the succession of glancing collisions as the grains of one layer overtake those of the adjacent slower layer (Fig. 1). Both the change in momentum during a single collision and the rate at which collisions occur are proportional to the relative velocity of the two layers, giving rise to stresses that depend on the square of the shear rate. Sanders (1963) called this process the “Bagnold effect.” Through an analysis that was roughly analogous to a simple kinetic gas theory, he deduced that the normal stress in the 2direction was and that the grain shear stress was p12 = ~

2 tan 2 +D

,

(5)

where pp is the mass density of the individual particles, f is an unknown function of A, a is a constant, and +D is an unknown dynamic friction angle dependent upon collision conditions. Connecting the two limiting flow regimes was Bagnold’s transitional flow, in which the dependence of the stresses on shear rate varied from a linear one corresponding to the macroviscous regime to a square dependence predicted fGr the graininertia flow regime.

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From his experiments, Bagnold was able to define the various flow regimes in terms of N ; N < 40 corresponded to the macroviscous regime, N > 450 corresponded to the grain-inertia regime, and the transitional region occupied the intermediate range of N . The limiting experimental flow behavior was consistent with his predictions for the macroviscous and grain-inertia regimes. In the grain-inertia regime, stresses varied with the square of the shear rate, as in (4) and (5). For A < 14, f = A , and for A > 14, fincreased very rapidly with A. Large A corresponds to grains being close together; if A is large enough the grains lock together and very large stresses are required to shear the bulk. The angle 4~ was found to vary only slightly with A. One interesting feature common to all flow regimes was the presence of a substantial radial normal "grain" stress that was proportional to (and greater than) the shear stress. Such behavior is reminiscent of the Coulomb yield or failure criterion used to describe the stresses in certain soils under conditions of limiting equilibrium (Sokolovski, 1965; Schofield and Wroth, 1968). Yielding is influenced by the hydrostatic pressure in the Coulomb failure criterion, which states that yielding will occur at a point < on a plane element when (SI = c

+ P tan 4,

where S and P are, respectively, the shear stress and normal stress acting on the element, c is the cohesion, and 4 is the internal angle of friction of the bulk material. For dry, coarse materials, the cohesion c is negligible, and (6) takes on the same form as (5). Typical values for 4 obtained during quasi-static yielding at low stress levels are close to the angles of repose, i.e., about 24" for spherical glass beads and 38" for angular sand grains (Brown and Richards, 1970). It is interesting to compare the quasi-static 4 for typical spherical particles with the dynamic friction angle C # J ~obtained in Bagnold's experiments with spherical wax beads sheared in the graininertia regime. For A > 12 (v > 0.58) he found that 4D= 22", and for A < 12, +D tends to approach an angle of about 18" [see the discussion in Bagnold (1973)l. Bagnold (1954) applied his cylindrical shear cell results and his analyses for the stresses to study the problems of gravity flow of particulate matter down inclines as might occur in rock falls and debris flows. His predictions for the surface velocity in such a flow in the grain-inertia regime were about 50% higher than measurements he made of the velocity of quartz sand flowing down a simple flume. He also investigated the flow of gravel, whose interstices were filled with mud of slightly lower mass density; the predicted flow behavior was compared with some rough field observations of Professor K. Terzaghi. The grain-inertia analysis was

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used to explain the sorting of grains of mixed sizes that may be observed in bed load sediment transport and submarine grain flows (Middleton and Hampton, 1976) and is used to advantage in mineral processing. In this type of sorting, called “inverse grading,” the large particles drift toward the surface and the finer particles toward the bed. However, Bagnold’s (1954) explanation for the grading process is not above criticism, and Middleton (1970) [see Middleton and Hampton (1976)l has suggested an alternative mechanism. Bagnold further applied the results of his 1954 paper to study the bulldozing of a mass of dry sand and the “singing” mechanism in desert sand dunes (Bagnold, 1966) and to problems of bed load sediment transport rivers (Bagnold, 1956).

B. MODESOF FLOWFOR DRYCOHESIONLESS GRANULAR MATERIALS The work of Bagnold just discussed considered the various regimes of flow that occur in a fluid-solid mixture when the Bagnold number N is varied over a wide range. It happens that there are regions in “granular flow regime space” in addition to those noted by Bagnold. For example, in some flows of dry (gaseous interstitial fluid), coarse, particulate solids, the fluid phase can be neglected for essentially all values of the shear rate. Here, the Bagnold number alone is insufficient to characterize the types of flow behavior that are possible. In one limit, for large shear rates, we do expect the grain-inertia type of flow corresponding to large N . But for very small shear rates and high concentrations of coarse solids, one expects the kind of behavior characteristic of the quasi-static deformation of a cohesionless soil and not the behavior associated with Bagnold’s macroviscous flow. A transitional region joining the quasi-static type of flow to the grain-inertia flow must exist. In this section we shall examine these flow regimes in more detail using dimensional analysis and discuss in physical terms the various modes of flow. The discussion is a somewhat more general version of that given by Sayed (1981). For simplicity, electrostatic forces as well as the interstitial fluid phase will be neglected, although clearly there are situations where these effects are of considerable importance.

1 . Dimensional Analysis During the deformation of a bulk or dry granular material, mean stresses may be generated by a number of different mechanisms. In general, the instantaneous motions of particular grains, their translational

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velocities and spins, are different from the mean motion of the bulk. Individual particles may interact with one another in various ways; in rigid clusters of particles which generate a network of contact forces through sustained rolling or sliding contacts, or by nearly instantaneous collisions during which linear and angular momentum are exchanged and energy is dissipated because of inelasticity and friction. The relative importance of these mechanisms may be used as the characteristics which define various flow regimes. Some insight into the functional dependence of the stresses on the flow and material properties may be obtained through dimensional analysis. It is convenient to work in terms of the deviatoric stress where is the stress tensor (with compressive stresses being taken as positive following the convection used in soil mechanics) and p is the mean normal stress or pressure. For simplicity let us consider a particular flow situation; for example, a simple shear flow in which the only nonzero component of the velocity gradient is ul,2 (Fig. 2). We anticipate that, in general, the deviatoric stress fijand the pressure p may be expressed as and where pp is the mass density of the individual solid particles, (u2)Ii2 and (w2)II2 are the root mean squares of the translational and rotational velocity fluctuations arising from interparticle collisions. PB is some reference or characteristics value of the normal stress applied at the boundary of the sheared region; v is the solids fraction (volume of solids + total volume); u is the particle diameter; His the length scale for the width of shear flow; g is the gravitational acceleration; e , p , and E are the coefficient of restitution, the surface coefficient of friction, and the modulus of elasticity of the solid particles, respectively, and s is a shape factor defining the angularity or sphericity of the solid particles. If we were to consider a Couette flow in which the material was sheared between two parallel plates a distance H apart, then we should also include in (8) and (9) information about the wall properties; for example, the coefficient of restitution e , , the surface coefficient of friction p, , the modulus of elasticity E , , and some measure of the wall roughness or irregularity. Note that the pressure p has been included in the functional relationship for fii to explicitly recover the dependence of shear stress on normal stress that is characteristic of frictional Coulomb-type materials familiar in soil mechanics. Dimensional

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homogeneity requires that

We shall first examine these relationships for two limiting cases and then provide a tentative proposal to describe the microscopic physical behavior of the flow in the transition region between these two limiting flows.

(a) Fully Dynamic, Fluidlike Grain-Inertia Regime (high shear rates, moderate stress levels, low u ) . Here the shear rates and stress levels are high enough such that the gravitational forces are negligible, i.e., g1441,2)2+ 0 ,

(12)

but low enough such that the particles can be regarded as rigid, thus

p l E + 0. (13) The group pp~T’(u1,2)~/PB,which is proportional to the ratio of the dynamic or collisional stresses to the total boundary stresses, is the primary parameter that distinguishes the various flow regimes for dry granules. In PB one. If the width of the the fully dynamic regime, ~ P U ~ ( U ~ , ~is) ~of/ order shear layer H i s large such that we can neglect the effects of finite particle size to shear layer thickness ratio, i.e., u l H +-0, then Eqs. (10) and (1 1) reduce to

and

In this case, the stresses, including those at the boundaries, are completely determined by the flow dynamics; hence, it is redundant to include P B ~ [ ~ , ( T ~ (in U IEq. . Z (15). )~I This flow regime corresponds to the grain-inertia type of flow proposed by Bagnold (1954). Stresses are proportional to the square of both the particle diameter and the shear rate. The shear stress depends on the

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normal stress as found for dry frictional rate-independent Coulomb materials. At somewhat lower stress levels than those just mentioned, gravitational forces are important, and we must include the parameters g/[uCu1,2)~] and u / H in Eqs. (14) and (15). Equations (14) and (15) exhibit behavior characteristic of a nonlinear viscous fluid, such as a Reiner-Rivlin fluid. In the present limiting case, bulk stresses are developed as a result of two mechanisms. At the very lowest concentrations, when particles are widely dispersed and large void spaces exist, momentum transport can occur through the translation of particles from one shear layer to another. The flow is analogous to a dilute gas described by the kinetic theories, in which the mean free path is large compared to the particle diameter (Chapman and Cowling, 1970). In the kinetic theories at higher concentrations, voids large enough to accept a particle occur less frequently and momentum occurs primarily by the continuous action of intermolecular forces (Temperley et al., 1968; Faber, 1972; Hanson and McDonald, 1976). One might expect the granular flow mechanics in this regime to be similar to the “hard sphere models” used in statistical mechanical theories of the liquid state, where the intermolecular forces are impulsive forces associated with particle collisions. (b) Quasi-Static, Rate-Independent Plastic Regime (vanishing shear rate, high v , moderate to high stress levels). Let us first consider the stresses to be high enough that the elasticity and deformation of the individual grains are important. Deformations are slow, and the parameter p p 2 ( ~ l , t ) * /+ P ~0. Neglecting the effects of velocity fluctuations and eliminating the shear-rate dependence such that the stresses depend at most on the sign of u1.2 (or, for more general deformations, on D/(tr DD)l/Z,where D is the rate of deformation tensor), Eqs. (10) and (11) may be reduced to

The stresses here are governed not by the magnitude of the applied shear rate, but by the stresses applied to the boundaries, of which P B is a characteristic value. This case corresponds to the plastic behavior of a frictional Coulomb material of the kind that has been studied extensively in the context of soil mechanics (Schofield and Wroth, 1968; Mandl and FernBndez-Luque, 1970; Salencon, 1977; Spencer, 198 1). Particles can stick together, roll, or maintain sliding contact with one another for extended periods during the bulk deformation. Inertia forces are negligible

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and stresses are transmitted from one region to another through a network of contact forces. The yield stress depends on the normal stressp, and p is related to v as in the critical-state theories of soil mechanics (Roscoe et al., 1958). Although the velocity gradient does not appear in Eqs. (16) and (17), it is implied that the bulk is undergoing large and continuous deformation. During the initial failure or yielding, soils can experience an increase or a decrease in volume depending on the initial state of the material (Scott, 1963). But with continued deformation or flow the material tends towards an asymptotic state in which no further volume change occurs. The voids ratio (volume of interstitial voids + volume of solids) associated with this state is called the critical voids ratio. For a particular soil, its value depends on the mean stress level p . Note that a bulk material would not necessarily have a unique p-v relationship if not continuously sheared. Consider, for example, a vertical cylinder containing particles in a static state of loose random packing. The overall solids concentration v could be increased by vibrating the cylinder for a short period. Now the pressure p at a section near the base, for example, might be much the same as before, although v has been increased. At stress levels low enough that we can take p l E + 0, the particles can be considered rigid, and changes in v come about solely because of particle rearrangements. (c) Transitional Regime. In the two limiting flow regimes just discussed, the fully dynamic grain-inertia and the quasi-static plastic regimes, bulk stresses are generated by very different kinds of mechanisms. In the general case [described by Eqs. (10) and ( l l ) ] the stresses may be developed by some combination of these rate-dependent and rate-independent mechanisms. If we started with a flow in the fully dynamic graininertia regime and gradually reduced the shear rate while increasing the concentration, it is expected that the flow would pass through some transitional flow regime before reaching the quasi-static flow as D + 0. A tentative proposal for the microscopic flow behavior that might occur while passing through this transition regime is now presented. For discussion purposes, suppose that a Couette flow has been generated in the granular material contained between two rough parallel plates, the upper one moving relative to the lower one to generate a flow, as shown in Fig. 2. Let us consider how the normal and shear stresses could be developed at the plate surfaces. In the fully dynamic grain-inertia regime (Fig. 2a) v is low, the particles have a random fluctuating velocity component in addition to the mean shear u I ( x 2 )and , momentum is transferred almost entirely by binary interparticle collisions. The mean (time-averaged) stresses are proportional to

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clusters

(a)

(b)

(C)

(d)

FIG.2. Proposed microscopic description of transition as mean bulk density is increased and shear rate is increased. (a) Fully dynamic grain-inertia regime with viscous behavior, similar to molecules in a dense gas; (b) and (c) depict transitional regimes, and (d) depicts the quasi-static regime with rate-independent behavior, where almost all particles are in contact with near neighbors (after Sayed, 1981).

the square of the shear rate as described by Eqs. (14) and (15). Because of the random velocity fluctuations there also exist fluctuations in the bulk density. With increasing the mean bulk density (Fig. 2b) the bulk density fluctuations are such that clusters of particles (which are in rubbing contact) form and break up in random fashion. If the clusters remain rigid for a time they have the effect of increasing the effective shear rate locally in the adjacent sheared region; thus, they increase dynamic stresses. The stresses associated with deformation of the clusters also have a dry-friction, rate-independent component. A further increase in mean concentration increases the frequency of formation as well as the size of the particle clusters (Fig. 2c). Some of the clusters are large enough to span the width of the shear region, forming columns between the two walls. Columns or chains of particles in contact, which form force networks that are lined up primarily along the major principal stress direction, have been observed in quasi-static photoelastic studies by Drescher and de Josselin de Jong (1972). It is proposed that similar chains form in the dynamic case. Deformation and collapse of these columns adds a Coulomb-type stress component to the dynamic part associated with the regions adjacent to the columns where there are strong particle fluctuations. Increasing the mean bulk density still further (Fig. 2d) will force almost all the particles to be in continuous contact with their near neighbors (i.e., the whole region is occupied by the columnlike structures described earlier). Reducing the shear rate such that the accelerations generated during particle overriding are negligibly small leads to the quasi-static rate-independent behavior. The stresses result from surface friction and inter-

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locking between particles. The internal friction angle 4 associated with this Coulomb-type behavior depends upon the individual particle characteristics p and s and the solids concentration v.

c. S U M M A R Y O F LIMITING FLOWREGIMES We have discussed some of the possible flow regimes for both fluidsolid mixtures and dry granular materials. Each of these flow regimes is characterized by the relative magnitudes of interstitial fluid viscosity p , solids fraction v, and rate of deformation D and of the associated parameters such as the Bagnold number N = A1/2p~cr*(u1,2)/p and the group ppcr2(u1,2)2/PB.This information is summarized in Table I. Also included are the values of the parameter T J T T , which is the fraction of some time interval TT that particles are in contact with one another. This ratio is obviously related to the group p p ~ 2 ( ~ l , z ) In 2 /the P ~grain-inertia . regime, collisions are almost instantaneous, and T,/TT + 0. In the quasi-static regime, particles experience enduring contacts, and T J T T + 1. It should be emphasized that Table I is incomplete. We have not included regimes where there is a large differential velocity between the solid and the fluid phases (the extreme example being fluidized bed flows), regimes where electrostatic forces are important, etc. Table I is also coarse in the sense that the flow regimes listed may be further subdivided. For example, there are subregimes within the grain-inertia regime, characterized by the dissipative properties of the materials. In a shear flow involving smooth, nearly elastic grains, ( T U ~ , ~ / ( must U ~ ) ~be ~ ~small to generate the dissipation necessary to maintain a steady flow, whereas for rough inelastic particles ( T U , , ~ / ( U ~and ) ~ / ~u , , ~ / ( o * )are ~ / *large (Savage and Jeffrey, 1981; Jenkins and Savage, 1982).

111. Flows in Vertical Channels and Inclined Chutes Many industrial processes that involve the transportation, handling, and storage of granular materials make extensive use of devices such as bins, hoppers, channels, and inclined chutes (Reisner and Eisenhart Rothe, 1971). The flows in bins, vertical channels, and the major parts of hoppers are usually fairly slow and more likely to be in the quasi-static than in the grain-inertia flow regime. Extensive reviews of the slow flows in bins and hoppers have been given recently by Nedderman et al. (1982)

TABLE I CHARACTERISTICS OF SOMELIMITING GRANULAR FLOWREGIMES

Dimensionless groups Physical quantities Bagnold no.

Flow regimes Macroviscous Quasi-static Grain-inertia

Viscosity large small small

fi

Solids fraction v

Deformation rate D

small large small to moderate

small small large

N =

h”2pfd~1,2 CL

small usually small large

pp~2(~l,2)z

PB

7T

small small

-0 = I small

O(1)

Limiting prototype dilute suspension cohesionless soil dense gas or molecular fluid

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Stuart B . Savage

and Tiiziin et a f . (1982), and we shall only touch on some of the fluidlike aspects of the behavior of these flows here. On the other hand, because free surface flows of granular materials are in, or border on, the fully dynamic grain-inertia regime, we shall discuss these flows in some detail.

A. VERTICAL PIPESAND CHANNELS As a preliminary, it is instructive to review the classical analysis of Janssen (1895) for the stresses on the walls of a tall cylindrical bin of diameter D containing granular material of bulk-specific weight y. Consider a cylindrical slice of thickness dz located at a depth z measured from the upper stress-free surface. The vertical stress p , is assumed constant over the cross section. In static equilibrium, the weight of the cylindrical slab y(7rD2/4)dz, is balanced by the difference in normal force on the two horizontal faces ( v D 2 / 4 )dp, and the shear force acting on the periphery of the cylindrical slice r7rD dz, where T is the shear stress mobilized at the wall and acts upwards when the granular material is about to slide downwards. The shear stress T = Ph tan + w , where P h is the horizontal normal stress at the wall and +w is the friction angle for the granular material mobilized at the wall. Writing P h = K p , , where K is a constant independent of depth, the equation for equilibrium is dpvldz + PpV = y , where

p

=

(18)

(4K tan +w)/D.

Integrating Eq. (18) and taking the stress to be zero at the surface z yields p,

=

(y/P)(l

-

e-pZ),

=

0

(20)

For small depths the vertical stress behaves hydrostatically; i.e., p v = yz, and for large depths the vertical stress asymptotically approaches a constant value p , + y / P = y D / ( 4 K tan +W). Janssen’s analysis for static granular material contains some coarse assumptions, and numerous more refined analyses have been developed [see, for example, Pariseau and Nicolson (1979)l. Nevertheless, it gives the correct overall behavior and, with the proper choice of K and + w , can give reasonably good predictions of the wall stresses. It is natural to expect that the simplest kind of granular flow would be one corresponding to the Poiseuille flow of a fluid in a vertical tube, i.e., the dynamic analog of Janssen’s analysis just described. Because the shear stress is proportional to the normal stress during rapid shearing

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305

conditions as well as in the quasi-static state, we can expect the gross features of the stresses for granular flow in a pipe to be similar to Janssen’s quasi-static predictions. However, it is not always easy to set up a granular flow of this kind. If a long vertical tube is attached to the base of a large container, the lower end of the tube capped, the system filled with coarse granular material, and the tube cap then removed, one sometimes finds that the granular material accelerates down the tube and loses essential contact with the tube sidewalls. The tube does not flow full. The flow rate is controlled by the orifice at the bottom of the large container, i.e., the junction between the container base and the upper end of the vertical tube. In some cases a steady flow in the tube can be established by placing a constriction at the lower end of the long tube. Here the flow is not controlled so much by shear-rate-dependent friction at the side walls, as would be the case for a viscous fluid, but by the conditions in the hopperlike region near the bottom of the tube. It has been known for some time (Hagen, 1852) that the discharge rate of coarse granular material from a hopper or from an orifice in the base of a bin is essentially independent of the head and other conditions in the material some distance above the aperture [also, see Nedderman et a f . (1982)l.In this instance, there is virtually no similarity in terms of flow mechanics to the Poiseuille flow of a fluid. In the absence of the constriction, steady full flow in the tube can result when there are sizable interstitial fluid effects. Bingham and Wikoff (1931) measured the gravity flow of a fine dry sand through circular glass tubes of small diameter and obtained the surprising result that the mass flow rate increased with tube length. However, for coarse sands the flow rate was nearly independent of tube length. Richards [see Brown and Richards (1970,p. l85), de Jong (1969),Yuasa and Kuno (1972),and McDougall and Pullen (1973)l obtained similar results. The increased flow rate was attributed to the acceleration of the solid particles in the tube. The particles tend to drag air with them and create a partial vacuum beneath the orifice, thereby increasing the solids flow rate. Yuasa and Kuno (1972) found that, by introducing air to the upper end of the tube and hence reducing the partial vacuum there, the solids flow rate was reduced. Attempts to analyze the development of these interstitial pressure gradients were made by McDougall(1969/1970), Yoon and Kunii (1970)and Leung et a f . (1978).More general flows in hoppers and flows in standpipes with gas injection have been reviewed by Nedderman et al. (1982)and Leung and Jones (1978). A number of workers have measured the flow properties for cases in which the interstitial pressure gradient effects are small. Delaplaine (1956) made extensive tests of sand, glass beads, and bead catalyst in several

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cylinders of different diameters, materials, and wall roughness. Normal and shear stresses were measured with diaphragm gages mounted in the pipe wall. The stresses in the interior of the flowing granular material were determined by measuring the forces on small thin plates suspended in the flow. The internal and wall friction coefficients and the voids ratio were determined along with the stress ratio K = ph/pvappearing in Eq. (19). It was found that K was not constant as assumed in the Janssen (1895) analysis but varied somewhat with depth. The measured stresses were of the general form predicted by Eq. (20) and were independent of flow rate over the range of velocities tested. Strong time-dependent fluctuations in the wall stresses were observed, peak-to-peak variations being about 3050% of the mean stress. Toyama (1970/1971), Takahashi and Yanai (1973), Savage (1979), and Nedderman and Laohakul (1980) measured the granular flow velocity profiles in vertical parallel-sided channels of rectangular cross section. Toyama (1970/1971) observed the flow of sand, ore, and vermiculite through a transparent Plexiglas front wall of his channel. Tests with and without the presence of sandpaper to roughen the two outer side walls gave velocity profiles with a rigid plug flow core and shear zones or boundary layers adjacent to these sidewalls that were about 4 to 8 particle diameters thick. Similar results were found by Takahashi and Yani (1973) for glass, silica, and alumina spheres about 4 mm diameter. By using fiber optic probes, Savage (1979) measured the velocities of 1.2-mm diameter polystyrene beads in a rectangular channel having two smooth glass sidewalls and two walls lined with rough rubber sheet. The particles slipped along the glass wall, but the walls lined with rubber sheet were sufficiently rough to generate a two-dimensional shear flow approximating a no-slip condition at the rough walls. The velocity profiles became fully developed in a short distance and were found to be independent of the flow rate, which was varied by a factor of almost 10. In these tests the channel was narrow (-- 28a), the shear zones spanned nearly the full width between the rough walls, and the plug flow region was not readily apparent. In the wider channels, plug flows are commonly observed; thus, these tests, in which the plug flows were absent, are probably the exception to the norm. Fig. 3 shows photographs of flow visualization tests of glass beads (= 300 pm dia.) flowing down vertical rectangular channels. The front and back walls were glass plates spaced 25 mm apart, and the sidewalls were roughened with sandpaper and spaced 10 mm, 20 mm, and 45 mm apart in the tests shown in Fig. 3. Flow rates were approximately the same in all three tests. The dark bands correspond to “time lines” and are made up of dyed beads originally placed in horizontal layers at the start of the run. The central plug flow region is evident in the two wider

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307

FIG. 3. Development of central plug flow regions and boundary layers next to rough walls during the flow of glass beads (a 300 pm) down channels of widths of (a) 10 mm, (b) 20 mm, and (c) 45 mm (unpublished work of S. B. Savage and M. Sayed).

-

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Stuart B . Savage

channels, and the thickness of the boundary layers next to the rough walls shows only small variations with changes in channel width. Nedderman and Laohakul (1980) used a cinema camera to determine the velocities of glass ballotini, kale, and mustard seeds of about I to 2 mm diameter in a wood-and-glass-walled bin. Particles slipped readily at the glass walls but were retarded to some extent by planed wood sidewalls; the slip velocity at these wood walls was about 2/3 of the plug flow velocities. When the sidewalls were roughened by sticking particles to them, no slip occurred there. After a short entry length, the shear layer thickness remained constant at about 6 to 8 particle diameters and was not dependent upon the plug flow velocity, which was varied from 1.5 to 15 cmls. In all of these tests the shear rates are probably too small for grain inertia effects to be dominant. Bridgwater (1980) has proposed an analysis based on kinematic and statistical mechanical arguments to explain why the boundary layer, or “failure zone,” is about 10 particle diameters in thickness.

B. EXPERIMENTAL OBSERVATIONSOF FLOW DOWN INCLINED CHUTES Some studies of flows in chutes have been directed toward the solution of specific technological problems; examples are the work of Wolf and von Hohenleiten (1945) on the design of chutes for the handling of coal, experiments by Trees (1962) to determine flow rates for particulate iron oxide in open-ended sloping pipes, and that of Choda and Willis (1967) on the determination of the optimum profile and cross section of curved chutes for the transport of granular materials. Other studies have been aimed toward obtaining a better understanding of the mechanics and constitutive behavior of granular materials. The steady, two-dimensional, free-surface flow down a rough inclined plane can be regarded as a simple viscometric flow. For such a flow down a plane inclined at an angle 5 to the horizontal (Fig. 4), the normal and shear stresses on a plane parallel to the bed are given by the linear momentum equations as

where the bulk-specific weight y = ppvg. Thus 7 1 2 1 ~ 2=~

throughout the depth.

tan

5

=

const.

The Mechanics of Rapid Granular Flows

309

For such a flow to exist, the constitutive behavior of the granular material, in particular the dynamic friction angle + D , must be consistent with Eq. (23). For beds that are made fully rough (for example, by fixing a layer of particles to the bed) no flows are possible for values of 6 less than some critical angle that is close to the angle of repose for the granular material. Bagnold (1966) pointed out that the apparent limiting static friction angle +i of initial yield to an applied shear stress, exceeds the residual angle +r measured during very slow shear by the amount A+ = +i - + r , which he termed the dilation angle. In a study related to the avalanching of sand dunes, Allen (1970) measured A+ for a number of materials and found that it typically was between 1 and 4 degrees. In other words, one would have to tilt the bed such that 5 = 4i to initially get the material to flow, but 6 then could be reduced as much as A+ before flow would cease. Bagnold’s (1954) experiments on sheared spherical wax beads in water showed that the ratio of shear stress to normal stress (i.e., tan +D) depended on flow conditions but was limited to a finite range of values. Tests with dry granular materials in an annular shear cell by Savage and Sayed (1980, 1982) and Sayed (1981) have shown that +D is weakly dependent upon the concentration v and shear rate, and that it is restricted to a rather narrow band of values, i.e., +r < +D < + M , where +M is some upper limit to $q,. For a steady, nonaccelerating flow over fully rough beds, compatibility of material behavior with the equilibrium condition (23) means that 4 7 2 2 = tan 6 = tan +D and the bed slope is restricted to pr < 6 < pM.If 6 > p ~the, flow will accelerate down the chute. If the bed is not roughened, or is partially roughened, flow will begin when 5 exceeds the wall friction angle, which is related to the coefficient of friction between the bed and granular material. Further increase in 6 usually results in accelerated flow, but this matter is not settled and there may be a small range of bed slopes which permit nonaccelerating flows. One of the earliest investigations of granular flows in chutes was performed by Takahasi (1937) in an attempt to gain some understanding of geophysical phenomena such as snow avalanches and land slides. He

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Stuart B . Savage

performed laboratory tests with dry sands of various sizes flowing down a straight, rectangular, cross-sectioned wooden channel which could be set at various angles of inclination 5. The free-fall trajectories of the sand grains were used as a means to determine the mean velocity of the particles leaving the downstream end of the chute. Detailed velocity profiles were not obtained, but it was seen that the particles were retarded at the channel sidewalls and flowed fastest in the middle of the channel. Takahasi detected two flow regimes which could be clearly distinguished on plots of mean velocity versus chute inclination angle 5. The first regime was characterized by a very rapid increase in velocity with 5. A stationary layer of particles occurred above the chute bed and only a thin layer of particles near the surface was in motion. In the second regime the mean velocity increased much more gradually with 5. All the particles were in motion, and because of the vigorous interparticle collisions the bulk density was low and the upper free surface was indistinct. In materials handling operations, chutes are often used to direct and transfer material from one place to another, but if the chutes have an enclosed cross section (in the form of a tube) they can also be used to control the flow rate. Roberts (1969) has described experiments and analyses of such flows in straight chutes and circular, parabolic, and cycloidal curved chutes of rectangular cross section. The experiments employed millet seed flowing in a clear Plexiglas channel; particle velocities were determined through the use of a high-speed cinema camera (1200-2000 frameds). Although the particles were found to slip at the smooth walls of the chute, a small velocity gradient developed across the depth of the flow. Roberts (1969) developed a simple analysis to predict the mean velocity and depth profiles along the longitudinal chute axis for the cases of straight and curved chutes. The analysis was analogous to the sliding of a block down a rough inclined surface. The effective angle of friction between the granules and the channel walls included the effects of friction on the vertical sidewalls as well as the friction on the channel bed. The effective friction angle was determined empirically under quasi-static conditions and was assumed to be applicable during flow conditions. The analysis gave reasonably good predictions of depth profiles for the cycloidal chutes in which the flow depth first decreased to a minimum and then increased with downstream distance as the local bed slope decreased. Roberts used his analysis to predict the theoretical optimum chute profile for given practical applications. It is worth noting that because of the assumption of a single velocity-independent effective friction angle, Roberts’ analysis predicts that the flow accelerates, decelerates, or remains uniform when the effective friction angle is respectively less than, greater than, or equal to the chute angle of inclination.

The Mechanics of Rapid Grunular Flows

311

In a review of bulk solids flow through transfer chutes, Roberts and Scott (1981) reported further work on flows in straight inclined and curved chutes of rectangular and circular cross sections. Some results were presented for variable-diameter, straight-inclined chutes having tapered middle sections. Roberts and Scott (1981) also described experiments of Parlour (1971) in which the stresses developed at the bottom and sidewalls of a straight, rectangular, cross-sectioned chute were determined. During flow, the normal stress on the vertical sidewall increased approximately linearly with depth over the upper portion of the material (millet); but at greater depths, the stress increased nonlinearly at a more rapid rate reaching a peak in the bottom corner of the rectangular cross section. These sidewall normal stresses increased with flow velocity to as much as about twice those measured under static conditions. This kind of behavior is consistent with our previous discussions of the various granular flow regimes. Under static conditions the ratio of the horizontal to the vertical normal stress, K , is very roughly that which is predicted by the Rankine theory (Wieghardt, 1975, pp. 93-95) for a semi-infinite granular medium, K = ( 1 - sin +)/( 1 + sin 4). For an internal friction angle 4 of 30”, this gives K = 1/3. During flow approaching the fully-dynamic grain-inertiu regime, we expect that the three normal stresses are more nearly equal. For example, the theory of Savage and Jeffrey (1981) gives in the case of a simple shear flow u I ( x z )when ( T U ~ , ~ / ( U ~is) around ”~ I (probably a typical value for most materials), 711 = 722 = 7 3 3 ; i.e., in this case K = 1 . Thus, from these arguments we might anticipate that the dynamic normal stress at the sidewall could be as much as three times those measured in the static state. Suzaki and Tanaka (1971) suggested that the inclined chute be used as a viscometer and attempted to apply it to determine the constitutive behavior of glass, crushed calcite, and sand particles. The variations of flow depth with flow rate in a straight rectangular chute were measured for various bed inclinations. Their results showed that the materials did not behave as a Newtonian fluid. They assumed that a better approximation to the bulk solid behavior would be that of a Bingham fluid having a constant yield stress and a constant viscosity. This is still a rather crude assumption since bulk solids behave as Coulomb materials in which yielding is influenced by the mean pressure, and because the stresses are related to the deformation rates in a nonlinear way as described in Section 11, dealing with granular flow regimes. Ridgway and Rupp (1970) used a polished brass chute of rectangular cross section to study the flow of sand of three different size ranges (250355, 420-500, and 758-850 pm) and four different sphericities. Particle velocities were measured by photoconductive cells and by a high-speed

3 12

Stuart B . Savage

cinema camera. Through the use of mirrors and a transparent glass section in the bed, particles at the free surface and the bed could be filmed. Since the cinema film showed the particle velocities to be the same at the free surface and at the bed, Ridgway and Rupp inferred that a plug flow with no velocity gradients existed throughout the depth of flow. The flow was found to accelerate down the full 1-m length of the chute for 30" < 6 < 60". A horizontal knife edge was placed at the end of the chute and positioned at different depths to act as a flow divider. By weighing the flows collected in a given time above and below the splitter plate, and assuming a uniform flow velocity, the density profiles over the depth were calculated. In general, the bulk density decreased with downstream distance; this effect was more pronounced for the larger particles than for the smaller ones. The interesting feature of the bulk density profiles was that although there apparently was no shear, all the density profiles had maxima near the mid-depth. The densities were quite low near the bed (typically about 200 kg/m3, or 15% of the stationary poured bulk density). As a result of particle saltation, a "cloud" of particles developed at the upper surface and the bulk density dropped gradually to zero. Since the particles were not small compared to the gap between the bed and the splitter plate or to the total flow depth, the splitter plate may have induced a flow interference, and the calculated density profiles may be unreliable. In some experiments using the same splitter plate technique, Knight (1983) calculated maximum bulk densities considerably in excess of the static loose random packing. As Knight (1983) noted, such results are unlikely and the splitter plate definitely interferes with the flow. However, the low-density region near the bed also has been observed in the experiments of Augenstein and Hogg (1978), Bailard (19781, and Knight (1983). Similar lowdensity regions are evident in some computer simulations of Campbell (1982). Campbell (1982) performed numerical calculations of the flow of equal, circular, inelastic, rough disks down a frictional plane, thus providing a "two-dimensional" computer model of granular chute flow. Calculations were performed for bed inclination angles 5 of 20", 30", and 40". The velocity profiles were blunt with slip at the bed. The bulk density profiles showed a maximum around mid-depth. Increasing the bed slope resulted in a decrease in the bulk densities at a given fraction of the overall flow depth and made the decrease in density near the bed more pronounced. At a bed slope of 40", the flow was accelerating, but at 5 = 20" and 30" the profiles appeared to be closer to a steady state. Since the low-density bed region was apparent in these numerical simulations, it is likely to be a real effect and not merely an artefact of the splitter plates used in the experiments.

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313

Ridgway and Rupp (1970) observed that, for a given bed slope 5 and particle size, the particles which were more angular (less spherical) had much greater variations in bulk density over the depth than those which were more rounded and spherical. During flow, the more spherical particles developed a density profile which showed a relatively small density reduction over the lower half of the depth, a nearly uniform density over the upper half and a more distinct upper free surface. A possible explanation of the low-density region near the bed is as follows. As the mass of particles slides down the chute, the particles nearest the bed acquire spin, and as a result of their collisional interactions with their neighbors and the bed, develop random translational and rotational velocity fluctuations. The bed, in a sense, acts as a source of translational and spin fluctuation kinetic energy. Due to the collisions, normal stresses are generated in the interior, causing the bulk to dilate to values appropriate for equilibrium. Because of the inelastic and frictional nature of the particles, the fluctuation kinetic energy generated at the bed is damped as it flows towards the free surface. If sufficient fluctuation energy reaches the upper surface, particles there occasionally acquire sufficient energy to saltate and develop the low density “free surface cloud.” Thus one anticipates a bulk density profile having a maximum somewhere near the mid-depth. It is also likely that the sharp angular particles are more effective than the rounded ones in acquiring spin at the bed and transmitting the fluctuation energy to the interior. This could explain the more pronounced bulk density variations with increased particle angularity. By estimating the velocities from the free-fall trajectories of particles leaving the downstream ends of chutes having different flow lengths, Augenstein and Hogg (1974) inferred how the velocity would vary with distance along chutes set at particular inclinations. Then using these “measured” velocity variations in a simple analysis analogous to that given by Roberts (1969), they determined the effective friction angles for smooth and sand-roughened stainless steel inclined chutes. In an extension of this work, Augenstein and Hogg (1978) determined the velocity distribution over the depth for sand flow down chutes of different bed roughnesses. Significant slip occurred at the smooth steel bed but a small amount of shear did occur in the interior of the flow. When the bed was roughened with sand grains of the same size or larger than that of the flowing materials, no slip occurred. The velocity profiles for the fine sand flowing on the rougher wall were the same as when the wall roughness grain size was equal to that of the flowing sand. When the bed was roughened with sand of smaller size than the flowing sand, some slip occurred at the bed. In these tests, the bed angles were high enough and the chutes were short enough that the flow accelerated over the full length of the

3 14

Stuart B. Savage

chute in all cases. For the flow on the roughened bed, it was estimated that a layer of low bulk density of about 310 kg/m3occurred near the bed. The bulk density in the upper part of the flow was estimated to be approximately 1100 kg/m3, about the same as in the flow on the smooth stainless steel chute where slip occurred at the bed. The static bulk density was 1400 kg/m3. Similar but more extensive tests were performed by Baillard (1978). He determined the velocity variations with depth in the middle region away from sidewalls by using vertical and horizontal splitter plates and measurements of the free-fall particle trajectories. Bulk density profiles were also obtained using the approach of Ridgway and Rupp (1970). Experiments were performed with sand grains of nominal diameters of 0.21, 0.42, and 0.84 mm. The 2-m-long aluminum chute was roughened by attaching a layer of sand grains to “contact paper” fixed to the channel bed. The nominal size of the roughness grains was the same as the flowing material for each test. While the tests of Ridgway and Rupp (1970) and Augenstein and Hogg (1974, 1978) were performed at high bed slopes such that the flows always accelerated, Bailard’s tests spanned a narrow range (5 = 34”, 36.5”, and 39”) and were low enough such that the flow reached a terminal velocity and became fully developed as discussed in the beginning of this section. While the bed roughness was sufficient to prevent slip at the bed in the case of the large, 0.84-mm grains, some slip occurred for the finer 0.21- and 0.42-mm grains. Shear was present over the full depth in all cases. Bailard also found the particle solids fraction profiles to be similar to those discussed earlier for the smooth chutes, with a maximum near mid-depth and low solids fractions of about 0.2 near the bed. Some typical velocity and concentration distributions determined by Bailard (1978) are shown in Fig. 5 . These data are for sand of 0.84-mm diameter flowing down a chute for which 5 = 36.5”, for flow depths h of 7.27, 7.77, and 9.18 mm, and surface velocities U of 0.502, 0.607, and 0.672 m/s. The low bulk densities near the top of the flow are associated with the “cloud” of saltating particles which occurs there during rapid flows. The low density region near the bed is rather more difficult to explain in this case where there was no slip at the bed and no obvious “source” of fluctuation kinetic energy near the bed. Savage (1979) performed two series of experiments in straight inclined chutes. The first study used 0.42-0.59 mm spherical glass beads flowing in smooth and sand-grain-roughened aluminum channels. The velocity profiles at the free surface across the width of the flow were determined with a high-speed camera, In the smooth-walled channel at lower inclinations a plug flow with slip at the walls developed; at higher inclinations a slight shear occurred. A blunt velocity profile with little slip at the walls was

The Mechanics of Rapid Granular Flows

Nondirnensional velocity (9)

u,/u

315

Solids fraction V

(b)

FIG.5. Typical (a) velocity and (b) concentration profiles for 3 experiments of flow of 0.84 mm diameter sand down a rough chute, 5 = 36.5". Solid line in (b) depicts data for which h = 7.27 mm; dashed line h = 7.77 mm; dotted line, h = 9.18 mm (after Bailard, 1978).

observed in the rough-walled chute. By dropping a thin line of colored particles on the free surface and observing its development with time, it was inferred that secondary flows existed. These were similar in form to those predicted by Green and Rivlin (1956) for non-Newtonian fluids consisting of two cells in which the flow moved downward at the centerline and upward at the vertical sidewalls. Detailed depth and velocity profiles were measured in Savage's (1979) second series of experiments which tested spherical, 1.2 mm diameter polystyrene beads in a glass-walled channel. The channel bed was roughened by rubber sheets having cylindrical protuberances in an effort to generate a two-dimensional shear flow. For a range of bed inclinations between about 26" and 39", flows of uniform depth were observed. It is interesting to note that Roberts' (1969) analysis, which relies upon an empirical effective friction coefficient determined in the channel under quasi-static conditions, predicted accelerating flows with decreasing depth for these bed inclinations. Velocity profiles were determined by the use of two fiber optic probes held in a traversing gear mounted in the channel sidewall. The flow of a mass of beads generated a fluctuating signal from the upstream sensor and a similar but time-delayed signal from the downstream sensor. The cross correlation of these two outputs yielded the mean transit time between the two probes and thus the mean velocity. Velocity profiles were measured at three streamwise stations for 6 = 32.6", 35.3", and 39.3". For these cases the flow appeared to be approximately fully developed; there were only slight streamwise variations of the flow depth and velocity profiles. Figure 6 shows the observed

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Stuart B . Savage

FIG. 6. Nondimensional velocity profiles for flow of 1.2 mm polystyrene A,0 spheres down an inclined chute. 0, depict streamwise stations 330, 635, and 940 mm, respectively, from entry (after Savage, 1979).

velocity profiles. The velocity profiles have an inflection point and a shape reminiscent of a laminar boundary layer near separation. With increasing bed slope the velocity profile becomes more full. Velocity profiles have also been measured by Ishida and Shirai (1979) using a fiber optic probe (Ishida et al., 1980a) which was placed in the interior of the flow along the vertical center-plane. The sidewalls of the 0.85-m-long channel were smooth glass and the bed was roughened with very coarse sandpaper. The aspect ratios (widthldepth) of the channel flows were larger than in Savage's tests. Velocity profiles were measured near the downstream end of the channel. Three types of beads, 0.35-0.5 mm diameter glass, 0.21-0.3 mm diameter alumina, and 0.044-0.063 mm diameter fluid catalytic cracker (F.C.C.) were tested. Typical velocity profiles for the glass beads are shown in Fig. 7. The general characteristics of these profiles are similar to those of Savage (1979) for polystyrene beads; no slip occurs at the bed, the profiles at low 5 are concave and with increasing 5 they fill out towards a linear profile having a constant shear rate. Knight (1983) has also measured chute flow velocity and concentration profiles using fiber optic probes and the splitter plate technique. The channel was rectangular in cross section with Plexiglas sidewalls and a coarse carborundum paper lined bed. Knight's experiments using 0.3 1 mm sodium perborate particles and spray dried detergent of 0.48 mm diameter were performed for bed inclinations of between 39" and 70°, and thus included accelerating flows as well as fully developed ones. Although the velocity profiles showed some scatter, the general trends were the same as those observed by Savage (1979) and Ishida and Shirai (1979).

The Mechanics of Rapid Granular Flows

FIG. 7. Velocity profiles for flow of glass beads

317

-EE

(0.35-0.5 mm diameter) down an inclined channel (af-

ter Ishida and Shirai, 1979).

Types of Inclined Chute Flow Patterns One apparent anomaly, which as yet we have not explained, is the difference between the velocity profiles measured by Savage (1979) and Ishida and Shirai (19791, which were linear or had a “separation” type shape with an inflection point, and the velocity profiles measured by Augenstein and Hogg (1978) and Bailard (l978), which in general were much fuller (blunter) in shape. One possible cause is channel sidewall friction. Savage’s (1979) velocity profiles were measured at the sidewall and might not be representative if a velocity gradient across the width of flow was present. However, Ishida and Shirai’s (1979) profiles were measured with fiber optic probes placed along the central plane between the sidewalls and similar profile shapes were observed, so this explanation seems inadequate. But, even if the flow was two-dimensional and sidewall slip occurred, the sidewall friction could still exert a restraining influence on the flow. Because the pressure increased with depth, the sidewall friction increases with depth, tending to decrease the velocities near the bed from what might occur in the absence of sidewall friction. However, an analysis by Sayed and Savage (1983) indicates that while increasing sidewall friction reduces the flow rate, the change in the shape of the velocity profile is not large until one has very large sidewall friction effects or, equivalently, very narrow channels. Another possible cause is the different materials: Savage (1979) and Ishida and Shirai (1979) used spherical polystyrene and glass particles, whereas Augenstein and Hogg (1978) and Bailard (1978) used angular sand grains. Very strong differences in the flows of these kinds of materials have been observed in bin and hopper flows. For example, Nguyen (1979) found that during flow in wedge-shaped hoppers, spherical beads flow steadily in fairly straight, radial paths, but a bulk composed of fine angular sand tends to divide into rigid blocks separated by thin bands where shear takes place [also see Tuzun et al. (1982) for other references

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to these phenomena]. The fine sand often flows in an unsteady intermittent fashion. It is not clear to what extent these observations are relevant, because the chute flows generally occur at higher shear rates and lower bulk densities than those typical in the upper parts of hoppers. Furthermore, in a later study of chute flows in which the particles were fluidized by air flow through a porous bed, Ishida et al. (1980b) found the velocity distributions for angular sand particles and alumina beads were quite similar to those measured for spherical glass beads. The essential explanation for the different kinds of velocity profiles observed seems to be that there are a number of different types of flows. In their fluidized-bed channel flows, Ishida et al. (1980b) were able to discern five distinct types of flow patterns depending on the bed slope and the strength of the fluidizing air velocity. When the velocity of the fluidizing air through the porous bed was zero (the case of primary interest here) they observed three types of flow. We list them in order of increasing bed slope, imagining that the mass flow rate is kept constant. (a) Immature slidingflow In this case the bed slope 5 is close to the angle of repose of the material. There can be a stationary layer just on top of the bed, thicker on the upstream than on the downstream side. In other words, an effective (or zero-velocity) bed surface occurs within the material itself, its slope being greater than the slope of the solid plane surface of the actual channel. The velocities near the bed are quite low, and the profiles have the inflection points, as shown in Figs. 6 and 7. The flow is probably in the transitional regime between the quasi-static and the graininertia regime. (b) Sliding flow For a slightly increased bed slope, there is no dead region next to the channel bed, all the particles are in motion, the shear rate approaches a constant value over the depth, and the velocity approaches the straight linear distribution shown in Fig. 7. The flow is probably approaching the grain-inertia regime. The free surface is fairly distinct; saltation of particles may be present, but individual particle jumps are not large. (c) Splashing flow With a further increase in bed slope the velocity profile becomes blunter, with a higher shear rate near the bed than near the top of the flow. Saltation is vigorous, and the top of the flow consists of a low-density cloud of particles rather than a distinct free surface. The motions of the saltating particles and those near the upper layer are affected by fluid drag. The preceding sequence of flow types was observed when the bed inclination angle was increased while the flow rate and particle size re-

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mained constant. In other tests at a fixed inclination angle, increasing the mass flow rate resulted in an increase of the flow depth and a change in the shape of the velocity profile from one with a nearly straight linear shape to blunter profiles having a higher shear rate near the bed than near the surface. Although data were not presented for the case of zero-fluidizing air through the bed one might anticipate a similar behavior for the case in which mass flow rate is increased and 6 is held constant. It seems likely that Augenstein and Hogg's (1978) and Bailard's (1978) flows were of the type that Ishida et al. (1980b) called splashingjows, whereas those of Savage (1979) and Ishida and Shirai (1979) were of the sliding and immature sliding type. Although there was no consistent change in Bailard's velocity profiles when the bed slope was increased from 34" to 39", there were significant differences between flows with different-sized sands. The flows with the finer 0.21 mm diameter sands, corresponding to a larger flow depth to particle diameter ratio h l a , had blunter velocity profiles than those with the medium (0.42 mm) and coarser (0.84 mm) sands that corresponded to smaller nondimensional depth h l a . Finally it should be mentioned that the velocity and concentration profiles of Augenstein and Hogg (1978) and Bailard (1978) were not determined directly but were inferred from the trajectories of layers of particles leaving the end of the chute after being segregated by splitter plates. The splitter plates may cause obstructions to the flow, distorting the velocity profiles from those that might otherwise be present. The trajectories of particles in a given layer can be affected both by air drag and by interactions with particles in adjacent layers. The magnitude of these effects is uncertain. Nevertheless, the gross behavior exhibited in these measurements is consistent with the simulation of Campbell (1982) and the classification of chute-flow types by Ishida et al. (1980b). For cases in which the particles were fluidized by air flow through the porous bed, Ishida et a/. (1980b) found that the bulk solids would flow for bed slopes less than the angle of repose for the nonfluidized material; i.e., they behaved more like a normal viscous fluid. For the fluidized material, they observed two further flow types: (d) Bubbling $ow These flows occurred for very low bed inclination angles, when the air flow was increased beyond the minimum fluidizing velocity UMF(Davidson and Harrison, 1963). Large bubbles similar to those that may be seen in static fluidized beds were evident. The velocity profiles were significantly curved and blunt. (e) Glidingflow When the fluidizing velocity was greater than U M Fand the bed slopes were increased, the bubbles were extinguished, the

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velocity gradients became more uniform, and the velocity profile tended toward the linear one observed in the sliding-flow type.

A review of flow of fluidized solids in channels is beyond the scope of the present article, but further discussion of these flows may be found in Singh et al. (1978) and a series of interesting papers by Botterill and his colleagues (Botterill et al., 1972; Botterill and Bessant, 1976; Botterill and Abdul-Halim, 1979).

C. FLOWSAROUND OBSTACLES Nedderman et al. (1980) investigated the two-dimensional flow of mustard seed around cylinders having circular, square, and triangular cross sections. The cylinders were placed in a vertical glass-walled channel with their axes oriented perpendicular to the direction of flow. The streamline patterns and velocity profiles were measured for mean flow velocities of a few cm/s. Stagnant flow regions were observed upstream of the bodies, and voids regions and wakes were observed behind them. Ishida et al. (1980a) studied the flow around rectangular plates placed perpendicular to the flow direction in their free-surface inclined-channel experiments. When a plate having a width about one-third of that of the channel and a height about equal to the granular flow depth was placed into flows of the immature sliding or sliding type, the upstream flow depth increased in height and the granular material flowed over the plate, the plate acting rather like a weir in an open-channel water flow. For flows of the splashing, bubbling, and gliding types, the granular material passed around the plate without causing a noticeable change in upstream depth. Wieghardt (1975) has measured the drag forces of vertical cylinders placed in a rotating bed of sand. The drag was found to depend only slightly on the flow velocity and cylinder cross section. Bow waves were present at the free surface of the sand, and troughs, behind the cylinders and a few diameters deep, were observed at the higher speeds.

D. GRANULAR JUMPS In his inclined-chute tests, Savage (1979) observed that surge waves and granular jumps analogous to hydraulic jumps could be generated when the upstream Froude number FI was supercritical, i.e., when FI > 1 (where FI = iI/(gh1)”*, hl is the upstream depth and E l is the upstream depth-averaged velocity). The possibility of such jumps had been previ-

The Mechanics of Rapid Granular FZows

32 1

FIG.8. Granular jump generated during the flow of polystyrene beads down inclined chutes (flow is from right to left); Froude number = 2.5, conjugate depth ratio = 6, 5 = 35" (Savage, 1979).

ously predicted by Morrison and Richmond (1976). A photograph of a typical jump in flowing polystyrene beads is shown in Fig. 8. Savage (1979) developed an analysis for the change in flow properties across the jump that was somewhat more general than that presented by Morrison and Richmond (1976). The ratio of the downstream to upstream depth h2/hIversus the upstream Froude number F, predicted by this analysis agreed reasonably well with the measurements. Brennen et al. (1982) have also performed experiments on granular jumps, using glass beads and mustard seed. These tests were consistent with those of Savage (1979) but were more extensive and examined flows with much higher upstream Froude numbers. Brennen et al. (1982) were able to distinguish two types of jump: a smooth expansion that occurred at lower FI, and one with a recirculating eddy at higher FI. The structure of the flow downstream of the jump and the presence or absence of downstream stagnant flow regions were found to depend on the nature and size of the downstream obstruction used to initiate the jump.

IV. Rheological Test Devices and Experiments A number of devices analogous to fluid viscometers have been developed to obtain experimental data on bulk solids concerning the failure and deformation modes during shear, the stress-strain and stress-strain-rate behavior, and the dependence of these characteristics on bulk density,

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and microscopic material properties and texture. These experiments have been designed solely to discern the rheological behavior of bulk solids, in contrast to many of the experiments described in the last section, which had direct applications to engineering problems of bulk solids transport and handling in addition to their rheological content.

A. QUASI-STATIC SHEARDEVICES Shear tests are commonly used in soil mechanics and powder technology to characterize the failure and yield behavior of granular materials. Such tests are typically carried out under relatively low shear rates, corresponding to the quasi-static flow regime described earlier, and they often involve only relatively small strains. Hence, the results such devices can obtain are not of great relevance to the case of rapid shear flows. However, details of the mechanical design and test procedures may be useful to someone interested in the design of a device with a high shear-rate capability, and we review these quasi-static testers primarily for that reason.

1 . Simple Shear Apparatus This device was originally devised by Roscoe (1953) to study the stress-strain behavior of soils. Over the years it has undergone significant development through numerous versions of the basic apparatus (Roscoe, 1970). It consists of a rectangular box that can be distorted to generate a simple shear in the granular material contained within. Normal and shear stresses at the boundary of the soil sample are measured by load cells mounted in the walls of the shear apparatus. Overall strain is determined from the motion of the boundary walls. In the interior of the sample the strains, rupture zones, and bulk densities can be determined by the use of X rays and y rays. A similar simple shear apparatus has been developed by Schwedes (1973, 1975) to characterize the flowability of granular materials and powders.

2. The Jenike Shear Cell Jenike (Jenike et al., 1960) developed this cell to obtain information that could be used for the design of storage hoppers for industrial powders. The apparatus consists of a shallow circular cylindrical cell having its axis oriented vertically. The cell is split horizontally into upper and lower

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323

parts with the base half fixed. A shear force is applied to the upper lid, driving it at constant speed in an attempt to impose a constant strain rate on the specimen. Samples are prepared by the application of a consolidating normal load prior to the application of the shear force. One carries out series of tests in which the shear force for failure is measured for various normal loads and for various consolidating conditions, resulting in a number of yield loci, each corresponding to a different degree of consolidation. Williams and Birks (1965) have discussed methods for sample preparation, and Schwedes (1973) has made a critical comparison of results obtained from the Jenike shear cell with those obtained from a simple shear cell. 3. Annular Shear Cell

The typical annular shear cell consists of cylindrical upper and lower halves, one or both of which have ringlike cutouts or annular troughs to contain the test material. After the application of vertical normal loads, a torque is applied zbout the vertical cylindrical axis to rotate half of the cell relative to the other half to generate a shear within the test sample (the shear surfaces being horizontal planes). There is no limit to the strain that the material can experience. Thus, this kind of device is appropriate for the study of material behavior under conditions of continued flow, as opposed to the simple shear cell and the Jenike shear cell which, because of geometrical constraints, can be used only to investigate the initial yielding behavior. Hvorslev (1936, 1939) developed the first annular shear cell to study the behavior of soils. In soil mechanics, the initial failure criteria are of more interest than the soil behavior under continued flow, and probably for this reason the annular shear cell has not come into widespread use as a standard piece of soil mechanics test apparatus. However, in problems of bulk solids handling, situations involving continued deformation are of great interest, and a number of annular shear cells have been designed by engineers working in this area. Novosad (1964) and Carr and Walker (1967/1968) determined the internal friction angles, the wall friction angles, and the yield strength characteristics of granular materials during slow, continued shear. At these low strain rates the stresses were found to be rate independent. Scarlett and Todd (1968, 1969) and Scarlett et al. (196911970) measured stresses, critical porosities, and the dilation characteristics of sand sheared in an annular shear cell. The lid of their cell was split into three concentric annuli. To minimize the effects of friction on the vertical side-

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Stuart B . Savuge

walls of the annular trough the test results were based upon measurements of normal force and torque on the middle one of the three lid annuli. The thickness of the shear layer formed during failure was determined by using a y-ray attenuation technique as well as tracer particles. Mandl et al. (1977) developed a very sophisticated ring shear apparatus to investigate the development of shear zones and the accompanying changes in texture and stress state in granular materials undergoing continued shearing. Stephens and Bridgwater (1978a, 1978b) studied details of the failure zones; the zone depth, the velocity distribution, and the diffusion of particles across the shear zone.

B. DENSESUSPENSIONS Cheng and Richmond (1978) have used the term “dense suspensions” to denote solid-liquid mixtures in which the solid phase is present in concentrations close to that corresponding to the densest possible packing of the particles, i.e., well in excess of the lower limit often associated with “concentrated suspensions.” Literature dealing with concentrated suspensions has been reviewed recently by Jeffrey and Acrivos (1976), Goddard (1977), Buyevich and Shchelchkova (1978), and Gadala-Maria (1979). Most of the work these researchers discussed pertains to small particles at low Reynolds numbers, and few papers have dealt with dense suspensions of relatively large particles at high shear rates. The earliest and still the most extensive investigation of the shearing of dense suspensions at high Reynolds numbers is the classical work of Bagnold (1954) that was discussed in detail in Section 11. As noted there, these flows behave very differently from dilute suspensions at low Reynolds numbers. Their rheological behavior is more akin to that familiar in soil or powder mechanics when a bulk made up of dry granular materials is deformed, and, for this reason, Cheng and Richmond (1978) have termed the behavior of dense suspensions as “granulo-viscous.” Until recently, Bagnold’s (1954) work seems to have gone largely unnoticed by workers concerned with suspension flows, even though some of their experiments produced results similar or related to those observed by Bagnold. Metzner and Whitlock’s (1958) experiments with suspensions of Ti02 spheres of 1 pm diameter in a Storrner rotational viscometer indicated Newtonian, shear-thinning, or shear-thickening behavior depending upon shear rate and concentration. The shear-thickening behavior was prevalent at the higher concentrations and higher shear rates. Rheologists sometimes use the term dilatancy to designate shear-thickening behavior

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325

in which the viscosity increases with shear rate. This rheological dilatancy is not the same thing as the volumetric dilatancy defined by Reynolds (18851, in which the bulk volume of an initially static, closely packed assemblage of solid particles expands when the bulk is sheared. One purpose of Metzer and Whitlock’s (1958) work was to distinguish between these two types of dilatancy. To explain the rheological dilatancy, they proposed some simple physical mechanisms based on the idea that increasing the shear rate progressively increases the tendency for particles to become aligned into distinct laminae. Shear-thickening behavior similar to that noted by Metzner and Whitlock has also been observed by Krasheninnikov et al. (1967). Hoffman (1972) observed discontinuous jumps in viscosity with increasing shear rate in tests with 1.25 pm monosized polyvinyl chloride (PVC) spheres for solids concentrations greater than 50%. Through diffraction studies, Hoffman was able to associate the increase in viscosity at the critical shear rate with an abrupt transition in the types of motion of the particles, from the regular-ordered motion in layers at shear rates less than critical to a disordered flow above the critical shear rate. Using an orifice viscometer, Chong et al. (1971) investigated the viscosities of highly concentrated suspensions of both monodisperse particles and various mixtures of particles having bimodal size distributions. For a given total solids fraction of the mixture, the suspension viscosity varied as the proportions of the two sizes were changed. The viscosity was at a minimum when the proportion of the two sizes was about equal and increased as the proportion of one size became very large or very small. At solids concentrations near 60%, the monosized particles showed an unusual hysteresis behavior. Viscosities were measured in a series of tests in which the flow (shear) rate was increased in steps to a maximum value and then decreased. The viscosities at a given flow rate for an experiment in which the flow rate was decreased from test to test was greater than that in which it was increased from test to test. In a paper that reviews a good deal of his own extensive work on solidliquid systems, Umeya (1978) has distinguished six flow regions, which exhibit Bingham yielding, Newtonian, shear-thinning, or shear-thickening behavior, depending on the value of the shear rate. Cheng and Richmond (1978) have reviewed the rheology of dense suspensions and described experiments carried out at the Warren Spring Laboratory with a number of different types of viscometers. They describe the distinct and sometimes apparently chaotic and anomalous features that may be observed in typical viscornetric experiments with these “granulo-viscous” materials:

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(a) the phenomena of yield stress and slip-stick behavior, (b) the development of normal stresses proportional to the shear stresses, (c) dilation and compaction of the particles depending on the normal stresses and shear rates, (d) formation of rigid no-flow zones and shear bands, (e) fluctuations in the measured stresses associated with the jamming, locking, and release of particles, (f) steplike changes in flow stresses, most likely arising from sudden changes in the arrangement of particle arrays, (g) different stress versus strain-rate curves depending on whether the stress measurements correspond to a loading or an unloading sequence. In an effort to extend Bagnold’s (1954) investigation, Savage and McKeown (1983) performed experiments on dense concentrations of large (0.97-1.78 mm mean diameters), neutrally buoyant, spherical particles sheared in a concentric-cylinder, Couette-flow apparatus in which the inner cylinder rotated and the outer one was fixed. The variations of shear stress with apparent shear rate, concentration, particle diameter, and wall roughness were studied, and the results were compared with Bagnold’s (1954) experiments. Generally, the shear stresses measured in these experiments were larger than those of Bagnold. The differences were attributed to differences in the experimental arrangements; Bagnold’s flexible walled inner cylinder was fixed and the outer cylinder rotated. A strong effect of wall roughness was observed. The higher stresses generated with rough walls implied that particle ‘‘slip’’ may have occurred in the smoothwall tests. The larger stresses might also have been due to an increase in strength of the interparticle collisions (and thus the velocity fluctuations) caused by the roughness. No dependence of stress on particle diameter cr was observed for solids concentrations of about 0.3, but a strong dependence (> cr2) was found at the highest concentrations with the rough walls.

C. HIGHSHEAR-RATE DEVICES FOR DRYMATERIALS The only devices known to the present writer that have been designed specifically to test dry, coarse granular materials at high shear rates are those developed by Novosad (1964), Bridgwater (1972), and Savage (1978). All are annular shear cells, and this geometry seems best suited for testing dry materials. The study by Kuno and Kurihara (1965) of the rheological behavior of powders using a vaned rotational viscometer has

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been critically discussed by Cheng and Farley (1966). Rotating, concentric-cylinder, Couette-flow devices do not appear to be too promising for such tests. Unless the particles are fluidized by an upward air flow, it is likely that gravitational forces would tend to jam coarse granules between the cylinders, making it extremely difficult to develop a shear flow. Although fluidized particle flows will not be discussed here, we mention in passing that Botterill (1975, pp. 105-1 15) has reviewed a number of fluidized bed viscometric studies using Stormer viscometers, rotating paddle wheels and dumbbell elements, Brookfield viscometers, torsion pendulums, etc. Although some of the tests of Novosad (1964) did depart from the quasistatic flow regime, Novosad did not notice any effect of shear rate. In an effort to investigate the matter further, Bridgwater (1972) developed an annular shear cell capable of achieving higher shear-rates than were possible in Novosad’s apparatus. Several glass and plastic particulate materials of different shapes and sizes were tested. Loading weights were used to maintain a constant vertical normal stress at the top of the granular material contained in the annular trough, and torques were measured while the lower half of the shear cell was driven at various rates of rotation. Although the results, as presented, initially suggest only a small “velocity dependence” of the stresses, further consideration reveals that they confirm in part Bagnold’s (1954) results. In the grain-inertia region, Bagnold found that both the dispersive normal stress and the shear stress increased with concentration and shear rate but that the ratio of the shear stress to the normal stress remained almost constant. In Bridgwater’s apparatus, the normal stress was fixed, and, because the bulk solid could expand vertically (decreasing the concentration), little variation of the shear stress with the shear rate is to be expected. The same remark applies as well to Novosad’s (1964) experiments. Savage (1978) realized that if one were to obtain stress data analogous to those obtained by using dense suspensions (Bagnold, 1954), it was important to keep the concentration fixed while the shear rate was varied. His annular shear cell (Figs. 9 and 10) consisted of two concentric, circular, disk assemblies mounted on a fixed shaft. The bottom disk assembly was driven via a belt and pulley by a variable-speed dc motor. The top disk assembly was restrained from rotating by a torque arm connected to a force transducer, but it was free to move vertically. The granular material was contained in an annular trough in the bottom disk and capped by a lipped annular ring on the top disk. The vertical walls of the trough were made as smooth as possible to permit slip there, and the bottom of the trough and the face of the annular ring of the top disk were lined with very coarse sandpaper to generate no-slip conditions on the bottom of the

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Stuart B . Savage @>[Fixed

Steel Base Plate

1 1

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FIG.9. Schematic diagram of annular shear cell of Savage (1978) (see also Sayed, 1981; Savage and Sayed, 1982).

FIG.10. Photograph of annular shear cell of Savage (1978).

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329

trough and the top plate when the material was sheared by rotating the lower disk. A displacement transducer was mounted so as to measure the displacement of the top disk and, hence, to determine the overall bulk density of material in the annular shear region. The apparatus was thus designed to determine shcat' and normal stresses as functions of solids concentration and shear rate. Because of centrifugal effects at the higher shear rates, the shear and normal stresses are not uniform over the width of the annular region, and corrections were applied to the measurements to account for these effects. Care was taken to insure that the shear occurred over the full depth of granular material in the tests that were reported. Typically, the shear layers were about 10 particle diameters thick. Some preliminary tests with 1.8 mm diameter glass beads were described by Savage (1978). Savage and Sayed (1980, 1982) and Sayed (1981) carried out more extensive and more accurate tests that clarified some of the anomalous behavior described by Savage (1978). The extremely rapid increase in stress with shear rate at high concentrations that was observed in Savage's (1978) preliminary experiments was thought to be due to particle jamming. The lack of stress dependence on concentration in the preliminary tests at low concentration was believed to be due to the formation of a free surface adjacent to the inner trough wall (i.e., granular material did not completely fill the rectangular cross section of the annular trough). The later tests were more consistent and presumably were free from these effects. The more recent tests of Savage and Sayed (1980, 1982) and Sayed (1981) were performed with a number of different types of particles: (1) spherical glass and polystyrene beads of diameters between 1 and 2 mm, each sample having a different mean diameter but a fairly uniform size distribution, (2) spherical polystyrene beads having a bimodal size distribution (30% by weight of 0.55 mm particles and 70% by weight of 1.68 mm particles), (3) angular particles of crushed walnut shells, 1.19 mm mean diameter. In the experiments with monosized spherical particles at lower concentrations and higher shear rates, both shear and normal stress depended on the square of the shear rate. These tests with particles of different diameters and materials also indicated that the stresses were dependent on particle mass density and the square of the diameter as proposed by Bagnold (1954) and Eqs. (14) and (15) for the grain-inertia regime. At

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higher concentrations and lower shear rates, dry friction between particles becomes more important, and the stresses were found to be proportional to the shear rate raised to a power less than two. The flow evidently corresponded to the transitional regime described in the introductory discussion of the present review dealing with modes of flow for dry granular materials. All tests showed a strong dependence upon solids concentration. Some typical results for polystyrene beads are shown in Figs. 1I and 12. In Fig. 1 1 the nondimensional normal stress q2/ppgc is plotted versus a nondimensional apparent shear rate &/g U/H ( U is the linear velocity of the lower disk assembly at midannular radius and H is the height of sheared granular material in the annular trough). Figure 12 shows the ratio of shear to normal stress (tan 40) versus &/g U/H for several values of solids concentration Y. Except for a few tests at small shear rate, the angle $0 is larger than the quasi-static friction angle 4. There are not extreme variations in tan C#JDwith either Y or shear rate, but there is a small but clear increase in tan 4Dwith decreasing Y. This latter trend is opposite to what is usually observed in quasi-static tests on granular soils, but it is consistent with the experiments on free surface flow down inclined chutes. These steady, "uniform" flows are observed for a range of bed I ' '

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The Mechanics of Rapid Granular Flows

33 1

inclination angles 5 , which for equilibrium must equal 4 ~An. increase in 5 (i.e., &), in these experiments is accompanied by a decrease in solids concentration. It was also observed that the flow in the shear cell was not as twodimensional as might have been hoped, but that the primary shear flow was accompanied by a weak secondary flow in which particles moved radially inward at the top of the material and radially outward at the bottom of the rotating trough. Typical stresses measured in tests with the beads having a binary size distribution are shown in Fig. 13. At the lower concentrations the stresses approach a square dependence on shear rate, but with increasing concentration and decreasing shear rate there appears to be a tendency for the sttebses t o become rate independent. Apparently, the smaller particles fit into the spaces between the larger ones, increasing the dry Coulomb friction and reducing the collisional momentum transfer, thereby increasing the rate-independent components of the total stresses. Also evident at the higher concentration shown in Fig. 13 is a hysteresis, in which the stresses at a given shear rate are higher during the loading sequence than 0.7

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(fYh0/~

FIG. 13. Nondimensional normal stress versus nondimensional shear rate for polystyrene beads having a bimodal size distribution (30% by weight of 0.55 mm spheres and 70% by weight of 1.65 mm spheres) (Sayed, 1981).

during unloading. Although the two sizes were well mixed at the beginning of a test sequence, it was observed at the completion that segregation had occurred. The fines driven by centrifugal forces had percolated through the coarse particles and concentrated at the outer radii. Note that the shear and normal stresses on the upper annular lid are not distributed uniformly in the radial directions but are higher at larger radii. Because the material at the outer radii changed during the course of a complete test from one that was a mixture of 30% fines and 70% coarse to one that was primarily fines, and because the stresses depend roughly on the square of the particle diameter, it might be expected that the stresses at a given shear rate would be higher at the beginning of a test sequence than at the end. This appears to be a plausible explanation for the hysteresis observed with the binary size mixture. However, it should be noted that there were some tests with the monosized particles in which the stresses were higher on unloading than on loading, and this behavior is harder to explain. The tests with the angular ground walnut shells showed the same general trends, stresses proportional to the square of shear rate at lower

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333

concentrations and a weaker dependence at higher concentrations and lower shear rates. Some tests (Savage and Sayed, 1980) were performed with lesser amounts of material in the trough such that the shear layer (gap height H ) was only about 5 to 6 particle diameters thick. Stresses in these tests at high concentrations were much more erratic; large increases or decreases in stresses occurred when H was changed by fractions of a particle diameter, even though the overall mean solids concentration was kept constant. Finite particle size effects were evidently significant. At a given concentration, the gap height H must be such that particles can form into distinct layers if they are to be sheared easily at low stresses. Otherwise, jamming or locking together of groups of particles will occur, resulting in an increase in the stresses developed during shear of the bulk.

D. SOMEREMARKSCONCERNING HIGHSHEARRATEV~SCOMETR~C EXPERIMENTS About the only detailed shear cell measurements of stresses for various concentrations of dense suspensions or dry granules at shear rates high enough to be in or to border on the grain-inertia regime are those of Bagnold (1954), Savage and McKeown (1983), Savage and Sayed (1980), and Sayed (1981). Although the general trends of all these data are in agreement, there are differences among all of them with regard to the specific levels of stresses that were measured. If we plot the stresses for a given solids concentration versus p p ~ 2 ( u 1 , 2as ) 2 is , suggested by Eqs. (4) and (5) (Bagnold, 1954), the data do not collapse. The stresses measured by Savage and McKeown (1983) for polystyrene spheres in water are higher than Bagnold’s (1954) data for lead sterate wax beads in water. The stresses determined for polystyrene, glass, and ground walnut shells in air (Savage and Sayed, 1980; Sayed, 1981) are all higher than those measured by Bagnold (1954) and Savage and McKeown (1983). To help explain these results we can make an analogy with the kinetic theory of gases, in which the viscosity increases with temperature. The magnitude of the granular particle velocity fluctuations corresponds to the “temperature.” It seems plausible that an interstitial liquid, with its high density and viscosity, is more effective than an interstitial gas in damping and dissipating the particle velocity fluctuations, resulting in lower stresses for the case of a liquid interstitial fluid. Evidently the interstitial fluid plays a more important role than Bagnold (1954) suggested.

334

Stuart B . Savage

The particle material properties, the coefficient of restitution e and surface friction coefficient p also affect the energy dissipation and thus the magnitude of the velocity fluctuations and stress levels in a given shear flow. It is quite likely that the wax beads used by Bagnold (1954) had a lower e than the polystyrene beads used by Savage and McKeown (1983), which would result in lower velocity fluctuations, and hence the lower stresses developed by the wax beads. These kinds of arguments follow from the microstructural theories of Kanatani (1979a), Ogawa et al. (1980), Savage and Jeffrey (1981), Shen and Ackermann (1982), Shen (1982), Jenkins and Savage (1982) and Lun et al. (1982) and were alluded to in the introductory discussion of granular flow regimes in the present article. Although the work of Shen (1982) and Shen and Ackermann (1982) does not treat the dynamics in the most sophisticated way, this theory is the most general in the sense that it accounts for the dissipation caused by the interstitial fluid, as well as the material properties of the solid particles e and p. Shen and Ackermann attempted to predict the experimental results of Bagnold (1954), the smooth-wall tests of Savage (1978), Savage and Sayed (1980), and Sayed (1981). Although their theory predicted stresses that were too low by about one order of magnitude, the’relative differences between the various sets of data were predicted with some degree of success. It is probably correct to say that all of the previously mentioned experimental devices were designed as viscometers to obtain data that could be applied directly to other flow situations as a rheologist might do after determining the shear viscosity. But both the experimental results and the microstructural theoretical analyses indicate that things are considerably more complicated than one might have initially guessed. A large number of factors besides just shear rate and solids concentration affect the generation of stresses: the inelastic and surface frictional properties of the particles as well as their diameter, particle angularity and size distributions, the degree of wall roughness and rigidity and their effects on the generation or damping of velocity fluctuations, the geometry of the test device and its effect on the likelihood of particle jamming and the formation of locked rigid zones, etc. In the future one should perform experiments, not so much with the intent to determine unique viscosity coefficients, but more to build up a catalog of flow “case studies.” In studying a particular flow, one should attempt to acquire information that can lead to a better understanding of the flow dynamics, which can be used for the further development of theories and perhaps to acquire data that can be extrapolated for the prediction or interpretation of flows under other conditions. The plain search for simple and distinct viscosity coefficients is naive, in the same sense that the idea of a definitive value for the eddy

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335

viscosity for a turbulent Newtonian fluid is naive. This is not to say that further experiments are not sorely needed; in fact, very little data presently exist. Although the theoretical work is now progressing rapidly and can indeed suggest some of the pertinent variables that should be measured, experiments are needed for verification of these theories as well as to reveal their faults and suggest areas for their improvement.

V. Theories for Rapid Granular Flows As described in Section 11, a number of possible modes of granular flow exist. Table I listed some of the limiting flow regimes; each is characterized by a single, dominant mechanism responsible for the generation of bulk stresses: (a) Macro-viscous regime. The viscous effects of the interstitial fluid and the interactions of the solid particles with this fluid determine the stresses.

(b) Quasi-static regime. Dry friction and interlocking between particles are of primary importance. By comparison with the interparticle contact forces, grain inertia effects are negligible. (c) Grain-inertia regime. The inertia associated with the individual grains is fundamental. At low concentrations portions of the stresses are due to the transport of momentum by particle translation from one shear layer to another. At higher concentrations, translation of individual particles between layers is less likely and momentum transport by collisions between particles is most important. In a general granular flow, stresses could be generated by all of the preceding mechanisms, and a complete theory should be able to describe this general flow as well as each of the limiting flow regimes. No one, as yet, has been so ambitious as to attempt to develop such a theory; most of the theoretical models proposed involve only one of the mechanisms for stress generation, but a few have included two. The emphasis in the present section will be on theoretical models to describe rapid granular flows in or close to the grain-inertia regime. Currently these theories are undergoing rapid development. Theoretical work dealing with effects usually associated with the quasi-static and macroviscous regimes will only be mentioned as it is included as a part of a particular rapid granular flow theory. In fact, there is very little theoreti-

336

Stuart B . Savage

cal work on macro-viscous flows at high concentrations available to be reviewed in any case. Although a large body of theoretical work on flows of suspensions exists, this work deals almost entirely with dilute suspensions having concentrations well below that typical in granular flows. For the quasi-static flow regime, quite a different situation exists; in this case an extensive body of theoretical work, stemming largely from soil mechanics, has been completed. Mroz (1980) and Spencer (1981) have reviewed this work, and there is little point in further review here. However, it is worth noting Spencer’s conclusion that none of the theories he reviewed were above criticism or could explain all the available experimental data. He suggested that different theories would be needed to explain the behavior of different materials or even the same material subjected to different conditions. Spencer’s comment was directed at theories dealing with quasi-static flows only; hence, a general theory capable of treating macro-viscous, quasi-static, grain-inertia flows, and all the transitional flows as well, seems beyond our present reach. For the sake of presentation, we shall begin with a discussion of the various continuum theories that were developed without any detailed reference to the microscopic particulate nature of the material. Microstructural theories that are based on the individual particle dynamics are discussed in the Section V,B.

A. CONTINUUM THEORIES I . Goodman-Cowin-Type Theories Goodman and Cowin (1971, 1972) developed a continuum theory intended for situations where the stress levels are less than 10 psi (Cowin, 1974a,b)and where interstitial fluid effects are negligible. The stress tensor was obtained by the linear superposition of two parts: TO, a rate-independent part that was related to the solids fraction v and its gradients, and T*, a rate-dependent part that was assumed to behave as a Newtonian fluid. On the basis of all the experimental evidence beginning with Bagnold’s (1954) experiments, this latter assumption is inappropriate, but as Cowin (1974a) pointed out, the theory can be extended without affecting the rateindependent part of the stress tensor. The theory has been linearized and specialized to the case of incompressible granules (Goodman and Cowin, 1971, 1972). For limiting equilibrium the theory exhibits certain characteristics common to Mohr-Coulomb materials in that the shear and normal stresses on a plane can be related in the form of Eq. (6). For cohesionless incompressible grains, the rate-independent part of the stress tensor has

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337

the form (retaining their notation in which tensile stresses are positive)

TO=F V = - P81J - 2 a V , I v 7 J r (24) where p is the pressure and a depends upon v , p , and vc, the value of v corresponding to the critical voids ratio. The theory has been applied to the problems of granular flow down an inclined plane and between vertical parallel plates (Goodman and Cowin, 1971). In addition to the assumption of Newtonian behavior for the rate-dependent part of the stress tensor, the solutions to these problems were based on a special case of the linear theory that contains certain inconsistencies [see Jenkins and Cowin (1979) and Savage (1979)l. Because of the Newtonian flow assumptions, Goodman and Cowin (1971) found that steady, fully developed, uniform depth, open-channel flows were possible for all channel inclinations greater than the angle of repose of the material. As described in Section 111, this is not observed in experiments. Their theory also required a plug flow in the central region of flow between parallel plates and a jump in velocity gradient at the edge of the plug region to satisfy the steady equations of motion. Savage and Cowin (Savage, 1979) extended the theory of Goodman and Cowin to incorporate the nonlinear shear-rate effects evident in Bagnold’s (1954) experiments. The theory was developed in a rather different manner to avoid the restrictions inherent in the original Goodman and Cowin approach. The stress tensor of cohesionless materials was assumed to be of the general form

T

=

aoI + a l D + a2M + a3(MD + DM)

+ a4D2 + as(MD2 + D2M) (25)

where M

= VJV,,

D =

4(14~,,

+ uJJ

is the rate of deformation, u, are the velocity components, vo is a reference value of the solids fraction v , and ao,al, ... , a5 are functions of YO,v , tr M, tr D, tr D2, tr MD, and tr MD2. The quasi-static part of the stress tensor Totook the same form as in the original theory of Goodman and Cowin, i.e., that of Eq. (24). The form assumed for the rate-dependent part, T*, was the simplest representation that would reasonably approximate Bagnold’s ( I 954) experimental data in the grain-inertia regime. Thus, they took for the case of isochoric flows T* = 4pol2I

+ 4/~11121~”D

(26)

where Z2 = &(tr2D- tr D-D) is the second principal invariant of the rateof-deformation tensor. The coefficients po and p l are functions of v

Stuart B . Savage

338

assumed on the basis of curve fits to Bagnold’s (1954) experiments to be of the form

Po

Po

=

PI

= PI

(s) -

8

uo (&)

8

uo;

where vmcorresponds to the densest possible packing of particles, urnis the maximum value of the solids fraction at which continued shearing can occur, uo is the solids fraction at which fluidity occurs (the concentration where the residual shear resistance at zero shear rate disappears), and Po and PI are constants. For a simple shear flow u I ( x 2 )Eq. , (26) yields a rate-dependent normal stress T3: in addition to the normal stresses TTl.and T&. Savage (1979) performed a simple experiment to indicate the presence of such stresses. A circular cylinder roughened by gluing glass beads to its surface was rotated to form a vortex flow in a mixture of glass beads and a solution of methanol and bromoform with the same density as the beads (Figs. 14a; the 3-direction is along the axis of the rotating cylinder). If the “fluid” behaved in a Newtonian fashion, then we should expect the free-surface profile gradually to decrease close to the rotating cylinder, as shown in Fig. 14b. However, when normal stress effects are present, then we might expect that they would be evidenced by an increase in the free-surface level near the cylinder, as shown. Assuming that the normal stresses in all three directions are of the same magnitude and using Bagnold’s (1954) data for A = 12 with a cylinder radius of 1 cm and rotation rate of 1000 rpm, we might expect that the free-surface level should be increased above that for Newtonian behavior by 1/2 cm next to the rotating cylinder. Figure 15 is a photograph of the experiment in operation. A bump in the free surface of the kind anticipated is evident. The resulting constitutive equations (24), (26), and (27) for the stresses were applied to solve two simple boundary value problems, the openchannel flow down a rough inclined chute and the flow down a roughwalled vertical channel. The general features of the predicted velocity profiles and flow characteristics were in agreement with the experiments carried out in the same study (Savage, 1979). Further modifications of the Goodman-Cowin theory have been used by Passman et al. (1978) and Nuziato et al. (1980) to consider flows in vertical channels and down inclined planes and by Passman et al. (1980)

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339

with normal stress effects

Glass beads in solution of bromoform and methanol fa)

FIG. 14. Experiment to indicate presence of normal-stress effects. (a) Rough cylinder rotating in beaker containing neutrally buoyant glass beads suspended in a solution of bromoform and methanol. (b) Anticipated velocity profile due to rotating cylinder and increase in free-surface height due to normal-stress effects. On the basis of Bagnold’s (1954) experiments it is estimated that Ah = 4 cm assuming A = 12, w = 1000, and r = 1 cm (Savage, 1979).

to study the Couette flow between parallel plates. Both of these two later analyses assumed forms for the coefficients appearing in the expressions for the free energy [including, for example, a in Eq. (24)] and the ratedependent stress that were somewhat different from those used by Goodman and Cowin (1971) and Savage (1979). Nunziato et af. (1980) and Passman et al. (1980) assumed that the rate-dependent stresses depended in a linear way on D but that the shear viscosity coefficient depended on v in the same manner as expressed by Eq. (27). This modification does not repair the basic fault of Goodman and Cowin’s T* that the ratio of shear to normal stresses acting on the shear surfaces is not limited. Thus the theory of Nuziato et al. (1980) still predicts fully developed “uniform” free-surface flows down chutes for any bed inclination, whereas they are observed experimentally for only a small range of bed slopes. The ratedependent part of the stress tensor should behave in a manner similar to Eq. (26) suggested by Savage and Cowin (Savage, 1979) if the observed behavior for flow down inclined chutes is to be predicted. It should be noted that even the form given by Eq. (26) may be oversimplified. For example, the microstructural theory of Savage and Jeffrey (1981) in gen-

340

Stuart B . Savage

FIG. 15. Photograph of experiment showing bump in free surface near rotating cylinder due to normal-stress effects (Savage, 1979).

era1 predicts normal stress differences. Whereas the general expression [Eq. (25)] given by Savage and Cowin for T yields normal stress differences, the simplified form [Eq. (26)] does not. On the other hand, for shear flows of smooth, nearly elastic particles the normal stress differences (Savage and Jeffrey, 1981) are not large, and they may not be an essential component of such flows. The fact that all of the preceding continuum theories have exhibited some gross features that are qualitatively similar to experimental observations should not be interpreted as a verification of Eq. (24) for the rateindependent part of the stress tensor. Such similarities that do exist could arise simply because the stress tensor has been divided into a rate-dependent and a rate-independent part. The physical arguments for the dependence of To upon gradients of v are at best obscure. Furthermore, it is easy to visualize situations for which Eq. (24) does not predict shear stresses that are most certainly anticipated on physical grounds. For ex-

The Mechanics of Rapid Granular Flows

34 1

ample, in a region of uniform bulk density, Eq. (24) can yield only an isotropic pressure and no shear stresses. Also, for fully developed, uniform flows down inclined surfaces (see Fig. 4) in which there are variations in u normal to the bed but no variations in the streamwise direction, Eq. (24) predicts no shear stresses on the shear surfaces that are parallel to the inclined bed. Savage (1979) has pointed out another related example of a Rankine stress state at limiting equilibrium in which this theory is incapable of providing the stresses that must exist for equilibrium.

2 . Rate-Dependent Plasticity Theories Because of the inconsistencies just described, McTigue (1979) and Sayed and Savage (1983) replaced Eq. (24) by expressions for the rateindependent part of the stress tensor resembling those used in the plasticity theories of ideal soils; thus, To depended upon v and not explicitly upon the gradients of u. McTigue’s (1979) stress tensor was written (again with tensile stresses taken as positive) in the form

To = -pa0

+ ( K . & ~ / ~ + 4Av2Z31/2) Do + 4Bv2DikDkj

(28)

where 1; = +Z: - Z2,11 = D j i ,Z2 = +(DjiDii- DikDki),A and B are material constants, and K M and p are functions of u and the particle velocity fluctuations. If the rate-dependent part of the stress tensor vanishes, then the principal axes of stress (the rate-independent component now comprises all the stress) coincide with the principal axes of strain rate. Sayed and Savage (1983) used a form for the rate-independent stress that was slightly different from that of McTigue (1979). They assumed a form based on the quasi-static theory of Spencer (1964) in which the principal axes of stress and strain rate need not coincide but may be inclined at any angle between ?4/2, where 4 is the internal friction angle of the granular material. Both theories used nonlinear forms for the ratedependent part of the stress tensor that were consistent with Bagnold’s (1954) experiments. Both McTigue (1979) and Sayed and Savage (1983) applied their theories to solve for the free-surface two-dimensional flow down a rough inclined chute. These analyses gave results that were similar in form to both the analysis of Savage (1979) based upon a Goodman-Cowin-type theory and the experimental measurements of Savage (1979). Uniform flows were possible only over a limited range of bed inclinations and u was predicted to increase with depth. At the lower bed inclinations the velocity profile had a separation-type inflection point and it filled out and be-

342

Stuart B . Savage

came more blunt with increasing bed slope in both analyses. Sayed and Savage (1983) investigated the effects of sidewall friction during the flow down an inclined chute of rectangular cross-section. The vertical sidewalls were considered to be sufficiently rough to generate finite shear stresses at the wall but not rough enough to cause the material to shear on surfaces other than those parallel to the bed. Thus the stress field was three-dimensional but the velocity field was only two-dimensional. Similar flows can be observed experimentally (see Section 11). Using their annular shear cell experiments (Savage and Sayed, 1980; Sayed, 1981) to determine appropriate material constitutive coefficients, Sayed and Savage (1983) were able to obtain reasonably good predictions of the chute experiments of Savage ( 1979). Increasing the sidewall friction decreased the flow-rate, but there were relatively small changes in the velocity profile unless the sidewall friction was large or equivalently the channel was very narrow. However, the shape of the velocity profile was found to be quite sensitive to the form of the coefficients POand pl analogous to Eq. (27).

3. Multitemperature Theories Ogawa and Oshima (Ogawa and Oshima, 1977; Ogawa, 1978; Oshima, 1978; Oshima, 1980) have derived theories using the usual formal procedure of continuum mechanics, but they specifically recognized the discrete nature of granular materials. They defined different kinds of temperatures, one that is the usual temperature associated with the thermal fluctuations of the molecules making up each grain, and another that is related to the “random” translational and rotational fluctuations of the individual grains making up the mass of granular material. The two temperatures of the second kind are thus proportional to (v2) and (o*), which were discussed in Section I1 and appeared in Eqs. (8) and (9). The equations of conservation of mass, momentum, and energy were derived along with the forms of the constitutive equations. As is usual, the constitutive equations, in their most general form, are extremely complex, and particular assumptions (for example, linearization) must be made to simplify them to the extent that the solution of boundary value problems is at least manageable. Blinowski (1978) proposed an approach related to that of Ogawa and Oshima, in which he regarded the material as a continuum and expressed each of the flow field variables-the bulk density p , the velocity components ui, and the stress components TU-as the sum of a mean and a random fluctuating part. The approach closely parallels the statistical

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343

description of the turbulent flow of fluids involving Reynolds’ stresses and similar correlations. Balance laws similar to those obtained by Ogawa and Oshima are derived, but, again, some assumptions about the constitutive equations are necessary before the theory can be applied further. In their review of granular flow theories, Jenkins and Cowin (1979) discussed balance laws of the kind contained in the theories of Ogawa, Oshima, and Blinowski and proposed simple forms for the constitutive equations for stress tensor, the flux of fluctuation energy, and energy dissipation. The stress tensor was a function of both mean shear rate and particle velocity fluctuations. The evolution equation for the kinetic energy associated with the particle velocity fluctuations expressed a balance between creation, diffusion, and dissipation of fluctuation energy. Because of the possibility of diffusion of fluctuation energy, this constitutive theory is a nonlocal one.

B. MICROSTRUCTURAL THEORIES None of the foregoing continuum theories can yield the detailed form of the constitutive equations; one must turn to experiments or other means to deduce the explicit form of these equations as well as the phenomenological coefficients associated with them. Possible sources for such information are microstructural theories that analyze in detail the dynamics of individual particle collisions. By suitable statistical averaging one can, in principal, determine all the continuum properties explicitly.

1 . Simple Particle Dynamics Theories The first rudimentary attempt of this kind was Bagnold’s (1954) theory for the grain-inertia regime that was described in Section 11. Bagnold’s (1954) analysis was modified in an attempt by Shahinpoor and Siah (1981) to account for rotations arising from collisions between rough particles. McTigue (1978, 1979) endeavored to improve on Bagnold’s simple model by considering collisions arising from the shear of a random arrangement of particles. His approach follows Marble’s (1964) suggestive analysis of particle collisions in a one-dimensional flow of a gas containing solid particles of two different sizes and Soo’s (1967, pp. 197-216) closely related analysis of stresses due to collisions in a particle cloud subjected to shear. McTigue’s model is highly simplified in that he ignores particle fluctuations, constrains the particles to translate with the mean motion, and neglects higher-order effects of concentration upon collision fre-

344

Stuart B . Savage

quency. Although he shows that the stresses depend upon the square of the shear rate, the predicted stresses are between one and two orders of magnitude lower than Bagnold’s measurements, and the dependence of the stress on the solids concentration is poorly represented. Ogawa, Umemura, and Oshima (Ogawa, 1978; Ogawa et al., 1980) employ a rather idealized model of the particle fluctuations and collision dynamics to determine the stress tensor Tijand energy dissipation rate y for sticky, rough, inelastic particles. Dissipation is determined by considering the collisions of spherical particles (having random but isotropic velocity fluctuations) with the walls of an imaginary collision sphere of radius b,

= (T(v*/u)’/3

(29)

where Y* is said to correspond to a “packed state” (for example, v* = 0.7405 for a regular array of closely packed spheres). The equation for conservation of fluctuation specific kinetic energy k = 4 (v2)was written as p dkldt = TijDij

- y - 9i.i

(30)

where q is the flux of k . This equation relates the local rate of change of fluctuation energy to its production by the mean flow, its dissipation into molecular thermal energy, and the diffusive energy flux from neighboring points of the flow. Although flux terms appear in Eq. (30), they were later neglected in the determination of the constitutive equations. By employing a statistical averaging, they eventually obtained the constitutive equations for stress and dissipation

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345

and (u2)”’ is the rms fluctuation velocity, e is the coefficient of restitution, p is the surface friction coefficient, and CY is an accommodation coefficient, defined as the fraction of particles that adhere to a surface during collisions. For dry, nonsticky particles we can take CY = 0. This constitutive model was used to investigate the free surface granular flow down a rough inclined surface. Kanatani (1979a, 1979b, 1980) has developed a model that is in some respects similar to that of Ogawa et al. (1980) but that includes the particle rotational as well as translational fluctuation kinetic energy. He introduced an undetermined constant relating the fractions of energy that are partitioned into rotational and translational fluctuation kinetic energy. The model only accounts for energy dissipation due to the friction coefficient p , and also has the unrealistic characteristic that the shear stresses vanish for smooth particles (i.e., when p = 0). The resulting constitutive equations were again applied to the problem of gravity flow down a rough incline. The theories of Ogawa et al. (1980) and Kanatani (1979a, 1979b) as applied to the inclined gravity flow problem were incomplete in the sense that all the flow field variables were not explicitly determined. Kanatani assumed a linear velocity profile and determined the bulk density profile. Ogawa et al. (1980) assumed a linear bulk density profile and determined the mean and fluctuation velocity profiles. Ackermann and Shen (Ackermann and Shen, 1982; Shen and Ackermann, 1982; Shen, 1982) proposed a theory that not only considers the energy dissipation due to the inelasticity and surface friction of the particles, but also includes the dissipation in the viscous interstitial fluid by accounting for the drag of each spherical particle as it moves through the fluid contained within an “effective cylinder” formed by the neighboring spheres. They considered the case of a simple shear flow ul(x2)and in the later versions of the theory (Shen, 1982; Shen and Ackermann, 1982) they determined the magnitude of the velocity fluctuations by using the balance equation (30) for fluctuation kinetic energy. For the case of simple and the rate of shear this reduces to a balance between shear work T12~I.Z dissipation per unit volume y . To simplify the collision analysis a number of approximations were made. First, it was assumed that tan-I[p(l + e ) ] << 1 and that ( ~ ( 1+ ~ / A ) u ~ , ~ (<< u ~ I) , ~where ~ ~ A is defined by Eq. (2). In the dissipation calculation they neglected the presence of the mean shear and only considered the rms fluctuation velocity (u2)Il2.To determine the shear stress T12they considered the collisional momentum transfer, assuming the particles move with the mean shear velocity, and neglecting the fluctuation velocities. To determine the normal stress, the momentum transfer was calculated by neglecting the mean shear and considering only the rms fluctuation velocity. The following expressions for the stresses

346

Stuart B. Savage

are given by Shen (1982) for the case of simple shear A T12 = C0ppu23 [(l

A

3

+ eY(0.053 + 0.081p)) (m) /T]

112

utq2 (34)

where

pf is the interstitial fluid density, and CD is the drag coefficient for the

spheres moving in the interstitial fluid. Shen took CD = I , and Co = voc= 0.7405 [see Eq. (3)] corresponding to the densest possible packing for identical spheres. Shen (1982) also considered the two-dimensional shear flow of circular disks in a more general type of analysis that relaxed the l / A ) ~ , , ~ / ( v ~<< ) ’ / ~1 and treated the effects of assumption that a(1 surface friction during collisions in a more accurate way. When the theory for spherical particles was compared with the experiments of Bagnold (1954), Savage (1978), and Sayed (1981), the theoretical predictions were found to be about one order of magnitude low. Shen suggested that the difference might be due to clustering or grouping together of particles such that the effective diameter of the particles might be acu instead of merely u. Hence, in the expressions for the stresses ) ~ .best fit to the given by Eqs. (33) and (34), u2was replaced by ( a , ~ The experimental data was achieved by taking ac = 3.16 (see Fig. 16). Although clustering of particles is probable at high concentrations, it seems far less likely at some of the lower concentrations shown on Fig. 16, and the discrepancies between theory and experiment may be due to approximations made in the analysis. For example, Shen calculated u(l + I / A ) u I , ~ / ( u ~ to ) ” ~be between 0.68 and 3 for conditions corresponding to the experiments shown in Fig. 16. This is inconsistent with the assumption that ~ ( +l l/h)~1,2/(v~)”~ << 1 used in the analysis. It should be noted that the theories of Jenkins and Savage (1982) and Lun et al. (1982), which treat the collision dynamics in a more sophisticated way, predict stresses that are about one order of magnitude higher than the theory of Shen and Ackermann. Nevertheless, one should not underrate the degree of success that has been achieved in Fig. 16 in collapsing the data from these disparate experiments which involved particles of different densities, inelastic and frictional properties, and vastly different interstitial fluids.

+

The Mechanics of Rapid Granular Flows I I 1 1111 Wax i n Watc [e=.Z, P=O.Z]

I

104

----

I I I1111

A

4

’

I 1 1 I1111 I I I I IIll P o l y s t y r e n e i!, A i r (Sayed 1981) [ e = o . 9 , u=o.z]

0

7.1 = A

o

6.5

(Bagnold 1954)

7.6 = X 6.7 4.1 3.1

A

a

I

347

-

-

2.1

P o l y s t y r e n e i n h’ater N

E

\

z

0

Q)

c

0

---

-

-

D 6.8 = X

rn 4 . 5 0

3.0

G l a s s i n A i r (Sayed

Ce=i.o,

1981)

0

u=.zI

2 . Dissipationless Dense Gas Approach

Savage and Jeffrey (1981) made the first attempt to make more substantial use of the ideas contained in the previous theoretical work that has dealt with dense gases [for example, see Chapman and Cowling (1970)l. They proposed a theory to determine the stress tensor for granular material in a rapid, simple shear flow by supposing that binary collisions between smooth, perfectly elastic, uniform spheres were responsible for most of the momentum transport. The single-particle velocity distribution function was taken to be locally Maxwellian. They assumed a plausible

348

Stuart B . Savage

modification to the radial distribution function of Carnahan and Starling (1969) to account for anisotropies in the spatial distribution of the particles. The components of the stress tensor were expressed as integrals ~’~. soinvolving the nondimensional parameter R = U U I . ~ / ( U ~ )Asymptotic lutions were obtained for both small and large values of R, and the integrals were evaluated numerically for intermediate values of R. For small R the normal stresses were isotropic and depended upon (v2).With increasing R normal stress differences appeared, and for R + 33 both normal stress and shear depended explicitly upon the square of the shear rate. The substance of this work is that it suggests a relatively simple, but more appropriate and more accurate, treatment of the particle collisional dynamics than the microstructural theories just described. As Savage and Jeffrey (1981) noted, the weak link in their analysis was the inability to calculate the appropriate value of R. The parameter R is directly related to the dissipative properties of the system, and the determination of R requires, for example, the consideration of inelastic and/or rough particles that can provide mechanisms for energy dissipation. If we consider the case of a granular material subjected to a shear stress T I 2such as to generate a simple shear u I ( x z ) ,equilibrium will be established when the fluctuations (v2) are such that X = T12ul,2.For smooth, nearly elastic particles e = I , the velocity fluctuations would increase to quite large values before the required dissipation was achieved, whereas for quite inelastic particles e << I , or for large interstitial fluid viscous losses, the resulting (u2) would need to be much less. Thus, slightly dissipative systems with e = I correspond to small values of R; highly dissipative systems with small e correspond to relatively large values of R. Jenkins and Savage (1981) drew an analogy between granular and turbulent flows and wrote down an equation for the kinetic energy of the particle velocity fluctuations in the form of Eq. (30), i.e., similar to those proposed by Blinowski (1978) and Ogawa et al. (1980). The possibility of calculating the detailed form of the various terms in this energy equation by following the approach of Savage and Jeffrey (1981) was noted. 3 . Dense Gas Approach for Inelastic Particles

Extensions of the Savage and Jeffrey analysis to incorporated energy dissipation have been performed by Jenkins and Savage (1982) and Lun et al. (1982). Jenkins and Savage (1982) developed a theory applicable to general deformations of a granular material made up of smooth, nearly elastic spherical particles. As in the work of Savage and Jeffrey, it follows the general approach used for the theory of dense gases involving collisional transfer of various properties (Chapman and Cowling, 1970), but in addition it includes energy dissipation due to inelastic collisions.

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349

The analysis considers binary collisions between particles of mass m, located at rl and r2, and moving with velocities cI and c2. The change in kinetic energy during a single collision is found to be

AE = (-m/4)(1

-

e2)(k-c12)2,

(36) where c12 = cI - c2 and k is the unit vector along the line of centers at contact from the first to the second particle. The ensemble average of some function $ was defined as

where n is the number density of particles and f(l) (rl, cl; t ) is the usual single-particle velocity distribution function from kinetic gas theory. A general form of the Maxwell-Chapman transport equation (Present, 1958) appropriate for the case of inelastic collisions was derived. This was used to generate the continuum hydrodynamic forms of the conservation equations for mass, linear momentum, and translational fluctuation kinetic energy.

dpldt = -p V-u p(duldt) = pb - V*p gp(dT/dt) = -p: V u - V*q - 7

(38) (39)

(40)

with

where p is the bulk mass density, u is the bulk velocity, 3T is the fluctuation-specific kinetic energy, b is the external force per unit mass, p is the pressure tensor made up of a kinetic or diffusional part pk and a collisional transfer part pc, q is the flux of fluctuation energy consisting of a kinetic part qk and a collisional part qc, and y is the collisional rate of dissipation per unit volume. The expressions for pc, qc, and y involve integrals analogous to the expression given by Savage and Jeffrey for the stress tensor. The pressure tensor p in this theory includes momentum transport by translation of particles between shear layers Pk as well as a contribution pc owing to the instantaneous collisions between particles. A general granular flow could also involve stresses arising from enduring contacts be-

Stuart B . Savage

350

tween particles, so in the most general kind of flow we might regard p = pij as making up a part of the total stress tensor T~ appearing in Eq. (7). The single-particle velocity distribution functions are assumed to be Maxwellian about the mean velocity; for example,

and the complete collisional pair distribution function was expressed as the product of a pair distribution function g(rl,r2; t) and the two Maxwellian velocity distribution functions

Because of the presence of a mean shear, the spatial distribution of particles is anisotropic; a particle experiences more frequent collisions on its “upstream”-facing quadrants in which the mean shear acts to “bring particles together,” than on its downstream-facing ones. Part of the spatial anisotropy is incorporated with the expression for g(rl, r2) and part is included by taking the Maxwellians about the local mean velocity. By /~ 1 , assuming e = I , which is equivalent to taking crkk:V U / ( C ~ ) I<< dimensional arguments yield an expression for g ( r l , r2) of the form

If the coefficient a is taken to be unity and go(u) is the radial distribution function at contact proposed by Carnahan and Starling ( 1969)

then Eq. (44) is the same as a linearization of the g(rl, r2) used by Savage and Jeffrey (1981). After expanding the Maxwellians about the point of contact and taking the spatial gradients to be small (i.e., again taking crkk :V u / f l < < I , etc.) the integrals for qc,y, and pc were evaluated to yield the constitutive equations g,

= -K

y = 6(1

pc =

vT -

(46)

e)K

[T -

[KU-I(TTTT)I12

-

( 4 +~ $a)c~(T/~r)’/~ tr D ] / d bK(2

(47)

+ a)tr D]I - gK(2 + a ) D ,

(48)

+ e)pcT(T/7r)1’2

(49)

where K

= 2ugo(u)( I

The Mechanics of Rapid Granular Flows

35 1

and

D

+ Uj,i).

= k(~i,j

For concentrations of interest, the kinetic terms Pk and q k are assumed small compared with the collisional contributions pc and qc. After substituting the expressions (46), (47), and (48) for q,, y , and pc into the conservation equations (38), (39), and (40), there result five equations for the five unknowns, p (or v), ui,and T (or (u2)). Note that in this method, no attempt was made to determine the distribution functions by solving the Boltzmann equation or generalizations of, or approximations to, it as is done, for example, in the Chapman-Enskog approach. The distribution functions themselves are not of real practical interest; it is the average properties such as v, ui,and T that one seeks in the solution of boundary value problems. Thus, Savage and Jeffrey (1981) and Jenkins and Savage (1982) used simple but plausible assumptions for the distribution functions in which v , u, and T were regarded as parameters. These assumed forms for the distribution functions were used for the calculation of the constitutive equations, which were subsequently used in the overall conservation equations for the solution of boundary value problems. In a sense, this approach is analogous to that used in the momentum integral analyses of boundary layers, and the kinetic theory approach of Lees (1959) based on the Maxwell transfer equation. The expressions for the collisional energy flux qcand stress pc given by Eqs. (46) and (48) are similar both in form and in their numerical values to the corresponding collisional terms calculated in the kinetic theory of dense gases as given by Chapman and Cowling (1970). The solution to the problem of Couette flow between parallel plates was outlined by Jenkins and Savage (1982). Complete solutions were not obtained because of uncertainty about the appropriate specification of the boundary conditions. The determination of the slip velocity and the energy flux at a boundary requires an analysis of the collisional transfer at the wall that is similar to, but perhaps more complicated than, that just explained for particles in the interior of the flow. Such an analysis has not yet been carried through, but it is necessary to properly complete the theory . It is interesting to consider what the theory yields for the case of a simple shear flow u,(x2)in which there are no gradients of (u2).Neglecting P k and using Eqs. (47), (48), and (49) in the energy balance (40) yields

352

Stuart B . Savage

Putting e = I for perfectly elastic particles, it is seen that Eqs. (50) and (51) give stress components that are the same as those given for small R in the paper of Savage and Jeffrey (1981), after correcting an error they made in defining their Maxwellian ((vz)in their paper should be replaced by 3(v2)).However, the work of Jenkins and Savage gives the relationship between e and R, which for this simple example is R =

[Y

C T U ~ , ~ / ( U= ~ ) ~ / ~(1

-

(52)

Of course, similar relationships may be obtained from the theories of Ogawa et al. (1980) and Shen (1982). They, in fact, also include the effects of particle friction coefficient p and, in the theories of Ackermann and Shen, the effects of the interstitial fluid. For a given material having a particular value of e, the value of R is fixed by Eq. (52) and it is seen that both normal shear stresses are proportional to vg0p&4:,2 and that pc,2/pc22= -(9/5)(37r)-Il2 R = const.

(53)

The theory of Jenkins and Savage (1982) was an asymptotic analysis based upon the assumption of nearIy elastic particles e = 1 (or, equivalently, small mean flow gradients), and there is the obvious question related to the value of e at which this asymptotic analysis departs from one that involves a more exact evaluation of the integrals involved in the collisional transfer of the various properties. Lun et al. (1982) have dealt with this question for an example involving simple shear flow with no gradients of (v2).The collisional integrals 1 .o

0.8

-

0.8

Shen (1982)

0.4

0.2

-

Lun et el. (1 982)

0 0.1

0.3

0.5

1

3

5

10

R J ~ )coeffi~ ~ FIG.17. Variation of mean shear to fluctuation velocity ratio R = U U ~ , ~ / ( Lwith cient of restitution e as predicted by various theories for the case of simple shear.

The Mechanics of Rapid Granular Flows

353

were evaluated both numerically and by asymptotic analyses involving direct and inverse series expansions in terms of the parameter R . These results were compared with the results for a simple shear flow obtained from the theories of Ogawa et al. (19801, Jenkins and Savage (1982), and Shen (1982) and the experiments of Savage and Sayed (1980). Some comparisons taken from Lun et al. (1982) are shown in Figs. 17-20. The theories of Ogawa et al. (1980) and Shen (1982) are shown for the case of

3.0

1.o

0.3

ln

E

0.1

c

0.03

0.01

e

FIG.18. Variation of nondimensional shear stresspl2/@vgecr2ul,:) with coefficient of restitution e for the case of simple shear. Theories of Ogawa er a/. (1980) and Shen (1982) shown for Y = 0.5.

Stuart B . Savage

354

3

1.o

0.3

0.1

0.03

0.01

0

e FIG. 19. Variation of nondimensional normal stress p221(pvgou2uI,:) with coefficient of restitution e for the case of simple shear. Theories of Ogawa er a / . (1980) and Shen (1982) shown for v = 0.5.

p = 0 and for negligibly small interstitial fluid density pf. In these theories increasing p generally increases R and decreases the stresses. All the

analyses show the same trends and similar values in Fig. 17 for the variation of R with e . If inelasticity is the only dissipative mechanism, there is an upper limit of about 2 to 3 for R corresponding to e = 0. The nondimensional shear and normal stresses from the four theories are plotted versus e in Figs. 18 and 19. Stresses decrease with decreases in e , and the values predicted by the theories of Shen (1982) and Ogawa et al. (1980) are lower than those predicted by Lun et al. (1982) and Jenkins and Savage (1982).

The Mechanics of Rapid Granular Flows 10

355

0

3N h

N . r

0.3

I

0.3

0.4

0.5

0.6

Solids fraction, Y

FIG.20. Variation of shear stress with v for case of simple shear. Comparison of various theories with experiments of Savage and Sayed (1980) using (0)glass beads and (0) polystyrene beads.

The differences between the analysis of Lun et al., which is based on an accurate evaluation of the collision integrals, and the asymptotic theory of Jenkins and Savage are quite small. Figures 20 and 21 compare the variations of shear stresses and normal stress with v that are predicted by the four theories (taking p = 0, pf/pp = 0). Experimental values from Savage and Sayed (1980) for glass and polystyrene beads in air are also shown. Although the theories of Jenkins and Savage and Lun et al. are close to the experimental data it is likely that a more complete theory that included particle rotational inertia and surface friction would predict stresses lower than those shown for p = 0. In a collisional interaction the inclusion of a nonzero p would probably increase the impulsive shear force and dampen the fluctuations. These two effects give opposing contributions to the averaged shear stress. Hence, we might anticipate that

356

Stuart B . Savage

solids fraction, Y

FIG. 21. Variation of normal stress with v for case of simple shear. Comparison of various theories with experiments of Savage and Sayed (1980)using ( 0 )glass beads and (0)

polystyrene beads.

consideration of frictional particles in the Jenkins and Savage approach would reduce the normal stress more than the shear stress. Such a tendency would correspond to the trends of the experimental data.

C. NUMERICAL MODELING There have been a few previous studies dealing with the numerical modelling of granular materials under quasi-static conditions [for example, Davis and Deresiewicz (1977), Cundall and Strack (19791, and Trollope and Burman (1980)], but the application of this approach to rapid granular flows is very recent.

The Mechanics of Rapid Granular Flows

357

Walton (1980) has developed a computer model to calculate the motion of large numbers (up to a few thousand) arbitrarily shaped two-dimensional polygonal blocks. The blocks can be made to interact in a linear or nonlinear elastic or plastic fashion under the action of both normal and tangential forces at the contact points. Unlimited particle displacements and rotations are permitted, and the collisions and contacts are created and destroyed as the bulk motion proceeds. The computer code has been used to study granular flows in hoppers, the collapse of explosively fractured rock, and flows of granular materials down inclined chutes. A number of fascinating computer-generated movies of these flows have been made. A typical “snapshot” of the output for a granular flow of circular disks of various diameters flowing down a chute inclined at 45” to the horizontal is shown in Fig. 22. It is possible to make use of the code to determine empirically constitutive equations and to verify and to help develop microstructural analytical models of granular flow and this work is in progress. Campbell (1982) has developed a somewhat simpler program involving instantaneous binary collisions between inelastic, frictional uniform, circular disks but has applied his program more extensively to investigate the constitutive behavior of Couette flows and flows down inclined chutes. Typical calculations involved, at any given instant, about 40 particles contained within a rectangular boxlike flow region. Periodic boundary conditions were applied; particles left one face of the rectangular region and re-entered the opposite face at the corresponding location with equal translational and rotational velocities. Starting with specified initial conditions, computations were continued (up to a few hundred thousand collisions) until the mean flow properties reached an equilibrium steady state (when the overall flow and boundary conditions were such that a steady state could in fact exist). The results obtained for the inclined chute calculations were described in Section 111. The results for the Couette flow calculations were converted to “equivalent” three-dimensional results by relating the two-dimensional solids fractions to three-dimensional equivalents. The general trend of the calculations were shown to be in agreement with the experiments of Bagnold (1954) and Savage and Sayed (1982). Campbell (1982) also used his model to investigate the form of the single particle velocity and the spatial pair distribution functions. The velocities were found to be quite well represented by Maxwellian distributions. Although the form of the spatial anisotropy at low concentrations was similar to that used in the analysis of Savage and Jeffrey (19811, at high concentrations noticeable spikes in the distribution function appeared. At the high concentrations the disks tended to arrange themselves in distinct

Stuart B . Savage

358

.lB

0.

-.18 -.28

FIG.22. Numerical modeling of "two-dimensional" granular flow of circular disks down a 45" incline (unpublished work of Otis Walton using computer code similar to that in Walton, 1980).

shear layers; the spikes were associated with glancing collisions between particles in adjacent layers and collisions between adjacent particles in a given layer. There is still much that can be learned from these kinds of numerical experiments; their full potential has yet to be exploited. Both Walton's and Campbell's computer codes are for disk flows; an analogous code for three-dimensional particles would be particularly illuminating.

VI. Concluding Remarks The present review of rapid shear flows of granular materials has concentrated on the mechanics that govern the development of the stress,

The Mechanics of Rapid Granular Flows

359

bulk density, and velocity fields. This sharp focus, as well as space limitations, has prevented the discussion of numerous interesting phenomena such as heat transfer, segregation and mixing of particles, and electrostatic effects. The many applications in various branches of engineering as well as in geophysics have hardly been mentioned. On the other hand, we have attempted to provide a complete, detailed, and up-to-date discussion of the relevant papers dealing with mechanics of relatively simple flows. A clear physical understanding of these simple flows and the development of the theoretical tools to predict them are necessary before we can begin to think sensibly about applications in complex practical engineering situations and realistic flows in nature. There is still a great need for further experiments to provide information on the constitutive equations. New instrumentation and experimental techniques for the measurement of mean and fluctuating velocities, bulk densities, and stresses would be extremely useful. It is possible that molecular-dynamics-type numerical modeling techniques will help to provide the kinds of information that are so difficult to obtain in physical experiments. The development of continuum theoretical models is still worthwhile and should be continued. Work on microstructural theories is active, and the incorporation of particle surface friction and interstitial fluid effects into the dense gas type theories seems possible. The effects of particle shape and size distributions should be investigated. The rateindependent stresses arising from enduring contacts between individual particles or groups of particles have not been considered. These effects are very likely to be important in many practical flows, but their inclusion in any theoretical framework is certain to be difficult. From the preceding remarks it should be evident that there is ample theoretical and experimental work yet to be done. Many challenging and fascinating problems of considerable practical significance remain; applied mechanics can contribute much toward their solution. ACKNOWLEDGMENT The preparation of this manuscript was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). 1 am also most grateful for their continued support of my own work that has been reviewed herein.

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Augenstein, D. A., and Hogg, R. (1974). Friction factors for powder flow. Powder Technol. 10,43-49. Augenstein, D. A,, and Hogg, R. (1978). An experimental study of the flow of dry powders over inclined surfaces. Powder Technol. 19, 205-215. Bagnold, R. A. (1954). Experiments on a gravity free dispersion of large solid spheres in a newtonian fluid under shear. Proc. R . SOC.London, Ser. A 225,49-63. Bagnold, R. A. (1956). The flow of cohesionless grains in fluids. Phil. Trans. R . SOC. London, Ser. A 249, 235-297. Bagnold, R. A. (1966). The shearing and dilation of dry sand and the ‘singing’ mechanism. Proc. R . SOC.London, Ser. A 295, 219-232. Bagnold, R. A. (1973). The nature of saltation and of ‘bed-load’ transport in water. Proc. R. SOC.,London, Ser. A 332,473-504. Bailard, J. (1978). “An Experimental Study of Granular-Fluid Flow.” Ph.D. Thesis. Univ. of Calif., San Diego, California. Balmer, R. T. (1978). The operation of sand clocks and their medieval development. Technol. Culture. 19, 615-632. Batchelor, G. K. (1974). Transport properties of two-phase materials with random structure. Annu. Rev. Fluid Mech. 6 , 227-255. Bingham, E. C., and Wikoff, R. W. (1931). The flow of dry sand through capillary tubes. J . Rheol. (NY) 2, 395-400. Blinowski, A. (1978). On the dynamic flow of granular media. Arch. Mech. 30, 27-34. Botterill, J. S. M. (1975). “Fluid-Bed Heat Transfer.” Academic Press, New York. Botterill, J. S. M., Elliot, D. E., van der Kolk, M., and McGuigan, S. J. (1972). Flow of fluidized solids. Powder Technol. 6, 343-51. Botterill, J. S. M., and Abdul-Halim, B. H. (1979). The open channel flow of fluidized solids. Powder Technol. 23, 67-78. Botterill, J. S. M., and Bessant, D. J. (1976). The flow properties of fluidized solids. Powder Technol. 14, 131-137. Brennen, C. E., Sieck, K., and Paslaski, J. (1982). Hydraulic jumps in granular material flow. Submitted to Powder Technol. Bridgwater, J. (1972). Stress-velocity relationships for particulate solids. Am. SOC. Mech. Eng. Pap. 72-MH-21. Bridgwater, J. (1980). On the width of failore zones. Geotechnique 30, 533-536. Brown, R. L., and Richards, J. C. (1970). “Principles of Powder Mechanics.” Pergamon Press, London. Buyevich, Y. A., and Shchelchkova, I. N. (1978). Flow of dense suspensions. Prog. Aeronaut. Sci. 18, 121-150. Campbell, C. E. (1982). “Shear Flows of Granular Materials.” Ph.D. Thesis. Calif. Inst. of Techn., Pasadena, California. Carnahan, N. F., and Starling, K. E. (1969). Equations of state for non-attracting rigid spheres. J . Chem. Phys. 51, 635-636. Cam, J. F., and Walker, D. M. (1967/1968). An annular shear cell for granular materials. Powder Technol. 1, 369-373. Chapman, S., and Cowling, T. G . (1970). “The Mathematical Theory of Non-Uniform Gases,” 3rd ed. Cambridge Univ. Press, London and New York. Cheng, D. C.-H., and Farley, R. (1966). Comments on the paper ‘Rheological Behaviour of Powder in a Rotational Viscometer’ by Hiroshi Kuno andKozo Kurihara. Rheol. Acta 5, 53-56, Cheng, D. C.-H., and Richmond, R. A. (1978). Some observations on the rheological behaviour of dense suspensions. Rheol. Acta 17, 446-453.

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Choda, A., and Willis, A. H. (1967). Flow regimes of grains in inclined chutes. Trans. Am. Soc. Agri. Eng. 10, 136-138. Chong, J. S . , Christiansen, E. B., Baer, A. D. (1971). Rheology of concentrated suspensions. J . Appl. Polymer Sci. 15, 2007-2021. Cowin, S. C. (1974a). A theory for the flow of granular materials. Powder Technol. 9,61-69. Cowin, S. C. (1974b). Constitutive relations that imply a generalized Mohr-Coulomb criterion. Acta Mech. 20, 41-46. Cox, R. G., Mason, S. G. (1971). Suspended particles in fluid flow through tubes. Annu. Reu. Fluid Mech. 3, 291-316. Cundall, P. A., Strack, 0. D. L. (1979). A discrete numerical model for granular assemblies. Ceotech. 29, 47-65. Davidson, J. F., and Harrison, D. (1963). “Fluidized Particles.” Cambridge Univ. Press, London and New York. Davis, R. A., Deresiewicz, H. (1977). A discrete probabilistic model for mechanical response of a granular medium. Acta Mech. 26, 69-89. de Jong, J. A. H. (1969). Vertical air-controlled particle flow from a bunker through a circular orifice. Powder Technol. 3, 279. de Josselin de Jong, G. (1959). “Statics and Kinematics of the Failure of a Granular Material.” Uitgeverij Waltman, Delft, The Netherlands. de Josselin de Jong, G. (1971). The double-sliding, free-rotating model for granular assemblies. Geotechnique 21, 155-163. de Josselin de Jong, G. (1977). Mathematical elaboration of the double-sliding, free-rotating model. Arch. Mech. 29, 561-591. Delaplaine, J. W. (1956). Forces acting on flowing beds of solids. AIChE J. 2, 127-138. Dresher, A,, and de Josselin de Jong, G. (1972). Photoelastic verification of a mechanical model for the flow of a granular material. J. Mech. Phys. Solids 20, 337-351. Erismann, T. H. (1979). Mechanisms of large landslides. Rock Mech. 12, 15-46. Faber, T. E. (1972). “An Introduction to the Theory of Liquid Metals.” Cambridge Univ. Press, London and New York. Gadala-Maria, F. (1979). “The Rheology of Concentrated Suspensions.” Ph.D. Thesis, Stanford Univ., Stanford, California. Goddard, J. D. (1977). A review of recent developments in the constitutive theory of particulate dispersions. In “Continuum Models of Discrete Systems” (J. W. Provan, ed.). Univ. Waterloo Press. Goodman, M. A., and Cowin, S. C. (1971). Two problems in the gravity flow of granular materials, J. Fluid. Mech. 45, 321-339. Goodman, M. A., and Cowin, S. C. (1972). A continuum theory for granular materials. Arch. Ration. Mech. Anal. 44, 249-266. Green, A. E., and Rivlin, R. S. (1956). Steady flow of non-Newtonian fluids through tubes. Quart. Appl. Math. 14, 299-308. Hagen, G. (1852). Druck und Bewegung des Trockenen Sandes. Eerl. Monafsb. Akad. d . Wiss. S35-S42. Hansen, J . P., and McDonald, I. R. (1976). “Theory of Simple Liquids.” Academic Press, New York. Herczynski, R., and Pienkowska, I. (1980). Towards a statistical theory of suspension. Annu. Rev. Fluid Mech. U ,237-269. Hill, H. M. (1966). Bed forms due to a fluid stream. J. Hydraul. Diu., Am. Soc. Ciu. Eng. 92, 127- 143. Hoffman, R. L. (1972). Discontinuous and dilatant viscosity behavior in concentrated suspensions. I. Observation of a flow instability. Trans. Soc. Rheol. 16, 155-173.

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Hopfinger, E. J. (1983). Snow avalanche motion and related phenomena. Annu. Reu. Fluid Mech. 15, 47-76. Hsu, K. J. (1975). Catastrophic debris streams generated by rock falls. Geol. SOC.Amer. Bull. 86, 129-140. Hvorslev, M. J. (1936). A ring shearing apparatus for the determination of the shearing resistance and plastic flow of soil. Pror.. Int. Conf. Soil Mech. Found. En#/. 2, 125129. Hvorslev, M. J. (1939). Torsion shear tests and their place in the determination of the shearing resistance of soils. Proc. ASTM 39, 999-1022. Ishida, M., and Shirai, T. (1979). Velocity distributions in the flow of solid particles in an inclined open channel. J. Chem. Eng. Jpn. 12, 46-50. Ishida, M., Shirai, T., and Nishiwaki, A. (1980a). Measurement of the velocity and directions of flow of solid particles in a fluidized bed. Powder Technol. 27, 1-6. Ishida, M., Hatano, H., and Shirai, T. (1980b). The flow of solid particles in an aerated inclined channel. Powder Technol. 27,7-12. Janssen, H. A. (1895). Tests on grain pressure silos. Z. Ver. Dtsch. Ing. 39, 1045-1049 (in German). Jeffrey, D. J., and Acrivos, A. (1976). The rheological properties of suspensions in rigid particles. AIChE J. 22, 417-432. Jenike, A. W., Elsey, P. J., and Woolley, R. H. (1960). Flow properties of bulk solids. Proc. ASTM 60, 1-14. Jenkins, J. T., and Cowin, S. C. (1979). Theories for flowing granular materials. In “Mechanics Applied to the Transport of Bulk Materials” (S. C. Cowin, ed.), pp. 79-89. Am. SOC.Mech. Eng. Jenkins, J. T., Savage, S. B. (1981). The mean stress resulting from interparticle collisions in a rapid granular shear flow. I n “Continuum Models of Discrete Systems’’ Vol. 4 (0. Brulin and R. K. T. Hsieh, Eds.), pp. 365-371. North-Holland, Amsterdam. Jenkins, J. T., and Savage, S. B. (1982). A theory for the rapid flow of identical smooth, nearly elastic particles. Submitted to J. Fluid Mech. Kanatani, K. (1979a). A micropolar continuum theory for the flow of granular materials. Int. J. Eng. Sci. 17, 419-432. Kanatani, K. (1979b). A continuum theory for the flow of granular materials. I n “Theoretical and Applied Mechanics” (Japan Natl. Comm. Theor. and Appl. Mech.), 27, 571578. Kanatani, K. (1980). A continuum theory for the flow of granular materials (11). In “Theoretical and Applied Mechanics” (Japan Natl. Comm. Theor. and Appl. Mech.), 28, 485492. Knight, P. C. (1983). The role of particle collisions in determining high strain rate flow behaviour. Proc. I n t . Symp. Role Particle Interactions Powder Mech. Eindhoven. Komar, P. D. (1976). “Beach Processes and Sedimentation.” Prentice-Hall, Englewood Cliffs, New Jersey. Krasheninnikov, A. I., Malakhov, R. A., and Fioshina, M. A. (1967). Sharp growth of viscosity with increasing deformation rate in some disperse systems. Dokl. Phys. Chem. 174, 360-363. Kuno, H., and Kurihara, K. (1965). Rheological behaviour of powder in a rotational viscometer. Rheol. Acta 4, 73-74. Leal, L. G . (1980). Particle motions in a viscous fluid. Annu. Reu. Fluid Mech. 12, 435-476. Lees, L. (1959). A kinetic theory description of rarefied gas flows. Hypersonics Res. Proj. Memo N o . 51, Guggenheim Aero. Lab., Calif. Inst. Tech.

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Author Index Numbers in italic refer to the pages on which the complete references are listed.

A Abdul-Halim, B. H., 320, 360 Ackermann, N. L., 334, 345, 352, 353, 354, 359, 365 Acrivos, A., 324,362 Agrawal, H. C., 58, 91 Akers, R. J., 323, 365 Allen, J. R. L., 309, 359 Amamoto, H., 216, 288 Aoki, K., 264, 288 Amtodo, A., 202,213 Asano, S., 216,288 Athanassoula, E., 173, 175, 179, 187 Augenstein, D. A., 312, 313, 314, 317, 319, 360

B

Birks, A. H., 323, 366 Blinowski, A., 342, 348,360 Botterill, J. S. M., 320, 327, 360 Brandstater, A., 212, 212, 213 Brennen, C. E., 321, 360 Bridgwater, J., 308, 324, 326, 327, 360, 366 Brillouin, L., 41. 91 Brown, R. L., 291, 295, 305, 360 Burman, B. C., 356, 366 Buyevich, Y. A., 324,360 C

Callcott, T. G., 320, 365 Campbell, C. E., 312, 319, 357,360 Campbell, L., 190,213 Carnahan, N. F., 348, 350, 360 Cam, J. F., 323,360 Chan, R. K. C., 264, 287 Chapman, S., 299, 347, 348, 351,360 Cheng, D. C.-H., 324, 325, 327,360 Chien, W.-Z., 94, 127, 131, 153 Choda, A., 308,361 Chong, J. S., 325, 361 Christiansen, E. B., 325, 361 Chung, B. T. F., 58, 91 Coullet, P., 202, 213 Cowin, S. C., 291, 336, 337, 339, 343, 361, 362 Cowling, T. G., 299, 347, 360 Cox, R. G., 290,361 Crutchfield, J. P., 212, 212, 213 Cundall, P. A , , 356, 361 Curry, J. H., 201, 213 Curry, J. S., 201, 213

Baba, E., 216,287 Baer, A. D., 325, 361 Bagnold, R. A., 292,293,295,296,298,309, 324,326,327,329,333, 334,336, 337, 338, 339, 341, 343, 346, 357, 360 Bailard, J., 312, 314, 315, 317, 319, 360 Bailey, P. B., 338, 339, 364 Balmer, R. T., 290, 360 Batchelor, G. K., 290, 360 Benson, S. V., 205, 207, 208, 210, 211, 213 Berry, M. V., 190,212 Bertin, G., 155, 157, 158, 159, 161, 163, 164, 165, 166, 170, 171, 17.5, 176, 179, 182, 183, 184, 187 Bessant, D. J., 320,360 Bingham, E. C., 305,360 Biot, M. A., 6 , 7 , 9 , 10, 15, 16, 17, 19,23,28, 31,33,34,35,36,40,41,43,44,45,48,51, D 54,55,57,58,59,62,63,64,68,69,70,71, 7 2 , 7 3 , 7 6 , 7 8 , 7 9 , 8 0 , 8 1 , 8 3 , 8 4 , 8 5 , 8 6 , 8 7 , DaCosta, L. N., 202, 213 89, 89, 90, 91 Daly, B. J., 264, 288 367

Author Index

368 Davidson, J. F., 319, 361 Davies, S. T., 320, 363 Davis, R. A., 356, 361 DeDonder, T., 91 deJong, J. A. H., 305,361 deJong, L. N . J., 324,363 deJosselin deJong, G., 291, 301,361 Delaplaine, J. W., 305, 361 Deresiewicz, H., 356, 361 Dresher, A., 301,361

E Eckert, E., 287 Eckmann, J.-P., 211, 213 Eggers, K., 216,287 Eisenhart Rothe, M., 302,364 Elliot, D. E., 320,360 Elsey, P. J., 322, 362 Erismann, T. H., 292,361

F Faber, T. E., 299,361 Farlev. R., 327. 360 Farm&, J. D., 212,212, 213 Fauve, S., 206,213 Feigenbaum, M. J., 208,213 FernBndez-Luque, R., 299,363 Fioshina, M. A., 325, 362 Fowler, R., 13, 91 Fromm, J. E., 270,287

G Gadala-Maria, F., 324,361 Garnett, W., 190, 213 Gibbs, J. W., 9, 91 Glansdorff, P., 85, 91 Goddard, J. D., 324,361 Goddman, M. A,, 336, 337, 339,361 Gollub, J. P., 205, 207, 208, 210, 211, 213 Gorman, M., 206,213 Gough, D. O., 201,214 Green, A. E., 315,361 Green, B. E., 153 Guggenheim, E. A., 13, 91

H Haass, J., 163, 178 Hagen, G., 290, 305,361 Hampton, M. A., 292, 296, 363 Hansen, J. P., 299, 361 Harlow, F. H., 264, 288 Harrison, D., 319,361 Hatano, H., 318, 319, 362 Hatsopoulos, G. N., 9, 91 Herczyiiski, R., 290, 361 Herring, J. R., 201,213 Herrman, L. R., 152, 153 Hill, H. M., 293, 361 Hinze, J. 0..211, 213 Hoffman, R. L., 325,361 Hogg, R., 312, 313, 314, 317, 319,360 Hopfinger, E. J., 292,362 Horton, D. J., 320,363 Houslby, G. T., 291,302,304,305,317,363, 366 Hoyle, Fred, 212, 213 Hsu, K. J., 292, 362 Hsuan, H. C. S., 59, 91 Hutton, R. E., 194,213 Hvorslev, M. J., 323, 362

I Inui, T., 216, 288 Ishida, M., 316, 317, 318, 319, 320, 362

1

Janssen, H. A., 304, 306, 362 Jeffrey, D. J., 302, 311, 324, 334, 339, 340, 346,347,348,350,351,352,353,354,357, 362, 363, 365 Jen, E., 212, 212, 213 Jenike, A. W., 322, 362 Jenkins, J. T., 291, 302, 334, 337, 338, 343, 346, 348, 351, 352, 353, 354, 362, 364 Jones, E. E., 153 Jones, P. J., 305,363 Jones, R. E., 153

Author Index K Kajitani, H., 242, 264, 287, 288 Kanai, M., 242, 264,288 Kanatani, K., 334, 345, 362 Kawamura, H., 242, 288 Kayo, Y.,216, 288 Keenan, J. H., 9, 91 Knight, P. C., 312, 316,362 Knobloch, E., 202. 213, 214 Knowlton, T. M., 305,363 Komar, P. D., 292,362 Krasheninnikov, A. I., 325,362 Kunii, D., 305, 366 Kuno, H., 305, 326,362, 366 Kurihara, K., 326,362

L Lamb, H., 86, 88, 91 Landau, L. D., 205,213 Lanford, O., 191, 213 Laplace, Pierre Simon, 188, 213 Lardner, T. J., 58, 91 Lau, Y. Y., 165, 179, 185, 187 Leal, L. G., 290,362 Lees, L.,35 1,362 Leung, L. S., 305,363 Libchaber, A., 206,213 Lichtenberg, A. J., 190, 198, 201, 206, 213 Lieberman, M. A., 190, 198, 201, 206,213 Lifshitz, E. M., 205, 213 Lin, C. C., 155, 157, 158, 159, 163, 165, 170, 175, 176, 179, 184, 185, 186, 187 Loahakul, C., 306, 308,363 Loncaric, J., 201,213 Lonngren, K. E. 59, 91 Lorenz, E. N., 190, 199,213 Lun, C., 334, 346, 348, 352, 353, 354,363

M McDonald, I. R., 299, 361 McDougall, I. R., 305,363 McFarland, 192, 214 McGuigan, S. J., 320, 360 McKay, R. W., I53 McKeown, S., 326, 327, 333, 334,364

369

McLaughlin, J. B., 201, 208, 209, 214 McTigue, D. F., 341, 343, 363 Malakhov, R. A., 325, 362 Malkus, W. V. R., 202, 213 Maltha, A., 324,363 Mandel, J., 291, 363 Mandelbrot, B., 191, 213 Mandl, G., 299, 324, 363 Manneville, P., 191, 211, 213 Marble, F. E., 343,363 Marcus, P. S., 201,213 Martin, P. C., 201,214 Mason, S. G., 290,361 Masuko, A., 264,288 Matsui, M., 216, 288 Maurer, J., 213 Meixner, J., 47, 91 Mellor, M., 292, 363 Metzner, A. B., 324, 325,363 Middleton, G. V., 292, 296, 363 Miles, J. W., 194, 195, 1%. 197, 198, 199, 214 Mindlin, R. D., 72, 91 Miyata, H., 242, 264, 287, 288 Monin, A. S., 192, 214 Moore, D. R., 201, 202, 214 Morrison, H. L., 321, 363 Mroz, Z., 291, 336,363

N Nagahama, M., 288 Nagashima, T., 211, 214 Nedderman, R. M., 291, 302, 304, 305, 306, 308, 317, 320,363, 366 Newhouse, S., 205,214 Nguyen, T. V., 317,363 Nicholson, D. E., 304, 364 Nishiwaki, A., 316, 320, 362 Nito, M., 287,288 Novosad, J., 323, 326, 327,363 Nuziato, J. W., 338, 339, 364

0 Odb, H., 80, 91 Ogawa, S., 334, 342, 344,345,348,352,353, 354,364

370

Author Index

Ohkubo, K., 216,288 Onsager, L., 26, 35, 48, 91 Orr, W., 192, 214 Orszag, S. A., 201, 208, 209,213, 214 Oshima, N., 334, 342, 344, 345, 348, 352, 353, 354,364

P Pariseau, W. G., 304,364 Parkinson, J. S., 323, 365 Parlour, R. P., 311,364 Paslaski, J., 321, 360 Passman, S. L., 338, 339,364 Pian, T. H. H., 94, 104, 116, 153 Pienkowska, I., 290, 361 Poincart, H., 190, 214 Pomeau, Y., 191, 211,213 Prasad, A., 58, 91 Present, R. D., 349, 364 Prigogine, I., 85, 91 Pullen, R. J. F., 305,363

R Raudkivi, A. J., 292, 364 Rayleigh, L., 200, 214 Reisner, W., 302,364 Reith, L. A., 206, 213 Reynolds, O., 191, 214, 290, 325, 364 Richards, J. C., 291, 295, 305, 360 Richmond, O., 321,363 Richmond, R. A., 324, 325,360 Ridgway, K., 311, 313, 314,364 Rigby, G. R., 320,365 Rivlin, R. S., 315, 361 Roache, P. J., 270, 288 Robbins, K. A., 201, 214 Roberts, A. W., 310, 311, 313, 315,364 Roscoe, K. H., 300, 322,364 Rothrock, D. A., 292, 364 Rowlinson, J. S., 299,366 Ruelle, D., 191, 203, 205,214 Rupp, R., 311, 313, 314,364 Rushbrooke, G. S., 299,366 Russel, W. B., 290,364

5

Salencon, J., 299,364 Saltzman, B., 190, 199, 200,214 Sanders, J. E., 294,364 Savage, S. B., 291, 302, 305, 309, 311, 317, 326,327,328,329,333, 334,337, 338, 339, 340,341,342,346,347,348,350, 351,352, 353,354,355,356,357,362,363,364,365, 366 Sayed, M., 296, 301,309,317,328, 329, 330, 331,332,333,334,342,346,353, 355,356, 357,365 Scarlett, B., 323, 365 Schofield, A. N., 295, 299, 300,364,365 Schwedes, J., 322, 323,365 Scott, 0. J., 311, 364 Scott, R. F., 300,365 Senf, L., 59, 91 Shahinpoor, M., 343,365 Shannon, J. P., 264,288 Sharma, S. D., 287 Shchelchkova, I. N., 324,360 Shen, H., 334, 345, 346, 352, 353, 354, 359, 365 Shimada, I., 211, 214 Shirai, T., 316, 317, 318, 319, 320, 362 Siah, J. S. S., 343,365 Sieck, K., 321, 360 Singh, B., 320, 365 Sodhi, D. S., 292, 365 Sokolnikoff, I. S., 22, 91 Sokolovski, V. V., 295, 365 Soo, S. L., 365 Sparrow, C., 201, 202,214 Spencer, A. J. M., 291, 299, 336, 341, 365, 366 Spiegel, E. A,, 201, 214 Starling, K. E., 348, 350,360 Stephens, D. J., 324,366 Strack, 0. D. L., 356, 361 Street, R. L., 264, 287 Strom, K. M., 158, 187 Strom, S. E., 158, 187 Strome, D. R., 153 Suzuki, A., 264,288, 311,366 Swift, J., 212, 212, 213 Swinney, H. L., 206, 212,212, 213

Author Index T Takahashi, H., 306,366 Takahashi, M., 242,288 Takahasi, K., 309, 366 Takahasi, T., 292,366 Takekuma, K., 216,287,288 Takens, F., 191, 205, 214 Tanaka, T., 31 I , 366 Taneda, S., 216,288 Taylor, G. I., 192, 193,214 Temperley, H. N. V., 299,366 Thomas, J. P., Jr., 338, 339, 364 Todd, A. C., 323,365 Tong, P., 94, 104, 105, 116, 153 Toomre, A., 165, 167, 168, 173, 176, 179, 180, 181, 184, 187 Toomre, J., 201, 202, 214 Toyama, S., 306,366 Trees, J., 308, 366 Tresser, C., 202, 213 Trollope, D. H., 356, 366 Tsuchiya, Y., 287, 288 Tsuda, T., 216,288 Tsuruoka, M., 242,288 Tsutsumi, T., 216,288 Tiiziin, U., 291, 302,304,305, 317,363,366

U Umemuna, A., 334, 344, 345, 348, 352, 353, 354,364 Umeya, K., 325, 366 Ushio, T., 288

37 1 V

van der Kolk, M., 320,360 Veronis, G., 202, 214 von Hohleiten, H. L., 308, 366

W Walker, D. M., 323, 360 Walton, 0.. 357, 358, 366 Washizu, K., 16, 91 Weiss, N . O., 201, 202, 213, 214 Welander, P., 202, 214 Welch, J. E., 264, 288 Whitlock, M., 324, 325, 363 Wieghardt, K., 291, 311, 320, 366 Wikoff, R. W., 305,360 Williams, J. C., 323, 366 Willis, A. H., 308, 361 Wolf, A., 212, 212, 213 Wolf, E. F., 308,366 Woods, L. C., 88, 91 Woolley, R. H., 322, 362 Wroth, C. P., 295, 299, 300, 364, 365

Y Yani, H., 306,366 Yeh, L. T., 58, 91 Yoon, S. M., 305,366 Yuasa, Y., 305, 366 L

Zagaynov, L. S., 291,366

Subject Index

A

experimental results, 308-3 16 pattern types, 317-320 Clausius’ uncompensated heat, expression for, 24 Configuration space, Lagrangian equations in, 89 Continuum theories for granular Rows, 336-343 Convective entropy, notation for, IS Corotation resonance in galactic dynamics, 166-167, 168 Couette flow(s), 192, 193, 197, 205, 212 in granular material, 300, 351, 357 Coulomb friction, 87 Coupling of subsystems, principle of interconnection and, 86-88 Courant condition, 270 Creep of layered viscous solid, 85-86

Actual physical energy flux, expression for, 29 Annular shear cell for studies of granular flows, 323-324 Associated fluence fields. 58

B Bagnold effect, 294, 343 Bagnold number, 293 Bending plate elements of, generalized variational principle of, 127-153 of thin plate, matrix formulation of incompatible elements for, 147- 152 Bifurcation sequences in turbulence, 204-205 Bifurcations Lagrangian formulation of, 82 Boussinesq approximation, 193 Bulk solids, examples of, 290

D

d’Alembert’s principle of classical mechanics, 3 generalization to irreversible mechanics, 2, 25, 33 Darcy’s law, 81 De Donder’s formula, 12 Deflection-incompatible elements, matrix equations of, 131-139 Deformable solids with thermonuclear diffusion and chemical reactions, irreversible thermodynamics of, 59-62 Density wave theory, 157 of spiral structures, 164-171 Density waves, propagation of, 171-172 Dirac functions, 71

C

C modes, I81 Carnot cycle of thermal well, 34 Cartesian mass fluence, expression for. 37 Channels, vertical, granular flows in, 302-321 Chaotic trajectory on Lorenz attractor, 190-191 Chemical thermodynamics, new approach to, 9-13 Chutes inclined, granular flows in, 302-321 372

373

Subject Index Disk galaxies, density wave theory applied to, 165 Dissipative structures, near unstable equilibrium, 78-80

E Edge modes, 181 Elasticity plane problems in, generalized variational principle of, 118-126 small-displacement type, problems of,

94-96 Elastoviscous stresses in solids, dynamics of, 44-51 Energy flux theorem, derivation of, 40 Entropy, nature of production of, 23-30 Entropy flux across an area, definition of,

Roberts, Roberts, and Shu survey of,

161 single-mode type, 163 spiral arms in, 159-161 Gas motions in M81 galaxy, 159 Generalized free energy, 3 Generalized variational principle, derivation from minimum complementary energy principle,

139- I44 Gibbs-Duhem theorem, reformulation of,

2, 13-16, 41 Gibbs paradox, 2,9 avoidance of, 15 Global generalized variational principle, derivation from the minimum complementary energy principle,

Ill-112 Goodman-Cowin-type theories of granular ROWS,

23 Euler’s theorem, application of, 27 Exergy, 3

F Finite strain and stress, nontensorial virtual work approach to, 16-19 Fluence concept, 21-23 Fluid dynamics, strange attractors in,

189-2 I4 Fluid-saturated deformable porous solid, dynamics of, 64-68

G

336-341

Granular flow mechanics, 289-366 around obstacles, 320 Bagnold’s papers on, 293-296 for dry cohesionless granular materials,

296-302 of granular jumps, 320-321 limiting flow regimes in, 302 preliminary discussion of, 292-302 reviews of, 291 rheological test devices for, 321-335 dense suspensions, 324-326 dry materials, 326-333 quasi-static shear devices, 322-324 theories of, 335-358 continuum theories, 336-343 microstructural theories, 343-356 in vertical channels and inclined chutes,

302-321 Galaxies absence of continuous spectrum in,

168-169 classification of, 156 disk-type, 165 discrete spiral modes in, 175-182 dual spiral structures in, 185 dynamics of classification by, 182-185 gravitational plasmas and, 155-187 mechanisms of, 171-175 interaction of, 170-171

Gravitational plasmas, galactic dynamics and, 155-187 Green’s definition of finite strain, 17 Green’s function, 167

H

Heat conduction by inhomogeneous viscous fluid, 51-56 thermoelasticity and, 44-51 Heat conduction law, 48

314

Subject Index

Heat of transport, expressions for, 26 Heat transfer, Lagrangian equations of, 56-59 Heaviside operators, 5 Helmholtz definition of generalized free energy, 31 Hemholtz’s theorem, application of, 86 Holonomic field variations, 35 Homogeneous mixtures, new restructured thermodynamics of, 13-16 Hopf-bifurcation point, 196 Howard-Malkus-Welander convection model in fluid dynamics, 202-204 Hubble time scale, 175 Hybrid incompatible elements, generalized variational principle of, 108-1 I I Hypersystem, concept of, 3

I Incompatible elements, generalized variational principles and, 93-153 Inhomogeneous viscous fluid with convected coordinates and heat conduction, 51-56 Interconnection principle, coupling of subsystems and, 86-88 Intermittent scenario for strange attractors in fluid dynamics, 21 1-212 Intrinsic heat of reaction, new concept of, 9

Irreversible processes, thermodynamics of, 1-91

L Lagrangian equations for bifurcations, 82 in configuration space, 89 of creeping motion, 85-86 derivation for irreversible thermodynamics, 1-91 general, 35-36 for finite element methods, 88-89 of heat transfer, 56-59 Lagrangian multipliers for continuity conditions of field variables on the interelement boundaries, 94 Landau “dissipative” phenomena, 181, I86 Landau’s model of turbulence, 205 Layered viscous solid, creep and folding instability of, 85-86 Lindblad resonances in galactic dynamics, 167, 173, 180 Linear thermodynamics, 30 dissipative structures near unstable equilibrium and, 78-80 near equilibrium, 68-72 of a solid under initial stress, 73-78 Lin-Shu dispersion relation of stars, I 65- 167 Liouville’s partial differential equation in study of stellar system, 156 Lorenz’s convection model in fluid dynamics, 199-202 Lyapunov exponent, 212

M J Jenike shear cell, for studies of granular BOWS, 322-323

K Kelvin’s linear dispersive wave system from ships, 216 Kirchkopf‘s formula, generalization of, 3, 10

Kuzmin-Toomre sequence, 183

M5 I galaxy, radio continuum observations of, 159 M81 galaxy gas motions in, 159 sprials in, 185 Manneville-Pomeau intermittancy in fluid dynamics, 2 I I Marker-and-cell method for nonlinear ship wave studies, 264-286 boundary conditions, 271 computational procedure, 264-271 Mass conservation equation, 21, 37

375

Subject Index Maxwell relations, 7 Maxwell transfer equation, 156, 351 Minimum complementary energy principle derivation of generalized variational principle of compatible elements from, 112-1 14 generalized variational principle for incompatible elements derived from, 115-118, 124-126, 139-144 global generalized variational principle derived from, I 11-1 12 Minimum potential energy principle generalized variational principle used for compatible elements derived from, 96- I03 generalized variational principle of displacement-incompatible elements Of, 118-126 generalized variational principle used for hybrid incompatible elements derived from, 108-1 I I generalized variational principle for incompatible displacement elements derived from, 103-108 Mixed collective potential, expression for, 33 Modal approach to galactic dynamics, 169- I70

N Navier-Stokes equations, 265, 268 NGC 2841 galaxy, spirals in, 185-186 Nontensorial virtual work approach to finite strain and stress, 16-19

0 Open deformable solids, thermodynamic functions of, 19-21 Open systems, restructured thermodynamics of, 6-9 Onsager's principle, 3, 26, 38, 76 Onsager's reciprocity relations. 27, 48, 54 Orr-Sommerfeld equation, 192 Ostriker-Peebles criterion, 183

P Penetration depth, heat fluence derivation from, 58 Period-doubling scenario for strange attractors in fluid dynamics, 208-21 I Piola stress, 53, 54, 55 Piola tensor, nine components of, 18 Plate elements of bending, generalized variational principle for, 127-153 Poisson equation, 156, 270, 271 Pseudofluence concepts, 28

Q Quasi-reversible processes, thermodynamics of, 30

R Radio continuum observations of M5 I galaxy, 159 Rayleigh-Benard convection problem, 193, 197, 199, 201, 202, 204, 211, 212 Resonances, roles in maintenance of spiral modes, 180-182 Rheological test devices for granular flows, 32 1-335 Ruelle-Takens scenario for strange attractors in fluid dynamics, 205-208

S

Ship waves (nonlinear), 215-288 discontinuity and energy deficit of, 228-242 formation of, 219-228 marker-and-cell method in studies of. 264-286 from wedge models, 242-264 computation, 274-286 configuration, 243-25 I velocity and pressure distribution, 25 1-264 Spherical pendulum a s two degree-offreedom oscillator, 193-199

Subject Index

376

Spiral arms in galaxies, 156 Spiral modes barred, 177-179 methods of analysis of. 179-180 normal, 177 resonance role in maintenance of, 180- 182 theory of discrete, 175-182 in transition, 177-179 Spiral patterns, dynamic theory of, 167- I7 1 Spirals density wave theory of, 164-171 in galaxies, 156-158 Stability criteria for time-dependent evolution far from equilibrium, 82-85 Stars, Lin-Shu dispersion relation of, 165-167 Stellar systems, basic dynamic considerations of collective nature of, 164-165 SUMMAC method for studying wave hydrodynamics, 264

T Thermobaric potential, notation for, 32 Thermobaric transfer chemical thermodynamics and, 9 concept of, 6-9 principle of, 30 Thermodynamics, linear, near equilibrium, 68-72 Thermoelastic creep buckling, dynamics of, 81 Thermoelastic potential, derivation of, 5 I Thermoelasticity, heat conduction and, 44-5 I Thin-plate bending, matrix formulation of incompatible elements for, 147-152 Time-dependent physical evolution of a system, generalized stability criteria for, 82-85

TUMMAC computational procedure of, 265 method for wave hydrodynamics, 271 Turbulence, mathematical routes to, 204-2 12

U UGC 2885 galaxy, spiral structure in, 186

V Variational-Lagrangian irreversible thermodynamics (new), 1-91 Variational principles, incompatible elements and, 93-153 of small-displacement linear elasticity, 94-1 I8 Virtual dissipation, principle of, 30-34 Viscodynamic operators, 78 Viscoelasticity, nonlinear, thermodynamics Of, 62-64 Viscosity, entropy production and, 27 Viscous fluid mixtures, dynamics of, 36-44 Vlasov equation, 156

W WASER mechanism for mode excitation, I80 Wave packets, propagation of, 173 Wave trains, sustained, propagation of, 173-175 Waves made by ships, see Ship waves (nonlinear) Wedge models of ships wave studies on, 242-264 computation, 274-286 Welander’s convection loop, 202