Statistical Microhydrodynamics

Emmanuil G. Sinaiski and Leonid I. Zaichik

Statistical Microhydrodynamics

Emmanuil G. Sinaiski and Leonid I. Zaichik Statistical Microhydrodynamics

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Emmanuil G. Sinaiski and Leonid I. Zaichik

Statistical Microhydrodynamics

The Authors Prof. Dr. Emmanuil Sinaiski An der Kotsche 12 04207 Leipzig Germany Prof. Dr. Leonid I. Zaichik Nuclear Safety Institute Russian Academy of Science B. Tulskaya 52 115191 Moscow Russia

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek Die Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at . # 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Typesetting Thomson Digital, India Printing betz-druck GmbH, Darmstadt Book Binding Litges & Dopf GmbH, Heppenheim

Printed in the Federal Republic of Germany Printed on acid-free paper ISBN:

978-3-527-40656-2

V

Contents Preface IX Nomenclature

XIII

Basic Concepts of the Probability Theory 1 Events, Set of Events, and Probability 1 Random Variables, Probability Distribution Function, Average Value, and Variance 4 1.3 Generalized Functions 5 1.4 Methods of Averaging 8 1.5 Characteristic Functions 12 1.6 Moments and Cumulants of Random Variables 14 1.7 Correlation Functions 16 1.8 Bernoulli, Poisson, and Gaussian Distributions 18 1.9 Stationary Random Functions, Homogeneous Random Fields 22 1.10 Isotropic Random Fields. Spectral Representation 25 1.11 Stochastic Processes. Markovian Processes. The Chapman–Kolmogorov Integral Equation 28 1.12 The Chapman–Kolmogorov, Chapman–Feller, Fokker–Planck, and Liouville Differential Equations 31 1.12.1 Derivation of the Differential Chapman–Kolmogorov Equation 31 1.12.2 Discontinuous (‘‘Jump’’) Processes. The Kolmogorov–Feller Equation 35 1.12.3 Diffusion Processes. The Fokker–Planck Equation 35 1.12.4 Deterministic Processes. The Liouville Equation 40 1.13 Stochastic Differential Equations. The Langevin Equation 43 1.13.1 The Langevin Equation 43 1.13.2 The Diffusion Equation 44 1.13.2.1 The Diffusion Equation with Chemical Reactions Taken into Account 45 1.13.2.2 Brownian Motion of a Particle in a Hydrodynamic Medium 46 1.14 Variational (Functional) Derivatives 48 1.15 The Characteristic Functional 53 1 1.1 1.2

VI

Contents

2 2.1 2.2 2.3 2.3.1 2.3.2 2.4 2.5

Elements of Microhydrodynamics 59 Motion of an Isolated Particle in a Quiescent Fluid 61 Motion of an Isolated Particle in a Moving Fluid 70 Motion of Two Particles in a Fluid 78 Fluid is at Rest at the Infinity (v1 ¼ 0) 78 Fluid is Moving at the Infinity (v1 6¼ 0) 92 Multi-Particle Motion 95 Flow of a Fluid Through a Random Bed of Particles 98

3 3.1 3.1.1 3.1.2 3.1.3 3.2 3.3 3.4 3.5 3.6 3.7 3.8

3.11 3.11.1 3.11.2 3.12

Brownian Motion of Particles 109 Random Walk of an Isolated Particle 110 Isotropic Distribution 113 Gaussian Distribution 114 An Arbitrary Distribution tðrÞ in the Limiting Case N  1 115 Random Walk of an Ensemble of Particles 116 Brownian Motion of a Free Particle in a Quiescent Fluid 117 Brownian Motion of a Particle in an External Force Field 122 The Smoluchowski Equation 124 Brownian Motion of a Particle in a Moving Fluid 126 Brownian Diffusion with Hydrodynamic Interactions 130 Brownian Diffusion with Hydrodynamic Interactions and External Forces 136 High Peclet Numbers: Peij  1 139 Small Peclet Numbers, Peij  1 140 Particle Sedimentation in a Monodisperse Dilute Suspension 142 Particle Sedimentation in a Polydisperse Dilute Suspension, with Hydrodynamic and Molecular Interactions and Brownian Motion of Particles 151 Transport Coefficients in Disperse Media 157 Infinitely Dilute Suspension with Non-interacting Particles 161 The Influence of Particle Interactions on Transport Coefficients 164 Concentrated Disperse Media 167

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.10.1

Turbulent Flow of Fluids 183 General Information on Laminar and Turbulent Flows 183 The Momentum Equation for Viscous Incompressible Fluids 184 The Equations of Heat Inflow, Heat Conduction and Diffusion 187 The Conditions for the Beginning of Turbulence 189 Hydrodynamic Instability 190 The Reynolds Equations 192 The Equation of Turbulent Energy Balance 197 Isotropic Turbulence 202 The Local Structure of Fully Developed Turbulence 212 Turbulent Flow Models 223 Semi-empirical Theories of Turbulence 224

3.8.1 3.8.2 3.9 3.10

Contents

4.10.2 4.11 4.12

The Use of Transport Equations 230 Use of the Characteristic Functional in the Theory of Turbulence Intermittency in a Turbulent Flow 244

5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.8.1 5.8.2 5.8.3 5.9

Particle Motion in a Turbulent Flow 251 The Eulerian and Lagrangian Approaches to the Description of Fluid Flow and Particle Motion 251 Lagrangian Statistical Characteristics of Turbulence 256 Turbulent Diffusion 267 A Semiempirical Model of Turbulent Diffusion 277 Models of Two-phase Disperse Turbulent Flows 282 Deposition of Particles from a Turbulent Flow 292 Interaction of Particles in a Turbulent Flow 304 Chemical Reactions in a Turbulent Flow 306 Concepts of Chemical Kinetics 309 Method of Moments 312 Approximations for Chemical Reaction Rates 319 The PDF Method 322

6 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Coagulation and Breakup of Inertialess Particles in a Turbulent Flow 335 Kinetic Equations of Coagulation 335 Fundamental Features of the Coagulation of Particles 344 A Model of Turbulent Diffusion 349 Hydrodynamic, Molecular, and Electrostatic Forces 357 Conducting Particles in an Electric Field 365 Coagulation of Particles in a Turbulent Flow 370 Breakup of Particles 384

7 7.1 7.2 7.3 7.4

Motion and Collision of Inertial Particles in a Turbulent Flow 395 Motion of Particles without Mutual Collisions 396 Motion of Particles with Mutual Collisions 413 Frequency of Collisions of Particles 421 Preferential Concentration of Particles in Isotropic Turbulence 433

Author Index 455 Subject Index 459

234

VII

IX

Preface Practically all fluids in our everyday surroundings are to some degree disperse systems one of the following types: liquid-solid particles, liquid-gas particles, gasliquid particles, liquid-liquid particles. They are called accordingly suspensions, liquid-gas and gas-liquid mixtures, emulsions and consist of continuous or carrying phase, which could be liquid or gaseous, and dispersed phase, which contains solid particles of different size and form, bubbles and droplets. The latter will be henceforward called simply particles, adding when needed words solid, liquid or gaseous. Particle size changes in broad range from microscopic (submicron, micron) to macroscopic (millimeter, centimeter). Particles, which sizes do not exceed several microns, are called Brownian, since they are subjected to Brownian (thermal) motion. Small particles, which density slightly differs from that of surrounding medium, are called inertialess and passive when in addition their volume concentration is negligible small, so that they do not affect the motion of carrying flow. Increasing of particle size and (or) particle density relative the outer medium leads to rise of particle inertia, while buildup of volume concentration enhances particlefluid and particle-particle interactions, formation of particle aggregates and in the presence of external oriented force field production of oriented particle structures. The latter exerts influence on rheological properties and sometimes causes the nonNewtonian behavior of the medium. Particle shape may also be multiform. Sometimes it could be approached by form close to canonical: spherical, ellipsoidal with different ratio of semi axes (in limiting case oblate disk-shaped, prolate cigar-shaped) and cylindrical. Prime interest in engineering applications attracts determination of properties of dispersive media, separation of multiphase multicomponent mixtures subjected to different force fields (gravitational, centrifugal, electrical, magnetic) and motion of dispersive media in tubes, channels and through porous medium. To solve these problems it is necessary to know behavior of separate particle as well as ensemble of particles. One of the basic parameter affecting both properties and dynamic characteristics of medium is volume content (volume concentration) w of disperse phase. Disperse medium with w  1 are called dilute. For such a medium average spacing between particles is much more than the particle mean size and each particle in limiting case

X

Preface

of infinite diluted solution, that is at w ! 0, behaves as a single one, its motion is completely determined by external forces, including forces acting on particles from surrounding medium. To the latter belong regular (mean) viscous drag force and random force due to collisions of molecules of surrounding medium with the particle, which causes Brownian motion of the particle. Brownian motion is noticeable only for Brownian particles and does not substantially affect movement of larger particles. Random force arises besides by chaotic fluctuations of carrying fluid, when particle moves in turbulent flow. Enhancement of w reduces the mean distance between particles and requires to take into account interactions between particles. The motion of particle subject to interaction forces is called hindered motion. Among interaction forces are distinguished hydrodynamic, molecular and electrostatic forces. The first one is characterized by long-range interaction and depends on hydrodynamic parameters, geometrical properties of particles (size, shape, orientation in space) and mutual arrangement of particles in space (configuration). Hydrodynamic forces are most pronounced when the distance between particle surfaces (clearance) is equal or less than particle linear size. The molecular interparticle force (Van der Waals attractive force) manifests itself distinctly only when the clearance between approaching particles becomes much less than the particle size. This force keeps particles together and promotes coagulation of rigid or coalescence of liquid and gaseous particles. Electrostatic force is repulsion force due to thin charged double layers on particle surfaces. This force prevents particle collisions and stabilizes dispersive medium. The range of action of electrostatic force is small compared with hydrodynamic force, and so it is short-range as well as molecular force. When particle volume concentration exceeds 10% the average interparticle spacing is not great and combined action of all mentioned forces can lead to formation of ordered structures of particles causing anisotropy of transport coefficients and non-Newtonian properties of dispersive system. Great influence on particle behavior exert flow conditions of carrying phase. Particle in quiescent fluid settles under gravity with constant velocity called sedimentation velocity. At w ! 0 this velocity is determined by Stokes formula for rigid spherical particle and Hadamar-Rybczynski formula for liquid one. Increasing of w leads to noticeable influence of surrounding particles on sedimentation velocity. If besides the particle size is enough small, the Brownian motion also affects the sedimentation velocity. Inhomogeneity of velocity distribution in laminar flow can also influence particle motion. Since the particle size is small compared to characteristic linear scale of the flow region, the flow in the vicinity of the particle could be considered as sheared. Such a flow induces translation and rotation particle motions, which with regard to interaction forces brings to mutual approach and further collision and coagulation of particles. Particle motion in turbulent flow is more complicated problem, while particle random motion due to effects of chaotic turbulent fluctuations, which enhances collision frequencies, is superimposed on regular transport with mean velocity under the action of carrying medium and external forces. Hence the rate of particle coagulation increases and processes of heat and mass exchange become more intensive.

Preface

The size of particles and ratio of densities of particle and carrying medium determine the particle inertia. Dimensionless parameter that allows to distinguish inertial particle from inertialess one, is Stokes number St equal to the ratio of particle dynamic relaxation time to characteristic time of exposure to environmental factors on the particle. Inertialess particles (St  1, small particles which density differs slightly from that of carrying medium) are fully involved in the movement of the carrying flow and its motion is on the whole determined by the characteristics of continuous phase. Inertial particles (St  1, relative big particles which density considerably deviates from the density of carrying medium) are only partially involved in the motion of continuous phase. All this makes difficult to investigate dynamics of such particles, since it requires to take into account interparticle collisions (for high concentrated disperse medium this concerns inertialess particles too) and to recruit kinetic theory of gases. Presence of enormous amount of particles in a unit volume of disperse medium, action of random fluctuations of environmental factors and inverse influence of random motion of particles on the surrounding medium makes impossible the description of dispersive medium behavior through deterministic method. The most fruitful and productive method is statistical method. This method examines not the behavior of each particle but the behavior of particle ensemble by means of probability distribution function (PDF), which, is able to describe the change of particle ensemble configuration in space-time with regard for particle relative motion under action of different forces. Statistical characteristics of PDF permit to determine the macroscopic properties of dispersive medium. The content of the book stems from lecture courses given to students of Moscow State University of Oil and Gas. The aim of the book is to give foundation of statistical methods used in hydrodynamics of micro particles that is hydrodynamics of suspension, which contains suspended in fluid micro particles. The first two chapters provide an introduction to probability theory and microhydrodynamics. The theory of Brownian motion of micro particles taking into account particleparticle and particle-fluid interactions are described in chapter 3. The fourth chapter contains necessary information about turbulent flow and its statistical description. The motion of micro particles in turbulent flow forms the subject of chapter 5. Chapters 6 and 7 deals with interactions of inertialess and inertial particles. It should be made a remark about the title of the book. The notion of microhydrodynamics was first introduced by G.K. Batchelor (see Batchelor G.K. Developments in Microhydrodynamics/In theoretical and applied Mechanics. Ed. W.T. Koiter. - Amsterdam: North Holland, 1976. P. 33-55) and was defined as a part of hydrodynamics, which studies the motion of particles in fluid under low Reynolds numbers. In the book this notion is extended not only to small but also to finite Reynolds numbers, in order to cover micro particle motion in turbulent flow. It seems that this naturally reflects the increasing interest to the topic of the book. December 2007

Emmanuil G. Sinaiski and Leonid I. Zaichik

XI

XIII

Nomenclature a a0 a1 ai aF ae A A A A A A,B, . . . A þ B!products A þ B!R, R þ B!P Au b bij bijk(r, t) bk1 k2 ...kN bLij(r, t) bLL(r, t) bLLL(r) bNN(r, t) buu(M1, M1) B kBk B B(0) B(M1, M2)

particle radius coefficient in the expression for correlation between acceleration fluctuations empirical constant semi-axes of an ellipsoid empirical constant in the e-F model constant in the k-e model set of events parameter in the equation of motion of a particle reaction constant drift vector (absolute) complement of a set A components (reactants) of a chemical reaction second order reaction two-step reaction particle acceleration particle radius mobility tensor third order structure function central moment Lagrangian two-point structure function longitudinal second-order structure function longitudinal third-order scalar function transverse second-order structure function two-point central moment of the second order; correlation function of fluctuations parameter in the equation of motion of a particle mobility matrix mobility tensor total energy of the field correlation tensor

XIV

Nomenclature

B(r) B1, B2, B3 Bab

correlation function of an isotropic random field empirical constants; dimensionless parameters correlation of concentration fluctuations at points Xa and Xb Bij(M1, M2) correlation function; two-point second-order moment Bijk(r, t) third-rank correlation tensor of isotropic turbulence Bk1 k2 ...kN moment of the order k1 þ k2þ . . . þkN BLa,b triple correlation of longitudinal velocity and concentration fluctuations at points Xa and Xb B0Li j ðtÞ Lagrangian single-point correlation function BLij(r, t) Lagrangian two-point correlation function BLL(r, t) longitudinal correlation function of isotropic turbulence transverse correlation function of isotropic turbulence BNN(r, t) BLL,L(r, t) scalar function of isotropic turbulence BLN,N(r, t) scalar function of isotropic turbulence BNN,L(r, t) scalar function of isotropic turbulence Buu. . .u(M1, M2, . . . MN) k-point moment Buv(M1, M2) mutual correlation function; two-point mixed moment c parameter of the distribution pðecr ; eÞ cp specific heat capacity at constant pressure specific heat capacity at constant volume cv C concentration of passive impurity C0 (X, t) concentration fluctuation h(C0 )2i intensity of concentration fluctuations C0 characteristic value of passive impurity concentration C1, C2 empirical constants; constants in the k-e model Ca0 concentration fluctuation at point Xa Ca resistance factor in the Archimedes force viscous resistance factor of a particle Cd Cd empirical constant CE ; CE0 empirical constants hCi0 DCi0 i moment that characterizes micromixing hðCi0 Þ2 i variance of concentration distribution of i-th component CL resistance factor in the lifting force arising due to the velocity shear Cm average mass concentration N binomial coefficient Cm Cp resistance coefficient of a particle Cw surface concentration Cz molar concentration dCz/dt specific rate of increase/decrease of matter concentration in a chemical reaction Ce concentration measurement error Cm parameter in the Reynolds stress equation Ca modified capillary number

Nomenclature

d d˜ d02 hdc2 i dcc(r) dij(t) ðrÞ

di j

dp D D D D Dbr Dbr D0br Drbr DF DE Dij Dij ð0Þ

Di j

Dri j DrLL Dm DrNN D pi j Dt ð0Þ Dt De DF DF Da DNS e

particle diameter ratio of particle diameter to pipe diameter dispersion of the average concentration distribution relative to the initial position of the source dispersion of the concentration distribution relative to the center of gravity structural concentration function of isotropic turbulence components of the fluid particle’s displacement dispersion tensor; components of the correlation tensor of displacement fluctuations components of the tensor of relative dispersion of two fluid particles particle diameter drag force on a particle diffusion coefficient dispersion tensor pipe diameter tensor of Brownian diffusion coefficient of Brownian diffusion coefficient of unhindered Brownian diffusion coefficient of rotational Brownian diffusion effective diffusion coefficient in equations for the e-F model effective diffusion coefficient in equations for the k-e model components of the dispersion tensor tensor of hindered relative (mutual) diffusion of two particles tensor of unhindered relative (mutual) diffusion of two particles tensor of relative diffusion of two fluid particles longitudinal component of the relative diffusion coefficient coefficient of molecular diffusion transverse component of the relative diffusion coefficient components of the tensor of turbulent diffusion of particles coefficient of turbulent diffusion coefficient of unhindered mutual turbulent diffusion function in the k-e model parameter in the three-parametric model of turbulence parameter in the three-parametric model of turbulence Damko¨hler number direct numerical simulation coefficient of restitution (COR)

XV

XVI

Nomenclature

e e (e1, e2, e3)  ei ¼ u0k v0k =2 ek = ru  kuk/2  ek ¼ u0k u0k =2   e0k ¼ r u0k u0k =2 e pk ¼ v0k v0k =2 e p ¼ v0k v0k =2 es ¼ rhui ihui i=2 E E E E(k, t) E(o) E1 EA E1 Ec(k) Ec(k, t) Ecr Eij Ei1j hEik(x, t)Ejn(x þ r, t)i E, F, . . . f f f f f (Xp(t), t) hfi f (V) h f (X, t)i h f (X, t)iT h f (X, t)iV h f (X, t)iTV f 1, f 2 _ _ f1 ; f2 fA fA fB feQ1

unit vector unit vector along the line of centers of two particles basis of a Cartesian coordinate system fluctuation energy of interfacial interaction kinetic energy density in the continuous phase intensity of turbulence; mean kinetic energy of fluctuational motion per unit mass in the continuous phase kinetic energy density of fluctuational motion turbulent energy of the disperse phase fluctuational energy of a particle kinetic energy density of averaged turbulent flow internal energy vector of electric field strength rate-of-strain tensor spectrum of average energy energy distribution over the frequency spectrum maximum principal value of rate-or-strain tensor activation energy rate-of-strain tensor at the infinity spectral density of concentration fluctuations intensity spectrum of concentration fluctuations critical value of external electric field strength rate-of-strain tensor components rate-of-strain tensor components at the infinity correlation function of rate-of-strain tensor components of the product of chemical reaction coefficient of particle’s involvement into fluctuational motion of the carrier flow (involvement coefficient) stochastic force external force density random force exerted by the flow on a fluid particle random Brownian force acting on a particle average (mean) value particle breakup frequency expectation value; ensemble average time average spatial average spacetime average functions in the k-e model dimensionless coefficients force acting on a particle due to the particle acceleration dimensionless force of molecular attraction between spherical particles Basset force resistance coefficient for particle’s translational motion

Nomenclature

fer fsr fsY h fAi0 pi h fBi0 pi fi fj fL h fLi0 pi fr

fu fm(Re) F F F F F(k) F[w] F(Xp(t), t) F2x F2z kF0 k FE FB Fe Fe

Ffl Fg Fm FAs

resistance coefficient for particle’s translational motion resistance coefficient for particle’s translational motion resistance coefficient for particle’s translational motion correlation between Archimedes force fluctuations and probability density function (PDF) of particles correlation between Basset force fluctuations and PDF of particles dimensionless coefficients in the expression for the force of electrical interaction between particles j-th component of the force acting on the unit surface area of a particle lifting force acting on a particle due to the velocity shear of the carrier flow correlation between lifting force fluctuations and PDF of particles coefficient of involvement of a pair of particles separated by the distance r into fluctuational motion of the continuous phase coefficient of involvement of a particle into fluctuational motion of the turbulent carrier flow parameter in the equation for Reynolds stresses in twoparametric turbulence models function in the e-F model systematic force particle interaction force generalized force vector spectrum of a homogeneous random field functional external force acting on a particle component of interaction force between two conducting charged particles perpendicular to their line of centers component of interaction force between two conducting charged particles along their line of centers generalized force matrix external force stochastic Brownian force external force component of the force exerted on a particle by the surrounding fluid; hydrodynamic force resulting from the particle’s proper motion fluctuating force gravitational force molecular interaction force force of molecular attraction between two spherical particles

XVII

XVIII

Nomenclature

Fab(k, t) Fcap Fg Fh Fh Fij(k) Fij,k(k) Fik(r, t) ðiÞ

Fk FLL(k, t) FMi FNN(k, t) F sR Fs Fih Fv g g gi(v/o) g1 gi1j jg  jkj  g jk   T g jk  gfl gu G G G G G(C, C0 , C00 )

S(v, o)

spectrum of concentration fluctuations surface tension force (capillary force) gravitational force force exerted on a particle by the surrounding fluid hydrodynamic resistance force acting on a particle spectral tensor spectrum of the third-rank correlation tensor of isotropic turbulence integral of two-point structure function of velocity fluctuations in a continuous medium, taken along the trajectory that describes the relative motion of a pair of particles force acting on i-th particle longitudinal spectrum of isotropic turbulence migration force caused by the interaction of particles with turbulent eddies of the carrier flow transverse spectrum of isotropic turbulence electrostatic repulsion force between two spherical particles Stokesian component of the force exerted on a particle by the carrier flow thermodynamic force viscous friction force acceleration of gravity coefficient of involvement of particles into fluctuational motion of the carrier flow single-modal density distribution of particle volume ratios in the course of particle breakup velocity gradient tensor at the infinity components of velocity gradient   tensor at the infinity determinant of the matrix g jk  matrix transposed matrix fluctuation of the rate of matter production or consumption in the course of a chemical reaction coefficient of involvement of a particle into fluctuational motion of the carrier turbulent flow dimensionless parameter Gibbs free energy spatial gradient of a physical parameter empirical function kernel of the micromixing operator in coalescencedispersion models; distribution density of intermediate concentrations that are formed during an elementary mixing event collision cross section of two particles of volumes V and o; normal cross-sectional area of the limiting flow tube

Nomenclature

hGi1 Gfl(X, t) Gik(r, t)

Gj Ga h h h h0 h0 hi hu hd H H(X) H(X) i (i1, i2, i3) i, j I[p] I I Ik IR j j(R1, R2) J0 J0(0) jfl J J J Jc

conditional average fluctuational term in the diffusion equation integral of the two-point structure function of velocity fluctuations in the continuous phase, taken along the trajectory of the relative motion of two particles parameter in the expression for coagulation kernel Galilei number coefficient of hydrodynamic resistance coefficient of hydrodynamic resistance for particles approaching each other along their line of centers distance between two planes minimum gap between surfaces of spherical particles hydrodynamic resistance coefficient for unhindered particle motion hydrodynamic resistance coefficient for i-th particle coefficient of particle’s involvement into fluctuational motion of a turbulent carrier flow asymptotic expression for the hydrodynamic resistance coefficient of a particle at small clearances between particles enthalpy Saffman function Heaviside function parameter of the gamma distribution basis of a local Cartesian coordinate system mutually orthogonal unit vectors in a plane perpendicular to the line of centers of two particles Boltzmann collision operator in the Enskog form unit tensor number of particle collisions in the absence of the force of electrostatic repulsion collisional term in the kinetic coagulation equation number of particle collisions, with electrostatic repulsion taken into consideration diffusion flux diffusion flux of particles of radius R2 toward a particle of radius R1 diffusion flux of unhindered particles diffusion flux per unit solid angle vector of diffusion flux fluctuation displacement vector dimensionless diffusion flux of particles of radius R2 toward a particle of radius R1 particle flux toward a test particle collision operator in the kinetic equation for PDF of velocity of a single particle

XIX

XX

Nomenclature

Jf Ji Ji Jij Jiaj Jik Jr Jw Jw k k k k k kB kc kf k kcab krab ktab kb kk km

K K K K(V,o)

K K c0 K r0 Kt K c 0 Kt K ti jk Kr

incident particle flux flux density of impurity along the Xi-axis moment of inertia of i-th particle term in the balance equation for turbulent stresses in the disperse phase, which are caused by particle collisions components of the Oseen tensor (stokeslet) for a-th particle i-th component of the diffusion flux of k-th component of the impurity particle flux reflected from the wall flux of matter toward the wall flux of particles depositing on the wall ratio of particle radii wave number permeability of a porous medium reaction rate constant unit vector along the line of centers of two colliding particles parameter in the equation of motion of a particle in a turbulent flow wave number of a scalar field coefficient of aerodynamic resistance the Boltzmann constant conjugate mobility tensor rotational mobility tensor translational mobility tensor wave number corresponding to the Batchelor scale wave number corresponding to the Kolmogorov scale wave number corresponding to the maximum size of inhomogeneity (the right end of the inertial-convective range, starting from which micromixing commences) empirical coefficient transport coefficient moment’s order kernel of the kinetic coagulation equation; collision frequency of two particles of volumes v and o in a unit disperse phase volume at unit particle concentrations; coagulation constant micromixing operator conjugate translational resistance tensor rotational resistance tensor translational resistance tensor conjugate rotational resistance tensor third-rank tensor of translational shear resistance components of translational shear resistance tensor third-rank tensor of rotational shear resistance

Nomenclature

kK0 k  K t;r;c  0 Keff Kg Kirjk Kpr Ks Kt Ktg Kts l l l l1 li j lm lu l L

L L L L L Lc L ðiÞ

Lk Lki LL LN ðrÞ Lq ð j+jÞ Ls LES m m m m

generalized resistance matrix resistance matrices macroscopic (effective) transport coefficient collision kernel in the gravitational field components of third-rank rotational shear resistance tensor permeability matrix collision kernel in a shear flow collision kernel in a turbulent flow collision kernel in a turbulent flow with gravity taken into account collision kernel in a turbulent shear flow step length for a particle’s random walk Prandtl mixing length Prandtl–Nikuradze mixing length Taylor mixing length components of symmetric scale tensor mean free path of molecules coefficient of particle’s involvement into fluctuational motion of turbulent carrier flow unit vector characterizing particle’s orientation average distance between particles; characteristic linear size; spatial macroscale of turbulence; Eulerian spatial macroscale; integral spatial scale of turbulence linear operator torque linear macroscale of concentration field empirical function parameter in the equation of motion of a particle in a turbulent flow integral scale (macroscale) of concentration field hydrodynamic torque exerted on a particle by the surrounding fluid torque acting on i-th particle associated Laguerre polynomial longitudinal integral scale transverse integral scale Lagrange interpolation polynomial Stokesian component of torque exerted on a fixed particle by the surrounding fluid large eddy simulation mass empirical constant in the e-F model particle surface mobility parameter dimensionless parameter

XXI

XXII

Nomenclature

m _ m m0 m1 mi mi mk mu M Mi Ni n(X, a) n(v, X, t) n n n n n0 n0(v) ni N N N(t) N N hNi N Na hN c i ND

Nt

NW

Oh p p

virtual mass of a particle dimensionless moment of distribution zero order moment; number concentration of particles first order moment; volume concentration of particles mass of i-th particle mass of i-th component produced or consumed in the course of chemical reaction k-th order moment of the volume distribution of particles coefficient of particle’s involvement into fluctuational motion of turbulent carrier flow migration coefficient molecular mass of i-th component number of moles of i-th component distribution of particles over their radii distribution of particles over their volumes empirical constant in the e-F model total reaction order; kinetic order of reaction normal vector number concentration of particles concentration of particles far from the test particle initial distribution of particles over volumes order of i-th reaction spacetime point number of particles in a given volume number of particles in a unit volume (number concentration) number of steps (displacements) in particle’s random walk dimensionless parameter dissipation of concentration inhomogeneity scalar dissipation of concentration inhomogeneity number of particles of type a in a unit volume conditional average rate of scalar field dissipation at fixed concentration C ratio between the characteristic time of a process and the characteristic time of impurity transport via molecular diffusion ratio between the characteristic time of a process and the characteristic time of convective transport of impurity via turbulent motion ratio between the characteristic time of a process and the characteristic time of matter production or consumption in a chemical reaction Ohnesorge number pressure dimensionless parameter

Nomenclature

p p p0 p p(rjj1, r0, t, t0) p(u, C1, C2, . . ., CN; X, t) C1, C2, . . ., CN p(X) p(X1, X2, . . ., XN) pðX ; V jj1 ; j2 ; . . . ; jn ; t1 ; t2 ; . . . ; tm Þ pðX; Vjj1 ; j2 ; . . . ; jn ; t1 ; t2 ; . . . ; tm Þ pðX ; V ; ujj1 ; j2 ; . . . ; jn ; X 1; X 2; . . . ; X n; t1 ; t2 ; . . . ; tm Þ p(u, tju0, t0) pðV 1 ; V 2 ; . . . ; V n j j1 ; j2 ; . . . ; jn ; t1 ; t2 ; . . . ; tn Þ p(Yjt; j, t0) pðeÞ  Dt _ Dt _ p e + e; e  eÞ 2 2 _ pðe; eÞ ˙ pðecr ; eÞ hp0 u0 i p1 pc(u, C, X) peq pij(r) pp(X, v, t) P P = hpi P(AjB) PðA \ BÞ

unit vector dynamic probability density for particle ensemble perturbation (fluctuation) of pressure pressure averaged over the cross section PDF of distance between two particles; function of distance between neighbors joint single-point PDF of velocity and scalar quantities (concentrations of different passive impurities) at time t probability density function, PDF joint multi-particle PDF joint multi-dimensional (multi-particle) PDF of coordinates X and Lagrangian velocities V for a given set of time values t1, t2, . . ., tm and initial particle positions j1, j2, . . ., jm; the main Lagrangian statistical characteristic of turbulence joint PDF of Lagrangian (V) and Eulerian (u) quantities

conditional probability density joint PDF of Lagrangian velocities V(j1, t1), V(j2, t2), . . . V(jn, tn) of n fluid particles at different moments of time probability density distribution of particle displacements from the initial point j during the time t one-dimensional distribution of random specific energy dissipation joint distribution density of random values e and e_ at different moments of time joint distribution density of random values e and e_ at one and the same moment of time joint distribution density of ecr and e_ mixed correlation (correlator) of pressure and velocity fluctuations static pressure at the infinity joint PDF in a turbulent mixing layer equilibrium PDF pair PDF dynamic probability density in the phase space of particle coordinates and velocities probability velocity PDF of a system of particles conditional probability probability of a joint event

XXIII

XXIV

Nomenclature

P(r, w, t) P(v, o)

P(wr) P(YN) P0(v) Pr0 P2(v, v1) Pnm Pt Pe PeD PeT Pr Prp Prt q(r, t) q q q qe Q Q Qc Q aij;n r rw r (r, Y, z) (r, Y, F) r0 r0 (r0, Y0, F0) ri R R R R

PDF of a particle pair (probability of the relative velocity w separated by the distance r at the time t) probability of formation of a droplet with volume in the range (v, v þ dv) in the process of breakup of a droplet with volume in the range (o, o þ do) probability distribution of fluctuational component of the relative radial velocity probability distribution of a certain N-particle configuration Maxwellian velocity distribution associated vector of dynamic pressure in rotational motion PDF of velocities of two particles associated Legendre polynomial associated vector of dynamic pressure in translational motion Peclet number diffusional Peclet number thermal Peclet number Prandtl number Prandtl number of the disperse phase turbulent Prandtl number of the carrying phase Richardson function particle migration coefficient heat flux vector particle charge density of internal heat sources dynamic pressure directed flux of matter intensity of fluctuational energy dissipation via inelastic collisions of particles n-th order moment (multipole) of surface forces acting on the particle a distance between the centers of two particles pipe radius unit vector along the line of centers of two particles cylindrical coordinates spherical coordinates initial radius vector of the particle center radius vector of a point relative to the center O. initial coordinates of the particle center radius vector of the center of i-th particle gas constant resistance tensor tensor of generalized resistance due to the translational and rotational motion of the particle grand resistance matrix

Nomenclature

R R R R R/(R þ P) R12 Ra Rc Rij RFE RFU RFV RLE RLU RLV RSE RSU RSV Raij Rm Rp Re Recr Rep Ret Rel s s s s s hs 2i S S S S S(k) dS SA Sij Sij Saij

particle’s center of reaction particle radius pipe radius correlation coefficient of colliding particles selectivity parameter of a two-stage chemical reaction velocity correlation coefficient of two particles at the point of impact radius of a particle of type a coagulation radius components of the resistance tensor component of the resistance tensor of a particle component of the resistance tensor of a particle component of the resistance tensor of a particle component of the resistance tensor of a particle component of the resistance tensor of a particle component of the resistance tensor of a particle component of the resistance tensor of a particle component of the resistance tensor of a particle component of the resistance tensor of a particle rotlet of particle a minimum radius of a breaking droplet coordinate vector of the particle’s center Reynolds number critical value of the Reynolds number Reynolds number of a moving particle Reynolds number corresponding to the spatial Taylor microscale local Reynolds number for fluctuations having scale l. dimensionless distance between particle centers minimum separation from a plane distance traveled by a particle in time t variable in the image domain of the Laplace transformation unit vector along the particle’s displacement vector r mean-square particle displacement surface entropy sedimentation coefficient symmetric part of the velocity gradient tensor element of sphere surface area jkj = k oriented surface element parameter of molecular interaction sedimentation coefficients intensity of force dipoles distributed over the particle surface surface force dipole (stresslet) of particle a

XXV

XXVI

Nomenclature

SBij SG ij SIi j Sk

Sk1 ;k2 ;...;kN SR SBT SET SH T Sc Scbr Sct St St0 t0 t12 t12 t21 tA tc tcom tD tE tew tew1 tsw ttE g

tE teff tik tL

contribution of Brownian diffusion to the sedimentation coefficient contribution of gravity to the sedimentation coefficient contribution of molecular interaction to the sedimentation coefficient specific source term of k-th component produced/ consumed in a chemical reaction in the mass conservation equation for passive impurity components cumulant (semi-invariant) electrostatic interaction parameter contribution of particles’ Brownian motion to the stress tensor contribution of particles’ motion driven by non-hydrodynamic forces to the stress tensor contribution of particles’ motion driven by interparticle hydrodynamic interactions to the stress tensor Schmidt number Schmidt number for Brownian diffusion turbulent Schmidt number Stokes number calculated with respect to the integral macroscale Stokes number calculated with respect to the Kolmogorov microscale initial moment of time time required for a perturbation to go through the wave number range (k1, k2) times of collisions of particle 1 with particles 2 times of collisions of particle 2 with particles 1 characteristic time of particle enlargement that takes into account molecular interactions between particles effective time between particle collisions characteristic time of change of system configuration characteristic time of a diffusion process characteristic enlargement time for conducting droplets in an electric field rotational resistance coefficient of a particle rotational resistance coefficient of a particle rotational resistance coefficient of a particle characteristic time of enlargement of conductive droplets in a turbulent flow in an external electric field characteristic time of enlargement of conductive droplets undergoing gravitational sedimentation in an electric field effective relaxation time of a particle components of the Maxwellian stress tensor characteristic non-stationarity time for an averaged flow

Nomenclature

tT tu tv tV tl T T T T0 T (E) Tfl T (L) ðLÞ T1

Ts Te

p

Tij TL0 ðLÞ

TL p ðtÞ ðLÞ

TN p ðtÞ ðLÞ

Tp

ðLÞ

Tr u u

u u(X, t) (u, v, w) u0 u0 u0 (X, t) u u0 u0

characteristic thermal relaxation time for a particle effective relaxation time that takes into account the effect of virtual mass characteristic time of dynamic relaxation for a particle (depends on Rep) characteristic time of viscous relaxation for a particle characteristic period of fluctuations having the scale l characteristic time scale stress tensor turbulent fluctuation period _ parameter of the distribution pðecr ; eÞ Eulerian time scale fluctuational component of the stress tensor Lagrangian correlation time; Lagrangian integral time scale of turbulence asymptotic expression for the Lagrangian integral scale of turbulence at high Re systematic component of the stress tensor component of hydrodynamic torque exerted on a particle by the surrounding fluid and caused by the particle’s proper motion contribution of particles suspended in the fluid to the stress tensor parameter in the parabolic exponential approximation of the autocorrelation function Lagrangian longitudinal (parallel to the average relative velocity of particles) integral time scale of fluid velocity fluctuations along the particle trajectory Lagrangian transverse (perpendicular to the average relative velocity of particles) integral time scale of fluid velocity fluctuations along the particle trajectory Lagrangian integral time scale two-point integral time scale velocity vector relative velocity of two particles approaching each other along their line of centers carrier flow velocity Eulerian velocity field at the point (X, t) velocity components velocity perturbation fluctuational component of the carrier flow velocity Eulerian field of velocity fluctuations mean flow rate velocity initial velocity characteristic turbulent velocity

XXVII

XXVIII

Nomenclature

u0 u0c(t) u02 ¼ hu0k u0k i=3 u u1 Dui(r, t) ui(S) uL um uN ul ul0 0 hu0i fBk i hu0i fLk0 i hu0i u0k i hu0i u0j i rhu0i u0j i hu0i q0 i U u (U, V, W) Ui U U ku0 k U ra U ta Um Uv v v v1 vp v p ; v p1 v0p ; v0p1 ˜v pi ðsÞ V V0 Vav

velocity of Stokesian flow around a particle effective path length of a particle in fluctuational motion turbulence intensity; intensity of velocity fluctuations of the continuous phase dynamic velocity velocity at the infinity difference of velocity fluctuations at two points Laplace transform image of carrying phase velocity projection of velocity onto r average molecular velocity projection of velocity onto a direction perpendicular to r rate of fluctuations of scale l rate of fluctuations on the Kolmogorov microscale l0 correlation between fluid velocity fluctuation and Basset force fluctuation correlation between fluid velocity fluctuation and lifting force fluctuation components of the correlation tensor of velocity fluctuations specific Reynolds stresses of the carrying phase Reynolds stresses in the carrying phase turbulent heat flux in the carrying phase velocity; averaged velocity; average velocity of turbulent flow generalized velocity vector, which includes both translational and rotational components of velocity velocity components degree of immiscibility of i-th reagent degree of immiscibility of the total reaction mean-flow-rate velocity matrix of generalized velocity angular velocity of particle a translational velocity of particle a average mass velocity sedimentation velocity of a particle of volume v velocity translational velocity of the particle center velocity at the infinity particle velocity velocities of two particles before the collision velocities of two particles after the collision Laplace transform image of particle’s velocity particle volume parameter of the volume distribution of particles average particle volume

Nomenclature

Vk Vm (vr , vy , vF) V V DV V V(j, t) V0 V 1, V 2 V r0 Vt p VA s VA Vij ð0Þ

Vij Vimj hViVji hVi0 V 0j i hVi0 q0j i Vp VRs w ¼ v p  v p1 w w

wk0 wr wr0 hwr0 2i W W W jW0j Wg hWii

propagation speed of perturbations along the wave number axis minimum volume of breaking droplets components of the velocity vector in a spherical coordinate system spatial volume total interaction potential of two spherical particles volume element Velocity Lagrangian velocity particle velocity fluctuation potentials of charged particles rotational velocity tensor translational velocity tensor molecular interaction potential of two planes molecular interaction potential of two spherical particles relative velocity of two particles relative velocity of unhindered motion of two particles Lenard–Jones potential Lagrangian correlation function of components of the vector V turbulent stresses in the disperse phase turbulent heat flux in the disperse phase particle velocity electrostatic interaction potential of two particles relative velocity of colliding particles relative velocity of two particles term in the particle’s equation of motion that helps account for the hydrodynamic (collisionless) interactions of particles; represents a continuous random process random field of hydrodynamic interactions radial component of relative velocity of a particle pair fluctuational part of the radial component of relative velocity of a particle pair intensity of fluctuations of the radial relative velocity of a particle pair stability factor of a dispersed system collisional term in the Langevin equation of motion of a particle; discontinuous random process distribution of discontinuous jumps characteristic rate of chemical reaction difference of two particle’s sedimentation velocities average chemical reaction rate of i-th component

XXIX

XXX

Nomenclature

Wi Wr We x1 (X, Y, Z) X X(X1, X2, . . ., Xn) X(j, t) X 0a X sa ~ rab ; x~rab X ~ t ; x~t X ab ab X0(t) hXc2 i Xp(t) yþ ~ cab ; ~ycab Y ~ r ; ~yr Y ab ab ~ t ; ~yt Y ab ab Y(t) Y0 Z1, Z2, Z3 a a a aE a aj aL at ae b

source term in the diffusion equation, which results from a chemical reaction radial component of the average relative velocity of a particle pair Weber number dimensionless distance from the particle surface Cartesian coordinates generalized vector that includes spatial coordinates and orientation angles relative to a given coordinate system point in space random function – Eulerian coordinates of a fluid particle vector of spatial coordinates (position vector) of the center of particle a in a given coordinate system vector of orientation angles of particle a in a given coordinate system dimensionless resistance coefficients for rotational motion of particles about the line of centers dimensionless resistance coefficients for translational motion of particles along the line of centers coordinate of the center of mass variance of concentration distribution relative to a source particle coordinates at the moment t dimensionless distance to a wall dimensionless resistance coefficients for rotational motion of particles about the j-axis dimensionless resistance coefficients for rotational motion of particles about the i-axis dimensionless resistance coefficients for translational motion of particles perpendicular to the line of centers vector of particle’s displacement from the initial position during the time interval t fluctuation of particle displacement conservative variables; Schwab–Zeldovich variables empirical constant weight function _ parameter of the distribution pðecr ; eÞ constant in the three-parametric turbulence model empirical constant in the e-F model exponent in the coagulation kernel expressed as a power function of particle volumes constant dissipation rate of hydrodynamic field; inverse relaxation time constant in the three-parametric turbulence model constant in the k-e model correction factor

Nomenclature

b b b b1 b2 bE bj bL bt g g g g(t) g g˙ g˙  gf g0 gij(t) g˙ t gt gF G G(x) GF d d d d(X) d(X) dD D D(t) D1, D2 e e e e e e e e

dimensionless parameter empirical constant parameter defining the condition for heterogeneous reaction square of skewness excess (kurtosis) of the distribution constant in the three-parametric turbulence model exponent in the coagulation kernel expressed as a power function of particle volumes constant dissipation rate of a scalar field; inverse relaxation time constant in the three-parametric turbulence model empirical constant dimensionless parameter density ratio between a particle and the surrounding fluid Green’s function of the particle’s equation of motion function in the e-F model shear rate dimensionless hydrodynamic interaction parameter particle shape parameter constant in the three-parametric turbulence model components of velocity fluctuation gradient tensor average shear rate in small-scale fluctuations constant in the three-parametric turbulence model parameter in the three-parametric turbulence model Hamaker constant complete gamma function constant in the three-parametric turbulence model gap (clearance) between particle surfaces dimensionless parameter thickness of the viscous boundary layer delta function of a scalar quantity delta function of a vector quantity thickness of the diffusion boundary layer dimensionless gap (clearance) between particle surfaces effective diameter of a particle cloud dimensionless parameters dielectric permittivity relative fluctuation of distributed particle density rate of turbulent energy dissipation dissipation function small parameter permutation symbol parameter in the Lennard–Jones potential specific dissipation of turbulence energy

XXXI

XXXII

Nomenclature

e0 ecr ðVÞ heki eijk es z h q q0 hq0 u0 i q qp(t) y y(F) k k k k kp l l l0 lc lð0Þ c (lc)i lD lL lL lN lt L L L1 L1 Lij Lij L1 ij

specific dissipation rate of a scalar field critical value of specific dissipation of energy mean specific dissipation of energy of fluctuational motion components of the permutation (Levi–Civita) symbol specific dissipation of energy of averaged motion drift parameter characterizing the effect of intersection of particle trajectories porosity of a medium absolute temperature temperature perturbation (fluctuation) mixed correlation of temperature fluctuations and velocity fluctuations temperature of the carrying medium particle temperature orientation angle of the particle pair relative to the external electric field angle between the radius vector r0 of the particle center and a point on the limiting trajectory coefficient of heat conductivity empirical constant dimensionless parameter in the expression for the coefficient of mutual turbulent diffusion dimensionless parameter characterizing droplet stability in an external electric field coefficient of turbulent thermal conductivity dimensionless resistance coefficient scale of velocity fluctuations inner scale of turbulence; Kolmogorov spatial microscale differential scale of fluctuations inner scale of concentration; Batchelor microscale microscale of i-th component concentration Debye radius London wavelength longitudinal differential scale transverse differential scale Taylor spatial microscale vorticity tensor Loitsyansky invariant vorticity tensor at the infinity antisymmetric part of the undisturbed velocity gradient tensor components of the tensor of relative dispersion of particle cloud components of the vorticity tensor components of the vorticity tensor at the infinity

Nomenclature

hLik(x, t)Ljn(x þ r, t)i m m me meff mi mt n nA, . . ., nF nAA þ nBBþ. . .! nEE þ nFF. . . ne ni np nt j j(t) j jj Xi P P(X, t) r0 re ri rp s s s s2 sij YN S S S t t t t

correlation function of the vorticity tensor chemical potential ratio of viscosity coefficients of the internal and external fluids dynamic viscosity coefficient of the external fluid effective dynamic viscosity coefficient dynamic viscosity coefficient of the internal fluid turbulent dynamic viscosity number of steps in particle’s random walk stoichiometric coefficients kinetic scheme of chemical reaction kinematic viscosity coefficient of the external fluid kinematic viscosity coefficient of the internal fluid turbulent viscosity coefficient of the disperse phase turbulent viscosity coefficient of the continuous phase dimensionless gap (clearance) between particles degree of completeness of a chemical reaction ratio of particle diffusion fluxes with and without electric field Lagrangian coordinates; initial coordinates of a particle coefficient of virtual turbulent diffusion of a particle cloud in the Xi direction triadic (third-rank) tensor intermittency function density ratio of the carrying medium and the particle external fluid density internal fluid density particle density surface charge density parameter in the Lennard–Jones potential root-mean-square deviation distribution variance components of the viscous stress tensor element of the set of configurations of an N-particle ensemble specific surface area of the boundary between regions of different concentrations surface tension coefficient parameter characterizing the relative influence of gravity and turbulence on the collision kernel characteristic time of a process stress dimensionless time ratio of the particle radius to the double layer thickness

XXXIII

XXXIV

Nomenclature

tE t0 tc (tc)1 (tc)2 tD tf ti(ri) tij ð1Þ

ti j tm tt tt tw

t l0 tL u w w(r) w1 wm f f(s) f1, f2 fij kFk F(p)

F[(r, x)] Fp(r, t) x x x xt c c

characteristic time of decrease of the rate-of-strain tensor turbulent component of stress relaxation time of a concentration field; characteristic time of micromixing time of micromixing in the inertial-convective region time of micromixing in the viscous-convective region characteristic time of impurity transport via molecular diffusion friction force per unit surface area probability density distribution of i-th displacement components of the stress tensor components of the Reynolds stress tensor empirical constant characteristic time of turbulent convective impurity transport Taylor time microscale; characteristic time of hydrodynamic relaxation characteristic time of matter production/consumption in a chemical reaction characteristic period of fluctuations on scale l0; the Kolmogorov time microscale characteristic time of decrease of the vorticity tensor particle mobility volume concentration (content) of particles characteristic function; moment generating function empirical constant limiting volume concentration of closely packed particles force field/electric field potential function in the Laplace transform image of particle’s velocity particle surface potentials molecularinteractionpotentialbetweeni-thandj-thparticles shear resistance matrix correction factor introduced into the expression for the molecular attraction force in order to account for electromagnetic retardation characteristic functional probability density of particle’s displacement by the distance r during the time t reflection coefficient of a particle thermal diffusivity inverse Debye radius turbulent thermal diffusivity cumulant-generating function stream function

Nomenclature

c(t)

C(t) C(E)(r) C(E)(t) C(L)(t) Ccc ðEÞ CL ðr; tÞ

CL(t) CLp(t  t1) ðLÞ

CL p ðtÞ

CLr(tjr) ðEÞ

CN ðr; tÞ ðLÞ

CN p ðtÞ

Cu(j) Cuu Cuv Cv CTv Cvt C kCk v v hvi v(X) v12 vl

function characterizing the particle’s effective free path resulting from its involvement into fluctuational motion of the carrier flow autocorrelation function Eulerian spatial autocorrelation function Eulerian time autocorrelation function Lagrangian autocorrelation function correlation coefficient of concentration fluctuations longitudinal (parallel to the average relative velocity vector) Eulerian spacetime autocorrelation function of fluid velocity fluctuations along the particle trajectory dimensionless autocorrelation function characterized by Lagrangian integral time scale two-time correlation function of carrier flow velocity fluctuations along the particle trajectory longitudinal (parallel to the average relative velocity vector) Lagrangian time autocorrelation function of fluid velocity fluctuations along the particle trajectory Lagrangian autocorrelation function characterizing the relative motion of two particles initially separated by the distance r = jrj transverse (perpendicular to the average relative velocity vector) Eulerian spacetime autocorrelation function of fluid velocity fluctuations along the particle trajectory transverse (perpendicular to the average relative velocity vector) Lagrangian time autocorrelation function of fluid velocity fluctuations along the particle trajectory autocorrelation function of fluid velocity fluctuations along the particle trajectory correlation coefficient correlation coefficient two-time autocorrelation function of fluid velocity fluctuations along the particle trajectory two-time autocorrelation function of fluid temperature fluctuations two-time autocorrelation function of fluid velocity fluctuations function appearing in the e-F and k-e models shear resistance matrix element of the event set vorticity vector average vorticity vector weight function frequency of collisions between particles of types 1 and 2 frequency of fluctuations on scale l

XXXV

XXXVI

Nomenclature

v0l V X X Vi X1 V1 i Vu

frequency; time period of fluid particle velocity recurrence set of all events angular velocity vorticity vector vorticity vector component vorticity vector at the infinity vorticity vector component at the infinity inertness parameter for a particle

Some mathematical notations [ union of sets \ intersection of sets 1 empty set 5 gradient operator r divergence operator 5 curl operator 5d gradient of the delta function D = jqXi/qjjj determinant of a Jacobian rr = rirj dyadic ab. . .c = aibj. . .ck polyadic product of tensor and vector Aa = Aijak Aa = Aijaj scalar product of tensor and vector A:B = AijBij double-dot scalar product of tensors 2 dX d X time derivatives ; X¨ ¼ 2 X˙ ¼ dt dt D @ @ substantial derivative ¼ +uk Dt @t @Xk dF½w functional derivative dwðX Þ dN F½rðX Þ N-th order functional derivative drðX1 ÞdrðX2 Þ . . . drðXN Þ R ðquÞ ¼ qðMÞuðMÞdM scalar product in the functional space d Dk ðMÞ ¼ dqk ðMÞ i Im erf(z)

functional derivative operator imaginary unit imaginary part probability integral

j1

1 Basic Concepts of the Probability Theory In order to formulate the theoretical concepts that will be crucial for the subsequent chapters, we must first mention some basic notions of the probability theory and their further ramifications. This chapter provides a brief outline. The interested reader will find a more detailed discussion of the relevant topics in textbooks and monographs on the probability theory and statistical physics (see the list of references at the end of the chapter).

1.1 Events, Set of Events, and Probability

Some of the typical situations considered in this chapter are as follows: a particle is in small volume (elementary volume) that includes a point X; N particles are in a small region of space; N1 particles of type 1 and N2 particles of type 2 are in a certain region. Speaking more generally, we shall be dealing with situations having to do with particle positions in space at different instants of time. Every such case can be thought of as a specific realization of some event. We shall define an event as an element of a certain space of events. In what follows, we will most often be using vector spaces defined by vectors X (X1, X2, . . ., Xn), where Xi are real numbers. The probability theory introduces the notion of a set of events. Then the condition that an event o belongs to a set of events A is written as o 2 A. Consider an event of finding a particle in some volume element DV centered at a point X. Let all such events form an ensemble of events denoted as A(DV, X). It turns out that it is possible to determine whether the particle is in the vicinity of X, but it makes no sense to determine whether the particle is exactly at the point X. Physically, the probability of finding a particle exactly at some given point is zero. As an idealization one can consider a probability distribution given by a delta function, in which case the probability can be a finite number, and it is associated with an infininesimal interval around this point as the interval tends to zero. Let o(X) denote the event that a particle is found in infinitesimal volume element centered at X. We can then ascribe a certain probability to the condition o(X) 2 A(DV, X). This is just the probability for the particle to be in the volume element DV centered at X.

2

j 1 Basic Concepts of the Probability Theory The probability P(A) of a set of events is defined as a function of A which satisfies the following probability axioms: 1. P(A) 0 for all A; 2. P(O) ¼ 1; 3. if {Ai} is a finite or countable sequence ofX non-overlapping sets, that is, (Ai \ Aj ¼ f), then Pð[ Ai Þ ¼ PðAi Þ. i

i

The two consequences of these axioms are:  ¼ 1PðAÞ; 1. PðAÞ 2. P(f) ¼ 0. Here \ and [ denote, respectively, the intersection and union operations on sets;  – the complement of a set A, that is set O is the set of all events; f – the empty set; A of all events not belonging to A. Coexisting with the notion of probability is the notion of frequency of an event. To understand the difference between probability and frequency of events, consider an event o selected at random from the full set O. The number of occurrences of o 2 A in N trials gives us the relative frequency of realization of the event o 2 A. When N is increased, the relative frequency goes to the limit P(A), which is defined as the probability of the event A. At N  1, it is safe to assume the relative frequency of an event to be equal to the probability of occurrence of this event presuming that relative frequency has been normalized. The axiom 3 is given for a finite or countable number of sets. But often it is necessary to deal with uncountable infinite number of sets. For instance, when studying the motion of particles under the action of external forces, one has to deal with sets of particle positions in spacetime. Let X is the position of a particle in space. The probability for the particle to be exactly at point X (it would then belong to a set consisting of only one element) is equal to zero, while the probability to find the particle in the vicinity of that point (that is, in the finite volume element DV centered at X) is nonzero. The region DV can be visualized as a union of an infinite number of one-element sets of the type X. A direct application of axiom 3 to this case would produce an uncertainty of the type 0  1. Therefore the axiom 3 is unsuitable for infinitive sequences of sets, and the probability for the event to belong to the set DV cannot be obtained as the sum of such probabilities for the sets X  DV. Axiom 3 is applicable only to incompatible events, that is, mutually exclusive events that belong to non-overlapping sets. Consider now the case of intersecting sets and overlapping events, that is, events belonging to two or more sets at the same time. Such events are called joint. Consider two sets A and B, whose intersection A \ B is not empty. We say that o belongs to the intersection (o 2 A \ B) if o 2 A and o 2 B. Then the probability of the joint event o can be written as PðA \ BÞ ¼ Pðw 2 A and

w 2 BÞ:

ð1:1Þ

As examples of joint events, consider two situations, which will prove to be of interest further on:

1.1 Events, Set of Events, and Probability

1. At a given time, the volume element DV centered at the space point X contains N1 particles of type 1 (first event) and N2 particles of type 2 (second event). The probability of this happening is given by the joint probability of both events. 2. A volume element DV centered at a space point X contains N1 particles of type 1 and N2 particles of type 2 at the time t1 (first event) and n1 particles of type 1 and n2 particles type 2 at the time t2 (second event). The probability of the joint event is the joint probability of both events at times t1 and t2. Sometimes one is interested in the probability of an event given the occurrence of some other event. For example, we may want to know the probability of finding a particle in a volume element DV centered at the point X at the time t given that at the time t0 < t, it was located in a volume element DV0 centered at the point X0 6¼ X. Actually, we consider the set of all events C, where C denotes an event of finding the particle in the volume element DV at the time t. The particle could get into this element from any initial spatial position (with different probabilities), but we are interested only in some of those positions, that is, in a subset B of the set A. The probability of such an event is called a conditional probability. Conditional probability is defined as the probability of realization of an event o 2 A under the condition that o 2 B and is equal to PðAjBÞ ¼ PðA \ BÞ=PðBÞ:

ð1:2Þ

The theory of stochastic processes is based (to a considerable degree) on the notion of joint probability. In this context, let us mention an important property of the joint probability. Suppose the full set O is divided into non-overlapping subsets Bi, that is, Bi \ B j ¼ f

[ Bi ¼ W:

and

i

As far as UðA \ Bi Þ ¼ AIðU Bi Þ ¼ A \ W ¼ A i

i

and (see axiom 3) X PðA \ Bi Þ ¼ Pð [ ðA [ Bi ÞÞ ¼ PðAÞ i

i

i

we find from (1.2): X PðAjBi ÞPðBi Þ ¼ PðAÞ:

ð1:3Þ

i

Thinking of the subset Bi as a variable, one can see from the last relation that summation over all mutually exclusive possibilities (i.e. over all sets Bi) eliminates this variable from the outcome.

j3

4

j 1 Basic Concepts of the Probability Theory Yet another important notion is the notion of independent events. Two sets of events A and B are called independent if the probability for an event to belong to set A and the probability to belong to set B are not correlated. Then PðA \ BÞ ¼ PðAÞPðBÞ:

ð1:4Þ

1.2 Random Variables, Probability Distribution Function, Average Value, and Variance

The concept of a random variable is of primary importance in stochastic processes. A random variable F(X) is defined as a function of the element X of the space of probabilistic events X. An event is specified by X, so X now stands for the event previously denoted by o. The examples of random variables include position, momentum, and spatial orientation of a particle driven by random external forces (Brownian motion, motion in a turbulent flow). The introduction of a random variable notation simplifies operations with functions of random variables, calculations of random variable distributions, of averages and other statistical characteristics of distributions. Furthermore, the introduction of continuous random variables enables us to operate with stochastic differential equations and study the change of random variables in space and in time in the same way as we study deterministic systems by using differential equations. The frequency (or probability) of realization of a definite event is equal to some value between zero and one. If the events are mutually exclusive, the sum of probabilities must be equal to one. This means that one of the events will realize. Statistical mechanics is usually concerned with continuous random variables, that is, variables that can assume a continuous range of values. As far as the probability to get any given value from a continuum of possible values is zero, and the sum of all probabilities is one, it is necessary to look at the probability of realization of an event that is associated with an infinitesimal interval (set) of values rather than a single value. This probability is also an infinitely small quantity having the same order as the length of the interval (measure of event) and so is proportional to the measure of events, that is, to dX. Thus the probability that a random variable is contained in the interval (X, X + dX) can be represented as PðX 2 ðX ; X þ dX ÞÞ ¼ pðX ÞdX :

ð1:5Þ

The function p(X) is called the probability density function (PDF) or simply the probability density. The condition that the sum of probabilities for a continuous random variable is equal to one can be written in the integral form: ð pðX ÞdX ¼ 1;

ð1:6Þ

X

where X is the domain of the n-dimensional space in which X varies. The relation (1.6) can be interpreted as the normalization condition for the PDF.

1.3 Generalized Functions

The introduction of a PDF enables us to find statistical characteristics of the distribution of a random variable X. The most important of them is the average value (aka mean value, or expectation value) of a random variable or random function: ð h f i ¼ f ðX Þ pðXÞdX :

ð1:7Þ

X

If X is a vector in an n-dimensional coordinate space, then (1.6) and (1.7) can be written in the coordinate form: ð¥

ð¥ ð¥ ... ¥¥

f ðX1 ; X2 ; . . .; Xn Þ pðX1 ; X2 ; . . .; Xn ÞdX1 ; dX2 ; . . .; dXn ¼ 1: ¥

ð¥ ð¥ hf i ¼

ð¥ ...

¥¥

f ðX1 ; X2 ; . . .; Xn Þ pðX1 ; X2 ; . . .; Xn ÞdX1 ; dX2 ; . . .; dXn : ¥

Another statistical characteristic is the variance s2. For a one-dimensional space, the variance is defined as ð s2 ¼ ðf h f iÞ2 pðXÞdX:

ð1:8Þ

X

The square root s of the variance is called the standard deviation. Sometimes the PDF has the form of a function with a sharp peak at the point X ¼ X0. In limiting case it is infinite at X ¼ X0 and zero at X 6¼ X0. Such a case arises when we idealize a process. For example, we can choose to regard a mass that is continuously distributed in a small volume element centered at X0 as localized at one space point X0 (i.e. as ‘‘point mass’’). Than the density of the substance differs from zero only at this point and the integral (1.7) has the meaning of the total mass. A similar reasoning leads to the concept of a point force – the net force with which we replace a force that is continuously distributed over a small volume element. To ensure the existence of integrals of such functions, we have to extend the notion of a function, what is achieved by the introduction of generalized functions.

1.3 Generalized Functions

The simplest and most extensively used generalized function is Diracs delta function d(X  X0), which can be defined as the limit of following sequence (sometimes referred to as ‘‘delta sequence’’):

j5

6

j 1 Basic Concepts of the Probability Theory  dðX X 0 Þ ¼ lim

m!¥

m pffiffiffi p

n expðm2 ðX X 0 Þ2 Þ:

ð1:9Þ

Here n is the number of dimensions and, accordingly, X is an n-dimensional vector with components X1, X2, . . ., Xn. Eq. (1.9) can also be written for one-dimensional sequences of Xi Xi0 . Then the following identity will hold: dðX1 X10 ÞdðX2 X20 Þ. . .dðXn Xn0 Þ ¼ dðX X 0 Þ:

ð1:10Þ

The limit on the right-hand side of Eq. (1.9) is 0 at X 6¼ X0 and þ1 at X ¼ X0. Therefore Diracs delta function is not a function in the usual sense and should not be interpreted as giving the value of the dependent variable at each point. What is important, however, is that this function is still integrable, and behaves similarly to ordinary functions in its capacity as an integrand. In particular, the integral of the scalar product of Diracs delta function d(X  X0) and an ordinary function j(X) equals ð ðdðX X 0 Þ; jðX ÞÞ ¼ dðX X 0 ÞjðX ÞdX ð  n m pffiffiffi expðm2 ðX X 0 Þ2 ÞjðX ÞdX ¼ jðX 0 Þ m!¥ p

X

¼ lim

X

provided the domain contains the point X0. Thus, by its definition, the Delta function has two basic properties:  dðX Þ ¼

0; 1;

for X ¼ = 0; for X ¼ 0;

ð1:11aÞ

ð dðX X 0 ÞjðX ÞdX ¼ jðX 0 Þ:

ð1:11bÞ

X

In the particular cases j(X) ¼ 1 and j(X) ¼ X one gets: ð dðX X 0 ÞdX ¼ 1:

ð1:12Þ

X

and ð dðX X 0 ÞX dX ¼ X 0 :

ð1:13Þ

X

Hence, according to Eq. (1.7), d(X  X0) can be taken as a PDF such that the random variable X has the average value X0. For the one-dimensional case, the following equality can be written:

1.3 Generalized Functions

ðXX0 ÞdðX X0 Þ ¼ 0 or X dðXX0 Þ ¼ X0 dðX X0 Þ:

ð1:14Þ

Taking j(X) ¼ (X  X0)2, we can write ð dðX X 0 ÞðX X 0 Þ2 dX ¼ ðX 0 X 0 Þ2 ¼ 0:

ð1:15Þ

X

The left-hand side of (1.23) coincides with the definition of the variance for the PDF d(X  X0). Thus, its variance is zero, and the delta function describes the case when one knows for sure that hXi ¼ X0. The (one-dimensional) Cauchy sequence is not the only sequence converging to the delta function. For example, the sequence e

dðX X0 Þ ¼ lim

e ! 0 pðX X0 Þ2

ð1:16Þ

þ e2

can be used as an alternative representation of the delta function. The delta function is an infinitely differentiable function. Its derivative can be defined by differentiating the integral ð jðX Þ

d dðX X0 ÞdX dX

by parts and using the property (1.11): ð

ð d 0 ðX X0 ÞjðX ÞdX ¼  dðX X0 Þj0 ðX ÞdX ¼ j0 ðX0 Þ;

X

ð1:17Þ

X

from which there follows a useful symbolic equality d 0 ðXÞ ¼ 

dðX Þ ; X

ðX ¼ = 0Þ;

In the more general case, one can write d ðrÞ ðXÞ ¼ ð1Þr

dðX Þ ; Xr

ðX ¼ = 0;

r ¼ 0; 1; . . .Þ:

ð1:18Þ

If a function Y ¼ f(X) is single-valued, that is, if it can be solved with respect to X in a unique way, then X ¼ f1(Y) and

j7

8

j 1 Basic Concepts of the Probability Theory dðY f ðX ÞÞ ¼

dðX  f 1 ðYÞÞ : jd f =dX j

ð1:19Þ

A similar relation takes place for a vector function Y ¼ f (X): dðY f ðX ÞÞ ¼

dðX f 1 ðY ÞÞ : D

ð1:190 Þ

where D is the determinant of the Jacobian ||@fi/@Xj||. The delta function can be connected with the unit step function (Heaviside function) defined as  HðX Þ ¼

0; 1;

for X < 0; for X > 0

through the symbolic relation dðX Þ ¼

dH : dX

ð1:20Þ

If there is more then one independent variable, one has to use partial derivatives of the delta function. For example, if we take the delta function as a generalized vector function d(X  X0), its gradient is defined as qdðX X 0 Þ rd ¼ ¼ qX



 qd qd qd ; ; . . .; : qX1 qX2 qXn

ð1:21Þ

1.4 Methods of Averaging

When looking at the hydrodynamic characteristics of a turbulent flow or at the motion of particles under the action of random external forces, we notice one distinguishing feature shared by these two types of motion: the presence of random fluctuations. Because of fluctuations, the dependences of hydrodynamic field parameters on spacetime coordinates, and the configuration of particles in space at different moments look irregular and have a confusing pattern. If a process is repeated multiple times under the identical set of initial and boundary conditions, the observed values of field parameters and particle positions will be different. This necessitates the use of averaging methods in any study of random motions. Averaging allows us to make a transition from irregular characteristics to much more smooth and regular mean values. In practice the mean value is determined by averaging over the time interval,

1.4 Methods of Averaging

h f ðX ; tÞiT ¼

1 T

T=2 ð

wðX ; tÞ f ðX ; t þ tÞdt;

ð1:22Þ

T =2

or by averaging over the considered spatial region, h f ðX ; tÞiV ¼

ð 1 wðx; tÞ f ðX þ x; tÞdx V

ð1:23Þ

V

or, most generally, by spacetime averaging, 1 h f ðX ; tÞiTV ¼ VT

ð ð T=2 wðx; tÞ f ðX þ x; t þ tÞdxdt;

ð1:24Þ

V T=2

where o(x, t) is the weight function. We can also introduce the autocorrelation function C(t), which is defined as follows: take a random function f(t) at one and the same point of space but at instances of time t and t + t, form the product, and find its average value over the time interval (0, T) for T ! 1: 1 T !¥ T

ðT

YðtÞ ¼ lim

f ðtÞ f ðt þ tÞdt:

ð1:25Þ

0

This function plays an important role in many applications. The three types of averaging mentioned above have one drawback, namely, they apply to only one instance of the process under consideration (turbulent velocity field, etc.). Another shortcoming is that one is faced with the problem of choosing the most convenient weight function. If the process is repeated multiple times under the same initial conditions, we are dealing with many instances of the same process. In this case one can talk about a statistical set of identical processes (flows, particle motions etc.) taking place under fixed initial and external conditions. Let one and the same experiment be replicated N times under the same conditions, yielding different values ui of one and the same parameter, for example, velocity u. By averaging the velocities ui observed in a discrete set of similar tests, we obtain the mean value huðX ; tÞi ¼

N 1X ui ; N i¼1

which is called the ensemble average. In many cases the ensemble average proves to be stable enough, in other words, the outcomes of a sufficiently large set of experiments show a very small variance.

j9

10

j 1 Basic Concepts of the Probability Theory Let a continuous random variable u (1 < u < 1) be characterized by the PDF p(u). If we are interested in the value of u at one and the same space point M, then p(u)du signifies the probability for u to be found in interval (u, u + du). Then the ensemble average of u is ð¥ huðX ; tÞi ¼

u pðuÞ du:

ð1:26Þ

¥

By analogy, the ensemble average of any function F is equal to ð¥ hF ðuðX ; tÞÞi ¼

FðuÞ pðuÞdu:

ð1:27Þ

¥

Now, let u be measured at different spacetime points M1 ¼ (X1, t1), M2 ¼ (X2, t2), . . . , MN ¼ (XN, tN). The resulting values of u are denoted by u1, u2, . . . , uN. We then introduce the N-dimensional PDF p(u1, u2, . . . , uN), where p(u1, u2, . . . , uN)du1du2 . . . duN means the probability of finding ui in the interval (ui, ui + dui) . We have used a common convention where the index stands for all N variations, that is ðui ; ui þ dui Þ ¼ ðu1 þ du1 ; . . . ; un þ dun Þ. The average of any function will be written as ð¥ ð¥ hF i ¼

ð¥ ...

¥¥

Fðu1 ; u2 ; . . . ; uN Þ pðu1 ; u2 ; . . . ; uN Þdu1 du2 . . . duN :

ð1:28Þ

¥

By introducing an N-dimensional vector u(u1, u2, . . . , uN), we can rewrite the relation (1.28) in a more compact form: ð¥ hF i ¼

FðuÞ pðuÞdu:

ð1:29Þ

¥

Multidimensional PDFs are especially important for studying the behavior of an N-particle system in a random field of external forces. If ui denotes the coordinate of the i-th particle, then the above-introduced PDF is called a multiparticle PDF. Oneparticle and two-particle PDFs are of particular interest in applications. Sometimes the two-particle PDF is also called ‘‘pair PDF’’ or ‘‘pair distribution’’. These PDFs can be derived from multidimensional PDFs by integrating them over all possible positions of the remaining particles. For instance, a single-particle PDF is obtained as ð¥ ð¥ pðX1 Þ ¼

ð¥ ...

¥8

pðX1 ; X2 ; . . .; XN ÞdX2 . . .dXN ¥

Such PDFs are also called marginal PDFs.

ð1:30Þ

1.4 Methods of Averaging

If we consider spherical particles of different radii ai, than the PDF p(X1, . . ., XN, a1, . . . , aN) will be associated with the radius distribution in addition to the coordinate distribution, and p(X1, . . . , XN, a1, . . . , aN) dX1 . . . dXN da1 . . . daN will have the meaning of probability to find the N-particle system in the volume element (dX1 . . . dXN) with particle radii lying in the interval (a1 + da1), . . . , (aN + daN). The corresponding single-particle PDF is ð¥ ð¥ pðX 1 ; a1 Þ ¼

pðX1 ; X 2 ; . . .; X N ; a2 ; . . .; aN ÞdX 2 . . .dX N da2 . . .daN :

ð1:31Þ

¥ 0

If the radius a must be the same for all particles, then it is convenient to operate with particle distribution over the radius nðX ; aÞ ¼ N pðX ; aÞ;

ð1:32Þ

such that n(X, a)da is the probabilistic numerical concentration (aka number concentration) of particles with radius in the interval (a + da) in the volume element dX. The multidimensional PDF should satisfy the following properties: 1. Ðp(u) 0; 1 2. 1 pðuÞdu¼ 1; 3. pðu1 ; u2 ; . . . ; uN Þ ¼ pðui1 ; ui2 ; . . . ; uiN Þ, where the set {i1, i2, . . ., iN} is formed from the set {1, 2, . . ., N} by changing the order. 1 ð ð 1

4. pðu1 ; u2 ; . . . ; un Þ ¼

1 ð

... 11

pðu1 ; u2 ; . . . ; 1

un ; unþ1 ; . . . ; uN Þdunþ1 . . . duN for n < N; 5. For independent random variables u1 ; u2 ; . . . ; uN , there holds: pðu1 ; u2 ; . . .; uN Þ ¼ pðu1 Þ pðu2 Þ; . . .; pðuN Þ

ð1:33Þ

Property 3 is known as the symmetry property and property 4 – as the consistency property. It is now time to discuss the connection between different types of averaging. In practice, we use time or space averaging rather than ensemble averaging, because the latter requires a large number of experiments. In Statistical Mechanics, ensemble averaging, that is, averaging over the set of all possible states, is often replaced by time or space averaging, with the implicit assumption that by increasing the averaging interval we can always make the average values converge to the corresponding ensemble averages. This assumption is called the ergodic hypothesis, or, in those special cases when it can be rigorously proved, the ergodic theorem. When studying such problems as the flow of a disperse medium or the filtration of a fluid through a porous medium, one often uses the so-called Saffman step

j11

12

j 1 Basic Concepts of the Probability Theory function:  HðX Þ ¼

0; 1;

for X for X

in rigid body; in fluid:

ð1:34Þ

This function depends on statistical parameters of the distribution of moving particles of the disperse phase or fixed particles of the porous medium. After averaging over the particle ensemble, we get hHðX Þi ¼ 1j;

ð1:35Þ

where j is volume concentration of particles. One can use the Saffman function to perform space averaging of hydrodynamic parameters. For example, the velocity of the fluid u will be averaged as  ¼ hHui=hHi ¼ hui=ð1jÞ; u

ð1:36Þ

where u is the mean-flow-rate velocity through the microcapillaries of the porous medium. It should not be confused with the ensemble average hui, although for a highly permeable medium (j  1), the two velocities are equal: hui¼ u.

1.5 Characteristic Functions

Instead of using the PDF p(u1, u2, . . . , uN), it is often convenient to use its Fourier transform: ð¥ ð¥ jðr1 ; r2 ; . . .; rN Þ ¼

ð¥ ...

¥¥

¥

(

N X exp i rk uk

) pðu1 ; u2 ; . . .; uN Þdu1 du2 . . .duN

k1

or, in the vector form, ð¥ jðrÞ ¼

eiru pðuÞdu:

ð1:37Þ

¥

Here r is an N-dimensional vector with components (r1, r2, . . . , rN). The function j(r) is called the characteristic function or the moment-generating function. Because of Eq. (1.29), it can be represented as   jðrÞ ¼ eiru :

ð1:38Þ

1.5 Characteristic Functions

If the characteristic function is known, then the PDF is obtained as the inverse Fourier transform: pðuÞ ¼

ð¥

1 ð2pÞN

eiru jðrÞdr:

ð1:39Þ

¥

So, the knowledge of the characteristic function is tantamount to the knowledge of the PDF. Hence the properties of the PDF are readily obtained from those of the characteristic function. The normalization condition for the PDF means that wð0Þ ¼ 1:

ð1:40Þ

For independent random variables we have, according to (1.33): jðrÞ ¼ jðr1 Þjðr2 Þ. . .jðrN Þ:

ð1:41Þ

The symmetry and consistency properties of characteristic function follow from properties 3 and 4 of the PDF: jðr1 ; r2 ; . . .; rN Þ ¼ jðri1 ; ri2 ; . . .; riN Þ;

ð1:42Þ

jðr1 ; r2 ; . . .; rn Þ ¼ jðr1 ; r2 ; . . .; rn ; 0; 0; . . .; 0Þ;

ð1:43Þ

where i1, i2, . . . , iN is any combination of non-repeating numbers 1, 2, . . . , N. In the last relation, n < N and the number of zeros is equal to N  n. The property (1.57) allows us to obtain the characteristic function for a smaller number of dimensions (smaller number of particles) from the N-dimensional (N-particle) characteristic function, and then to get the corresponding marginal PDF by using the inverse Fourier transform (1.39). Therefore one can specify all PDFs, describing random variables at all possible points, through a single characteristic function known as the characteristic functional. In particular, for one-dimensional random function u(X) defined on a finite interval a X b, the characteristic functional is 8b 9+ <ð = exp i rðX ÞuðXÞdX ; : ;

* FðrðX ÞÞ ¼

ð1:44Þ

a

where rðXÞ is a function selected in such a way that the integral in the exponent converges. The left-hand side is a function of a function, which is why it is called a functional. If rðX Þ ¼

N X ri dðX Xi Þ; i¼1

j13

14

j 1 Basic Concepts of the Probability Theory then Eq. (1.44) gives us: * ( FðrðX ÞÞ ¼

N X exp i rk Xk

)+ ¼ jðr1 ; r2 ; . . .; rN Þ;

ð1:45Þ

k¼1

Thus the characteristic functional turns into a characteristic function of the multidimensional PDF for u(X1), u(X2), . . . , u(XN). Additional information about the characteristic functional can be found in Section 1.15.

1.6 Moments and Cumulants of Random Variables

To solve a specific problem in a rigourous way, one has to specify a multidimensional (multiparticle) PDF. However, one run into difficulties with this approach because the PDF cannot be determined with a sufficient accuracy. Furthermore, it is inconvenient to use because it results in cumbersome expressions. In practice, when solving applied problems, one usually considers only the more simple parameters that characterize specific statistical properties of the process. The most important of these parameters are moments. Let us consider a set of N random variables with N-dimensional PDF p(u1, u2, . . . , uN). The moments are defined as follows: D E Bk1 k2 ...kN ¼ uk11 uk22 . . .ukNN ð¥ ð¥ ð¥ . . . uk11 uk22 . . .ukNN pðu1 ; u2 ; . . .; uN Þdu1 du2 . . .duN ; ð1:46Þ ¼ ¥¥

¥

where k1, k2, . . . , kN are non-negative integers, whose sum K ¼ k1 þ k2þ . . . þkN is called the moments order. It is evident that moments of the first order are simply the mean values of random variables u1, u2, . . . , uN. In addition to ordinary moments (1.61) one often uses some combinations of moments. In particular, central moments are defined as moments of fluctuations (deviations of random variables u1, u2, . . . , uN from their mean values): D E ð1:47Þ u1 Þk1 ðu2  u2 Þk2 . . .ðuN  uN ÞkN : bk1 k2 ...kN ¼ ðu1  For N ¼ 1 and k1 ¼ 2, we get the second-order central moment b2 ¼ s2. If ui are the velocities of a turbulent flow at spatial points xi, then the differences ui  huii have the meaning of velocity fluctuations at these points. Thus central moments characterize statistical properties of random variables – velocity fluctuations. In the case when ui are the random positions of particles driven by a random external force, central moments characterize the statistical properties of the disperse phase in the suspension. By removing brackets in Eq. (1.47) and making use of Eq. (1.46), we can obtain connections between central and ordinary moments. The case of N ¼ 1 yields

1.6 Moments and Cumulants of Random Variables

b1 ¼ 0; b2 ¼ B2 B21 ; b3 ¼ B3 3B1 B2 þ 2B31 ; b4 ¼ B4 4B1 B3 þ 6B21 B2 3B42 etc:

ð1:48Þ

i ¼ 0, the central and ordinary moments coincide. The two combinations of When u central moments, pffiffiffiffiffi b3 b1 ¼ 3=2 ; b2

b2 ¼

b4 3 b22

ð1:49Þ

serve as statistical characteristics of a random quantity and are called, repectively, the asymmetry and the excess. The moments of random variables u1, u2, . . . , uN can be expressed through a corresponding characteristic function j(r1, r2, . . . , rN) by comparing the relations (1.46) and (1.37):

Bk1 k2 ...kN ¼

qK jðr1 ; r2 ; . . .; rN Þ ðiÞK ; qrk11 qrk22 . . .qrkNN r ¼r ¼...¼r ¼0 1 2 N

ð1:50Þ

from which one can see that moments can also be thought of as coefficients in the Taylor expansion of the characteristic function: jðr1 ; r2 ; . . .; rN Þ ¼

X

iK

k1 ;k2 ;...;kN

Bk1 k2 ...kN rk1 rk2 . . .rkNN : k1 !k2 !. . .kN ! 1 2

ð1:51Þ

Thus, if the moments are known, Eq. (1.51) gives us the characteristic function, and then the PDF follows from Eq. (1.39). It means that the PDF is uniquely defined by the moments of the distribution. The other category of combinations of central moments are the so-called cumulants (aka semi-invariants) Sk1 k2 kN . Let us introduce the logarithm of the characteristic function yðr1 ; r2 ; . . .; rN Þ ¼ ln jðr1 ; r2 ; . . .; rN Þ;

ð1:52Þ

which is called the generating function of cumulants. Cumulants can then be defined as the coefficients in the Taylor expansion of c in just as the moments were defined as the coefficients in the expansion (1.66): yðr1 ; r2 ; . . .; rN Þ ¼

X k1 ;k2 ;...;kN

iK

Sk1 k2 ...kN rk1 rk2 . . .rkNN ; k1 !k2 !. . .kN ! 1 2

ð1:53Þ



Sk1 k2 ...kN ¼

qK jðr1 ; r2 ; . . .; rN Þ ðiÞK ; qrk11 qrk22 . . .qrkNN r ¼r ¼...¼r ¼0 1 2 N

ð1:54Þ

j15

16

j 1 Basic Concepts of the Probability Theory Recalling that j(0, . . . , 0) ¼ 1 and taking N ¼ 1, we can express cumulants in terms of ordinary and central moments: S1 ¼ B1 ; S2 ¼ B2 B21 ¼ b2 ; S3 ¼ B3 3B1 B2 þ 2B31 ¼ b3 ; S4 ¼ B4 4B1 B3 3B22 þ 12B21 B2 6B41 ¼ b4 3b22 ; S5 ¼ b5 10b2 b3 etc:

ð1:55Þ

By the same token, moments could be expressed in terms of cumulants: B1 ¼ S 1 ;

B2 ¼ S2 þ S21 ;

B3 ¼ S3 þ 3S2 S1 þ S21 ;

etc:

ð1:56Þ

In the case of one-dimensional PDF p(u) we have the following expressions for moments and cumulants: ð¥ Bn ¼

pðuÞun du ¼ ¥

  1 d n jðrÞjr¼0 ; i dr

ð1:57Þ

  1 d n yðrÞjr¼0 : Sn ¼ i dr

ð1:58Þ

There is a recurrent relation between moments and cumulants. In one-dimensional case it has the following form: B0 ¼ 1;

Bn ¼

n X

ðn1Þ! Sk Bnk ; ðk1Þ!ðnkÞ! k¼0

ðn ¼ 1; 2; . . .Þ:

ð1:59Þ

From this relation, one can derive Eq. (1.56).

1.7 Correlation Functions

In statistical mechanics of disperse media as well as in the turbulence theory, one often encounters random fields described by a random function u(M) of a spacetime point M. Following the definition (1.46), let us call expressions Buu...u ðM1 ; M2 ; . . .; Mk Þ ¼ huðM1 ÞuðM2 Þ. . .uðMk Þi

ð1:60Þ

the k-th order moments of such a field. Generally speaking, some points may coincide. The number of different points is called the type of the moment. In this context, one can talk about single-point moments, two-point moments, and so on. The average values of products of several correleted random functions of different fields are called mixed moments.

1.7 Correlation Functions

Consider, for example, the field of velocities in a turbulent flow given by the velocity vector u(u1, u2, u3). Components of this vector can be regarded as different mutually correlated random functions. Since the values of velocity components could be taken at the same point or at different points, there are moments of various types and orders. Of primary importance in statistical mechanics are two-point second-order moments known as correlation functions (or simple correlations):   Bi j ðM1 ; M2 Þ ¼ ui ðM1 Þu j ðM2 Þ :

ð1:61Þ

Bij are the components of a second rank tensor B. It is obvious that the relation (1.61) could be written in a matrix form:   BðM1 ; M2 Þ ¼ uðM1 ÞuT ðM2 Þ ; where the superscript T stands for transpose. When M1, M2, . . . , MK are points in spacetime, the corresponding moments and correlations are called spacetime moments/correlations. In statistical mechanics one usually has to deal with correlations of random functions at different points at one and the same instant of time or with correlations at one and the same point but at different instants of time. The former correlations are called spatial correlations and the latter – time correlations. Let us metion some important properties of correlation functions. A correlation function Buu(M1, M2) ¼ hu(M1)u(M2)i is symmetric with respect to the pair of points M1, M2: Buu ðM1 ; M2 Þ ¼ Buu ðM2 ; M1 Þ:

ð1:62Þ

A quadratic form with coefficients Buu (Mi, Mj) is always non-negative, that is, n X n X

Buu ðMi ; M j ÞXi X j > 0;

ð1:63Þ

i¼1 j¼1

for all real Xi, non-negative integer n and any selection of points M1, M2, . . . , Mn. In particular, at n ¼ 2 the expression (1.63) becomes jBuu ðM1 ; M2 Þj jBuu ðM1 ; M1 Þj1=2 jBuu ðM2 ; M2 Þj1=2 :

ð1:64Þ

In addition to the above-mentioned two-point moments of one random function at different points, u(M1) and u(M2), one can consider two-point moments of different random functions, u(M1) and v(M2). A mixed two-point moment Buv (M1, M2) is called the mutual correlation function. Its properties are similar to those of ‘‘ordinary’’ moments. For example, the symmetry property still holds: Buv ðM1 ; M2 Þ ¼ Bvu ðM2 ; M1 Þ:

ð1:65Þ

j17

18

j 1 Basic Concepts of the Probability Theory Two-point moments of orders higher than two are referred to as correlation functions of higher order. By analogy, one can define two-point central moments of the second order: buu ¼ hðuðM1 ÞhuðM1 ÞiÞðuðM2 ÞhuðM2 ÞiÞi ¼ Buu ðM1 ; M2 ÞhuðM1 ÞihuðM2 Þi; buv ¼ hðuðM1 ÞhuðM1 ÞiÞðvðM2 ÞhvðM2 ÞiÞi ¼ Buv ðM1 ; M2 ÞhuðM1 ÞihvðM2 Þi

ð1:66Þ

ð1:67Þ

The variances of distributions of random variables u and v can be expressed through central moments: s2u ðMÞ ¼ buu ðM; MÞ;

s2v ðMÞ ¼ bvv ðM; MÞ:

ð1:68Þ

Two-point central moments of the second order relate the deviations of random functions from their mean values (i.e. fluctuations) at two different points. This is why they are also called correlation functions of fluctuations. Another important statistical parameter is the correlation coefficient, defined as Yuu ðMÞ ¼

buu ðM1 ; M2 Þ ; su ðM1 Þsu ðM2 Þ

Yuv ðMÞ ¼

buv ðM1 ; M2 Þ : su ðM1 Þsv ðM2 Þ

ð1:69Þ

As a consequence of the Schwartz inequality, these coefficients satisfy |cuu| 1 and |cuv| 1. If the correlation coefficient vanishes, the correlation between fluctuations at different spatial points is absent. An important property that follows from physical considerations is the damping of correlation between random variables at different spacetime points as we increase the distance between the points. As the distance goes to infinity, the correlation function will tend to zero. Of course, the ‘‘distance’’ that goes to infinity can be the geometrical distance, |X2  X1| ! 1 at t2 ¼ t1, the temporal distance (time interval), |t2  t1| ! 1 at X2 ¼ X1, or both.

1.8 Bernoulli, Poisson, and Gaussian Distributions

Let us consider the three distributions that are most frequently used in physical applications – the Bernoulli, Poisson, and Gaussian (normal) distributions. Bernoulli distribution To begin, let us formulate the random walk problem, which will be considered in more detail farther on, in the chapter dedicated to Brownian motion. A particle undergoes a sequence of random displacements along a straight line. Each displacement is a step of the same length 1, and each step can be directed

1.8 Bernoulli, Poisson, and Gaussian Distributions

either forward or backward with the same probability of 0.5. The origin of the reference system is coincident with the initial position of the particle. Then the particles coordinate can assume only integer values . . . N, N þ 1, . . . , 0, 1, . . . , N  1, N . . .. The probability of finding the particle at a point m after N steps is given by the Bernoulli distribution, N Pðm; NÞ ¼ CðNþmÞ=2

N ¼ where CðNþmÞ=2

 N 1 ; 2

N!

ð12ðNþmÞÞ!ð12ðNmÞÞ!

ð1:70Þ are binomial coefficients.

The mean and root-mean-square displacements of the particle are, respectively, pffiffiffiffiffiffiffiffiffiffi pffiffiffiffi hm 2 i ¼ N :

hmi ¼ 0;

In the limiting case where N  1 and m  N this reduces to an asymptotic formula  Pðm; NÞ

2 pN

1=2

  m2 exp  : 2N

ð1:71Þ

Poisson distribution Let N particles be randomly distributed in a volume V. Then the probability of finding n particles in a volume element v, where v is a small part of V, is given by the Bernoulli distribution

PN ðnÞ ¼

v n

N! v Nn 1 : n!ðNnÞ! V V

ð1:72Þ

For a given N, V and v, the mean value of n equals hni ¼ N

v V

 n:

In the limiting case where N ! 1 and V ! 1 but n remains finite, the distribution (1.72) tends asymptotically to the Poisson distribution PðnÞ ¼

nn en : n!

ð1:73Þ

If n is large and n is of the same order as n, the Poisson distribution is close to the distribution  PðnÞ ¼

1 2pn

1=2

! ðnnÞ2 exp  : 2n

ð1:74Þ

j19

20

j 1 Basic Concepts of the Probability Theory Gaussian (normal) distribution The distributions (1.88) and (1.92) are both special cases of the Gaussian (aka normal) distribution. In the general N-dimensional case this distribution has the following normalized form:

(

) N X N 1X pðu1 ; u2 ; . . .; uN Þ ¼ C exp  g jk ðu j a j Þðuk ak Þ ; 2 j¼1 k¼1

ð1:75Þ

Here aj are real numbers; gjk are the elements of the positive definite matrix ||gjk||; C ¼ g1/2/(2p)N/2 is a constant that is given by the normalization condition for probability density (see Eq. (1.6); g ¼ |gjk| is the determinant of the matrix ||gjk||. The constants aj and gjk are related to the first and second moments of the distribution (1.75) (see Eq. (1.46) and Eq. (1.47):   u j ¼ a j;

    g jk b jk ¼ ðu j  u j Þðuk huk iÞ ¼ g

ð1:76Þ

Here Gjk ¼ @g/@gjk is the algebraic complement of the element gjk in the determinant g. It means that the matrices ||gjk|| and ||bjk|| are mutually inverse. The Gaussian distribution can also represented in the matrix form: pðuÞ ¼

1 ð2pÞN=2 b1=2



1  ðuhuiÞT b1 ðuhuiÞ ; 2

ð1:77Þ

where b ¼ ||bjk|| and b ¼ |bjk|. The ordinary second-order moments Bjk can be expressed in terms of the normal distribution parameters according to Eq. (1.76):   G jk þ a j ak : B jk ¼ u j uk ¼ G

ð1:78Þ

We see from Eq. (1.76) and Eq. (1.77) that the first two moments completely determine the PDF, and thereby the entire statistics of random variables, for a normal distribution. Hence the knowledge of mean values and correlation functions provides a complete statistical description of a random Gaussian field u(M) ¼ [u1(M), u2(M), . . . , uN(M)]. Central moments can be obtained from the property of normal distributions, which states that all central moments of an odd order are zero, whereas central moments of an even order are expressed through central moments of the second order: D E bk1 k2 ...kN ¼ ðu1 hu1 iÞk1 ðu2 hu2 iÞk2 . . .ðuN huN iÞkN X ¼ bi1 i2 bi3 i4 . . .bi2K1 bi2K ;

ð1:79Þ

where k1 þ k2þ . . . þkN ¼ 2K and subscript pairs are formed from numbers 1, 2, . . . 2K so that the first index is less than the second, for example,

1.8 Bernoulli, Poisson, and Gaussian Distributions

b1111 ¼ hðu1 hu1 iÞðu2 hu2 iÞðu3 hu3 iÞðu4 hu4 iÞi ¼ b12 b34 þ b13 b24 þ b14 b23 : When studying random variables described by a normal distribution it is convenient to use characteristic functions because of their simple form (see Eq. (1.37)): ð¥ ð¥ jðr1 ; r2 ; . . .; rN Þ ¼

(

ð¥ ...

¥¥

(

exp i

N X

) rk uk

pðu1 ; u2 ; . . .; uN Þdu1 du2 . . .duN

k¼1

¥

) N X N 1X ¼ exp i ak rk  b jk r j rk : 2 j¼1 k¼1 k¼1 N X

ð1:80Þ

Cumulants can be obtained from Eq. (1.53), using Eq. (1.80). For a Gaussian distribution, the cumulants of the first and second orders are respectively equal to aj and bjk whereas cumulants of higher orders are identically equal to zero. By using characteristic functions, one can prove that any linear combination of Gaussian random variables will also result in a Gaussian distribution. Gaussian distributions are of great importance in applications due to a number of reasons. First, the behavior of many random variables is well approximated by a Gaussian distribution. Secondly, according to the central limit theorem, a random variable that is a sum of a large number of independent components with arbitrary distributions (which is the most common situation in statistical mechanics) is Gaussian. Let us consider a one-dimensional Gaussian distribution    1 u2 pðuÞ ¼ pffiffiffiffiffiffi exp  2 ; ðs2 ¼ u2 Þ: ð1:81Þ 2s 2ps The characteristic function follows from the relations (1.37) and (1.52):  2 2 r s ; jðrÞ ¼ exp  2

r2 s2 2

ð1:82Þ

Sn > 2 ¼ 0:

ð1:83Þ

yðrÞ ¼ 

Then Eqs. (1.57)–(1.58) yield B1 ¼ S1 ¼ 0;

B2 ¼ S2 ¼ s2 ;

The recurrent relation (1.59) takes the form Bn ¼ ðn1Þs2 Bn2 ;

ð1:84Þ

from which there follows B2nþ1 ¼ 0;

B2n ¼ ð2n1Þ!!s2n :

ð1:85Þ

j21

22

j 1 Basic Concepts of the Probability Theory Consider yet another average value hXf (X)i, which is helpful in many applications. Here X is a Gaussian random variable given by Eq. (1.81) and f (X) is an arbitrary deterministic function. We make a further assumption that f (X) exp(X2/2s2) ! 0 for X ! 1 (i.e. the exponent dominates at large values of X). Then

1 hX f ðXÞi ¼ pffiffiffiffiffiffi 2ps

ð¥

 X2 X f ðX Þexp  2 dX: 2s

¥

Integrating by parts, we arrive at the following expression:

s hX f ðX Þi ¼ pffiffiffiffiffiffi 2p

𥠥



 d f ðX Þ X2 d f ðX Þ exp  2 dX ¼ s2 : 2s dX dX

ð1:86Þ

A similar expression can be obtained for a Gaussian random vector X ¼ (X1, X2, . . . , XN) with a multidimensional distribution given by Eq. 1.75:

h Xi f ðX Þi ¼ Bi j

 d f ðX Þ ; dX

ð1:87Þ

where Bij ¼ hXiXji are components of the correlation matrix.

1.9 Stationary Random Functions, Homogeneous Random Fields

When discussing the problem of random variable averaging in Section 4, we mentioned the ergodic hypothesis, which states that as we increase the temporal or spatial averaging interval to infinity, the corresponding mean values tend to the ensemble average. For the ergodic hypothesis to be valid, some necessary conditions should be satisfied. We thus arrive to a special class of random fields u(X, t) satisfying the ergodicity conditions. These fields are frequently encountered in Statistical Mechanics, in particular, in problems that involve Brownian motion and turbulence. Consider first the time averaging of a function u(t), written for simplicity as a function of one variable because its dependence on space coordinates X is of no relevance to the problem. The time average will be denoted by huiT. Then, in accordance with Eq. (1.22), 1 huðtÞiT ¼ T

T=2 ð

uðt þ tÞdt: T=2

ð1:88Þ

1.9 Stationary Random Functions, Homogeneous Random Fields

According to the ergodic hypothesis, hu(t)iT should tend to the ensemble average hu(t)i at T ! 1. For this to happen, the following simple relation must take place: huðtÞiT ¼ U ¼ const:

ð1:89Þ

This condition can be derived by considering the difference between the average values of the random variable calculated at different moments, t and t1, where t1 > t:

huðtÞiT huðt1 ÞiT ¼

8 > T=2 ð 1<

Tð=2

9 > =

uðt þ tÞdt uðt1 þ tÞdt > T> ; : T=2 T=2 9 8 T=2þt > >T=2þt ð 1 ð 1 = 1< ¼ uðsÞds uðsÞds : > T> ; : T=2þt

ð1:90Þ

T=2þt

At T ! 1 the right-hand side of Eq. (1.90) goes to zero, thus giving rise to the condition (1.89). Similarly, by time-averaging the product u(t)u(t1) ¼ u(t)u(t + s), where s ¼ t1  t, and letting T go to 1, we conclude that the time average of the correlation function (Buu(t, t1))T can be equal to its ensemle average (Buu(t, t1)) ¼ hu(t)u(t1)i only if for any two instants of time t1 and t2, where t2 > t1, the following condition is satisfied: Buu ðt2 ; t1 Þ ¼ Buu ðt2 t1 Þ:

ð1:91Þ

For a moment of N-th order, this condition takes the form Buu...u ðt1 ; t2 ; . . .; tN Þ ¼ Buu ðt2 t1 ; . . .; tN t1 Þ:

ð1:92Þ

In order for the ensemble averages of random values u(t1), u(t2), . . . , u(tN) to be obtainable by time averaging, it is necessary to consider only those random functions u(t) for which the N-dimensional PDF (at any N and t1, t2, . . . , tN) will depend on N  1 parameters t2  t1, t3  t1, . . . , tN  t1, rather than on N parameters t1, t2, . . . , tN. In other words, the PDF must satisfy the condition pi1 ;...;iN ðu1 ; u2 ; . . .; uN Þ ¼ pt2 t1 ;...;tN t1 ðu1 ; u2 ; . . .; uN Þ:

ð1:93Þ

It should be noted that the condition (1.93) leads to the conditions (1.98), (1.91) and (1.92), and if the random function is Gaussian, then from Eq. (1.89) and Eq. (1.91) one can derive the properties (1.92) and (1.93). The condition (1.93) describes a class of random functions whose PDF does not vary as we shift the time ti by any time interval. Such functions are called stationary random functions or stationary random processes. One example is a steady turbulent flow, whose average characteristics (velocity, pressure, temperature, etc.) do not change with time. Any hydrodynamic parameter u(u1(t), u2(t), . . . , uN(t)), for example, flow velocity

j23

24

j 1 Basic Concepts of the Probability Theory at different space points, whose PDF for any set of ui1 ðt1 Þ; ui2 ðt2 Þ; . . . ; uiN ðtN Þ does not vary as we shift all the instants of time t1, t2, . . . , tN by one and the same value, represents a multidimensional stationary random process. Then all the mixed moments of functions u(t) will also depend only on the differences between the corresponding instants of time. For example, all mutual correlation functions Bjk (t1, t2) ¼ huj(t1)uk(t2)i depend only on the time difference t ¼ t2  t1. Consider now the space averaging of a random function u(X), where X(X1, X2, X3) is a space point. The space average (recall the definition (1.23)) is equal to 1 huðX ÞiABC ¼ ABC

A=2 ð B=2 ð C=2 ð

uðX1 þ x1 ; X2 þ x2 ; X3 þ x3 Þdx1 dx2 dx3 :

ð1:94Þ

A=2B=2C=2

By analogy with time averaging, we can find the conditions that must hold in order for hu(X)iABC to coinside with the ensemble average hu(X)i at A ! 1, B ! 1, C ! 1 (or when at least one of the intervals A,B,C goes to the limit). It is evident that the necessary conditions would be relations similar to (1.89), (1.91)–(1.93) with t replaced by X: huðX Þi ¼ U ¼ const;

ð1:95Þ

Buu ðX 1 ; X 2 Þ ¼ Buu ðX 2 X 1 Þ;

ð1:96Þ

pX1 ;X2 ;...;XN ðu1 ; u2 ; . . .; uN Þ ¼ pX2 X1 ;...;XN X1 ðu1 ; u2 ; . . .; uN Þ:

ð1:97Þ

where Buu (X1, X2) ¼ hu(X1)u(X2)i. A random field u(X) satisfying the conditions (1.95)–(1.97) is called a statistically homogeneous field. Thus, in order for the space averaging of a function of random variables to produce the same results as ensemble averaging, it is necessary for the field u(X) to be homogeneous. Parameters of a homogeneous turbulent flow (velocity, pressure, temperature, etc.), which do not depend on spatial coordinates, are good examples. It is clear that homogeneity of the flow cannot be realized in the entire flow region, because any flow is always restricted by boundaries, and the flow near the boundary is essentially inhomogeneous. In reality, the property of homogeneity can be realized only far enough from the boundary. It should be mentioned that generally speaking, the conditions of stationarity and homogeneity are not sufficient for the convergence of time and space averages to ensemble averages. The necessary and sufficient conditions are formulated by ergodic theorem. Namely, it is necessary and sufficient to ensure the fulfillment of the following condition for the correlation function of fluctuations buu(t): ðT 1 buu ðtÞdt ¼ 0: T !¥ T lim

0

ð1:98Þ

1.10 Isotropic Random Fields. Spectral Representation

The necessary averaging interval T can be estimated from the corresponding correlation time T1, which is given by

T1 ¼

ðT 1 buu ðtÞdt: buu ð0Þ

ð1:99Þ

0

For sufficiently large T, the following asymptotic formula for root-mean-square deviation of the time average from the ensemble average is valid: D E hui hui 2 2 T1 buu ð0Þ: T T

ð1:100Þ

Eq. (1.99) allows us to determine the minimum averaging time for a given deviation of huiT from hui. In the case of spatial averaging, a similar estimation for the rootmean-square deviation of huiV from hui gives D E hui hui 2 2 V1 buu ð0Þ: V V

ð1:101Þ

Here huiV is the spatial (volume) average, and V1 is the correlation volume equal to 1 V1 ¼ buu ð0Þ

ð¥ ð¥ ð¥ buu ðrÞdr1 dr2 dr3 :

ð1:102Þ

¥¥¥

1.10 Isotropic Random Fields. Spectral Representation

A scalar random field u(X) is called isotropic when all finite-dimensional PDFs pX1 ;X1 ;...;XN ðu1 ; u2 ; . . . ; uN Þ corresponding to this field are invariant under rotations of points X1, X2, . . . , XN around the axis passing through the origin of the coordinate system and under mirror reflections of this set of points relative to planes passing through the origin. In applications, random fields that are both homogeneous and isotropic present the greatest interest. Henceforth these fields will be called simply isotropic. Thus the term ‘‘isotropic field’’ will imply a field whose PDF pX1 ;X2 ;...;XN ðu1 ; u2 ; . . . ; uN Þ is invariant under parallel translations, rotations, and specular reflections of the set of points X1, X2, . . . , XN. The homogeneity condition (1.95) for the field u(X) means that its average value hu(X)i should be constant. This constant is often made equal to zero by replacing the initial field u(X) with the field u0 (X) ¼ u(X)  hu(X)i. The correlation function B(X, X 0 ) ¼ hu(X)u(X 0 )i of an isotropic field has the same values at any pair of points (X, X 0 ) and ðX1 ; X01 Þ that would coincide after a combination of parallel translation and rotation. If the distance between the points X and

j25

26

j 1 Basic Concepts of the Probability Theory X 0 is the same as the distance between X1 and X01 , then BðX; X0 Þ ¼ BðX1 ; X01 Þ. Hence the correlation function B(X, X 0 ) depends only on the distance r between the points X and X 0 ¼ X þ r. Here r ¼ |X 0  X| ¼ |r|, and correlation function can be written as huðX ÞuðX 0 Þi ¼ BðrÞ:

ð1:103Þ

Application of harmonic (Fourier) analysis to random processes and random fields, that is, expansion of random functions as Fourier series (for functions defined on a finite domain) or Fourier integrals (for functions defined on an infinite domain) has proved to be a very successful approach. For any stationary random functions or homogeneous random fields, which by their definition cannot decay on the infinity, it is possible to carry out Fourier expansion (another common term is ‘‘spectral representation’’ or ‘‘spectral expansion’’). It has a clear physical meaning: superposition of harmonic oscillations (for stationary random processes) or plane waves (for homogeneous random fields). The integral representation of the correlation function of a homogeneous random field is ð BðrÞ ¼ eikr FðkÞdk; ð1:104Þ ð 1 FðkÞ ¼ 3 eikr BðrÞdr; 8p

ð1:105Þ

where F(k) is called the spectrum of the homogeneous field, and k is the wave vector. For an isotropic field, the condition (1.103) holds, so the spectrum depends on k ¼ |k| rather than on k. If we represent x, y, z in terms of spherical coordinates as x ¼ r sin y cos F, y ¼ r sin y sin F, z ¼ r cos y, the relation (1.105) will take the following form: FðkÞ ¼

¼

ð ð¥ ðp ðp 1 1 ikr BðrÞdr ¼ eikr cos BðrÞr 2 sin ddFdr e 8p3 8p3 1 2p2

𥠥

0 p 0

sinðkrÞ BðrÞr 2 dr ¼ FðkÞ: kr

ð1:106Þ

Similarly, ð¥ BðrÞ ¼ 4p ¥

sinðkrÞ FðkÞk2 dk: kr

ð1:107Þ

Instead of looking at F(k), we can consider the following statistical characteristic: ð ð FðkÞdSðkÞ; ð1:108Þ EðkÞ ¼ jkj¼k

where S(k) is a surface element of the sphere |k| ¼ k.

1.10 Isotropic Random Fields. Spectral Representation

Putting r ¼ 0 into Eq. (1.104) and recalling Eq. (1.103), we get D E ð¥ Bð0Þ ¼ ½uðX Þ 2 ¼ EðkÞdk:

ð1:109Þ

0

If u is the velocity (for instance, the velocity of a turbulent flow), then B(0) stands for the total energy of the field u(X). Therefore E(k)dk has the meaning of the energy of plane waves with wave numbers in the interval (k, k + dk). The derivations above can be generalized for the case of an isotropic multidimensional random field u(X) ¼ (u1(X), u2(X), . . . , uN(X)) characterized by the correlation matrix   jjBi j jj ¼ ui ðXÞu j ðX þ rÞ

ð1:110Þ

The components of such a matrix are functions of r ¼ |r|. Hence the spectral representation of this field will be written as ð¥ sinðkrÞ Bi j ðrÞ ¼ 4p Fi j ðkÞk2 dk; kr

ð1:111Þ

0

Fi j ðkÞ ¼

ð¥ 1 sinðkrÞ Bi j ðrÞr 2 dr: 2p2 kr 0

The above-formulated definition of an isotropic random field is valid for scalar random functions, for example, pressure p(X), temperature W(X), one-dimensional velocity u(X) and so on. In the case of vector random fields such as three-dimensional velocity, as well as for the fields given by a set of vector and scalar hydrodynamic parameters (for example, a field of three-dimensional velocity, pressure, and temperature), isotropy is defined in the following way. A random vector field u(X) is called isotropic if the PDF of the components of the vector u(X) taken at an arbitrary set of points X1, X2, . . . , XN is invariant under parallel translations, rotations, and mirror reflections of this set of points accompanied by rotation or mirror reflection of the coordinate system. Using the theory of invariants of the rotation-reflection group, we may conclude from this definition that the correlation tensor Bij(r) should be a linear combination of the constant invariant tensor dij (‘‘Kroneckers delta function’’) and the tensor rirj. The coefficients in this linear combination will depend on the only invariant that can be built from components of the vector r, that is, on the length r ¼ |r|: Bi j ðrÞ ¼ A1 ðrÞri r j þ A2 ðrÞdi j :

ð1:112Þ

j27

28

j 1 Basic Concepts of the Probability Theory 1.11 Stochastic Processes. Markovian Processes. The Chapman–Kolmogorov Integral Equation

The term ‘‘stochastic process’’ implies that the time evolution of a system is described probabilistically. This means that there is a certain time-dependent random variable. Examples of stochastic processes include Brownian motion of a particle driven by a random force and the motion of particles suspended in a turbulent flow. The random variable is the spatial position X of the particle at different instants of time. One can measure the values X1, X2, X3, . . . at the instants of time t1, t2, t3, . . . and assume that there should exist a joint PDF p(X1, t1; X2, t2; X3, t3; . . .) such that p(X1, t1; X2, t2; X3, t3; . . .)dX1, dX2, . . . would give the probability for the particle to be located in the interval (X1 þ dX1) at the instant of time t1, in the interval (X2 þ dX2) at the instant t2, and so on. When the particle moves under the action of a rapidly fluctuating random force (in the case of Brownian motion this random force is the sum of interaction forces between the particle and the molecules of the surrounding fluid, or, to use a more casual term, the sum of collision forces), it can change its direction millions times per second. In this context, when considering two successive particle positions Xi and Xiþ1 at the instants ti and tiþ1 such that the time increment Dti ¼ tiþ1  ti is much smaller than the characteristic time of the process but large as compared to the time between successive collisions of the particle with the surrounding molecules, it is natural to suggest a model where the particles position Xiþ1 at the instant tiþ1 is determined by its position Xi at the previous instant ti and does not depend on the earlier instants of time . . . ,ti2, ti1. In other words, in the proces of a chaotic small-scale random walk, the particle forgets its past very quickly. Such processes are known as Markovian processes. Hence a Markovian process is a stochastic process characterized by the independence of the future from the past, where the past is defined as the set of all events observed up to the present instant of time t. In other words, one has to deal with random functions whose variations are statistically independent from one another. A Markovian process can be described by using the concept of conditional probability. Let us consider an ordered sequence of times t1 t2 t3 . . . t1 t2 . . . , where t1, t2, . . . belong to the past and . . . t3, t2, t1 – to the future. Let Y1, Y2, . . . denote the values of a random variable at the past instants of time t1, t2, . . . and . . . X3, X2, X1 denote its values at the future instants of time . . . t3, t2, t1. The PDF of the events X1, X2, . . . under the condition that the events Y1, Y2, . . . have already occurred (the conditional PDF) is then written as pðX 1 ; t1 ; X 2 ; t2 ; X 3 ; t3 ; . . .jY1 ; t1 ; Y2 ; t2 ; . . .Þ: In accordance with the Markovian principle, we demand that the conditional probability must be completely determined by the state of the system at the most recent instant of time, that is, by the knowledge of the random variable at t1. Then the following equality must be valid:

1.11 Stochastic Processes. Markovian Processes. The Chapman–Kolmogorov Integral Equation

pðX 1 ; t1 ; X 2 ; t2 ; X 3 ; t3 ; . . .jY1 ; t1 ; Y2 ; t2 ; . . .Þ ¼ pðX 1 ; t1 ; X 2 ; t2 ; X 3 ; t3 ; . . .jY1 ; t1 Þ:

ð1:113Þ

This relation means that any conditional probability can be expressed through an ordinary conditional probability of the type p(X1, t1|Y1, t1). Indeed, from the definition (1.2) of conditional probability we have: pðX 1 ; t1 ; X 2 ; t2 jY1 ; t1 Þ ¼ pðX 1 ; t1 jX 2 ; t2 ; Y1 ; t1 Þ pðX 2 ; t2 jY1 ; t1 Þ: Applying the postulate (1.113) to the first factor on the right-hand side, we express the joint PDF through ordinary conditional PDFs: pðX 1 ; t1 ; X 2 ; t2 ; jY1 ; t1 Þ ¼ pðX 1 ; t1 jX 2 ; t2 Þ pðX 2 ; t2 jY1 ; t1 Þ:

ð1:114Þ

Continuing this procedure, we obtain for N successive events: pðX 1 ; t1 ; X 2 ; t2 ; . . .; X N ; tN Þ ¼ pðX 1 ; t1 jX 2 ; t2 Þ pðX 2 ; t2 jX 3 ; t3 Þ . . .  pðX N1 ; tN1 jX N ; tN Þ:

ð1:115Þ

As one could expect, the Markovian principle results in the independence of conditional pairs of successive events. From the consistency property (see property 4 in Section 4) of the PDF for two successive events X2 and X1 and from the relation (1.2) for the conditional probability there follows: ð pðX 1 ; t1 Þ ¼

ð pðX 1 ; t1 ; X 2 ; t2 ÞdX 2 ¼

pðX 1 ; t1 jX 2 ; t2 ÞpðX2 ; t2 ÞdX 2 :

ð1:116Þ

A similar relation can be written for the conditional probability: ð pðX 1 ; t1 jX 3 ; t3 Þ ¼

pðX 1 ; t1 ; X 2 ; t2 jX 3 ; t3 ÞdX 2 ð

¼

pðX 1 ; t1 jX 2 ; t2 ; X 3 ; t3 Þ pðX 2 ; t2 jX 3 ; t3 ÞdX 2 :

As far as t1 t2 t3, the Markovian principle allows to drop the dependence on X3 in the first factor of the integrand: ð pðX 1 ; t1 jX 3 ; t3 Þ ¼

pðX 1 ; t1 jX 2 ; t2 Þ pðX 2 ; t2 jX 3 ; t3 ÞdX 2 :

ð1:117Þ

The integral equation (1.117) is called the Chapman–Kolmogorov equation. This equation forms the basis of the theory of stochastic processes.

j29

30

j 1 Basic Concepts of the Probability Theory When considering a Markovian process, it is important to know whether the range of the random value is continuous or discrete and whether the trajectory X(t) is a continuous function of t. As an example, consider rarefied gas molecules characterized by the velocity V(t) and by the position X(t). In this example, the velocity range is obviously continuous, but the function V(t) can be discontinuious, which happens when the interactions between molecules are modeled by the elastic collisions of rigid spheres. However, even in such a model, the position of a gas molecule X(t) remains a continuous function. In reality, molecules do not interact as rigid spheres. There exists a molecular interaction between them that is characterized by some interaction potential (for example, the Lennard–Jones potential). If we account for this potential, we will find that the molecules trajectory deflects continuously in the process of collision. The characteristic time of molecular collisions is extremely short. It is much shorter then the time intervals that make up a Markovian chain. It can be said that the Markovian method circumnavigates the issue of continuity of a random variable by approximating the real process on a large-scale time grid. Hence, irrespective of how the collision process is modeled, on large-scale time grid, the collision will always be marked by a velocity jump. By the same token, the trajectories are not necessarily continuous on this time grid. Another example is a chemical reaction that involves production consumption of molecules of a certain substance. The characteristic time of a chemical reaction is also very short as a rule. Therefore the random value, for example, molecular concentration, changes discontinuously on the large-scale time grid during the reaction. In this context, the following continuity condition looks quite self-intuitive: if for any e > 0 uniformly in Z, t and Dt there holds 1 Dt ! 0 Dt lim

ð pðX ; t þ DtjZ; tÞdX ¼ 0;

ð1:118Þ

jX Zj > e

then the realization of X(t) is continuous function of t, with the probability 1. It means that the probability that the position X differs from Z by a finite amount at Dt ! 0 goes to zero faster than Dt. This is known as the Lindenberg continuity condition for a random function X(t). One can show that Einsteins solution of the Brownian motion problem, which is a Gaussian PDF written as ( ) 1 ðX ZÞ2 pðX ; t þ DtjZ; tÞ ¼ exp  ; ð1:119Þ 4DDt ð4pDDtÞ1=2 satisfies the condition (1.118). On the other hand, the PDF pðX ; t þ DtjZ; tÞ ¼

Dt p½ðX ZÞ2 þ Dt2

;

ð1:120Þ

which describes a Cauchy process, does not satisfy this condition. Both distributions tend to the delta function d(X  Z) at Dt ! 0 (see Eq. (1.9) and Eq. (1.16) and satisfy

1.12 The Chapman–Kolmogorov, Chapman–Feller, Fokker–Planck, and Liouville Differential Equations

Eq. (1.117). So the Chapman–Kolmogorov equation allows for both continuous and discontinuous solutions (PDFs).

1.12 The Chapman–Kolmogorov, Chapman–Feller, Fokker–Planck, and Liouville Differential Equations 1.12.1 Derivation of the Differential Chapman–Kolmogorov Equation

When solving concrete problems, one uses the differential form of the Chapman– Kolmogorov equation, which can be derived from the integral equation (1.117) under some additional assumptions. An additional assumption of continuity of the random process leads us to the Fokker–Planck equation. But discontinuous processes can also take place, as was mentioned in the previous section. Thus the Chapman–Kolmogorov differential equation should be able to describe both continuous and discontinuous processes. To satisfy this general requirement, we shall demand realization of the following conditions: 1: lim

1

Dt ! 0 Dt

pðX ; t þ DtjZ; tÞ ¼ W ðX jZ; tÞ

ð1:121Þ

should take place in the region |X  Z| e uniformly for all X, Z, t, and the limit should not depend on e; ð 1 ðXi Zi Þ pðX ; t þ DtjZ; tÞdX ¼ Ai ðZ; tÞ þ 0ðeÞ; ð1:122Þ 2: lim Dt ! 0 Dt jX Zj < e

1 Dt ! 0 Dt

ð ðXi Zi ÞðX j Z j Þ pðX ; t þ DtjZ; tÞdX

3: lim

jX Zj < e

¼ Di j ðZ; tÞ þ 0ðeÞ:

ð1:123Þ

The conditions 2 and 3 assume a uniform convergence with respect to Z, e, and t. Condition 1 is responsible for the continuity of the process. If W(X|Z, t) ¼ 0, the process can be described by continuous trajectories; otherwise the trajectories are discontinuous. To derive the differential equation, let us consider how the average value of some (twice differentiable) function f (X) varies with time. According to Eq. (1.7), the average is written as ð h f ðX Þi ¼

f ðX Þ pðX ; tjY ; t0 ÞdX :

j31

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j 1 Basic Concepts of the Probability Theory Then qh f i ¼ qt



1 Dt ! 0 Dt lim

ð

f ðX Þ½ pðX ; t þ DtjY ; t0 Þ pðX ; tjY ; t0 Þ dX :

Let us now put X1 ¼ X, t1 ¼ t þ Dt, X2 ¼ Z, t2 ¼ t, X3 ¼ Y, t3 ¼ t0 into the Chapman– Kolmogorov integral equation (1.110). It means that in addition to the point Y at the time t0 and the point X at the time t þ Dt, we take yet another point Z (between these two) on the trajectory at the intermediate time t (t0 < t ðð "X X X 1 q2 f ðZÞ 1< q f ðZÞ ðXi Zi Þþ ðXi Zi ÞðX j Z j Þ ¼ lim Dt ! 0 Dt> qZi 2 qZi qZ j : j i i jX Z < e

 pðX ; t þ DtjZ; tÞ pðZ; tjY; t0 ÞdX dZ ðð þ jX Zj2 RðX ; ZÞ pðX ; t þ DtjZ; tÞ pðZ; tjY ; t0 ÞdX dZ jX Zj < e

ðð

þ

f ðX Þ pðX ; t þ DtjZ; tÞ pðZ; tjY ; t0 ÞdX dZ

jX Zj > e

ðð þ

f ðZÞ pðX ; t þ DtjZ; tÞ pðZ; tjY ; t0 ÞdX dZ

jX Zj < e

ð

9 > =

f ðZÞ pðZ; tjY ; t0 ÞdZ : > ; ð1:126Þ

1.12 The Chapman–Kolmogorov, Chapman–Feller, Fokker–Planck, and Liouville Differential Equations

Consider the terms in the left-hand side of Eq. (1.126) in the consecutive order. Since p(X, t þ Dt|Z, t) is the PDF, we note that ð pðX ; t þ DtjZ; tÞdX ¼ 1: With this in mind, and using the condition of uniform convergence that allows us to take the limit of the integrand, the last term in Eq. (1.126) can be rewritten as ð f ðZÞ pðZ; tjY ; t0 ÞdZ ð

0

¼

f ðZÞ pðZ; tjY ; t ÞdZ ðð

ð pðX ; t þ DtjZ; tÞdX

f ðZÞ pðX ; t þ DtjZ; tÞ pðZ; tjY ; t0 ÞdX dZ:

¼

To transform the first term, it is necessary to exploit the conditions 2 and 3: 8 # > ðð "X X X 1 q2 f ðZÞ 1< q f ðZÞ ðXi Zi Þ þ ðXi Zi ÞðX j Z j Þ lim Dt ! 0 Dt> qZi 2 qZi qZ j : j i i jX Zj < e

pðX ; t þ DtjZ; tÞ pðZ; tjY ; t0 ÞdX dZg # ð "X q f XX1 q2 f pðZ; tjY; t0 ÞdZ þ 0ðeÞ: Ai ðZ; tÞ þ Di j ðZ; tÞ ¼ qZi 2 qZi qZ j j i i The second term tends to zero because of the condition R(X, Z) ! 0 at e ! 0, since |X  Z| ! 0. Carrying out integration in the last term over two subdomains, |X  Z| e and |X  Z| < e and taking into account Property 1, one can reduce the last three terms to 8 > ðð 1< lim f ðX Þ pðX ; t þ DtjZ; tÞ pðZ; tjY ; t0 ÞdX dZ Dt ! 0 Dt> : jX Zj > e

ðð þ

f ðZÞ pðX ; t þ DtjZ; tÞ pðZ; tjY ; t0 ÞdX dZ

jX Zj < e

ðð 

f ðZÞ pðX ; t þ DtjZ; tÞ pðZ; tjY ; t0 ÞdX dZ

jX Zj > e

ðð 

f ðZÞ pðX ; t þ DtjZ; tÞ pðZ; tjY ; t0 ÞdX dZ

jX Zj < e

ðð ¼

9 > = > ;

½ f ðX ÞWðX jZ; tÞ pðZ; tjY ; t0 Þ f ðZÞWðX jZ; tÞ pðZ; tjY ; t0 Þ dX dZ

jX Zj > e

j33

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j 1 Basic Concepts of the Probability Theory Nothing will be changed if we swap the variables X and Z in the first term: ðð

f ðZÞ½WðZjX ; tÞ pðX ; tjY ; t0 ÞWðX jZ; tÞ pðZ; tjY ; t0 Þ dX dZ:

jX Zj > e

Going to the limit e ! 0 in Eq. (1.126), one obtains the following relation: ð q f ðX Þ pðX ; tjY ; t0 ÞdX qt " # ð X q f XX1 q2 f þ pðZ; tjY ; t0 ÞdZ Ai ðZ; tÞ Di j ðZ; tÞ ¼ qZ qZ 2 qZ i i j j i ði ð þ f ðZÞdZ ½WðZjX ; tÞ pðX ; tjY ; t0 ÞWðX jZ; tÞ pðZ; tjY ; t0 Þ dX ; ð1:127Þ which is valid when the integral ð WðZjX ; tÞ pðX ; tjY ; tc ÞdX

ð1:128Þ

exists. The condition (1.121) determines the function W(Z|X, t) only if X 6¼ Z. But there are situations when X ¼ Z, for example, a Cauchy process given by Eq. (1.120). In this case the value of the integral (1.128) should be interpreted as the principal integral value. Because such singular cases are rare, we will not be using the principal integral value symbol in the discussion below. Integration by parts of the first term on the right-hand side of Eq. (1.127) yields ð

q pðZ; tjY ; t0 Þ dZ qt ( ð X q ½Ai ðZ; tÞ pðZ; tjY ; t0 Þ dZ ¼ f ðZÞ  qZi i

f ðZÞ

X X 1 q2 f ½Di j ðZ; tÞ pðZ; tjY ; t0 Þ qZ 2 qZ i j j i ð þ ½WðZjX ; tÞ pðX ; tjY ; t0 ÞWðX jZ; tÞ pðZ; tjY ; t0 Þ dX þ . . . ;

þ

where the dots denote the surface integrals over the boundary enclosing the considered region. As far as the function f(Z) was chosen arbitrarily with the only requirement that it should be at least twice differentiable, we can impose an additional requirement that this function should vanish on the regions boundary. Then all surface integrals also vanish, and finally, we obtain the Chapman–Kolmogorov differential equation:

1.12 The Chapman–Kolmogorov, Chapman–Feller, Fokker–Planck, and Liouville Differential Equations

q pðZ; tjY ; t0 Þ qt X q ½Ai ðZ; tÞ pðZ; tjY ; t0 Þ ¼ qZ i i X X 1 q2 f ½Di j ðZ; tÞ pðZ; tjY ; t0 Þ qZ 2 qZ i j j i ð þ ½WðZjX ; tÞ pðX ; tjY ; t0 ÞWðX jZ; tÞ pðZ; tjY ; t0 Þ dX :

ð1:129Þ

þ

Consider now some particular cases of the Chapman–Kolmogorov equation. 1.12.2 Discontinuous (‘‘Jump’’) Processes. The Kolmogorov–Feller Equation

This equation follows from the Chapman–Kolmogorov equation at Ai ¼ Dij ¼ 0: ð qpðZ; tjY ; t0 Þ ¼ ½WðZjX ; tÞ pðX ; tjY ; t0 ÞWðX jZ; tÞ pðZ; tjY ; t0 Þ dX : qt ð1:130Þ If we take p(Z, t|Y, t0 ) ¼ d(Y  Z) at t ¼ t0 as the initial condition, then for small values of Dt the solution will be approximately equal to ð pðZ; t þ DtjY; t0 Þ dðY ZÞ½1 WðX jZ; tÞDtdX þ WðZjX ; tÞDt:

It is implied by this solution that for any Dt, there is a finite probability ð 1 WðX jZ; tÞD td X

to find the particle at the initial position Y, and the distribution of particles leaving Y is given by the function W(Z|Y, t). Hence, the trajectory X(t) consists of linear segments X ¼ const alternating with jumps whose distribution is given by the function W(Z|Y, t). That is why the process has discontinuous character and the trajectories have discontinuities in a discrete set of points. 1.12.3 Diffusion Processes. The Fokker–Planck Equation

If the process is continuous, then W(Z|Y, t) ¼ 0, and the Chapman–Kolmogorov equation is reduced to the Fokker–Planck equation:

j35

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j 1 Basic Concepts of the Probability Theory X q qpðZ; tjY ; t0 Þ ½Ai ðZ; tÞ pðZ; tjY ; t0 Þ ¼ qt qZ i i X X 1 q2 þ ½Di j ðZ; tÞ pðZ; tjY ; t0 Þ 2 qZi qZ j j i

ð1:131Þ

Such a process is called the diffusion process. The vector A(Z, t) is called the drift vector. It is similar to the velocity vector in the convective term of the transport equation. The matrix D(Z, t) ¼ ||Dij(Z, t)|| is called the dispersion matrix. According to its definition (see Eq. (1.123)), it is non-negative, definite and symmetric. It can be shown that D is a tensor. It is known as the dispersion tensor. To understand the physical meaning of A and D, consider the initial phase of the process in the same manner as we just did for the Kolmogorov–Feller equation, with the same initial condition p(Z, t|Y, t0 ) ¼ d(Y  Z) at t ¼ t0 . Assuming that during a small time Dt  1 the values of Aj and Dij. will not change much as compared to p, the equation transforms to X qpðZ; tjY ; t0 Þ q Ai ðZ; tÞ ½ pðZ; tjY ; t0 Þ ¼ qt qZ i i XX1 q2 þ ½ pðZ; tjY ; t0 Þ ; Di j ðZ; tÞ qZi qZ j 2 j i

ð1:132Þ

where tt0 ¼ Dt  1. On this small time interval, Ai ðY;tÞ and Di j ðY;tÞ are regarded as dependent on the initial position Y but independent of time t. The solution of Eq. (1.132) has the form pðZ; t þ DtjY ; tÞ ¼

1 ð2pÞN=2 jDðY ; tÞj1=2 Dt1=2

(

1 ½ZY AðY ; tÞDt T ½DðY ; tÞ 1 ½ZY AðY ; tÞDt  exp  2 Dt

) ð1:133Þ

where D ¼ |D| is the determinant of the matrix D. Eq. (1.333) indicates that at the initial stage, the diffusion process is described by the Gaussian law (see Eq. 1.77) and that fluctuations with the correlation matrix D(Y, t)Dt are superimposed on the regular drift with the velocity A(Y, t). It means that at the initial stage, the systems trajectory can be represented as Zðt þ DtÞ ¼ Y ðtÞ þ AðY ðtÞ; tÞDt þ hðtÞðDtÞ1=2 ;

ð1:134Þ

where h(t) is a random vector with the mean value and the correlation matrix given by   hðtÞhT ðtÞ ¼ DðY ; tÞ: ð1:135Þ hhi ¼ 0;

1.12 The Chapman–Kolmogorov, Chapman–Feller, Fokker–Planck, and Liouville Differential Equations

In a diffusion process, trajectories are continuous everywhere because Z(t þ Dt) ! Z(t) at Dt ! 0. They are also non-differentiable at any point because of the term proportional to (Dt)1/2. Since Z(t þ Dt)  Y(t) ¼ DZ is a random increment of the particles position, we can divide both parts of Eq. (1.134) by Dt, obtaining DZ ¼ AðY ðtÞ; tÞ þ hðtÞðDtÞ1=2 : Dt

ð1:136Þ

Eq. (1.136) is a stochastic differential equation that has a fundamental role in describing the motion of particles driven by an external random force. In a three-dimensional case, the Gaussian distribution (1.133) can be written in a simpler form. Let us direct the Cartesian axes Z1, Z2, Z3, so that they would coincide with the principal directions of the dispersion tensor D. Let Dij be the principal values of the dispersion tensor matrix ||Dij||. In this coordinate system, the distribution (1.133) transforms into pðZ; t þ DtjY ; tÞ ¼

1 3=2

½D11 ðtÞD22 ðtÞD33 ðtÞ 1=2 ( ) ðZ1 Y1 Þ2 ðZ2 Y2 Þ2 ðZ3 Y3 Þ2   :  exp  2D11 ðtÞ 2D22 ðtÞ 2D33 ðtÞ

ð2pÞ

ð1:137Þ

Let us now introduce the probability flux with components Ji ðZ; tÞ ¼ Ai ðZ; tÞ pðZ; t þ DtjY; t0 Þ

1X q ½Di j ðZ; tÞ pðZ; tjY ; t0 Þ : 2 j qZ j

Then the Fokker–Planck equation can be written in a compact, universally accepted form of a conservation equation: qpðZ; tjY ; t0 Þ X qJi ðZ; tÞ ¼ 0: þ qt qZi i

ð1:138Þ

Introduction of the probability flux allows us to formulate the boundary conditions for the Fokker–Planck equation. Consider the process in a region R bounded by the surface S. One can see the following possible types of boundary conditions. a) Absorbing boundary

It is assumed that as soon as a particle reaches the boundary, it vanishes, that is, it leaves the system, for example, adheres to the surface or reacts with the boundary surface. Hence the probability to find the particle at the boundary is equal to zero, and the boundary condition for the PDF is pðZ; tjY ; t0 Þ ¼ 0

at Z 2 S:

ð1:139Þ

j37

38

j 1 Basic Concepts of the Probability Theory b) Reflecting boundary

If the particle cannot leave the region R, then the probability flux in the n direction at the boundary surface should be equal to zero, that is, nJðZ; tÞ ¼ 0

at Z 2 S;

ð1:140Þ

where n is the normal to the boundary. c) Surface of discontinuity

Suppose that the coefficients Ai and Dij experience a jump at the surface S, but particles can cross the surface freely. Such a behavior is possible, when the surface is an interface between two media with different properties. At such a surface, the probabilities and the normal components of probability fluxes should be equal at both sides of the boundary surface: pðZ; tjY ; t0 ÞjSþ ¼ pðZ; tjY ; t0 ÞjS ;

nJðZ; tÞjSþ ¼ nJðZ; tÞjS :

ð1:141Þ

d) Conditions at infinity

If the process is considered in an infinite region, then, depending on the problem under consideration, one of the following two boundary conditions must be valid: pðZ; tÞ ! 0 or

1

and

q pðZ; tÞ !0 qZ

at jZj ! ¥:

ð1:142Þ

Of special interest is the one-dimensional case of the Fokker–Planck equation, in which the drift and dispersion coefficients are become scalar quantities A and D: q pðZ; tjY; t0 Þ q 1 q2 ½DðZ; tÞ pðZ; tjY; t0 Þ : ¼ ½AðZ; tÞ pðZ; tjY; t0 Þ þ qt qZ 2 qZ2 This equation can be rewritten as   q pðZ; tjY; t0 Þ q 1 q 0 ¼ DðZ; tÞ pðZ; tjY; t Þ qt qZ 2 qZ    q 1 qD  AðZ; tÞ pðZ; tjY; t0 Þ : qZ 2 qZ

ð1:143Þ

Now it is possible to compare it with the molecular diffusion equation, which describes the change of concentration C of a substance in the solution due to the thermal motion of solvent molecules:

1.12 The Chapman–Kolmogorov, Chapman–Feller, Fokker–Planck, and Liouville Differential Equations

  qC qðvCÞ q qC ¼ þ D ; qt qZ qZ qZ

ð1:144Þ

where v is velocity of the substance under the action of external force. A comparison of equations (1.143) and (1.144) shows that D/2 has the meaning of diffusion coefficient D and the drift A A¼vþ

1 qD 2 qZ

has the meaning of average velocity of particle displacement. The latter consists of two terms. The first term is the drift caused by external forces, and the second term is the drift caused by the inhomogeneneity of the medium. Another difference between equations (1.143) and (1.144) is the difference between the unknown variables: probability density p in (1.143) and concentration C in (1.144). But concentration can be obtained from the probability density by multiplying p by the number of particles N in a unit volume. Therefore if C/N is taken instead of C, then we can also take C/N instead of p under the condition that N ¼ const. A process described by the one-dimensional Fokker–Planck equation with A ¼ 0 and D ¼ 1, q pðZ; tjY; t0 Þ 1 q2 ½ pðZ; tjY; t0 Þ : ¼ qt 2 qZ2

ð1:145Þ

is called the Wiener process. Under the initial condition pðZ; tjY ; tÞ ¼ dðY ZÞ at

t ¼ t0

its solution is ) ðZYÞ2 pðZ; tjY; t Þ ¼ exp  : 2ðtt0 Þ ð2pÞ1=2 0

1

(

ð1:146Þ

A multidimensional Wiener process is described by the multidimensional Fokker– Planck equation q pðZ; tjY ; t0 Þ 1 X q2 ½ pðZ; tjY ; t0 Þ ; ¼ qt 2 i qZi2 whose solution is

) ðZYÞ2 pðZ; tjY ; t Þ ¼ : exp  2ðtt0 Þ ð2pÞn=2 0

1

ð1:147Þ

(

ð1:148Þ

Sometimes the Wiener process is causally referred to as ‘‘Brownian motion’’.

j39

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j 1 Basic Concepts of the Probability Theory 1.12.4 Deterministic Processes. The Liouville Equation

The Liouville equation follows from the Chapman–Kolmogorov equation at W(Z|Y, t)¼ 0 and Dij ¼ 0: X q q pðZ; tjY ; t0 Þ ½Ai ðZ; tÞ pðZ; tjY ; t0 Þ : ¼ qt qZ i i

ð1:149Þ

Eq. (1.149) describes deterministic motion, which is given by a single-valued function and is determined by the initial conditions. Indeed, consider a trajectory X(t) that is a solution of the characteristic equation dX ðtÞ ¼ AðX ðtÞ; tÞ dt with the initial condition X(Y, t0 ) ¼ Y. Here A is a vector with components Ai. Let us show that pðZ; tjY ; t0 Þ ¼ dðZX ðY ; tÞÞ is the solution of Eq. (1.149) with the initial condition pðZ; t0 jX ; t0 Þ ¼ ðZX Þ: Substituting it into Eq. (1.149), one obtains: qdðZX ðY ; tÞÞ qt XqdðZX ðY ; tÞÞ XqdðZX ðY ; tÞÞ dXi ðY ; tÞ Ai ðX ðY ; tÞ; tÞ ¼ ¼ qZi dt qZi i i X q X q ¼ fAi ðX ðY ; tÞ; tÞdðZX ðY ; tÞÞg ¼  fAi ðZ; tÞdðZX ðY ; tÞÞg: qZ qZi i i i The example above considers the motion of a single particle. In the case of many particles, the Liouville equation has a somewhat different form. Statistical Mechanics uses the Liouville equation to describe the motion of a particle ensemble as a set of mass points moving in accordance with Newtons second law, 2

d X X€ i ¼ 2 ¼ F i dt

or

dV i ¼ Fi; V_ i ¼ dt

dX i ¼ V i; X_ i ¼ dt

ð1:150Þ

where Xi, Vi, Fi are, respectively, the radius vector, the velocity, and the force exerted on a unit mass of i-th particle by other particles and the environment. Let X i ð0Þ ¼ X 0i ;

V i ð0Þ ¼ V 0i :

ð1:151Þ

1.12 The Chapman–Kolmogorov, Chapman–Feller, Fokker–Planck, and Liouville Differential Equations

be the initial positions and velocities of the points. Then in order to describe the time evolution of the state of an N-particle system, one has to integrate the system of equations (1.150) with the initial conditions (1.151). For N  1, this is practically impossible, so one has to use statistical methods. Statistical Mechanics usually studies the dynamics of a mass point system by introducing generalized coordinates, which are either the ordinary coordinates Xi and velocities Vi or the ordinary coordinates Xi and momenta miVi of particles. The state of an N-particle system is characterized by the joint probability density taken at one given instant of time, which determines the chance to find particle 1 in the generalized coordinate interval (X1 þ dX1, V1 þ dV1), AND to find particle 2 in the interval (X2 þ dX2, V2 þ dV2), . . . , and to find particle N in the interval (XN þ dXN, VN þ dVN). If particle trajectories are known, that is, the functions X i ¼ X i ðtÞ;

V i ¼ V i ðtÞ

are given, then the probability density is equal to zero if Xi 6¼ Xi(t) or Vi 6¼ Vi(t) for even one value of i. It means that probability reduces to certainty and the probability density is given by a product of delta functions: pðX ; V ; tÞ ¼

N Y dðX i X i ðtÞÞdðV i V i ðtÞÞ:

ð1:152Þ

i¼1

It is easy to show that this PDF satisfies the Liouville equation. Taking the derivative of the product of functions and using properties (1.10), (1.18), (1.21) of the delta function, we write: q pðX ; V ; tÞ qt 0

1

N B Y N C X B C dðX k X k ðtÞÞdðV k X_ k ðtÞÞCX_ j ¼ B @ A j¼1 k¼1 k„ j

0 

1

N B Y N C X qdðX j X j ðtÞÞ B C dðV j X_ j ðtÞÞ B dðX k X k ðtÞÞdðV k X_ k ðtÞÞC @ A qX j j¼1 k¼1 k„ j

qdðV j X_ j ðtÞÞ : dðX j X j ðtÞÞX€ j  qV j

j41

42

j 1 Basic Concepts of the Probability Theory Now, using the relation (1.14) and the first equation (1.150), we get N X qdðX j X j ðtÞÞ q pðX ; V ; tÞ dðV k X_ k ðtÞÞ ¼  V j qX j qt j¼1

0

1

B Y C X _ B N C N _ k ðtÞÞC F j  qdðV j X j ðtÞÞ dðX X ðtÞÞdðV  X B k k k B C qV j @k ¼ 1 A j¼1 k„ j 0

1

B Y C B N C _ k ðtÞÞC: dðX dðX k X k ðtÞÞB X ðtÞÞdðV  X k k k B C @k ¼ 1 A k„ j It is easy to verify that 0

1

N B Y C q pðX ; V ; tÞ qdðX j X j ðtÞÞ B C ¼ dðV k X_ k ðtÞÞ  B dðX k X k ðtÞÞdðV k X_ k ðtÞÞC; @ A qX j qX j k¼1 k„ j

0

1

N B Y C q pðX ; V ; tÞ qdðV j X_ j ðtÞÞ B C ¼  dðX k X k ðtÞÞ  B dðX k X k ðtÞÞdðV k X_ k ðtÞÞC @ A qV j qV j k¼1 k„ j

The Liouville equation finally reduces to N N X q pðX ; V ; tÞ q pðX ; V ; tÞ X q pðX ; V ; tÞ  Fi : ¼  V i qt qX qV i i i¼1 i¼1

ð1:153Þ

If the generalized coordinates Zi are defined as coordinates Xi and momenta per unit particle mass Vi, then Eq. (1.153) will assume the following compact form: N X q pðZ; tÞ q pðZ; tÞ ; ¼  Ai  qt qZi i¼1

where A is the generalized vector that includes Vi and Fi.

ð1:154Þ

1.13 Stochastic Differential Equations. The Langevin Equation

1.13 Stochastic Differential Equations. The Langevin Equation

When a randomly and rapidly fluctuating function of time and (or) spatial coordinates appears in a differential equation, this equation is called stochastic. The presence of a random component in the equation means that the solution will also be a random function. One example of a stochastic differential equation is the Langevin equation describing random trajectories of a particle driven by a random force. Another example is the equation of diffusion, which takes into account chemical reactions that are responsible for fluctuations. Let us consider these equations in more detail. 1.13.1 The Langevin Equation

In the theory of Brownian motion one frequently encounters the following Langevin equation: dX ¼ aðX ; tÞ þ bðX ; tÞjðtÞ; dt

ð1:155Þ

where a(X, t) and b(X, t) are known functions and j(t) is a random fluctuating function. The problem of Brownian motion of a particle driven by a fluctuating external force (that is, the force resulting from collisions with molecules of the surrounding fluid) reduces to this equation. Every second, the particle experiences millions of collisions, each collision resulting in a microscopic motion in the direction of impact. Therefore the particles position at the time t is determined only by its position at previous instant of time and does not depend on the motion prehistory. The particles motion can be considered as a Markowian process that is taking place under the action of a random force j(t) with the average value of zero: hjðtÞi ¼ 0

ð1:156Þ

The force j(t) is a ‘‘pseudoforce’’ that models the real impact forces. There is no correlation between the forces at different instants of time, in other words, j(t) and j(t0 ) are mutually independent if t 6¼ t0 . Or, to use another term, they are deltacorrelated: hjðtÞjðt0 Þi ¼ jdðtt0 Þ:

ð1:157Þ

Equations (1.155) and (1.157) serve as idealizations of Brownian motion. The actual equation of motion of a particle is given by Newtons second law: m

d2 X dX ¼ h þFþf; 2 dt dt

ð1:158Þ

j43

44

j 1 Basic Concepts of the Probability Theory where hdX/dt is the drag force exerted on the particle by the surrounding fluid (see Example 4 for more details), F is a systematic force such as gravitational, centrifugal, or electrostatic force (in other words, an external force), and f is the stochastic force. The small size of the particle allows us to neglect its inertia in the first approximation, in other words, the left-hand side of Eq. (1.158) is set to zero. This immediately results in Eq. (1.155). The condition (1.157) is also an idealization, because at t ¼ t0 it gives an infinitely large dispersion, which is an obvious impossibility. The assumption (1.157) is similar to the idealization inherent in the concept of white noise in electrical engineering. The delta function representation of the correlation function is a natural idealization that helps us make the transition from a small time scale to a large scale typical for Brownian motion. Note, however, that differential equations that include perturbating terms given by random delta-correlated functions must be treated carefully, because the usual calculation rules will not always apply. 1.13.2 The Diffusion Equation

A deterministic description of processes in continuous media requires the use of conservation equations (conservation of mass, momentum, energy, and so on). But in order to get concrete solutions, the system of conservation equations must be complemented by constitutive relations or equations. As examples of such relations, we can mention Ficks law for diffusion (mass transport) processes, Fouriers law for heat transport processes, and the Navier–Stokes law describing the hydromechanics of viscous fluids. You may ask how these equations account for fluctuations of the relevant quantities. We shall answer this question by taking the diffusion equation as an example. According to Ficks law, the diffusive flux of the substance j(X, t) is proportional to the concentration gradient: jðX ; tÞ ¼ DrCðX ; tÞ:

ð1:159Þ

On the other hand, we have the equation of conservation of mass (the continuity equation): qC þ rjðX ; tÞ ¼ 0: qt

ð1:160Þ

Substitution of j(X, t) from (1.159) results in the standard diffusion equation: qC ¼ r½DrCðX ; tÞ : qt

ð1:161Þ

Fluctuations could be taken into account by adding a fluctuating term to the righthand side Eq. (1.159): jðX ; tÞ ¼ DrCðX ; tÞ þ j fl ðX ; tÞ:

ð1:162Þ

1.13 Stochastic Differential Equations. The Langevin Equation

Just as we did for the fluctuating term in the Langevin equation, we assume that this term has the following statistical properties: D E D E fl fl j fl ðX ; tÞ ¼ 0; j i ðX ; tÞ jk ðX 0 ; t 0 Þ

¼ KðX ; tÞd jk dðX X 0 Þdðtt0 Þ:

ð1:163Þ

The second property says that different components of the fluctuating flux vector j fl taken at one and the same spatial point, as well as the values of one and the same component taken at different instants of time and/or at different points are assumed to be statistically independent. Or, to say it in fewer words, this property states that fluctuations have local behavior. Eqs. (1.160) and (1.162) give us a diffusion equation where the additional term is expressed as the divergence of a vector: qC ¼ r½DrCðX ; tÞ rj fl ðX ; tÞ qt

ð1:164Þ

It can be shown that this term possesses the following statistical properties: D

E rj fl ðX ; tÞ ¼ 0; D E rj fl ðX ; tÞr0 j fl ðX 0 ; t0 Þ ¼ rr0 ½K1 ðX ; tÞdðX X 0 Þ dðtt0 Þ:

ð1:165Þ

1.13.2.1 The Diffusion Equation with Chemical Reactions Taken into Account The deterministic diffusion equation with a source/sink term arising due to chemical reactions has the form

qC þ rjðX ; tÞ ¼ F½CðX ; tÞ ; qt

ð1:166Þ

where the rate of substance production/consumption in the course of the chemical reaction appears in the right-hand side. This term usually depends on the substance concentrationl; depending on the reaction kinetics, it could be a linear or a nonlinear function of concentration. Production/consumption of matter gives rise to fluctuations, which can be taken into consideration by adding a fluctuating term gfl(X, t) to the systematic term F[C(X, t)]. The new term must satisfy the following statistical conditions:  fl    g ðX ; tÞ ¼ 0; g fl ðX ; tÞg fl ðX 0 ; t0 Þ 0 ¼ K2 ðX ; tÞdðX X Þdðtt0 Þ:

ð1:167Þ

The second condition expresses locality (lack of correlation between fluctuations at different points) as well as the Markovian character of chemical reactions. Eqs. (1.162), (1.166), and (1.167) give us the stochastic diffusion equation that

j45

46

j 1 Basic Concepts of the Probability Theory accounts for chemical reactions: qC þ rjðX ; tÞ ¼ F½CðX ; tÞ þ G fl ðX ; tÞ; qt

ð1:168Þ

where G fl ðX ; tÞ ¼ rj fl ðX ; tÞ þ g fl ðX ; tÞ: The fluctuating term has the following properties:  fl   fl  G ðX ; tÞG fl ðX 0 ; t0 Þ G ðX ; tÞG fl ðX ; tÞ ¼ 0; 0 0 ¼ fK2 ðX X 0 ÞdðX X 0 Þ þ rr ½K1 ðX ; tÞdðX X Þ gdðtt0 Þ: Eqs. (1.164) and (1.168) are both Langevin equations. The main shortcoming of these equations is that the functions K1(X, t) and K2(X, t) are not known in advance. Information about these functions can be obtained by studying the process on the microscopic level, using the same approach that helped us derive the Chapman– Kolmogorov equation. This approach allows to interpret diffusion coefficients as components of a dispersion tensor or a correlation matrix. The difference between equations (1.168) and (1.164) is that F[C(X, t)] that appears in Eq. (1.168) is not a function but a functional. 1.13.2.2 Brownian Motion of a Particle in a Hydrodynamic Medium Slow motion of a particle in a fluctuating hydrodynamic medium (the medium is assumed to be at rest at the infinity) under the action of the average viscous force exerted by the surrounding fluid hF(t)i and the fluctuating force F f l(t) (a force induced by thermal hydrodynamic fluctuations in the fluid or by some other source of random forces) is described by the Langevin equation (a stochastic analogue of Newtons second law):

m

dU ¼ hFðtÞi þ F fl ; dt

U¼

dX : dt

ð1:169Þ

To find the statistical properties of the random force F f l, it is necessary to use the hydrodynamic equations describing the fields of velocity u and pressure p in the fluid. The particle motion occurs at low Reynolds numbers, so one can use Navier– Stokes equations in the inertialess approximation, (i.e., Stokes equations): ru ¼ 0;

rT ¼ 0;

ð1:170Þ

where T(r, t) ¼ T s þ T f l is the stress tensor in the fluid, which consists of a systematic component T s ¼ pI þ 2meE and a fluctuating component T f l; pI is the spherical tensor (the isotropic part of the stress tensor); 2meE is the deviator (the deviatoric part of the stress tensor); E ¼ 0.5(ru þ ruT) is the symmetric rate-of-strain tensor, me is the coefficient of dynamic viscosity of the carrier (external) fluid.

1.13 Stochastic Differential Equations. The Langevin Equation

In a statistically homogeneous medium, the fluctuating stress tensor T f l is symfl metric and its trace is equal to zero, that is, T ii ¼ 0. The components Dof TEf l are fl usually assumed to have a Gaussian distribution with the average value T ik ¼ 0 and the correlation matrix D E 2 fl fl T ik ðr 1 ; t 1 ÞT lm ðr 2 ; t2 Þ ¼ 2kme ðd il d km þ dim dkl  d ik dlm Þ 3 ð1:171Þ  dðr 1 r 2 Þdðt1 t2 Þ: Equations (1.170) can then be rewritten in the coordinate form: qui ¼ 0; qXi

qTisj qX j

fl

¼

qTi j qX j

:

ð1:172Þ

Following the method of small perturbations, we shall be looking for the solutions in the form ~i ; ui ¼ hui i þ u

D E s Tisj ¼ Tisj þ T~ i j ;

ð1:173Þ

s where hui, hT si are the average values and u~ ,~ T are small fluctuating ‘‘additions’’ (perturbations). Substituting (1.173) into (1.172) and neglecting small terms of higher orders, we get the equations for the average values and for the fluctuating terms:

qhui i ¼ 0; qXi

D E q Tisj qX j

¼ 0;

q~ ui ¼ 0; qXi

~ qT ij s

qX j

qT~ i j

fl

¼

qX j

:

ð1:174Þ

Now, we must introduce two boundary conditions. First, the relative velocity at the ~i ¼ 0. interface between the fluid and the solid particle should be zero: huii ¼ Ui, u ~i ¼ 0. Secondly, the fluid must be at rest at the infinity: hui i ¼ u As a rule, the main purpose of hydrodynamic calculations is to find the force the surrounding fluid exerts on a moving particle. This force has systematic and fluctuating parts F and F f l, whose components are ðD E hFi i ¼ Tisj n j ds; s

ð

~ s n j ds; Fi ¼ T ij fl

ð1:175Þ

s

where ni are components of the outer normal vector and S is the surface of the particle. Solving the first two equations (1.174), we get the relation between the particles velocity and the drag force: hFi i ¼ Ri j U j ; where Rij are components of the resistance tensor R.

ð1:176Þ

j47

48

j 1 Basic Concepts of the Probability Theory ~ i , let us use In order to determine statistical characteristics of the random force F the relation that follows in a self-obvious way from Gausss theorem, equations (1.174), and the above-mentioned boundary conditions: ð V

D E1 ð D E q Tisj q s ~ s ~ @hui i A dV ¼ ~ ui ðhui iT i j ui Ti j ÞdV qX j qX j qX j 0

~ qT ij s

ð ¼

V

e s ~ ð h ui i T i j ui

s

ð ð D sE es qT ij e ~ s n j ds ¼ Ui F ~i; T i j Þn j ds ¼  hui i ds ¼ Ui T ij qX j s

s

where integration is carried out over the volume V occupied by the fluid. Using this relation together with the property (1.171), we finally get the two-time correlation between components of the fluctuating force: 

 ~ j ðt0 Þ ¼ ~ i ðtÞF F

ð   1 2kTdðtt0 Þ hui i Ti j n j ds ¼ 2kTRi j dðtt0 Þ: Ui U j

ð1:177Þ

s

1.14 Variational (Functional) Derivatives

When considering random processes and random fields, one can see that the PDF depends on random variables, which, in turn, are functions of time and (or) spatial coordinates. Therefore, the variables in the PDF are random functions rather than random variables. It was shown in Sections 1.5 and 1.6 that knowing the characteristic function j and the characteristic functional, one can determine statistical parameters of random quantities such as moments and cumulants from the formulas (1.50) and (1.54). These formulas contain derivatives of j and lnj. When considering random processes and random fields, we treat these derivatives as derivatives with respect to functions and not as derivatives with respect to random variables. Hence, instead of ordinary differentiation we have to perform functional, or, to use another term, variational, differentiation. The corresponding derivatives are called functional (variational) derivatives. Let us recal the general definition of a functional. We say that a functional is given when there exists a rule that assigns a definite number to each function belonging to some set of functions. Some examples are given below: 1. - linear functional Xð2

F½jðX Þ ¼

aðX ÞjðX ÞdX ; X1

where a(X) is a given function and the limits X1 and X2 can be either finite or infinite;

1.14 Variational (Functional) Derivatives

2. - quadratic functional Xð2 Xð2

F½jðX Þ ¼

Bðt1 ; t2 Þjðt1 Þjðt2 Þdt1 dt2 ; X1 X1

where B(t1, t2) is a given function; 3. - function of functional F½jðXÞ ¼ f ðF½j Þ where f(X) is a given function and F[j(X)] is a functional. Consider the difference in values of one and the same functional taken for two functions j(t) and j(t) þ dj(t), where t lies in the interval t 2 (X  0.5DX; X þ 0.5DX). The difference (more strictly, the linear (with respect to dj(t)) part of that difference) is called variation of the functional: dF½j ¼ fF½j þ dj F½j g: The variational (functional) derivative is defined as dF½j dF½j : ¼ lim Ð DX ! 0 djðX Þ DX jðtÞdt

ð1:178Þ

The variational derivative of a functional F[j] is itself a functional of j(t) that also depends on the point X as a parameter. Thus the variational derivative itself has two different derivatives. One can differentiate it in the ordinary way with respect to the parameter X; or one can take the variational derivative with respect to j(t) at the ~ . The latter would be the second variational derivative of the initial point t ¼ X functional F[j]:   d dF½j d 2 F½j ¼ : ~ Þ djðX Þ ~ ÞdjðXÞ djðX djðX

ð1:179Þ

The second variational derivative is also a functional of j(t), but, in contrast to the ~ . Variational first variational derivative, it now depends on two points: X and X derivatives of higher orders can be defined in a similar way. As examples, consider variational derivatives of the above-mentioned functionals. Xð2

dF ¼ F½j þ dj F½j ¼

X þ0:5DX ð

aðtÞdjðtÞdt ¼ X1

aðtÞdjðtÞdt: X 0:5DX

j49

50

j 1 Basic Concepts of the Probability Theory If the function a(t) is continuous on the interval DX, then, according to the mean value theorem, ð dF½j ¼ aðX 0 Þ djðtÞdt; DX 0

where X 2 (X  0.5DX; X þ 0.5DX). The definition (1.178) yields 0 1 ð ð dF½j djðtÞdtA ¼ aðXÞ: ¼ lim aðX 0 Þ@ djðtÞdt djðX Þ DX ! 0 DX

ð1:180Þ

DX

Applying the same approach to the quadratic functional, we get dF½j ¼ djðX Þ

Xð2

ðBðt; X Þ þ BðX ; tÞÞjðtÞdt;

ðX1 < t < X2 Þ:

ð1:181Þ

X1

Note that in many cases the function B(t, X) is a symmetric function of its arguments, that is, B(t, X) ¼ B(X, t). Finally, for a function of a functional, F½j þ dj ¼ f ðF½j þ dj Þ ¼ f ðF½j þ dFÞ qf ðF½j Þ qf ðF½j Þ ¼ f ðF½j Þ þ dF þ . . . ¼ F½j þ dF þ . . . qF qF and d qf ðF½j Þ dF½j f ðF½j Þ ¼ : djðX Þ qF djðX Þ

ð1:182Þ

Consider now some properties of variational derivatives. Let a functional be a product of two functionals: F[j] ¼ F1[j]F2[j]. The variation and variational derivative of this functional are: dF ¼ F½j þ dj F½j ¼ F1 ½j þ dj F2 ½j þ dj F1 ½j F2 ½j ¼ F1 ½j dF2 ½j þ F2 ½j dF1 ½j ; d dF2 ½j dF1 ½j ½F1 ½j F2 ½j ¼ F1 ½j þ F2 ½j : djðX Þ djðX Þ djðXÞ

ð1:183Þ

Of special interest is the functional of a Gaussian distribution, ( ) 1 ðtt0 Þ2 jðtÞdt: F½j ¼ pffiffiffiffiffiffi exp  2s2 2ps ð¥ 0

ð1:184Þ

1.14 Variational (Functional) Derivatives

Eq. (1.180) gives us its variatlional derivative: ( ) dF½j 1 ðtt0 Þ2 : ¼ pffiffiffiffiffiffi exp  2s2 djðtÞ 2ps

ð1:185Þ

Going to the limit s ! 0 in Eqs. (1.183), (1.184) and using the definition of the delta function Eq. (1.9) and its property (1.11), we obtain: lim F½j ¼ jðt0 Þ;

s!0

lim ¼

s!0

dF½j ¼ dðtt0 Þ: djðtÞ

Therefore, dj½t0 ¼ dðtt0 Þ: djðtÞ

ð1:186Þ

The relation (1.186) facilitates the derivation of formulas for some variational derivatives. As an example, let us derive the formula (1.181) for the quadratic functional: 8X X 9 ð2 ð2 = d < Bðt1 ; t2 Þjðt1 Þjðt2 Þdt1 dt2 ; djðX Þ: X1 X1

Xð2 Xð2

¼

Bðt1 ; t2 Þ

d ½jðt1 Þjðt2 Þ dt1 dt2 djðXÞ

X1 X1 Xð2 Xð2

¼

  djðt1 Þ djðt2 Þ Bðt1 ; t2 Þ jðt2 Þ þ jðt1 Þ dt1 dt2 djðX Þ djðXÞ

X1 X1 Xð2 Xð2

¼

Bðt1 ; t2 Þ½dðt1 X Þjðt2 Þ þ jðt1 Þdðt2 X Þ dt1 dt2 X1 X1 Xð2

¼

½Bðt; XÞ þ BðX ; tÞ jðtÞdt;

ðX1 < t < X2 Þ:

X1

Another example is the variational derivative of the functional Xð2

F½jðtÞ ¼ X1

_ dt; L½ðt; jðtÞ; jðtÞ

_ jðtÞ ¼

djðtÞ : dt

j51

52

j 1 Basic Concepts of the Probability Theory We have: Xð2    qL qL d djðt0 Þ qL qL d þ dt ¼ þ dðtXÞdt qj qj_ dt djðX Þ qj qj_ dt X1 X1  d q q _ ; X 2 ðX1 ; X2 Þ: þ L½ðt; jðtÞ; jðtÞ ¼  dX qj_ qj

dF½j ¼ djðX Þ

Xð2 

The functional F[j(t) þ Z(t)] can be expanded as a functional Taylor series in the vicinity of the point Z  0: ð dF½j hðX ÞdX F½jðtÞ þ hðtÞ ¼ F½jðtÞ þ djðXÞ ðð 1 dF½j hðX1 ÞhðX2 ÞdX1 dX2 þ . . .: þ 2! djðX1 ÞdjðX2 Þ

ð1:187Þ

Here and later, when the range of integration is not pointed out, it is assumed to be infinite. The Taylor series (1.187) could be written in the compact form ð F½jðtÞ þ hðtÞ ¼ exp dX hðX Þ

d F½jðtÞ djðX Þ

using the following operator notation: ð exp dX hðX Þ ðð

d djðX Þ



ð ¼ 1 þ dX hðX Þ

d djðX Þ

d2 þ. . . djðX1 ÞdjðX2 Þ ð 2 ð d 1 d ¼ 1 þ dX hðX Þ þ. . .: þ dX hðX Þ djðX Þ 2! djðX Þ 1 þ 2!

dX1 dX2 hðX1 ÞhðX2 Þ

Consider the transformation of variational derivatives as we change the functional variables. Let us replace the function j(t) by a new function c(t) given by the equality j(t) ¼ C[c(t); t], where c is the functional of a function c(t), which also depends on t. Then the functional F[j(t)] is a composite functional of c(t): F½jðtÞ ¼ F½Y½yðtÞ; t  F1 ½yðtÞ : For such functional, there exists the following expression for the variational derivative: ð dF1 ½yðtÞ dF½jðtÞ dY½yðtÞ; X 0 ¼ dt0 : djðtÞ djðt0 Þ dyðtÞ

ð1:188Þ

1.15 The Characteristic Functional

A functional change of variables plays an important role in Fourier transforms: ð jðXÞ ¼ yðwÞeiwX dw ¼ j½yðwÞ; X : According to the formula (1.173), we have dj½yðwÞ; X 0 ¼ eiw X dyðw0 Þ and from (1.180), there follows ð dF1 ½j½yðwÞ; X dF½jðXÞ iw0 X 0 0 ¼ e dX : djðw0 Þ djðX 0 Þ

ð1:189Þ

1.15 The Characteristic Functional

It was shown in Section 1.5 that a random value u is completely determined by its characteristic function j(r) ¼ hexp(iru)i, which allows us to use the inverse Fourier transform to get the PDF, pðuÞ ¼

ð 1 eiru dr 2p

the moments,  B n ¼ h un i ¼

 1 d n jðrÞjr¼0 ; i dr

the cumulants,   1 d n ½lnjðrÞ r¼0 Sn ¼ i dr and other statistical parameters of the PDF. For a multidimensional random quantity u ¼ (u1, u2, . . . , uN), the complete description is contained in the characteristic function * jðrÞ ¼ jðr1 ; r2 ; . . .; rN Þ ¼ hexpðiruÞi ¼

expði

X

+ r k uk Þ :

ð1:190Þ

k

The corresponding joint PDF of the random values u1, u2, . . . , uN is the Fourier transform of the characteristic function j(r1, r2, . . . , rN):

j53

54

j 1 Basic Concepts of the Probability Theory ð

1

eirX jðrÞdr ð2pÞ ð X 1 expði rk Xk Þjðr1 ; r2 ; . . .; rN Þdr1 dr2 . . .drN ¼ N ð2pÞ k

pðX1 ; X2 ; . . .; XN Þ ¼

N

ð1:191Þ

¼ dðu1 X1 Þdðu2 X2 Þ. . .dðuN XN Þ: Consider now a random function u(X). For its complete statistical description, it is sufficient to know the characteristic functional

F½r ¼

 ð  exp i rðtÞuðtÞdt ;

ð1:192Þ

where the function r(t) is an arbitrary function replacing the set of numbers r1, r2, . . . , rN in (1.190). Given F[r], we can find statistical characteristics of the random function u(X), for example, the average value hu(X)i, N-point moments Buu. . .u ¼ hu(X1) . . . u(XN)i, etc. To find the variational derivative, let us use the results obtained in Section 1.14, keeping in mind that the averaging operator commutes with the operator d/dr(X): dF½r d ¼ exp drðX Þ djðX Þ

 ð   ð  d i rðtÞuðtÞdt ¼ exp i rðtÞuðtÞdt djðXÞ  ð

 ¼ i uðX Þexp i rðtÞuðtÞdt :

Similarly, we write 

  

 ð  1 d 1 d ... F ¼ uðX1 Þ. . .uðXn Þexp i rðtÞuðtÞdt : i drðX1 Þ i drðXn Þ

Setting r ¼ 0 in the last relation, we obtain: 1 dn ¼ huðX1 Þ. . . uðXn Þi ¼ Buu...u : F n i drðX1 Þ. . .drðXn Þ r¼0

ð1:193Þ

So, it is possible to find the multi-point moments for a given characteristic functional. Expanding the functional F[r] into a functional Taylor series according to Eq. (1.187) and taking into account Eq. (1.193), one can express the characteristic functional in terms of moments: F½r ¼

¥ nð X i

ð . . . Buu...u ðX1 ; X2 ; . . .; Xn Þ

n!  rðX1 ÞrðX2 Þ. . .rðXn ÞdX1 dX2 . . .dXn : n¼0

ð1:194Þ

1.15 The Characteristic Functional

If the functional has the form F½r ¼ expðy½r Þ; then c is called the cumulant-generating function (see Section 1.6), and the cumulants themselves given by dn : S11...1 ðX1 ; X2 ; . . .; Xn Þ ¼ ðiÞ drðX1 Þ. . .drðXn Þ r¼0 k

ð1:195Þ

As an example, let us find the characteristic functional for a Gaussian random process u(X) with hu(X)i ¼ 0, under the additional assumption that the joint distribution for any two given values of X is also Gaussian. Let the random variable ð¥ hAi ¼

rðtÞuðtÞdt; ðrð¥Þ ¼ 0; ¥

have a Gaussian distribution ( ) 1 ðAhAiÞ2 pðAÞ ¼ pffiffiffiffiffiffi exp  2s2A 2psA with the parameters ð¥ A¼

rðtÞhuiðtÞdt ¼ 0; ¥

  s2A ¼ A2 ðhAiÞ2 ¼

ð¥ ð¥ Bðt1 ; t2 Þrðt1 Þrðt2 Þdt1 dt2 ; ¥¥

where B(t1, t2) ¼ hu(t1)u(t2)i is a two-point correlation function. The characteristic functional of the random quantity A is     1 expðeiA Þ ¼ exp  s2A : 2 Therefore the characteristic functional of a Gaussian random process is 8 9 < 1 ð¥ ð¥ = Bðt1 ; t2 Þrðt1 Þrðt2 Þdt1 dt2 : F½r ¼ exp  : 2 ;

ð1:196Þ

¥¥

Characteristic functionals and variational derivatives are widely used in problems pertaining to particle motion under the action of random fluctuating forces, for

j55

56

j 1 Basic Concepts of the Probability Theory example, in the theories of Brownian motion, diffusion, turbulence, etc. Such problems are the subject of the statistical theory of dynamic systems with fluctuating parameters. These parameters (e.g., coordinates and velocities of particles suspended in a fluid), are described by ordinary or partial differential equations (stochastic equations). The main challenge is to obtain and then solve a closed system of equations. It turns out that many processes can be treated as Markowian processes. Also, the distribution of fluctuating parameters (random variables) has proved to be Gaussian. As an example, consider a simplified form of the Langevin equation (see Section 1.13) that describes the time rate of change of velocity V(t) of a particle driven by a fluctuating force x(t): dV ¼ hV þ xðtÞ; dt

X ð0Þ ¼ 0;

ð1:197Þ

where h is a constant and x(t) is a random function. For a given x(t), the solution of Eq. (1.197) has the form ði ð1:198Þ

VðtÞ ¼ xðtÞexpðhðttÞÞdt: 0

From Eq. (1.198), one can derive all statistical characteristics of the random process V(t). On the other hand, all statistical characteristics are contained in the characteristic function * jðrÞ ¼ ðexpðirVðtÞÞ ¼

8 t 9+ < ð = exp ir xðtÞexpðhðttÞÞdt : : ;

ð1:199Þ

0

Taking into account the relation (1.192), the last equation can be rewritten as jðrÞ ¼ FfrexpðhðttÞÞg:

ð1:200Þ

Finally, we should mention the Furutsu–Donsker–Novikov correlation formula for the product of a Gaussian random function X(t) and a functional R[X(t)] that may depend (either explicitly or implicitly) on X(t): ð¥ hX ðtÞR½X ðtÞ i ¼

 dR½X ðtÞ dt1 : Bðt; t1 Þ dX ðt1 Þ

¥

This formula is the functional analogue of Eq. (1.86).

ð1:201Þ

References

References 1 Gardiner, C.W. (1985) Handbook of Stochastic Methods, 2nd ed., Springer– Verlag. 2 Klyatskin, V.I. (1975) Statistical Description of Dynamical Systems with Fluctuating Parameters, Nauka, Moscow, (in Russian). 3 Klyatskin, V.I. (1980) Stochastic Equations and Waves in Random Inhomogeneous Media, Nauka, Moscow, (in Russian). 4 Landau, L.D. and Lifshitz E.M. (1964) Statistical Physics, Nauka, Moscow, (in Russian). 5 Leontovitch, M.A. (1983) Introduction to Thermodynamics. Statistical Physics, Nauka, Moscow, (in Russian). 6 Monin, A.C. and Yaglom, A.M. (1971) Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. 1, MIT Press, Cambridge, MA.

7 Monin, A.C. and Yaglom, A.M. (1975) Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. 2, MIT Press, Cambridge, MA. 7 Feller, W. (1974) An Introduction to Probability Theory and Its Applications, Vol. I, II, Wiley. 8 Chandrasekhar, S. (1943) Stochastic Problems in Physics and Astronomy, Rev. Mod. Phys., 15, 1–89. 9 Cercignani, C. (1969) Mathematical Methods in Kinetic Theory, Macmillan. 10 Einstein, A. and Smoluchowski, M. (1936) Brownian Motion, ONTI, Moscow, (in Russian). 11 Pope, S.B. (2000) Turbulent Flows, Cambridge Univ. Press. 12 Saffman, P.G. (1971) On the Boundary Condition at the Surface of a Porous Medium, Studies in Appl. Math., 50 (2), 93–101.

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2 Elements of Microhydrodynamics The branch of hydrodynamics studying the motion of microparticles in liquid at low Reynolds numbers is called microhydrodynamics. The present chapter is devoted to the fundamentals of low Reynolds number hydrodynamics which will be a prerequisite for the subsequent discussion. Low Reynolds number hydrodynamics is based on Stokes equations [1–5] r  v ¼ 0;

r2 v ¼

1 r p: me

ð2:1Þ

The boundary conditions for these equations follow from the restrictions imposed on velocities at particle surfaces, at flow region boundaries (if any), and far away from the particles (‘‘at infinity ’’) if the fluid is unbounded. Exact solution of concrete boundary value problems presents considerable mathematical difficulties even for the single particle case. Still, there are some general properties common for this type of boundary value problems that prove to be very useful for derivation of solutions. For simplicity, we consider the behavior of particles in an unbounded fluid. The first property is called the reciprocity theorem. Let (v 0,T 0 ) and (v 00,T 00 ) be the velocity v and stress T fields corresponding to the flows of two fluids having different viscosities m0e and m 00e . Both flows are described by the same equations (2.1) and by the same boundary conditions. Then for any closed surface S bounding the fluid volume V (this surface could consists of several different surfaces, including surfaces of particles contained in the considered volume V ), there should hold ð ð m00e dS  T 0  V 00 ¼ m0e dS  T 00  V 0 : S

ð2:2Þ

S

Another important property, which is often used to determine the hydrodynamic force F and the torque L acting on a spherical particle that has translational velocity U and rotates with angular velocity V, is given by Faxen’s laws: F ¼ 6 pme aðv¥ j0 UÞ þ me pa3 ðrv¥ Þ0 ;   1 L0 ¼ 8pme a3 ðr  v¥ Þ0 V : 2

ð2:3Þ

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60

Here v¥ is the (translational) velocity of the liquid at the infinity. The subscript 0 indicates that the corresponding quantity is measured at the point occupied by the center of the sphere. When solving problems on slow motion of particles, it is important to estimate hydrodynamic field perturbations in the fluid that are induced by the moving particle. If the considered suspension is low-concentrated (dilute), then the effect of the particle on the fluid can be approximated by a point force applied at the particle center and equal to the drag force on the particle, but pointing in the opposite direction. The resultant problem is handled by writing Stokes equations and obtaining the fundamental solution. If we assume the volume concentration of particles j to be small, then the average distance between particles is large compared to their radius, and we need to consider only pair interactions between particles. Since the particles are far enough from each other, the hydrodynamic influence of particle B on particle A can be accounted for by replacing particle B with a point force FB applied at the particle center and equal to the drag force on particle B but having the opposite sign. The translational velocity of the fluid at the infinity is assumed to be constant: u¥ U0ex; we also assume that the line of centers of the particle pair is parallel to U0 and coincides with the X-axis. Then in order to find the drag force of particle A, it is necessary to solve the Stokes equations with the source term me Dur p ¼ FB dðX XB ÞdðYÞdðZÞex ;

r u ¼ 0

ð2:4Þ

and the boundary conditions u ¼ 0 on u ! U0

at

SA ; X ! ¥:

ð2:5Þ

The fundamental solution is ! FB 1 ðX XB ÞðX X B Þ ; u¼ ex þ 8pme jX X B j jX X B j3 p¼

! FB ðX XB Þ : 4p jX X B j3

ð2:6Þ

Owing to the linearity of the inhomogeneous boundary value problem, its solution can be obtained as a superposition of solutions of the corresponding homogeneous problems: Eqs. (2.4) at FB ¼ 0 with the boundary conditions (2.5); and Eqs. (2.4) with the boundary conditions (2.5) when U0 ¼ 0. The solutions of these particular problems are found as sums of spherical functions, while keeping in mind the axial symmetry of the problem. To determine the drag force acting on a particle of finite radius, one should average the velocity over the particle surface. The final expression for the force exerted on a single particle by the surrounding fluid will be given by the Stokes formula: FB ¼ 6pme bU0 :

ð2:7Þ

2.1 Motion of an Isolated Particle in a Quiescent Fluid

For a pair of particles, A and B, the resulting drag forces have the form   3 a3 a  FB ; DA ¼ 6pme aU0 þ 2 3r 3 r

  3 b3 b DB ¼ 6pme bU0 þ  FA ; 2 3r 3 r

ð2:8Þ

where a, b are the particle radii and r is the distance between their centers. For identical particles (a ¼ b), DA ¼ DB ¼ 6pme u0 lða=rÞ;

ð2:9Þ

where l is a dimensionless resistance coefficient. Up to the order a/r, it is equal to l ¼ 1

3a : 2r

ð2:10Þ

With further applications in mind, we shall focus our attention on the motion of one particle, two particles, and, finally, many (i.e., more then two) particles in a quiescent fluid or in a shear flow of viscous incompressible fluid.

2.1 Motion of an Isolated Particle in a Quiescent Fluid

Consider a solid particle of an arbitrary shape moving with translational velocity U0 and rotating with angular velocity V. By its nature, such a motion is unstable. But if the Reynolds numbers derived from translational and rotational velocities U0 and W are small, that is, if Ret ¼ aU0re/me  1 and Rer ¼ a2Wre/me  1 where a is the characteristic linear size of the particle and re and me are the density and viscosity of the fluid, then it is legitimate to derive the velocity and pressure fields v and p induced in the fluid by the moving particle by using quasi-stationary Stokes equations r  v ¼ 0;

r2 v ¼

1 r p: me

ð2:11Þ

Let 0 be a point attached to the particle, and let U0 be the instantaneous translational velocity of this point, and V – the instantaneous angular velocity of the particle. The particle is assumed to be solid, and thus the condition of zero velocity at the particle surface Sa yields v ¼ U0 þ V  r0

at

Sa ;

where r0 is the radius vector measured with respect to 0.

ð2:12Þ

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The second boundary condition is that fluid should be at rest at the infinity: v!0

at r0 ! ¥:

ð2:13Þ

The solution of thusly formulated problem gives the hydrodynamic force F and the torque L acting on the particle [6]: ð

ð dS  T;

F¼

L¼

Sa

r 0  ðdS  TÞ:

ð2:14Þ

Sa

Here dS ¼ n dS is a particle surface element (directed towards the fluid), T is the stress tensor of an incompressible Newtonian fluid: T ¼  pI þ me ðrv þ rvT Þ;

ð2:15Þ

I is the unit tensor with components d ij, and rv is the velocity gradient tensor with components rivj. Due to the linearity of the boundary value problem defined by Eqs. (2.11)–(2.13), it is possible to represent the velocity, the pressure, and the stress tensor as sums of translational and rotational terms, ðvt0 ; pt0 ; T t0 Þ and ðvr0 ; pr0 ; T r0 Þ, each term describing its corresponding type of particle motion: v ¼ vt0 þ vr0 ;

p ¼ pt0 þ pr0 ;

T ¼ Tt þ Tr:

ð2:16Þ

As a result, the boundary value problem separates into the translational problem defined by r  vt0 ¼ 0; vt0 ¼ U 0

r2 vt0 ¼

1 r pt0 ; me

Sa ;

vt0 ! 0

on

at

r0 ! ¥

ð2:17Þ

and the rotational problem defined by r  vr0 ¼ 0; vr0 ¼ V  r 0

r2 vr0 ¼ on

1 r pr0 ; me

Sa ;

vr0 ! 0

ð2:18Þ at

r0 ! ¥:

Using the formulas (2.14), we can obtained the hydrodynamic forces F t0 ; F r0 and torques Lt0 ; Lr0 induced by translational and rotational motions of the particle: ð ð t;r t;r F t;r ¼ dS  T ; L ¼ r 0  ðdS  T 0t;r Þ: ð2:19Þ 0 0 0 Sa

Sa

where T 0t;r are stress tensors corresponding to the translational and rotational particle motions: t;r t;r t;r T T t;r 0 ¼  p0 I þ ðrv0 þ ðrv0 Þ Þ:

ð2:20Þ

2.1 Motion of an Isolated Particle in a Quiescent Fluid

The resulting values of the hydrodynamic force F and torque L acting on the particle are F ¼ F t0 þ F r0 ;

L ¼ Lt0 þ Lr0 :

ð2:21Þ

Both translational and rotational fields, as well as forces and torques depend on the fluid’s viscosity me, on the length and orientation of vectors U0 and V relative to the particle’s coordinate system, and on the particle’s geometry. But such complex dependences are inconvenient to use. Let us try to separate these dependences. To this end, for each type of motion we shall introduce quantities that characterize only the geometrical properties of the particle, thus determining the geometrical dependence once and for all for any given class of geometrical shapes. For translational motion, there exists a velocity tensor V t (asymmetrical in the general case) and an associative dynamic pressure vector P t satisfying the relations vt0 ¼ V t  U 0 ;

p0 ¼ me P t  U 0

ð2:22Þ

and the equations r  V t ¼ 0;

r2 V t rPt ¼ 0;

Vt ¼ I

Sa ;

on

Vt !0

at

r0 ! ¥;

ð2:23Þ

and independent of the fluid’s viscosity (at any point in the fluid), of the choice of reference frame (i.e., of the origin 0), and of the length and orientation of the vector U0. They depend only on the geometry of the particle’s surface and on the radius vector (which can be taken relative to any origin of coordinates attached to the particle). Consider a translational Stokesian flow that goes around a spherical solid particle of radius a when the fluid’s velocity at the infinity is U ¼ const. The stream function for this flow is expressed in spherical coordinates as    1 a 3 a y ¼ Ur 2 sin2 u 3 : 4 r r The velocity components and the dynamic pressure are given by

a2 a 1 qy 1 r ¼  U cosu 3 ; sinu qu 2 r r a

a a2  1 qy 1 v ¼  þ3 ; ¼  U sinu r sinu qr 4 r r

vr ¼ 

r2

3 cosu p p¥ ¼ ame U 2 ; 2 r where p¥ is the static pressure at the infinity.

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64

The velocity tensor and the pressure vector for such a flow are equal to Vr ¼

 1 a3 3rr  3 a rr I ; þI  4r r 4 r r2

3 r Pr ¼ a 3 ; 2 r

ð2:24Þ

where r is measured from the center of the sphere. We see from Eq. (2.15) and Eq. (2.22) that the stress tensor can be written as T t ¼ me Pt U 0 ;

ð2:25Þ

where we have just introduced a tensor of the third rank (a triadic tensor) Pt ¼ I  P t þ ðrV t þ T ðrV t ÞÞ:

T

The superscript T to the left of the tensor (it is known as a polyadic) means that ðabc. . .Þ ¼ bac. . .. Let us introduce the tensors K t and K c0 characterizing the given particle: ð Kt ¼ 

ð dS  Pt ;

K c0 ¼ 

Sa

r 0  ðdS  Pt Þ: Sa

Then expressons for the force and torque acting on the particle undergoing a translational motion can be written in a compact form: F 0 ¼ me K t  U 0 ;

L ¼ me K c0  U 0 :

ð2:26Þ

The tensor K t is called the tensor of translational resistance. It depends only on the size and shape of the particle and does not depend on the particle’s velocity and orientation in space or on the properties of the fluid. The tensor of translational resistance has the dimensionality of length and characterizes resistance for the particle’s motion at low Reynolds numbers. Let us show that it is a symmetric tensor. Consider two motions of one and the same particle, in the same fluid, but with different velocities U 0 and U 00 . For these two motions we have the corresponding fields of velocities and stresses (v 0, T 0 ) and (v 00, T 00 ) in the surrounding fluid. As a consequence of the reciprocity theorem (2.2), we have for m0e ¼ m00e : ð S

ð dS  T 0  v00 ¼ dS  T 00  v0 :

ð2:27Þ

S

Suppose the surface of integration consists of a spherical surface S¥ of large radius (the whole sphere is still positioned inside the fluid) and the particle surface Sa which lies wholly inside the volume bounded by S¥. Then the integral (2.27) can

2.1 Motion of an Isolated Particle in a Quiescent Fluid

be written as ð

ð F dS ¼

ð F dS þ

S¥

S

F dS: Sa

2 Since dS  r2, and for large r, the fluid velocity is vÐ r1 Ð and the stress is T r , 1 we have vTdS  r for large r. Therefore at r ! ¥  , and Eq. (2.27) converts S Sa into

ð

dS  T 0  v00 ¼

Sa

ð

dS  T 00  v0 :

ð2:28Þ

Sa

On the particle surface S, the boundary conditions v0a ¼ U 0 and va00 ¼ U 00 must be satisfied. A substitution of these relations into Eq. (2.28) yields U 00

ð

dS  T 0 ¼ U 0

Sa

ð

dS  T 00 :

ð2:29Þ

Sa

Due to Eq. (2.14), we have ð

0

0

ð

dS  T ¼ F ; Sa

dS  T 00 ¼ F 00

Sa

and Eq. (2.29) reduces to U 00  F 0 ¼ U 0  F 00 :

ð2:30Þ

Let us now substitute (2.30) into (2.26). Then the forces acting on the particle in our two cases are equal to F 0 ¼ me K t  U 0 ;

F 00 ¼ me K t  U 00 :

ð2:31Þ

Note that one and the same tensor Kt enters the right-hand sides of both formulas, since it depends only on the size and shape of the particle. From Eq. (2.30) and Eq. (2.31) there follows U 00  K t  U 0 ¼ U 0  K t  U 00 : From the properties of scalar products of tensors and vectors we find that U 0  ðK t  U 00 Þ ¼ U 0  ðU 00  ðK t ÞT Þ ¼ U 00  ðK t ÞT  U 0

ð2:32Þ

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66

and, going back to Eq. 2.32, obtain the following: U 00  K t  U 0 ¼ U 00  ðK t ÞT  U 0 : Thus, K t ¼ ðK t ÞT :

ð2:33Þ

This equality means that the tensor K t is symmetric, that is, Kitj ¼ K tji . The tensor K c0 in the expression (2.26) for the torque L is called the conjugate tensor. Its dimensionality is the square of length, and it depends only on the particle’s geometry and on the position of the particle’s center 0. In general the tensor K c0 is neither symmetric nor antisymmetric. For rotational motion, we can also introduce the velocity tensor V r0 and the associative pressure vector P r0 satisfying the relations vr0 ¼ V r0  V;

pr0 ¼ me P0r  V

ð2:34Þ

and the equations r  V r0 ¼ 0; V r0 ¼ «  r0

r2 V r0 rP r0 ¼ 0; on

Sa ;

V r0 ! 0 at

r0 ! ¥;

ð2:35Þ

where « is the so-called permutation symbol (Levi–Civita symbol) – a tensor with components

ei jk

8 1; > > < ¼ 1; > > : 0;

if

ði; j; kÞ are

cyclic;

if

ði; j; kÞ are

anticyclic;

otherwise:

These tensors do not depend on me and W at any point inside the fluid; they depend only on the choice of the origin 0 and on the surface geometry of the particle. If the origin 0 coincides with the geometrical center of the particle, then V r0 ¼

a3 r

«  r;

Pr0 ¼ 0;

where r is measured relative to the particle’s center. The stress induced in the fluid by the particle’s rotational motion is determined by the tensor T 0r ¼ me P0r  V; where we introduced a tensor of the third rank Pr0 ¼ I  Pr0 þ ðrV r0 þ T ðrV r0 ÞÞ:

2.1 Motion of an Isolated Particle in a Quiescent Fluid

The force and torque acting on the particle during its rotational motion are ð ð F r0 ¼ dS  T r0 ; Lr0 ¼ r 0  ðdS  T r0 Þ: Sa

Sa 

Now, if we define the tensors K c0 and K r0 that are invariable for the given particle by the relations 

ð

K c0 ¼ 

ð dS  Pr0 ;

Sa

K r0 ¼ 

r 0  ðdS  Pr0 Þ; Sa

then expressions for the forces and torques experienced by the particle during its rotational motion can be written as 

F ¼ me K c0  U 0 ;

L ¼ me K r0  V:

ð2:36Þ 

The tensor K r0 is called the rotational resistance tensor and K c0 – its conjugate tensor. Their dimensionalities are, respectively, L3 and L2. The reciprocity theorem  implies their symmetry, that is, K c0 ¼ ðK c0 ÞT . Thus it follows from Eqs. (2.21), (2.26), and (2.36) that an isolated particle undergoing translational and rotational motion of in a fluid that is quiescent at the infinity experiences the force and the torque that are given by F ¼ me ðK t  U 0 þ ðK c0 ÞT  VÞ;

L0 ¼ me ðK c0  U 0 þ K r0  VÞ:

ð2:37Þ

They depend on three independent resistance tensors K t, K r0 , and K c0 characterizing the particle’s inherent properties: surface shape and size. The tensors Kt and K r0 are symmetric, that is, K t ¼ ðK t ÞT and K r0 ¼ ðK r0 ÞT . If a is the characteristic linear size of the particle, then, introducing dimensionless resistance tensors by dividing each dimensional resistance tensor by its corresponding ak, we can rewrite the relations (2.37): ~ t  U 0 þ aðK ~ c0 ÞT  VÞ; F ¼ me aðK

~ c0  U 0 þ aK ~ r0  VÞ: L0 ¼ he a2 ðK

ð2:38Þ

~ r and K ~ c are called particle shape tensors be~ t; K The dimensionless tensors K 0 0 cause, in contrast to K t ; K r0 ; and K c0 , they do not depend on the particle’s size. The relations (2.37) imply two remarkable properties. The first one is that translational and rotational motions are related in the sense that rotational motion induces translational motion and vice versa. Secondly, the resistance matrices corresponding to tensors K c0 and ðK c0 ÞT , which are responsible for the connection between rotational and translational motions, consist of the same elements. The latter property results from the reciprocity theorem, which can be considered as a

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68

manifestation of the general mechanical reciprocity principle (another manifestation of this general principle is the well-known Onsager reciprocity principle). The relations (2.37) can be represented in a compact matrix form. Let us write the vectors of force, torque, translational and rotational velocities as 3  1 matrices, whose elements are the corresponding components in the chosen coordinate system with the vector basis e1, e2, e3:    F1       jjFjj ¼   F 2 ;    F3 

   L10       jjL 0 jj ¼  L20  ;    L30 

   U10       jjU 0 jj ¼  U20  ;    U30 

   W1       jjVjj ¼  W2  :    W3 

From these matrices, we can compose new matrices of generalized force and generalized velocity:      kU 0 k   kF k      kF0 k ¼  : ; ku0 k ¼   kVk   kL0 k  In a similar manner, we can construct 3  3 resistance matrices  t;r;c K  11  t;r;c   t;r;c K  ¼   K21 0   t;r;c K 31

t;r;c K12 t;r;c K22 t;r;c K32

 t;r;c  K13   t;r;c  K23    K t;r;c  33

and the grand 6  6 resistance matrix  T   t    kK kK c0    kK 0 k ¼      :  K c0 K r0   With these notations, the relation (2.37) takes the form kF0 k ¼ me kK 0 k ku0 k:

ð2:39Þ

Since the rate of energy dissipation is ku0 kT kK 0 k ku0 k, the condition that it must be positive implies that the grand resistance matrix is positive definite. Resistance tensors K c0 and K r0 depend on the choice of origin 0. A transition to another coordinate system with the origin P transforms these tensors according to the rule K cp ¼ K c0 r 0 p  K t ;

K rp ¼ K r0 r 0 p  K t  r 0 p þ K c0  r 0 p r 0 p  ðK c ÞT ;

where r0p is the vector from 0 to P.

2.1 Motion of an Isolated Particle in a Quiescent Fluid

These formulas follow from the requirement that the force of hydrodynamic resistance does not depend on the choice of origin, while the torque and the translational velocity change as L p ¼ L0 r 0 p  F;

U p ¼ U 0 þ V  r0 p:

For particles whose shape possesses central symmetry, resistance tensors have the simplest form when the origin 0 lies on the particle’s symmetry axis. It is also possible to simplify the expression for the resistance tensor for a body of an arbitrary shape. It turns out that there exists a point called the center of reaction R, relative to which the conjugate resistance tensor is symmetric, that is, K cR ¼ ðK cR ÞT . If the tensor K c0 is known relative to some point 0 inside the particle, the position of the reaction center R can be determined from the relation r 0R ¼ ½ðI : K t ÞIK t 1  « : K c0 For some axisymmetric bodies (spheres, ellipsoids), the symmetric tensor is K cR ¼ 0. In this case the relations (2.37) have an especially simple form: F ¼ me K t  U R ;

L0 ¼ me K rR  V:

ð2:40Þ

It follows from the last relation that if the reaction center is taken as the origin, then translational and rotational motions are mutually independent. For spherical particles, the reaction center coincides with the center of the sphere, and resistance tensors are equal to K t ¼ 6pme aI;

K rR ¼ 8pme a3 I;

K cR ¼ 0:

For a particle with ellipsoidal surface, x12 x22 x32 þ þ ¼1 a21 a22 a23

ð2:41Þ

the reaction center coincides with the center of the ellipsoid, and in the coordinate system whose origin is placed at the point R and whose coordinate axes are parallel to the ellipsoid’s axes, the resistance tensors are  K t ¼ 16p

K rR ¼

 1 1 1 e e þ e e þ e e ; 1 1 2 2 3 3 c þ a21 a1 c þ a22 a2 c þ a23 a3

  16p a22 þ a23 a23 þ a21 a21 þ a22 e e þ e e þ e e 1 1 2 2 3 3 ; 3 a22 a2 þ a23 a3 a22 a2 þ a21 a1 a21 a1 þ a22 a2

K cR ¼ 0;

ð2:42Þ (2.42)

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70

where ð¥ aj ¼ 0

ð¥ dX dX ; ð j ¼ 1; 2; 3Þ; c ¼ DðXÞ ða2j þ X ÞDðX Þ 0

DðX Þ ¼ ½ða21 þ X Þða22 þ X Þða23 þ X Þ 1=2 : If the force and torque acting on the particle are given, then Eq. (2.39) gives the particle’s velocity: 1 ku0 k ¼ m1 e kK 0 k kF0 k:

ð2:43Þ

The matrix kK 0 k1 ¼ kBk is called the mobility matrix and the corresponding tensor B – the mobility tensor.

2.2 Motion of an Isolated Particle in a Moving Fluid

Consider now the translational and rotational motion of a solid particle of an arbitrary shape in a viscous incompressible fluid whose velocity at the infinity is non-zero and equal to v¥. One encounters such problems when studying motion of fluids with suspended particles inside pipes. The velocity profile of the carrier fluid can be nonuniform over the pipe cross section; this is the case, for example, in Poiseulle and Couette flows. Since the particle size is small and the average distance between particles in low-concentrated mixtures is large as compared to the particle size, we assume a shear velocity profile of the fluid in the vicinity of the particle. Let us put the origin 0 of the coordinate system inside the particle. The translational velocity of the point 0 will be denoted by U0 and the rotational velocity of the particle about the axis passing through 0 will be denoted by W . The fields of velocity v and pressure p inside the fluid are described by Stokes equations (2.1): r  v ¼ 0;

r2 v ¼

1 rp me

ð2:44Þ

with inhomogeneous boundary conditions v ¼ U0 þ V  r0

on

Sa ;

v ! v¥

at

r0 ! ¥;

ð2:45Þ

where r0 is the radius vector of a point relative to the origin. The velocity field at the infinity, v¥(r), should also satisfy the appropriate hydrodynamic equations (for Poiseulle and Couette flows, these are Stokes equations

2.2 Motion of an Isolated Particle in a Moving Fluid

(2.44)). We assume that the unperturbed velocity profile at the infinity corresponds to shear flow. Let us introduce the tensor of velocity gradient at the infinity g¥ ¼ rv¥ with components gi¥j ¼ r j v¥i . It is evident that the symmetric part of this tensor E ¥ ¼ 0.5(g¥ þ (g¥)T) with components Ei¥j ¼ 0:5ðgi¥j þ g ¥ji Þ is the rate-of-strain tensor and the antisymmetric part L¥ ¼ 0.5(g¥  (g¥)T) with components L¥i j ¼ 0:5ðgi¥j g ¥ji Þ is the vorticity tensor. The latter is related to the vorticity vector V¥, which stands for the rotational velocity of the fluid in an unperturbed flow, by the expression V¥ ¼ 0.5e  L, or, in the component form, V¥i ¼ 0:5ei jk Lk j . Then the unperturbed velocity, that is, velocity at the infinity, is written as v ¥ ¼ v 0 þ V¥  r 0 þ E ¥  r 0

ð2:46Þ

or, in the component form, v¥i ¼ v0i þ ei jk W¥i xk þ Ei¥j x j ;

ð2:47Þ

where v0 is the velocity of point 0 as if it were in the fluid, r0 is the radius vector (relative to point 0) of the point under consideration, and xj are the radius vector components. It should be noted that W¥ and E¥ are constants and do not depend on the choice of origin 0. To solve the boundary value problem defined by Eqs. (2.44), (2.45), one must exploit the linearity of the problem and seek the solution as a superposition of particular solutions ðv00 ; p00 Þ; ðv000 ; p000 Þ; and ðv000 ; p000 Þ; the first two of which are solutions of the boundary value problems r  v00 ¼ 0; v00 ¼ E ¥  r 0

on

r2 v00 ¼ Sa ;

r  v000 ¼ 0; v000 ¼ ðU 0 v¥ Þ þ ðVV¥ Þ  r 0

1 r p00 ; me

v00 ! 0

r2 v000 ¼ on

at

r0 ! ¥;

1 r p000 ; me

Sa ;

v000 ! 0

at r0 ! ¥;

and the third one is equal to v000 ¼ v0 þ V¥  r 0 þ E ¥  r 0 ;

p000 ¼ 0:

in the entire region. It should be noted that the fields ðv00 ; p00 Þ and ðv000 ; p000 Þ depend on the choice of origin 0, while the field ðv000 ; p000 Þ and the net field v ¼ v00 þ v000 þ v000 ; p ¼ p00 þ p000 þ p000 do not depend on it. Having determined the velocity and pressure fields, one can find the stresses T 0, 00 T and T 000 . It is evident that T 000 ¼ 2me E ¥ in the whole region occupied by the fluid. It follows from Eq. (2.14) that the force and torque experienced by the particle can also

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72

be expressed as a sum of corresponding forces and torques: F ¼ F 00 þ F 000 þ F 000 ;

L0 ¼ L00 þ L000 þ L000 :

Their components are equal to F 00 ¼ me K t0 : E ¥ ;

L00 ¼ me K r0 : E ¥ ;

F 000 ¼ me ðK t  ðU 0 v¥ Þ þ ðK c0 ÞT  ðVV¥ ÞÞ; L000 ¼ me ðK c0  ðU 0 v¥ Þ þ K r0  ðVV¥ ÞÞ; F 000 ¼ 0;

L000 ¼ 0:

To summarize, the hydrodynamic force and torque acting on the particle are: F ¼ me ðK t  ðU 0 v¥ Þ þ ðK c0 ÞT  ðVV¥ Þ þ K t0 : E ¥ Þ; L0 ¼ me ðK c0  ðU 0 v¥ Þ þ K r0  ðVV¥ Þ þ K r0 : E ¥ Þ;

ð2:48Þ

A comparison with the relation (2.37) for the force and torque acting on a particle moving in a quiescent fluid shows that in a shear flow, additional terms appear in the relations for the corresponding force and torque. These terms are proportional to the rate-of-strain tensor of an undisturbed flow E¥, with third-rank tensors K t0 and K r0 being the proportionality factors. These tensors are called the tensors of translational and rotational shear resistance; they characterize the particle’s intrinsic properties. Currently, there exist exact solutions of the problem of motion of spherical and ellipsoidal particles in a shear flow [6]. Presented below are some results for the case when the origin 0 is made to coincide with the reaction center R. For an ellipsoidal particle (2.41), the tensors K tR and K rR are K tR ¼ 0; K rR

 8p a23 a22 ¼ ðe1 e2 e3 þ e1 e3 e2 Þ 2 3 a2 a2 þ a23 a3 þ

 a21 a23 a22 a21 ðe e e þ e e e Þ þ ðe e e þ e e e Þ : 2 3 1 2 1 3 3 1 2 3 2 1 a23 a3 þ a21 a1 a21 a1 þ a22 a2 ð2:49Þ

In the particular case of spherical particle, we have K tR ¼ K rR ¼ 0; and the other tensors are given in Section 2.1. Formulas (2.37) and (2.48) allow us to find the forces and torques acting on particles undergoing translational and rotational motion with given velocities. For a particle freely suspended in the fluid, the force and torque are equal to zero. From

2.2 Motion of an Isolated Particle in a Moving Fluid

these conditions, one can find the translational and rotational velocities of the particle: U 0 v¥ ¼ ½ðK t ðK c0 ÞT  ðK r0 Þ1  K c0 1  ½K t0 ðK c0 ÞT  ðK r0 Þ1  K r0 Þ : E ¥ ; VV¥ ¼ ½ðK t K c0  ðK t Þ1  ðK c0 ÞT Þ1  ðK r0 K c0  ðK t0 Þ1  K t0 Þ : E ¥ : The relation (2.48) can be written in a matrix form similar to Eq. (2.39). One complication is that K t0 and K r0 are third-rank tensors with elements K ti jk ; K ri jk and they cannot be represented in a matrix form as simply as other terms. Consider the components of vectors F 00 and L00 : F0i0 ¼ me Kitjk Ek¥j ;

L00i ¼ me Kirjk Ek¥j

ð2:50Þ

in more detail. For convenience’s sake, we have dropped the index 0 in the tensors K t0 and K r0 , and will follow this convention from now on. As far as (i, j, k) ¼ (1, 2, 3), each of the tensors Kt and Kr has 27 elements. Due to the symmetry of the rate-ofstrain tensor E¥, we have E ¥jk ¼ Ek¥j ; Kitjk ¼ Kikt j , and Kirjk ¼ Kikr j . Therefore each of the tensors Kt and Kr has only 18 independent components. To represent the last terms in Eq. (2.48) in the matrix form, let us proceed in the following way. First, we shall use the symmetry of tensors Kt and Kr with respect to the last two indices and the symmetry of the tensor E¥ to replace the pair of indices (jk) with one index l according to the rule below:

tensor indices matrix indices

( jk) (l)

11 1

22 2

33 3

23, 32 4

31, 13 5

12, 21 6

t t r r Furthermore, since K112 ¼ K121 ; K112 ¼ K121 ; and so on, let us denote 1 1 r r t t K112 ¼ K121 ¼ F16 ; K112 ¼ K121 ¼ t16 ; etc. 2 2 In the same way, we make a transition from the matrix E¥(Eij) to the matrix S. As a result, the tensor Kt with components Kitjk (enumerated in the table below)

t K111

i¼1 t t K112 K111 t K122

t K123 t K133

t K211

i¼2 t t K212 K213 t K222

t K223 t K233

turns into a matrix F with components Fij,

t K311

i¼3 t K312

t K313

t K322

t K323 t K333

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F11

1 F16 2

1 F15 2

F12

1 F14 2

F21

1 F26 2

1 F25 2

F22

1 F24 2

F13

F31

1 F36 2

1 F35 2

F32

1 F34 2

F23

F33

and the tensor E ¥ with components Eij, E11

E12

E13

E22

E23 E33

turns into a matrix S with components S1

S6

S5

S2

S4 S3

The transition from the tensor K r to a matrix t(tij) is handled in a similar manner. As a result, the relation (2.50) reduces to F 00i ¼ me Kitjk Ek¥j ¼ me Fil Sl ;

L00i ¼ he Kirjk Ek¥j ¼ me til Sl ;

ð2:51Þ

where i ¼ 1, 2, 3 and l ¼ 1, 2, . . ., 6. 0 Letusnow introduce  a generalized force vector F0 written in the column form, so 0 0 0 T   that F0 ¼ F 0 ; L0 ; a shear vector J, also written in the column form, so that || J||T ¼kS1, S2, S3, S4, S5, S6||; and a matrix of shear resistance ||Y||,   F11    F21     F     F31 kCk ¼   ¼  t   t11    t21    t31

F12

F13

F14

F15

F22

F23

F24

F25

F32

F33

F34

F35

t12

t13

t14

t15

t22

t23

t24

t25

t32

t33

t34

t35

 F16    F26    F36   :  t16    t26     t36

The relation (2.51) may now be written in the matrix form:  0 F  ¼ m Yk J k e 0

ð2:52Þ

2.2 Motion of an Isolated Particle in a Moving Fluid

Combining it with (2.39), we get kF0 k ¼ me ðkK 0 k ku0 k þ kCk k JkÞ:

ð2:53Þ

For a particle freely suspended in the fluid, we use (2.53) with F0 ¼ 0, obtaining both translational and rotational velocities of the particle: ku0 k ¼ kK0 k1 kCk k J k:

ð2:54Þ

Note that the matrices kK 0k and kFk depend only on the intrinsic properties of the particle, and for a given class of geometrical shapes, they can be determined once and for all. If the particle’s shape possesses some kind of symmetry, then the task of finding the elements of matrices K0 and F is considerably simplified. For example, consider an axisymmetric particle. Introduce a Cartesian coordinate system with the basis e1, e2, e3, where e3 is directed along the symmetry axis. For axisymmetric bodies, the tensors that were introduced in the previous discussion become K t ¼ aðIe3 e3 Þ þ be3 e3 ;

K c0 ¼ cð«  e3 Þ;

K t0 ¼ f e3 e3 e3 þ gððIe3 Þ þ ðIe3 ÞT Þ;

K r0 ¼ dðIe3 e3 Þ þ ee3 e3 ;

K r0 ¼ hðð«  e3 e3 Þ þ ð«  e3 e3 ÞT Þ;

where the scalar factors a, b, . . . depend only on the geometrical properties of the particle. Using the definitions of polyadics abc ¼ aibjck and Aa ¼ Aijak, of the scalar product A  a ¼ Aijaj and of the permuation symbol « ¼ e1 e2 e3 e1 e3 e2 þ e2 e3 e1 e2 e1 e3 þ e3 e1 e2 e3 e2 e1 we can rewrite these relations in the matrix form:  a    t K ¼ 0   0  d    r K0 ¼  0   0

0 a 0 0 d 0

 0    0 ;   b

  0    c K0 ¼   c    0

 0    0 ;   e

 0    c T ðK 0 Þ ¼  c   0

c 0 0

 0    0 ;   0 c 0 0

 0    0 :   0

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The grand resistance matrix ||K 0|| and the shear resistance matrix ||C|| are  a 0 0   0 a 0   0 0 b  kK 0 k ¼  d 0 0   0 d 0   0 0 e

0

c

c

0

0

0

0

c

c

0

0

0

 0   0   0  ; 0   0   0

  kqk  kCk ¼   k0k

 kCk   ; k xk 

where  0    kqk ¼  0   0

0 0 0

    0       0 ; kCk ¼  2g      0 f þ 2g 

2g

0

0 0

   2h 0       0 ; kxk ¼  0      0 0

0 2h 0

 0    0 ;   0

and ||0|| is a 3  3 zero matrix. The inverse resistance matrix (mobility matrix) is   m    0    0  1 kK 0 k ¼   0    cmd1    0

0

0

0

cna1

m

0

cna1

0

0

b1

0

0

0

n

0

0

0

0

n

0

0

0

0

cmd

1

 0    0    0   ; 0    0    1  e

where m¼

d ; adc 2

n¼

a : adc 2

Now, using Eq. (2.47), we can find the velocity of a freely suspended axisymmetric particle:    kS2      0  U 0 v ¼  kS1  ;    lS3  where k ¼

   nS4      ¥  VV ¼  nS5  ;    0 

2ðchgdÞ ; adc 2

l¼

f þ 2g ; b

ð2:55Þ

n¼

2ðgcahÞ : adc 2

2.2 Motion of an Isolated Particle in a Moving Fluid

It is readily seen that an axisymmetric particle freely suspended in a fluid rotates with the same angular velocity as the fluid in a shear flow at the infinity. Consider now an ellipsoidal particle with semi-axes a1, a2 and a3. Put the origin 0 into the center of the ellipsoid and direct the basis vectors {ei} along the axes of the ellipsoid. Using the corresponding tensors Kt, K rR ; K cR and the parameters a1, a2 and a3 (see Section 2.1), we get  0   0   0   16p  0 kCk ¼  3    0     0 

0

0

0

0

0

0

0

0

0

0

0

0

0

0

a23 a22 2 a3 a3 þ a22 a2

0

0

0

0

a21 a23 2 a1 a1 þ a23 a3

0

0

0

0

     0    0    0  :     0    2 2 a2 a1   a22 a2 þ a21 a1  0

ð2:56Þ Similar relations for a spherical particle could be obtained from Eq. (2.42) and Eq. (2.56) by taking a1 ¼ a2 ¼ a3 ¼ a. For circular disk, we would have to take a1 ¼ a2, a3 ¼ 0. As an example, consider a circular disk freely suspended in a fluid. The origin of coordinates is placed into center of the disk, the basis i1, i2, i3 is fixed in space, and the unperturbed fluid velocity is given by 1 v¥ ¼ ˙gðx1 i1 þ x2 i2 2x3 i3 Þ: 2 For such flow, we have 1 v0 ¼ 0;

W¥ ¼ 0;

1 _ E ¥ ¼  gðI3i 3 i3 Þ: 2

By determining the matrices ||K 0||1 and ||F||, one can convince himself that U0 ¼ 0 (the translational velocity vanishes) and that the angular velocity is V ¼ S4 e1 S5 e2 : The basis vector e3 of the local coordinate system is normal to the plane of the disk. If q is the angle between the vector i3 and the plane of the disk, and j is the angle between the projection of i3 on that plane and one of the local axes, then 3 S4 ¼ E23 ¼ g_ sin q cos q sin j; 2

3 _ S5 ¼ E13 ¼ gsin q cos q cos j: 2

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2.3 Motion of Two Particles in a Fluid

Consider slow motion of two particles a and b in an unbounded viscouss incompressible fluid. The particles’ translational and rotational velocities are, respectively, Ua, Ub and Va, Vb. The Reynolds numbers of the particles are determined by their parameters and are assumed to be small, so the fluid flow is Stokesian and can be described by Stokes equations r  v ¼ 0;

r2 v ¼

1 rp me

ð2:57Þ

with the boundary conditions v ¼ U a þ Va  r a v!v

¥

at

on

Sa ;

v ¼ U b þ Vb  r b

on

Sb ;

jr 0 j ! ¥:

ð2:58Þ

Here r0 is the radius vector of the considered point with respect to a local origin 0 that is fixed inside the fluid; ra and rb are the radius vectors of this point with respect to the centers 0a and 0b of particles a and b; v¥ ¼ v0 þ W¥  r0 þ E¥  r0 is the velocity of the fluid far from both the particles (at the infinity). Exact, approximate, and numerical solutions have been derived for slow translational and rotational motions of two spherical particles in a quiescent fluid and in a shear flow [7–31]. When one particle is much larger then the other, the motion of the smaller particle in close proximity to the larger one can be treated as motion near a plane. Particle motions in a quiescent fluid (v¥ ¼ 0) and in a moving flow (v¥ 6¼ 0) are considered separately. 2.3.1 Fluid is at Rest at the Infinity (v¥ ¼ 0)

The problem of obtaining the fields of velocity v and pressure p in the fluid that are induced by translational and rotational motions of two spherical particles of radii a and b reduces to the following boundary value problem: r  v ¼ 0;

r2 v ¼

v ¼ U a þ Va  r a

1 rp me on

Sa ;

v ¼ U b þ Vb  r b

on Sb ;

at jr 0 j ! ¥: ð2:59Þ

Thanks to its linearity, it can be treated as a superposition of simpler problems. If the vectors of translational velocity are represented in terms of two components – along the line of centers and perpendicular to it, the problem is separated into three

2.3 Motion of Two Particles in a Fluid

particular problems: translational motion of particles with different velocities along their line of centers (axisymmetric problem); translational motion of particles with different velocities perpendicular to the line of centers (non-axisymmetric problem); and rotation of particles with different angular velocities. Each of these motions, in its turn, can be divided into two submotions where one of the particles is at rest. It should be noted that by setting the radius of one of the particles to infinity one obtains the solution for the problem of particle motion near a plane wall, which can be considered as an asymptotic solution of the problem of relative motion of two particles with substantially different sizes. If the particles are drops of fluid whose viscosity differs from that of the surrounding fluid, then particle rotation is a meaningless term, and we should assume Wa ¼ Wb ¼ 0. Boundary conditions on the particle surface differ from (2.59), because we must satisfy the condition of equality of velocities and stresses at the interface between the internal and external fluids. As opposed to the surface of a solid particle, drop surface deform under the action of nonuniform stresses, which become especially noticeable when the minimal clearance between drops becomes much smaller than the drop size. All of this greatly complicates the problem. The discussion that follows will apply to solid particles only. Exact and approximate solutions exist for the three above-mentioned problems, making it possible to determine the forces and torques acting on moving particles for two limiting cases: when the particles are relatively far from each other (the ‘‘far asymptotic region’’), and when the minimal clearance between particle surfaces is small compared to particle radii (the ‘‘near asymptotic region’’). An approximate solution suitable for all interparticle distances can be found by using the method of asymptotic expansions. To derive the exact solution, we introduce the stream function, which allows to make a transition from equations (2.57) to a single fourth order partial differential equation for the stream function. In a curvilinear bi-spherical coordinate system, the region between the two spheres transforms into a region between two parallel plates, and the new boundary value problem is solved by using the method of separation of variables. The variables are given by infinite series whose convergence rate decreases as the gap between particles gets smaller. This is why using an asymptotic solution is the preferred way of finding the hydrodynamic forces and torques acting on a sphere in the vicinity of another sphere (or a plane wall). In the case of more then two particles, as well as for two particles of non-spherical shape, it is impossible to find the appropriate coordinate system. This is why an exact solution exists only for a system of two spherical particles. The distinctive feature of the solution for two particles is the singularity of some elements of the resistance matrix (resistance factors) when the minimal interparticle distance tends to zero. This means that most of the influence particles exert on each other is confined to the near asymptotic region. This is why hydrodynamic forces and torques (together with molecular and electrostatic interparticle forces) play a key role in the phenomena of aggregation and coagulation of particles. When the particles are relatively far from each other, that is, when the interparticle distance is equal to several particle radii, elements of the resistance tensor are nearly

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equal to the corresponding values for an isolated particle. To derive hydrodynamic forces and torques for the case when the particles are relatively far from each other, we can use the regular method of successive approximations, which allows to solve the boundary value problem (2.59) to any approximation by considering at each step only the boundary conditions on one of the spheres. This method is called the reflection method. It was first applied by Smoluchowski to the problem of hydrodynamic interaction of n particles separated by large distances. Let us look at translational motion of n identical spherical particles with different velocities Uk(k ¼ a, b, c,. . .,n). Our goal is to find the forces Fk, and torques Lk acting on the particles and ensuring their translational motion with the given velocities Uk. The boundary value problem reduces to Eqs. (2.50) with the boundary conditions v ¼ Uk

on

Sk ;

ðk ¼ a; b; c; . . .; nÞ;

v!0

at r ! ¥:

Linearity of the problem allows us to seek the solution in the form of infinite sums: v¼

¥ X vðiÞ ; i¼1

p¼

¥ X

pðiÞ ;

i¼1

where each term (v(i), p(i)) obeys Eqs. (2.57) and the zero boundary conditions at the ðiÞ ðiÞ infinity and in its turn is represented as a finite sum of fields ðvk ; pk Þ; vðiÞ ¼

n X ðiÞ vk ; k¼b

pðiÞ ¼

n X

ðiÞ

pk ;

k¼b

also obeying Eqs. (2.57) and the zero boundary conditions at the infinity. Let now choose some particle, for example particle a, as the test particle and obtain the field (v(1), p(1)) that satisfies the boundary condition v(1) ¼ Ua on Sa. The reflection of this field from particle b is derived by using the boundary condition v(2) ¼ Ub  v(1) on Sb. ð2Þ In general, the reflection of v(1) from the other n  1 particles is vk ¼ U k vð1Þ ; (1) where k ¼ b, c, . . ., n. Thus the total reflection of the velocity field v of particle Sa from the other n  1 particles is vð2Þ ¼

n X ð2Þ vk : k¼b

It is now possible to estimate (to the first approximation) the influence of (n  1) particles on the test particle via the velocity v(3) resulting from the reflection of the field v(2) from this particle. The field v(3) obeys the boundary condition v(3) ¼ v(2) on Sa. Therefore terms of the order a/l, where l is the distance between particle centers, appear in the expression for v(3). To summarize, the velocity in the vicinity of particle a is determined up to the terms of order a/l: v vð1Þ þ vð3Þ :

2.3 Motion of Two Particles in a Fluid

In the same way, one could find velocity to the first approximation for any other particle chosen as the test particle. The next approximation is derived by continuing the process of reflection, taking ð4Þ the field v(3) instead of v(1). Reflection vb from particle b, that is, from Sb, is equal to ð4Þ ð3Þ vb ¼ v , and so on. The net reflection is vð4Þ ¼

n X ð4Þ vk : k¼b

The second approximation of the velocity field v(5) in the neighborhood of particle a satisfies the boundary condition v(5) ¼ v(4) on Sa and ensures accuracy to the order of a2/l2. The next approximations give the velocity field as a power series of a/l. If the particles are far away from each other, one can get a good approximation by assuming the velocity field induced in the fluid by a particle to be the same as the velocity field induced by a point force and (or) by a point-pair force applied at the particle center. Then the resistance force exerted on the given particle by the reflected field can be obtained by considering this field as equivalent to a uniform velocity field with the same magnitude and direction as the velocity that would exist at the particle’s location if the particle itself were absent. The obtained approximations of velocity and pressure fields make it possible to find the force and torque on the particle from Eqs. (2.14) and (2.15): ð3Þ ð5Þ F a ¼ F ð1Þ a þ F a þ F a þ . . .;

ð3Þ ð5Þ La ¼ Lð1Þ a þ La þ La þ . . . :

In the case of two particles a and b separated by a large distance, forces and torques have the form of power series in l/a and l/b. Another limiting solution can be derived in the case when the gap between the surfaces is small as compared to the particles’ radii a and b. In this case there appears a small parameter equal to the ratio of the minimum clearance to the particle radius (  1 and  k ¼ b/a, b a), which allows to obtain a solution by employing the method of asymptotic expansions in a local cylindrical coordinate system (ar, q, az) with the origin at the intersection of the line of centers and the spherical surface Sb and the z-axis directed along the line of centers, while the r-axis lies in the plane tangential to Sb. In this coordinate system, the equations of spheres Sa and Sb become z ¼ 1 þ eð1r 2 Þ1=2 ;

z ¼ k1 þ k1 ð1k2 r 2 Þ1=2 :

If we now introduce new scaled coordinates Z ¼ r/e and R ¼ r/e1/2 instead of z and r, the equations of the spheres assume an asymptotic form: Z ¼ 1 þ 0:5R2 þ 0ðeÞ;

Z ¼ 0:5kR2 þ 0ðeÞ:

Stokes equations and boundary conditions at particle surfaces can be transformed in a similar way, after which we can extract the principal terms in asymptotic expansions of the velocity and pressure fields. The concrete form of these

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expansions will depend on the type of particle motion: translational motion parallel or perpendicular to the line of centers, or rotational motion around these axes. A distinctive feature of near asymptotic expansions is the singularity of almost all resistance coefficients in the grand resistance matrix, with the exception of coefficients corresponding to the rotation of the particles around their line of centers. Taking together the far asymptotic and near asymptotic solutions, we can use the method of matching of asymptotic expansions to get approximate solutions for all kinds of particle motions for the entire range of relative distances between the particles. In doing that, we make use of the fact that due to the linearity of Stokes equations, the general equation describing translational motion with velocities Ua and Ub and rotational motion with angular velocities Wa and Wb can be thought of as a superposition of simple motions in which one of the particles is assumed to be at rest. It was shown in Section 2.1 that forces Fa and torques La acting on the particles a ¼ 1, 2 (here and later, particles will be indicated by indices 1 and 2) are connected with their velocities through the relation    t  F1   K    11     F2   K t    21  ¼  L1   K c    11     L2   K c 21

t K12

c T ðK11 Þ

t K22

c T ðK21 Þ

c K12

r K11

c K22

r K21

 c T  ðK12 Þ   c T  ðK22 Þ    r K12   r K21 

   U1       U2     :  W1       W2 

ð2:60Þ

The grand resistance matrix contains tensors K tab ; K cab and K rab , whose elements (where a, b ¼ 1, 2 are used to index the particles and i, j ¼ 1, 2 indicate the tensor elements) obey the conditions of symmetry, some of which are valid for particles of an arbitrary shape and some follow from the geometrical properties of a system of two spherical particles. The former type of symmetry conditions follows from the reciprocity theorem (see Section 2.1): ðabÞ Kij

ðabÞt

Ki j

ðabÞt

¼ K ji

;

ðabÞc T

ðKi j

ðbaÞc

Þ ¼ K ji

;

ðabÞr

Ki j

ðbaÞr

¼ K ji

:

ð2:61Þ

Let us prove the first condition. Consider two problems that involve translational motion of two spherical particles S1 and S2 in one and the same fluid. In the first problem particle S1 moves with the constant velocity U1 and the second particle is at rest: U2 ¼ 0. In the second problem, the opposite is true: the first particle is at rest, U1 ¼ 0, and the second one moves with the constant velocity U2. The velocity and stress fields in the two problems are designated as (v 0, T 0 ) and (v 00, T 00 ). From the reciprocity theorem (2.2), there follows that at m0e ¼ m00e ; ð S

ð ds  v0  T 00 ¼ ds  v00  T 0 :

ð2:62Þ

S

Form the surface S from the surface of a sphere S¥ with a large radius and the surfaces of two spheres S1 and S2 placed inside the sphere S¥. Since ds  r2, while v 0  r1, v 00  r1, and T r2, we have for r ! ¥:

2.3 Motion of Two Particles in a Fluid

ð

ð ¼

S

ð þ

S¥

ð þ

S1

ð 

S2

ð þ

S1

:

ð2:63Þ

S2

The following boundary conditions are valid at the surfaces S1 and S2: v0 ¼ U 1 ;

v00 ¼ 0

on

S1 ;

v0 ¼ 0;

v00 ¼ U 2

on

S2 ;

Therefore the relations (2.62) and (2.63) first reduce to ð ð U 1  ds  T 00 ¼ U 2  ds  T 0 S

S

and then, due to Eq. (2.14), to ð2Þ

ð1Þ

U 1  F1 ¼ U 2  F2 :

ð2:64Þ

ð2Þ

where F 1 is the resistance force acting on particle S1 and induced by the motion of ð1Þ particle S2, and F 2 – the resistance force acting on particle S2 and induced by the motion of particle S1. From Eq. (2.60), it follows that in the case of translational motion (W1 ¼ W2 ¼ 0) the forces acting on the particles S1 and S2 are ð2Þ

F 1 ¼ K t12  U 2 ;

ð1Þ

F 2 ¼ K t21  U 1 :

ð2:65Þ

A substitution of Eq. (2.65) into Eq. (2.64) yields U 1  K t12  U 2 ¼ U 2  K t21  U 1 :

ð2:66Þ

Using the properties of the scalar product of a tensor and a vector, one obtains: U 2  ðK t12  U 1 Þ ¼ U 2  ðU 1  ðK t21 ÞT Þ ¼ U 1  ðK t21 ÞT  U 2 : and hence, U 1  K t12  U 2 ¼ U 1  ðK t21 ÞT  U 2 and K t21 ¼ ðK t21 ÞT :

ð2:67Þ ðabÞt

This means that the tensor K t21 is symmetric, and its elements K i j conditions ðabÞt

Kij

ðbaÞt

¼ K ji :

The other relations in (2.61) can be proved in the same way.

obey the

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Geometry of the system of two spherical particles gives rise to a dependence of tensors Kab on the relative position of the particles, that is, on the vector r ¼ x2  x1 connecting the particle centers x1 and x2, and on the particle radii a1 and a2. It is obvious that physical essence of the problem does not change if we interchange the particle indices 1 and 2 and replace r with r. So, the conditions K ab ðr; a1 ; a2 Þ ¼ K ð3a;3bÞ ðr; a2 ; a1 Þ:

ð2:68Þ

must hold. Introduce a new coordinate system with the basis vectors e ¼ r/r (parallel to the line of centers) and i, j (perpendicular to each other and to the line of centers). Then the symmetry conditions (2.68) allow us to express components of the tensor Kab in terms of two scalar functions Xab and Yab: ðabÞt

t t ¼ Xab ei e j þ Yab ðdi j ei e j Þ;

ðabÞc

¼ ðK ji

ðabÞr

r r ¼ Xab ei e j þ Yab ðdi j ei e j Þ:

Kij Kij Kij

ðabÞc T

c Þ ¼ Yab ei jk ek ;

ð2:69Þ

where d ij is the Kronecker symbol and eijk – the permutation symbol. Let us replace dimensional tensors Kab with dimensionless ones: ~t ¼ K ab

K tab ; 3pðaa þ ab Þ

~c ¼ K ab

K cab 3pðaa þ ab Þ

2

;

~r ¼ K ab

K rab 3pðaa þ ab Þ3

;

Also, we introduce the dimensionless distance s between the particles and the ratio k of particle radii: s¼

2r ; a1 þ a2

k¼

a2 : a1

The symmetry conditions (2.61) and (2.68) will then rearrange to 1 ~ t ðs; kÞ ¼ X ~t ~ t ðs; kÞ ¼ X X ab ba ð3a;3bÞ ðs; k Þ; 1 ~ t ðs; kÞ ¼ Y ~t ~ t ðs; lÞ ¼ Y Y ab ba ð3a;3bÞ ðs; k Þ; 1 ~t ~ c ðs; kÞ ¼ Y Y ab ð3a;3bÞ ðs; k Þ; 1 ~ r ðs; kÞ ¼ X ~r ~ r ðs; kÞ ¼ X X ab ba ð3a;3bÞ ðs; k Þ; 1 ~ r ðs; kÞ ¼ Y ~r ~ r ðs; kÞ ¼ Y Y ab ba ð3a;3bÞ ðs; k Þ:

ð2:70Þ

The conditions (2.70) mean that the forces and torques acting on the particles are ~ cab and 3 functions for each of the specified by 16 scalar functions (4 functions for Y

2.3 Motion of Two Particles in a Fluid

~ tab ; Y ~ tab ; X ~ rab ; Y ~ rab ) that depend on s 2 [2,¥) and k 2 [0,1]. Each set of funcfollowing: X tions defined by superscripts t, c and r is responsible for a specific type of particle ~ tab are the resistance coefficients for translational motion of particles motion. Thus, X ~ t – resistance coefficients for translational motion along their line of centers; Y ab ~ c – resistance coefficients for rotational perpendicular to the line of centers; Y ab r ~ – for rotational motion around the line of centers motion around the axis j; X ab ~ r – for rotational motion around the axis i. (axis e); Y ab

Thus, scalar functions provide a suitable description of all kinds of relative motions of two solid spherical particles. Therefore, Eq. (2.60) allows to determine the forces and torques that act on particles moving in an arbitrary way as functions of translational and rotational velocities Ua and Wa, relative distance s between the particles, and the ratio k of particle radii. Both exact and asymptotic solutions of all the above-formulated problems about motion of two spherical particles in an unbounded fluid that is quiescent at the infinity are currently well-known. Of utmost interest are the approximate solutions obtained by splicing the far asymptotic solution and the near asymptotic solution because their relatively simple form comes in handy when we try to solve more complicated problems that involve, for example, the hydrodynamic behavior of a particle ensemble. Listed below are the expressions for dimensionless resistance functions that appear in Eq. (2.60), derived for various types of particle motions. Translational motion along the line of centers (axis e): Far asymptotic region (s ! ¥): ¥ X 1 1 ~ t11 ¼ X f2k ðkÞ ; 2k s2k ð1 þ kÞ k¼0 ~ t12 ¼  X

¥ 1 X 1 1 f2kþ1 ðkÞ ; ð1 þ kÞ k¼0 ð1 þ kÞ2kþ1 s2kþ1

where f0 ¼ 1; f1 ¼ 3k; f2 ¼ 9k; f3 ¼  4k þ 27k2  4k3; f4 ¼  24k þ 81k2 þ 36k3; f5 ¼ 72k2 þ 243k3 þ 72k4; f6 ¼ 16k þ 108k2 þ 281k3 þ 648k4 þ 144k5; f7 ¼ 288k2 þ 1620k3 þ 1515k4 þ 1629k5 þ 288k6; etc. Owing to the first condition (2.70), the following equality is valid: f2kþ1 ðkÞ ¼ k2kþ2 f2kþ1 ðk1 Þ: Near asymptotic region (s ! 2 or x ! 0): It is convenient to introduce the dimensionless distance (clearance) between the particles x¼

ra1 a2 ¼ s2: 0:5ða1 þ a2 Þ

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At x  1 and x  k we have:   ~ t11 ¼ g1 ðkÞ 1 þg2 ðkÞln 1 þ 0ð1Þ; X x x ~ t11 ¼  X

   2 1 1 g1 ðkÞ þg2 ðkÞln þ 0ð1Þ; 1þk x x

where 2k2

g1 ðkÞ ¼

3

ð1 þ kÞ

;

g2 ðkÞ ¼

kð1 þ 7k þ k2 Þ 5ð1 þ kÞ3

:

Intermediate region (2 < s < ¥): ~ t11 ¼  X

g1 g2 ln ð14s2 Þg3 ð14s2 Þ ln ð14s2 Þ ð14s2 Þ

þ f0 ðkÞg1 þ

¥  X m¼2 ðmevenÞ

1 2g2 4g3 fm ðkÞg1  þ m m m1 2m ð1 þ kÞm

 m 2 ; s

    1 2g1 sþ2 sþ2 t 2 ~  ð1 þ kÞX 12 ¼ g2 ln þ g3 ð14s Þ ln sð14s2 Þ 2 s2 s2

þ4

  m ¥  X g3 1 2g2 4g3 2 þ þ f ðkÞg  ; m 1 m m s m m m1 2 ð1 þ kÞ s m¼2 ðmevenÞ

where m1 ¼ 2dm2 þ ðm2Þð1dm2 Þ; g3 ¼

1 ð1 þ 18k29k2 þ 18k3 k4 Þð1 þ kÞ3 : 42

Translational motion perpendicular to the line of centers (axis i): Far asymptotic region (s ! ¥): ~t ¼ Y 11

¥ X k¼0

~t ¼  Y 12

f2k ðkÞ

1

1 2k s2k

ð1 þ kÞ

;

¥ 2 X 1 1 f2kþ1 ðkÞ ; ð1 þ kÞ k¼0 ð1 þ kÞ2kþ1 s2kþ1

2.3 Motion of Two Particles in a Fluid

where f0 ¼ 1;

3 f1 ¼ k; 2

f4 ¼ 6k þ

81 2 k þ 18k3 ; 16

f6 ¼ 4k þ 54k2 þ

f7 ¼ 144k2 þ

9 f2 ¼ k; 4 f5 ¼

f3 ¼ 2k þ

27 2 k þ 2k3 ; 8

63 2 243 3 63 4 k þ k þ k ; 2 32 2

1241 3 k þ 81k4 þ 72k5 ; 64

1053 3 19083 4 1053 5 k þ k þ k þ 144k6 ; 8 128 8

etc:

Near asymptotic region (x  1 and x  k):   ~ t ¼ g2 ðkÞ ln 1 þ 0ð1Þ; Y 11 x    ~ t ¼  2 g2 ðkÞ ln 1 þ 0ð1Þ; Y 12 1þk x where g2 ðkÞ ¼

4kð2 þ k þ 2k2 Þ 15ð1 þ kÞ3

:

Intermediate region (2 < s < ¥): ~ t11 ¼ g2 ln ð14s2 Þg3 ð14s2 Þ ln ð14s2 Þ þ f0 ðkÞ Y

þ

¥  X

m¼2 ðmevenÞ

1 2g2 4g3 fm ðkÞ þ m m m1 2m ð1 þ kÞm

 m 2 ; s

    1 sþ2 sþ2 t 2 ~  ð1 þ kÞY 12 ¼ g2 ln þ g3 ð14s Þ ln 2 s2 s2 þ4

 m ¥  g3 X 1 2g2 4g3 2 þ þ f ðkÞ ; m m s m¼2 2m ð1 þ kÞ m m m1 s ðmoddÞ

where g3 ðkÞ ¼

2 ð1645k þ 58k2 457k3 þ 16k4 Þð1 þ kÞ3 : 375

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Rotation around the axis perpendicular to the line of centers and to the direction of transverse translational motion (axis j): Far asymptotic region (s ! ¥): ~c ¼ Y 11

¥ X

f2kþ1 ðkÞ

k¼0

~c ¼  Y 12

1

¥ X

4 2

ð1 þ kÞ

1

ð1 þ kÞ2kþ1 s2kþ1

k¼0

f2k ðkÞ

1

; 1

ð1 þ kÞ2k s2k

;

where f0 ¼ f1 ¼ 0; f2 ¼  6k; f3 ¼  9k; f4 ¼ 

27 2 k ; 2

f5 ¼ 12k

81 2 k 36k3 ; 4

243 3 k 72k4 ; 8 8409 3 k 243k4 144k5 ; f7 ¼ 189k2 þ 16 f6 ¼ 108k2 

etc:

Near asymptotic region (x  1 and x  k):   ~ c ¼ g2 ðkÞ ln 1 þ 0ð1Þ; Y 11 x    2 1 ~ c12 ¼  g ðkÞ ln Y þ 0ð1Þ; 2 x ð1 þ kÞ2 where g2 ðkÞ ¼ 

1 kð4 þ kÞ : 5 ð1 þ kÞ2

Intermediate region (2 < s < ¥):     ~ c11 ¼ g2 ln s þ 2 þ g3 ð14s2 Þ ln s þ 2 þ 4 g3 Y s s2 s2    ¥ X 1 2g2 4g3 2 m þ þ ; m fm ðkÞ m m m m1 s 2 ð1 þ kÞ m¼2 ðmoddÞ

1 ~ c ¼ g2 ln ð14s2 Þg3 ð14s2 Þ ln ð14s2 Þ  ð1 þ kÞ2 Y 12 4  m ¥  X 1 2g2 4g3 2 þ f ðkÞg  ; þ m 1 m m m1 s 2m ð1 þ kÞm m¼2 ðmevenÞ

2.3 Motion of Two Particles in a Fluid

where g2 ðkÞ ¼ 

1 ð3233k þ 83k2 þ 43k3 Þ : 250 ð1 þ kÞ2

Rotation around the line of centers (axis e): Far asymptotic region (s ! ¥): ~ r11 ¼ X

¥ X

f2k ðkÞ

k¼0

1 ð1 þ kÞ ¥ X

8

~ r12 ¼  X

1

ð1 þ kÞ3

2k s2k

;

f2kþ1 ðkÞ

k¼0

1

1

ð1 þ kÞ2kþ1 s2kþ1

;

where f0 ¼ 1; f1 ¼ f2 ¼ 0; f3 ¼ 8k3; f 4 ¼ f5 ¼ 0; f6 ¼ 64k3; f7 ¼ 0; f 8 ¼ 768k5; f9 ¼ 512k6; etc. Near asymptotic region (x  1 and x  k):     k k2 1 z 3; x ln  þ 0ðxÞ; 4ð1 þ kÞ 1þk x ð1 þ kÞ3     8k3 k 2k2 1 ¼ z 3; x ln þ þ 0ðxÞ; 4 6 1 þ k x ð1 þ kÞ ð1 þ kÞ k3

~r ¼ X 11 ~ r12 X

where zðz; aÞ ¼

¥ X ðk þ aÞz : k¼0

Intermediate region (2 < s < ¥): ~ r11 ¼ X

  k2 k2 1 sþ2 lnð14s2 Þ þ ln þ1 2ð1 þ kÞ 1þks s2

( ¥ X þ k¼1

1 ð1 þ kÞ2k

f2k ðkÞ2

2kþ1

1 k2 kð2k1Þ 4ð1 þ kÞ

)  1 2k ; s

  sþ2 8k2 ln lnð14s2 Þ þ 4 s2 ð1 þ kÞ ð1 þ kÞ4 ( )  ¥ X 8 1 1 k2 1 2kþ1 2kþ2  f ðkÞ2 : 2kþ1 kð2k þ 1Þ 1 þ k s ð1 þ kÞ3 k¼1 ð1 þ kÞ2kþ1

~ 12 ¼ þX r

4k2

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Rotation around the axis perpendicular to the line of centers and parallel to the direction of transverse translational motion (axis i): Far asymptotic region (s ! ¥): ~r ¼ Y 11

¥ X

f2k ðkÞ

k¼0

~r ¼ X 12

1

¥ X

8 3

ð1 þ kÞ

1

ð1 þ kÞ2k s2k f2kþ1 ðkÞ

k¼0

; 1

1

ð1 þ kÞ2kþ1 s2kþ1

;

where f0 ¼ 1;

f1 ¼ f2 ¼ 0;

f6 ¼ 27k2 þ 256k3 ;

f3 ¼ 4k3;

f4 ¼ 12k; f5 ¼ 18k4 81 f7 ¼ 72k4 þ k5 þ 72k6 ; etc: 2

Near asymptotic region (x  1 and x  k):   ~ r11 ¼ g2 ðkÞ ln 1 þ 0ð1Þ; Y x

  ~ r12 ¼ g4 ðkÞ ln 1 þ 0ð1Þ; Y x

where g2 ðkÞ ¼

2k ; 5ð1 þ kÞ

g4 ðlÞ ¼

4k2 5ð1 þ kÞ4

:

Intermediate region (2 < s < ¥): ~ r11 ¼ g2 ln ð14s2 Þg3 ð14s2 Þ ln ð14s2 Þ þ f0 ðkÞ Y þ

¥  X

1 2g2 4g3 þ m fm ðkÞ m m m m1 2 ð1 þ kÞ

m¼2 ðmevenÞ

 m 2 ; s

    1 ~ r12 ¼ g4 ln s þ 2 þ g5 ð14s2 Þln s þ 2 ð1 þ kÞY 8 s2 s2 þ4

  m ¥  g5 X 1 2g4 4g5 2 f ðkÞ ; þ þ m s m m m1 2m ð1 þ kÞm s m¼2 ðmoddÞ

where

g3 ðkÞ ¼

ð8 þ 6k þ 33k2 Þ ; 125ð1 þ kÞ

g5 ðkÞ ¼

4kð4324k þ 43k2 Þ 125ð1 þ kÞ4

:

2.3 Motion of Two Particles in a Fluid

When the forces and torques acting on the particles are known, one can find particle velocities via the mobility matrix:     U 1   kt    11     U 2   kt    21  ¼  V1   kc    11     V2   kc 21

kt12

ðkc11 ÞT

kt22

ðkc21 ÞT

kc12

kr11

kc22

kr21

    ðkc12 ÞT    F1    c T  F  ðK22 Þ   2   :   kr11    L1    kr21   L2 

ð2:71Þ

The mobility matrix is the inverse resistance matrix, so, according of the reciprocity theorem, its elements ktab ; kcab ; and krab must obey conditions similar to Eq. (2.61), namely, ðabÞt

ðabÞt

¼ k ji ;

ki j

ðabÞc T

ðki j

ðbaÞc

Þ ¼ k ji

;

ðabÞr

ki j

ðbaÞr

¼ k ji

:

ð2:72Þ

They can also be represented in the form similar to (2.69): ðabÞt

t t ¼ xab ei e j þ yab ðd i j ei e j Þ;

ðabÞc

¼ ðk ji

ðabÞr

r r ¼ xab ei e j þ yab ðdi j ei e j Þ:

ki j ki j ki j

ðabÞc T

ð2:73Þ

c Þ ¼ yab ei jk ek ;

If we now introduce the dimensionless mobility tensors ~kt ¼ 3pðaa þ ab Þkt ; ab ab

~kc ¼ pðaa þ ab Þ2 kc ; ab ab

~kr ¼ pðaa þ ab Þ3 kr ab ab

and parameters s and k in the same way as before, the resulting 16 dimensionless scalar functions x~tab ; ~ytab ; ~ycab ; x~rab ;, and ~yrab depending on s 2 [2, ¥) and k 2 [0,1] will characterize all kinds of translational and rotational motions of two solid spherical particles in a fluid. Since the grand resistance and mobility matrices are mutually inverse, their components’ representations (2.69) and (2.73) in combination with the fact that axisymmetric translational motion (i.e., along the line of centers) is not mutually correlated with either non-axisymmetric motion (i.e., perpendicular to the line of centers) or axisymmetric rotation (i.e., around the line of centers), lead to the following dependence between the elements of both matrices:        

t x11

2 xt 1 þ k 12

  2  t  t   X11 x12   1þk ¼  1 þ k 1 t   t x22   X12 k 2

 1 þ k t 1 X12   2  ;   t kX22 

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r x11

2 xr 1 þ k 12

   2 r  r x12   X11   1þk ¼  1 r    ð1 þ kÞ3 r x22   X12  k 8

  t  y12     2 t   1 þ k y12    3 c  y   2 11   6   yc  ð1 þ kÞ2 21

2 t y 1 þ k 12 1 t y k 22 6 ð1 þ kÞ2

6 ð1 þ kÞ2

6 ð1 þ kÞ2

  c  y 2 21  ð1 þ kÞ    3 c  y  22 2 2k  ¼  6 r  3 y12   ð1 þ kÞ   3 r  y22  3  4k 6

c y12

3 r y 4 11

c y12

3 c y 2k2 22

  t  Y12      2ð1 þ kÞY t  12  ¼  3 c  Y  2 11    ð1 þ kÞ2  c Y21  6

3 c y 2 11

1 ð1 þ kÞ3 r  X12   8   ;   r kX22 

r y12

1þk t Y12 2

2 c Y 3 11

t kY22

ð1 þ kÞ3 c Y12 6

ð1 þ kÞ2 c Y12 6

4 r Y 3 11

2 2 c k Y22 3

ð1 þ kÞ3 r Y12 6

1 ð1 þ kÞ2 c  Y21   6    2 2 c k Y22   3   : 3 ð1 þ kÞ r  Y12   6    4 3 r  k Y22  3

2.3.2 Fluid is Moving at the Infinity (v¥ 6¼ 0)

Let us now consider the motion of two spherical particles in an unbounded fluid whose velocity at the infinity is the one that is typical for a shear flow [32–38]: v ¥ ¼ v 0 þ V¥  r 0 þ E ¥  r 0 :

ð2:74Þ

The velocity and pressure fields are described by the boundary value problem (2.57), (2.58): r  v ¼ 0;

r2 v ¼

1 r p; me

v ¼ U 1 þ V1  r 1 on S1 ; v ! v¥ at jr 0 j!¥: where v¥ is given by Eq. (2.74).

ð2:75Þ v ¼ U 2 þ V2  r 2

on

S2 ;

ð2:76Þ

2.3 Motion of Two Particles in a Fluid

The boundary value problem is handled in the same manner as in the case of single-particle motion (see Section 2.2). Think of the fluid velocity as a sum v ¼ v¥ þ v, where v¥, v satisfy Eqs. (2.75) and v also obeys the boundary conditions v ¼ ðU 1  v¥1 Þ þ ðV1  V¥ Þ  r 1  E ¥  r1 

v ¼

ðU 2  v¥2 Þ

v ! 0

at

¥

þ ðV2  V Þ  r 2  E  r 2 ¥

on

S1 ;

on S2 ;

jr 0 j ! ¥:

Here v¥1 and v¥2 are the unperturbed velocities at the particle surfaces. The fluid velocity disturbance is, in its turn, represented as a sum, v ¼ vt1 þ vr1 þ vs1 þ vt2 þ vr2 þ vs2 ; whose components with subscript 1 vanish on the sphere S1 and components with subscript 2 vanish on S2. The superscripts correspond to the categorization of the flows that has been done in Section 2, so t signifies translational, r – rotational,and s – shear flows and these flows obey the boundary conditions at particle surfaces Si (i ¼ 1, 2): vti ¼ U i v¥i ;

vri ¼ ðVi V¥ Þ  ri ;

vsi ¼ E ¥  r i

on

Si :

Because of the linearity of the boundary value problem, the force and the torque acting on particles can be represented as sums Fi ¼

XX j

F kij þ K ti : E ¥ ;

Li ¼

XX j

k

Lkij þ K ri : E ¥ ;

ð2:77Þ

k

where i, j ¼ 1, 2; k ¼ t, r, and the summands are given by F ti j ¼ me ðK ti j  ðU j v¥j ÞÞ;

F ri j ¼ me ððK cji ÞT  ðV j V¥ ÞÞ;

Lti j ¼ me ðK ci j  ðU j v¥j ÞÞ;

Lri j ¼ me ðK ri j  ðV j V¥ ÞÞ:

For particles separated by large distances, the tensors K ti j ; K ri j ; K ci j and ðK cji ÞT are determined by the following relations:

K t11

 9 a1 a2  1 þ  64 h2   ¼ 6pa1  0      0

0 1þ

9 a1 a2 64 h2 0

      ; 0    9 a1 a2   1þ 64 h2 0

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K r11

K t21

K c11

 3   1 þ 3 a2 a1  64 h4    ¼ 8pa31   0      0 a  1  h   9  ¼  pa2  0  4    0

0 a1 h 0

  0  4 9 a1 a2   ¼  p 3  1 16 h    0

0

1þ

3 a2 a31 64 h4 0

  0     0 ;   a1    h

K r21

 0 0    0 0 ;   0 0

  0     ; 0     1

 1  3 3 1 a2 a1  ¼ p 3 0 2 h   0

K c21

0 1 0

 0     0 ;   2 

  0 1  3 3 a1 a2  ¼  p 2  1 0 2 h    0 0

 0    0 ;   0

where 2h is the distance between the particle centers and al, a2 are the radii of the particles. The other matrices are obtained from the ones above by permutation of indices 1 and 2 and (or) transposition of a matrix. Finally, one can obtain the grand resistance matrix       t   c T   K i j  ðK ji Þ      kK k ¼    :    r   c  K i j    K i j 

Let us introduce the shear resistance matrix, just like we did in Section 2.2:        kCk ¼      

kq1 k kq2 k k0k k0k

 kC1 k     kC2 k   : kx1 k     kx2 k 

For the particles spaced far away from each other, the elements of this matrix are equal to

2.4 Multi-Particle Motion

   0 0 0   0          kqi k ¼  0 1 0 ; kCi k ¼  2Gi        0 0 F0   0 i    2Hi 0 0       2Hi 0 ; kxi k ¼  0      0 0 0

2Gi 1 0

 0    0 ;   0 F i

where   15 a2 2 45 a1 3 3 a2 4 9 a1 5 þ þ þ ; 4 h 16 h 8 h 64 h   3 a2 4 9 a1 5 þ G1 ¼ pa1 a2 ; 16 h 128 h  5 2 a2 4 9 2 a1 5 þ H1 ¼ pa2 a a : 16 2 h 128 1 h F10 ¼ pa1 a2

The expressions for F20 and G2 are derived from F10 and Gi by interchanging the indices 1 and 2 and changing the sign, and the expression for H2 – by simply interchanging the indices 1 and 2. Introducing the generalized force vector F, the particle velocity u, and the shear rate J, as we did in Section 2.1, we can rewrite Eq. (2.77) in the matrix form: kFk ¼ me ðkK kkuk þ kCkkJ kÞ:

ð2:78Þ

2.4 Multi-Particle Motion

Let us now generalize the problems examined in Sections 2.1–2.3, for the case of motion of more than two particles in an unbounded fluid [39]. Let there be n solid particles with arbitrary surface shapes and spaced far apart. We will be indicating i-th particle (i ¼ 1, 2, K, n) with the symbol Si, where Si will also imply the particle’s surface. The symbol 0i will stand for the center (local origin) of i-th particle; Ui will denote the translational velocity of the point 0i and Wi – the particle’s angular velocity. The velocity of the fluid at the infinity is v ¥ ¼ v 0 þ V¥  r 0 þ E ¥  r 0 ;

ð2:79Þ

where 0 is an origin of coordinates placed into an arbitrary point inside the fluid, r0 is the radius vector from the origin 0 to the given point, and v0 is the velocity of an undisturbed flow at the point 0.

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The velocity and pressure fields are represented as v ¼ v¥ þ

n X

vi ;

p¼

i¼1

n X

pi ;

ð2:80Þ

i¼1

where v¥, vi, pi obey Stokes equations and vi also satisfies the boundary conditions vi ¼ U i þ Vi  r i v0 V¥  r 0 E ¥  r 0 vi ¼ 0

at

S j ð j „ iÞ;

vi ! 0

at

at Si ;

ð2:81Þ

jr i j ! ¥:

ð2:82Þ

Here ri is the radius vector connecting 0i with the given point. If v¥i is the value of the undisturbed velocity at the point 0i (i.e., the value the velocity would have if the particle were absent), then the condition (2.81) can be rewritten as vi ¼ ðU i v¥i Þ þ ðVi V¥ Þ  r i E ¥  r i

at Si :

ð2:83Þ

In the next step we decompose (vi, pi) into summands corresponding to shear 000 ðv0i ; p0i Þ; translational, ðv00i ; p00i Þ and rotational ðv000 i ; pi Þ motions: vi ¼ v0i þ v00i þ v000 i ;

pi ¼ p0i þ p00 þ p000 i

where each component must obey Stokes equation and satisfy the boundary conditions vi0 ¼ E ¥  r i ;

v00i ¼ U i v¥i ;

¥ v000 i ¼ ðVi V Þ  r i

on

Si :

ð2:84Þ

If the solutions of the corresponding boundary value problems are known, one can find the hydrodynamic force and the torque relative to the point 0 acting on i-th particle: ( ) n X ¥ ¥ t ¥ c T F i ¼ me ½K i j  ðU i vi Þ þ ðK ji Þ  ðVi V Þ þ Fi : E ; j¼1

( ) n X ¥ ¥ c ¥ r Li ¼ me ½K i j  ðU i vi Þ þ K i j  ðVi V Þ þ t i : E :

ð2:85Þ

j¼1

The tensors K ti j ; K ri j ; K ci j , and ðK cji ÞT are determined from the expression for the net stress at the particle surface Si induced by translational motion T 00i , rotational 0 motion T 000 i , and the shear flow T i:

2.4 Multi-Particle Motion

ð

ð

K ti j ¼  P00j  ds;

K ri j ¼  r i  ðP000j  dsÞ;

si

j¼1

K ci j ¼  r i  ðP00j  dsÞ

si

n ð X ds  P0j ; Fi ¼ 

ð si

n ð X ti ¼  r i  ðds  P0j Þ: j¼1

si

si

Third-rank tensors P00j ; P000j (triadics) and the fourth-rank tensor (tetradic) P0j 000 0 0 characterize the properties of fields ðv00i ; p00i Þ; ðv000 i ; pi Þ, and ðvi ; pi Þ. The first two were determined in Section 2.1, and the third one is defined by the expression T 0i ¼ me P0j : E ¥ . The dyadics K ti j ; K ri j ; K ci j and the triadics Fi, ti characterize geometrical properties of the momentary configuration of a particle system: sizes, shapes, and positions of the centers 0i. To write the relations (2.85) in a matrix form, let us introduce 3n  1 matrices (column vectors) ||F||, ||L||, kU  v¥k, and kW  W¥k, and such that kF k ¼ kkF 1 kkF 2 k. . .kF 2 kkT ; which includes in consecutive order the parameters of n particles, and 3n  3n matrices ||Kt||, ||Kr||, ||Kc||:   kK 11 k    kK k ¼  . . .    kK n1 k

kK 12 k . . . ... kK n2 k   

 kK 1n k     . . . :   kK nn k 

   t Each  of the elementary matrices ||Kij|| is a 3  3 matrix, and the matrices K i j   r and K i j  are symmetric. The matrices ||Fi|| and ||ti|| are 3  6 matrices, from which we can form a 6n  6 combined matrix (see Section 2.2):    kF1 k       .   .   .       kF k   n  : kCk ¼     kt1 k       .   .   .       ktn k  The above-mentioned matrices allow us to construct the 6n  6n grand resistance matrix:

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  t  kK k kK k ¼   c  kK k

   c T  ðK Þ   ;  r kK k 

as well as generalized matrices of forces (6n  1), velocities (6n  1), and shear (6  1):    kF k    kFk ¼  ;  kLk 

   kðUv¥ Þk    kuk ¼  ;  kðVV¥ Þk 

   S1     .   k Jk ¼   .. :     S6

Now the equations (2.85) can be written in a compact matrix form: kFk ¼ me ðkK k kuk þ kCk k J kÞ:

ð2:86Þ

If the particles are freely suspended in the fluid, then setting ||F|| ¼ ||0|| in Eq. (2.86), we obtain the velocities of particles: kuk ¼ kK k kCk kJ k:

ð2:87Þ

The grand resistance matrix ||K || is symmetric and non-singular, so there exists an inverse matrix and consequently, it is possible to obtain the velocities of suspended particles. The elements of matrices ||K || and ||C|| do not depend on dynamic parameters (velocities); they depend only on the geometric properties of the particles and on the configuration of a particle system. At the present time, they can be determined only by solving problems on hydrodynamic interaction of two spherical particles as a function of the relative distance between particles and the ratio of their radii. Some examples of pair interactions were given in Section 2.3.

2.5 Flow of a Fluid Through a Random Bed of Particles

Consider a slow Stokesian flow of viscous incompressible fluid through a layer of buoyant (cloud) or fixed in space (porous medium) rigid spherical particles with a given distribution of particles over radii [40–43]. Our goal is to determine the force of particle resistance and the effective viscosity of the disperse medium taking into account hydrodynamic interaction between particles, that is, the hindered motion of particles [44]. The velocity and pressure in the fluid are described by Stokes equations r  u ¼ 0;

ð2:88Þ

r  T ¼ me r2 urð prg  rÞ ¼ 0;

ð2:89Þ

2.5 Flow of a Fluid Through a Random Bed of Particles

where T is the stress tensor with components  ti j ¼ ð prg  rÞd i j þ me

qui qu j þ qX j qXi

 ¼ 0:

ð2:90Þ

There are also the boundary conditions u ¼ us that must hold on particle surfaces; us is the particles’ velocity. The fluid’s velocity at the infinity is given. If the particles are at rest or their positions in space are fixed, then us ¼ 0. For buoyant particles, their velocity us is a sum of translational and rotational velocities and is found from the relations for the force and torque exerted on the particle by the surrounding fluid (see Section 2.1). Hydrodynamic interactions between particles can be taken into account by averaging over the particle ensemble. Following [42,43], we assume that u, p and r are defined in the whole region under consideration, including the particles. We further assume that that u is a continuous function, whereas p, r, and derivatives of u are discontinuous at the particle surface S. Introducing the Saffman function ( HðX Þ ¼

1;

if X is in fluid;

0;

if X is in particle:

ð2:91Þ

and averaging it over the particle ensemble, we get hHðXÞi ¼ 1j ¼ h;

ð2:92Þ

where j is the volume concentration (volume fraction) of particles and h is the porosity (volume fraction occupied by fluid) of the disperse medium. From now on, averaging over the particle ensemble will be designated by angle brackets, and the mean-flow-rate values of hydrodynamic parameters – by a horizontal bar on the top. The Saffman function gives the relation between these averages; in particular, the mean-flow-rate velocity and pressure in the fluid are equal to uðX Þ ¼

hHðX ÞuðX Þi huðX Þi ¼ ; h hHðX Þi

pðX Þ ¼

hHðX Þ pðX Þi h pðX Þi ¼ : h hHðX Þi

ð2:93Þ

Since velocity is a continuous function in the entire region, we can introduce the  in addition to the ensemble average hui. For a average cross-sectional velocity u porous medium, this velocity is called the speed of filtration. For j  1 (a low hui. concentrated suspension or a highly permeable medium), u Before averaging Stokes equations, we must first average the stress tensor T. We have:

   qui qu j þ : Hti j ¼ hHð prg  rÞi > d i j þ me qX j qXi

ð2:94Þ

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Using the obvious equalities

H

qui qX j

 ¼

qui ð1HÞ qX j

 q qH ; hHui i ui qX j qX j 

 q qH ¼ hð1HÞui i þ ui qX j qX j

ð2:95Þ

ð2:96Þ

and the property qH=qX j ¼ dðX X s Þ of the Saffman function (where Xs is a point on the particle’s surface), and adding the relations (2.95) and (2.96), we can write

H

qui qX j



 qui q ui : þ ð1HÞ ¼ qX j qX j

Interchanging the indices,  

  q uj qu j qu j H ; þ ð1HÞ ¼ qXi qXi qXi and adding both relations, we obtain the following:  

    qui qu j qui qu j qhui i q u j H þ þ þ : þ ð1HÞ ¼ qXi qX j qXi qX j qXi qX j

ð2:97Þ

Inside rigid particles, the velocity remains constant, and in the fluid, H ¼ 1. Therefore the second term on the left-hand side vanishes and Eq. 2.97 becomes  

  qui qu j qhui i q u j H þ þ : ¼ qXi qX j qXi qX j

ð2:98Þ

At i ¼ j this equality reduces to hHr  ui ¼ r  hui, and we get the averaged continuity equation, r  hui ¼ 0:

ð2:99Þ

Hence, Eq. (2.94) reduces to      qhui i q u j Hti j ¼ ð1jÞðprg  rÞdi j þ me þ : qXi qX j

ð2:100Þ

2.5 Flow of a Fluid Through a Random Bed of Particles

The ensemble-averaged momentum equation (2.89) has the form hHr  T i ¼ 0:

ð2:101Þ

Using the well-known tensor equality r  ðHTÞ ¼ Hr  T þ T  rH; we rewrite Eq. (2.101) as r  hðHTÞihT  rHi ¼ 0:

ð2:102Þ

Let us substitute Eq. (2.100) into Eq. (2.102) and then use the averaged continuity equation (2.99) to get me r2 huihT  rHirð1jÞðprg  rÞ ¼ 0:

ð2:103Þ

According to Eq. (2.91), the function H undergoes a jump at the particle surface, so rH is a delta function. In order to understand the meaning of the second term in Eq. (2.103), let us take an integral of this term over a finite volume V, transform it into a surface integral according to Gauss’s theorem, and use the property of the delta function. After some manipulations, we get

ð  ð ð ð  hT  rHidr ¼  r  hHT idr ¼  hHT i  n dS ¼  T  n dS ;

ð2:104Þ

where n is the unit normal vector. The quantity hHTi nds is equal to the force exerted by the fluid on a particle surface element, and Eq. (2.104) is the average force exerted on the fluid by the particles in the given volume. Hence, the second term in Eq. (2.103) has the meaning of volume force exerted on the fluid by the particles. Now, consider a disperse phase consisting of N identical spherical particles. The positions of their centers will be defined by radius vectors r1, r2, . . ., rN. Let pN (r1, r2, . . ., rN) be the N-particle PDF. If we assume the velocity u and function H to depend only on r1, r2, . . ., rN, then the ensemble average of a function G (r1, r2, . . ., rN), where the averaging is carried out over the ensemble of particle configurations, is given by the formula ð hGi ¼ Gðr 1 ; r 2 ; . . .; r N Þ pN ðr 1 ; r 2 ; . . .; r N Þdr 1 dr 2 . . .dr N :

ð2:105Þ

Of utmost interest are the one-particle PDF p1 (r1) and the two-particle PDF p2 (r1, r2). These are marginal PDFs that can be obtained from a given N-particle PDF pN (r1, r2, . . ., rN) using the formulas

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ð

p1 ðr 1 Þ ¼

pN ðr 1 ; r 2 ; . . .; r N Þdr 2 . . .dr N ;

ð2:106Þ

ð p2 ðr 1 ; r 2 Þ ¼

pN ðr 1 ; r 2 ; . . .; r N Þdr 3 . . .dr N :

ð2:107Þ

In the case of random independent single-particle distributions, the N-particle distribution is the product of single-particle distributions: pN ðr 1 ; r 2 ; . . .; r N Þ ¼ p1 ðr 1 Þ p1 ðr 2 Þ. . . p1 ðr N Þ: For identical particles, one can introduced the distribution of the number of particles, nðr 1 Þ ¼ N p1 ðrÞ;

ð2:108Þ

such that nd r is equal to the probable (i.e., ensemble-averaged) number of particles whose centers are located in the volume element d r. The Saffman function H for this case is N X Hðr; r 1 ; r 2 ; . . .; r N Þ ¼ 1 Hða rr j Þ;

ð2:109Þ

j¼1

where H (X) is the Heaviside function. Then ð hHi ¼ 1w ¼ H pN ðr 1 ; r 2 ; . . .; r N Þdr 1 dr 2 . . .dr N ð ¼ 1N Hðajrr 1 jÞ p1 ðr 1 Þdr 1 ð ¼ 1N

ð2:110Þ

ð p1 ðr 1 Þdr 1 ¼ 1

jrr 1 j < a

nðr 1 Þdr 1 ; jrr 1 j < a

Hence, the volume concentration of particles is ð nðr 1 Þdr 1 ;

j¼

ð2:111Þ

jrr 1 j < a

and for identical particles with a uniform distribution n ¼ const, j ¼ 4pa3 n=3:

ð2:112Þ

From the ensemble of probable particle configurations, we can choose a subensemble containing those configurations in which particle 1 occupies one and the same position in space. We can then introduce the PDF of all other particles. This

2.5 Flow of a Fluid Through a Random Bed of Particles

PDF is called the conditional PDF and is equal to pN ðr 1 ; r 2 ; . . .; r N Þ= p1 ðr 1 Þ:

ð2:113Þ

Averaging of functions that relies on the use of such conditional PDFs will be designated as ð pN ðr 1 ; r 2 ; . . .; r N Þ dr 2 . . .dr N : hGi1 ¼ Gðr 1 ; r 2 ; . . .; r N Þ p1 ðr 1 Þ

ð2:114Þ

The conditional two-particle PDF and the corresponding distribution of the number of particles of type 2 are, respectively, p2 (r1, r2)/p1 (r1) and n (r2) ¼ N p2 (r1, r2)/ p1 (r1). For a random and independent distribution of identical particles, we can take p2 (r1, r2) ¼ 0 at |r1  r2| < 2a; p2 (r1, r2) ¼ p1 (r1) p1 (r2) at |r1  r2| > 2a; and n (r2) ¼ 0 at |r1  r2| < 2a; n (r2) ¼ const at |r1  r2| > 2a. Such an approximation is possible for non-interacting particles only. The PDF for interacting particles is determined from Liouville or Fokker–Planck equations (for more details, see Sections 3.7–3.9). Going back to Eq. (2.103), let us look at the second term. We have ! N N X X rr j rH ¼ r 1 Hða rr j Þ ¼ rr j dð rr j aÞ j¼1

j¼1

and hT  rHi ¼

N ð X j¼1

rr j dr 1 . . .dr N pN ðr 1 ; r 2 ; . . .; r N Þdð rr j aÞT  rr j

ð ¼N

pN dðjrr 1 jaÞT 

rr 1 dr 1 . . .dr N jrr 1 j

ð rr 1 dr 1 ¼ nðr 1 Þdðjrr 1 jaÞhT i1  jrr 1 j ð ¼ nðranÞhT i1  na2 dn: r1 ¼ran r fixed

ð2:115Þ The derivation relies on property 2 of the PDF (see Section 1.4) and also uses the fact that a unit normal vector to the particle surface is equal to n ¼ (r  r1)/|r  r1| and that dn is an element of a solid angle on a unit sphere. For n ¼ const, we have ð ð2:116Þ hT 1  rH i ¼ na2 hT i1  ndn: r1 ¼ran r fixed

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104

and now Eq. (2.103) takes the form ð me r2 huirð1jÞðprg  rÞna2

hT i1  ndn ¼ 0:

ð2:117Þ

r 1 ¼ran r fixed

hT i1 is obtained from the relation (2.100), which, when reformulated for the case of conditional averaging, will be written as   !   qhui i1 q u j 1 : ti j ¼ ð1jÞðp1 rg  rÞd i j þ me þ qX j qXi

ð2:118Þ

When deriving the last relation, we remembered that j1 ¼ 0 at the surface of the sphere. The subscript 1 implies that we are considering the situation when particle 1 is fixed. The equation for hui1 is similar to (2.103): me r2 hui1 rð1j1 Þðp1 rg  rÞhT  rHi1 :

ð2:119Þ

The derivation of hT  rHi1 is similar to the derivation of Eq. (2.115): ð hT  rHi1 ¼

n1 ðranÞhT i1;2  na2 dn;

ð2:120Þ

r 2 ¼ran r 1 and r fixed

with the only difference that in the right-hand side, n is replaced by n1 and hT i1 is replaced by hT i1;2. Here hT i1;2 is the outcome of conditional averaging of stress with particles 1 and 2 fixed, and the dependence of hT  rHi1;2 on hui1;2 has the same form as the dependence of hT  rH i1 on hui1 . We can continue this process by fixing (in consecutive order) the positions of particles 1, 2, 3, etc. The result is an infinite system of equations, where each finite subsystem is unclosed. Considering the first approximation only, we shall assume ð na ¼ 2

hT i1  ndn ¼ FðhuiÞ

ð2:121Þ

r 1 ¼ran r fixed

to close the equations (2.99) and (2.117). The functional dependence (2.121) generalizes the assumption made in [40]. A medium which obeys the condition (2.121) is called the Brinkman medium. The form of the function FðhuiÞ is different for the cases when the particles are fixed in space (as in a porous medium) and when they are suspended in the fluid (as in a suspension). In the first case the function is assumed to have the form FðhuiÞ ¼ Ahui þ Br2 hui and in the second case – FðhuiÞ ¼ Br2 hui þ Cg.

2.5 Flow of a Fluid Through a Random Bed of Particles

Let us fix particle 1 and consider the fluid flowing around it. The presence of other particles is accounted for by introducing the effective fluid viscosity meff and the  at the infinity. Both of these quantities have to be determined. upstream velocity u The velocity and pressure fields are described by the equations r  hui1 ¼ 0; rð p1 rg  rÞ þ meff r2 hui1 meff a2 hui1 ¼ 0;

ð2:122Þ

where meff a2 ¼ A/(1  j), meff ¼ (me  B)/(1  j), and the parameter a has to be determined. These equations should be solved under the following boundary conditions: hui1 ¼ 0

at

r ¼ r 1 þ an;

 hui1 ! u

at

r ! ¥;

ð2:123Þ

 must obey the equations similar to Eqs. (2.122): The unperturbed velocity u  ¼ 0; r u

meff a2 u  ¼ 0; rp0 þ meff r2 u

ð2:124Þ

where p0 ¼ prg  r. Solving equations (2.124) as discussed in [43], we get aa ¼

9j=4 þ 3ð8j3j2 Þ1=2 =4 13j=2

ð2:125Þ

and meff (j). The derivation of the latter dependence is lengthy and is not presented here. One can see from the second equation (2.122) that if the characteristic linear scale  is negligibly small as compared L of the problem exceeds 1/a, then the term meff r2 u . The equation then reduces to the Darcy equation for a low-permeable to meff a2 u porous medium: rp

meff k

 ¼ 0; u

ð2:126Þ

where k ¼ 1/a2 is the permeability of the porous medium. In the second case, the problem reduces to the system of equations r  hui ¼ 0; rð1jÞðprg  rÞ þ me r2 huiFðhuiÞ ¼ 0

ð2:127Þ

r  hui1 ¼ 0; r Þ þ me r2 hui1 Fðhui1 Þ ¼ 0 rð1jÞðp1 rg  _

ð2:128Þ

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106

with the boundary conditions r  V at hui1 ¼ V þ a _

r ¼ r1 þ a _ r ; hui1 ! hui at r ! ¥; ð2:129Þ

where r1 is the particle’s radius vector, and V and V are the translational and rotational velocities the test particle 1 freely suspended in the fluid. They are determined by the condition that the force and torque acting on the particle are both equal to zero: ð 2

a

4 r d_ r þ pa3 r p g ¼ 0; hT i1  _ 3

r¼r 1 þa _ r r 1 fixed

ð

_ rd_ r ¼ 0; r  hT i1  _

r¼r 1 þa _ r r 1 fixed

ð2:130Þ The functional FðhuiÞ is given by FðhuiÞ ¼ Br2 hui þ Cg: The corresponding equations look simpler than equations (2.122) and (2.124). In particular, the second equation (2.122) assumes the form of a standard Stokes equation   Cg  r þ meff r2 hui1 ¼ 0; r p1 rg  r þ 1j

ð2:131Þ

where meff ¼ (me  B)/(1  j) is the effective viscosity. Skipping the solution of this equation, we give only the formula for the coefficient of effective viscosity: meff me

¼

1 : 15j=2

ð2:132Þ

At j  1, the equation (2.132) gives the well-known Einstein’s formula: meff me

5

1 þ j: 2

References 1 Lamb, H. (1945) Hydrodynamics, Dover, New York. 2 Levich, V.G. (1962) Physicochemical Hydrodynamics, Prentice– Hall,Englewood Cliffs, NJ.

3 Happel, J. and Brenner, H. (1983) Low Reynolds Number Hydrodynamics, Martinus Nijhof, The Hague. 4 Kim, S. and Karrila, S.J. (1991) Microhydrodynamics, Butterworth– Heinemann, Boston.

References

5 Probstein, R.F. (1995) Physicochemical Hydrodynamics, Wiley, New York. 6 Brenner, H. (1966) Hydrodynamic Resistance of Particles at small Reynolds Numbers. Adv. Chem. Eng., 6, 287–438. 7 Stimson, M. and Jefferey, G.B. (1926) The Motion of two Spheres in a viscous Fluid. Proc. Roy. Soc. A, 111, 110. 8 Brenner, H. (1961) The slow Motion of a Sphere through a viscous Fluid toward a plane Wall. Chem. Eng. Sci., 16, 242–251. 9 Goldman, A.J., Cox, R.G. and Brenner, H. (1966) The slow Motion of two identical arbitrary oriented Spheres through a viscous Fluid. Chem. Eng. Sci., 21, 1151–1170. 10 Goldman, A.J., Cox, R.G. and Brenner, H. (1967) Slow viscous Motion of a Sphere parallel to a plane Wall. Part I. Motion through a quiescent Fluid. Part. II. Couette Flow. Chem. Eng. Sci., 22, 637, 653. 11 O’Neil, M.E. and Stewartson, K. (1967) On the slow Motion of a Sphere parallel a nearby plane Wall. J. Fluid Mech., 27, 705–724. 12 Cooley, M.D. and O’Neil, M.E. (1968) On the slow Rotation of a Sphere about a Diameter parallel to a nearby plane Wall. J. Inst. Math. Applics., 4, 163–173. 13 O’Neil, M.E. (1969) On asymmetric slow viscous Flows caused by the Motion of two Spheres almost in Contact. Proc. Camb. Phil. Soc. A, 65, 543–556. 14 Cooley, M.D. and O’Neil, M.E. (1969) On the slow Motion generated in a viscous Fluid by the approach of a Sphere to a plane Wall or stationary Sphere. Mathematika, 16 (1), 37–49. 15 Cooley, M.D. and O’Neil, M.E. (1969) On the slow Motion of two Spheres in Contact along their Line of Centers through a viscous Fluid. Proc. Camb. Phil. Soc., 66, 407–415. 16 Davis, M.H. (1969) The slow Translation and Rotation of two

17

18

19

20

21

22

23

24

25

26

unequal Spheres in a viscous Fluid. Chem. Eng. Sci., 24, 1769–1776. O’Neil, M.E. (1970) Exact Solutions of the Equations of slow viscous Flow generated by the asymmetrical Motion of two equal Spheres. Appl. Sci. Res., 21, 452–466. O’Neil, M.E. and Majumdar, S.R. (1970) Asymptotic Flow viscous Motions caused by the Translation or Rotation of two Spheres. Part I. The Determination of Exact Solutions for any Values of the Ratio Radii and Separation Parameters. Part II. Asymptotic Forms of the Solutions when the Minimum Clearance between the Spheres approaches Zero. ZAMP, 21, 164, 180. Goren, S.L. (1970) The normal Force exerted by creeping Flow on a small Sphere touching a Plane. J. Fluid Mech., 41, 619–625. Goren, S.L. and O’Neil, M.E. (1971) On the hydrodynamic Resistance to a Particle of a dilute Suspension when in the Neighborhood of a large Obstacle. Chem. Eng. Sci., 26, 325–338. Wakiya, S. (1971) Slow Motion in Shear Flow of a Doublet of two Spheres in Contact. J. Phys. Soc. Japan, 31, 158– 1587. Wacholder, E. and Weihs, D. (1972) Slow Motion of a Fluid Sphere in the Vicinity of another Sphere or a Plane Wall. Chem. Eng. Sci., 27, 1817–1828. Haber, S., Hetsroni, G. and Solan, A. (1973) On the low Reynolds Number Motion of two Droplets. Int. J. Multiphase Flow, 1, 57–71. Zinchenko, A.Z. (1978) To Calculation of hydrodynamic Interactions of Droplets by Low Reynolds Number. Appl. Math. Mech., (5), 955–959. Zinchenko, A.Z. (1980) Slow asymmetric motion of two Droplets in viscous Medium. Appl. Math. Mech., (1), 49–59. Schmitz, R. and Felderhof, R. (1982) Mobility Matrix for two spherical Particles with hydrodynamic Interaction. Physica A, 116, 163–177.

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27 Jeffrey, D.J. and Onishi, Y. (1984) The Forces and Couples acting on two nearly touching Spheres in LowReynolds-Number Flow. ZAMP, 35, 634–641. 28 Jeffrey, D.J. and Onishi, Y. (1984) Calculation of the Resistance and Mobility Functions for two unequal Spheres in Low-Reynolds-Number Flow. J. Fluid Mech., 139, 261–290. 29 Kim, S. and Mifflin, R.T. (1985) The Resistance and Mobility Functions of two equal Spheres in Low Reynolds Number Flow. Phys. Fluids, 28, 2033– 2044. 30 Fuentes, Y.O., Kim, S. and Jeffrey, D.J. (1988) Mobility Functions for two unequal viscous Drops in Stokes Flow. Part 1. Axisymmetric Motions. Phys. Fluids, 31, 2445–2455. 31 Fuentes, Y.O., Kim, S. and Jeffrey, D.J. (1989) Mobility Functions for two unequal viscous Drops in Stokes Flow. Part 2. Nonaxisymmetric Motions. Phys. Fluids A, 1, 61–76. 32 O’Neil, M.E. (1968) A Sphere in Contact with a Plane Wall in a slow linear Shear Flow. Chem. Eng. Sci., 23, 1293–1298. 33 Lin, G.L., Lee, K.J. and Sather, N.F. (1970) Slow Motion of two Spheres in a Shear Field. J. Fluid Mech., 43, 35–47. 34 Nir, A. and Acrivos, A. (1973) On the creeping Motion of two arbitrary-sized touching Spheres in a linear Shear Field. J. Fluid Mech., 59, 209–223. 35 Hetsroni, G. and Haber, S. (1978) Low Reynolds Number Motion of two Drops

36

37

38

39

40

41

42

43

44

submerged in an unbounded arbitrary Velocity Field. Int. J. Multiphase Flow, 4, 1–17. Zinchenko, A.Z. (1983) Hydrodynamic Interactions of two identical Liquid Spheres in linear Flow Field. Appl. Math. Mech., (1), 56–63. Martinov, S.I. (1998) Hydrodynamic Interactions of Particles. Fluid Dyn., (2), 112–119, (in Russian). Martinov, S.I. (2000) Interactions of Particles in Flow with parabolic Velocity Profile. Fluid Dyn., (1), 84–91, (in Russian). Brenner, H. and O’Neil, M.E. (1972) On the Stokes Resistance of multiparticle Systems in a linear shear Field. Chem. Eng. Sci., 27, 1421–1439. Brinkman, H.C. (1947) A Calculation of the viscous Force exerted by a flowing Fluid on a dense Swarm of Particles. Appl. Sci. Res., A1, 27–34. Tamm, C.K.W. (1969) The Drag on a Cloud of spherical Particles in low Reynolds Number Flow. J. Fluid Mech., 38, 537–546. Saffman, P.G. (1971) On the Boundary Condition at the Surface of a Porous Medium. Studies in Appl. Math, 30 (2), 93–101. Lundgren, T.S. (1972) Slow Flow through stationary Random Beds and Suspensions of Spheres. J. Fluid Mech., 51, 273–299. Brenner, H. (1974) Rheology of dilute Suspension of axisymmetric Brownian Particles. Int. J. Multiphase Flow, 1, 195–341.

j109

3 Brownian Motion of Particles Chaotic motion of microparticles (whose size can range from 1 nm to 10 mm) suspended in a fluid was first discovered by Brown in 1827 and is now called Brownian motion. A systematic study of Brownian motion was initiated only in the early 20th century by Einstein, Smoluchowski, Perrin, Langevin, and Lorentz. Their works showed that Brownian motion is caused by very frequent collisions of particles with molecules of the surrounding fluid undergoing incessant random motion (thermal motion). The motion of molecules is so irregular that Brownian motion can be described only probabilistically, under the assumption of frequent and statistically independent impacts of molecules against a particle. More recently discovered phenomena similar to Brownian motion, for instance, fluctuations of current within conductors, are also caused by thermal motion of molecules and electrons. Brownian motion of a particle in a fluid is taking place under the action of two forces: a random force caused by collisions with molecules, and a systematic force caused by viscous friction. The statistical average of kinetic energy of thermal motion of a particle is equal to 3kW/2, where k is the Boltzmann constant and W – the absolute temperature. This follows from the law of uniform distribution of kinetic energy over degrees of freedom (equipartition theorem). Thus, when a particle is moving along a straight line, this energy equals kW/2. The first works on Brownian motion were devoted to the motion of an isolated particle. The results obtained in these works are applicable for suspensions with very low volume concentration of particles (infinitely dilute suspensions), when the average distance between particles is large compared to particle sizes, so that interparticle interactions (hydrodynamic interactions, molecular interactions, collisions) can be neglected. Later works on Brownian motion took into account pair interactions between particles, which made their results applicable for higher (but not too high!) volume concentrations j. The more recent works attempt to take into account multiparticle interactions as well. We begin our discussion of Brownian motion with the analysis of random motion of an individual particle in a quiescent fluid, neglecting external forces and

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110

interactions with other particles and with the surrounding fluid. Such a motion is called random walk. Further on, we shall discuss particle motion under the action of systematic and random forces. Finally, we go to the case of particle motion affected by the flow of the surrounding fluid and by interparticle interactions. Works [1–12] will provide the reader with extra information about Brownian motion and the common mathematical techniques employed in research.

3.1 Random Walk of an Isolated Particle

The case of Brownian motion of an isolated particle boils down to the random walk problem, which can generally be formulated in the following way. A particle driven by some external influence undergoes successive displacements r1, r2, . . . , ri, . . . , each of which is independent from all previous ones (in terms of both absolute magnitude and direction). The probability for a displacement to lie in the interval (ri, ri þ dri) is specified by some probability density distribution ti(ri). (Possible types of distributions will be considered later on.) Our task is to find the probability that after N steps the particle will be located in the interval (R, R þ dR). Consider first the simplest variant of random walk – a series of steps of equal length along a straight line, where each step can be directed forward or backward with the same probability of 0.5. The step length is assumed to be 1, so after N steps the particle can wind up at any one of the points with coordinates N, N þ 1, . . ., 1, 0, 1, . . ., N  1, N. The probability P(m, N) that after N steps the particle will be found at the point m is given by the Bernoulli distribution Pðm; NÞ ¼

m CðmþNÞ=2

 N 1 ; 2

ð3:1Þ

m where CðmþNÞ=2 ¼

N! is the binomial coefficient, and the numbers m ð12ðNþmÞÞ!ð12ðNmÞÞ! and N have the same parity. The average displacement and the root-mean-square deviation of the particle are, respectively,

hmi ¼ 0;

pffiffiffiffiffiffiffiffiffiffi pffiffiffiffi hm2 i ¼ N :

In the case when N  1 and m  N, which is of primary interest to us, Eq. (3.1) gives the following asymptotic distribution:  Pðm; NÞ ¼

2 pN

1=2

 m2 : exp  2N

ð3:2Þ

Now, consider the same problem but with steps of length l. Let us introduce the particle displacement X ¼ ml and consider the intervals DX  l along the

3.1 Random Walk of an Isolated Particle

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straight line. Then the probability to find the particle in the coordinate interval (X, X þ DX) after N steps is  DX 1 X2 exp  DX: ¼ 2Nl2 2l ð2pNl2 Þ1=2

PðX ; NÞDX ¼ Pðm; NÞ

If the particle accomplishes n displacements in a unit time, the probability to find it in the interval (X, X þ DX) at the moment t is  X2 DX : exp  4Dt 2ðpDtÞ1=2 1

PðX ; tÞDX ¼

ð3:3Þ

where D ¼ vl22. It was assumed up to this point that the straight line along which the particle is moving is unbounded. Consider now the case of a bounded straight line. Suppose a boundary is placed at the point (X1 ¼ m1l > 0) which either reflects or absorbs the particle. If the boundary is reflective, the probability for the particle to be at the point m < m1 after N steps is Pðm; N; m1 Þ ¼ Pðm; NÞ þ Pð2m1 m; NÞ: The second term on the right-hand side is the probability to find the particle at the mirror-reflected point 2m1  m after N steps; keep in mind that the probability P(m,n) is still determined by the formula (3.1). In the limiting case of N  1 and m  N we arrive at the formula  Pðm; NÞ ¼

2 pN

1=2

( )!  m2 ð2m1 mÞ2 þ exp  : exp  2N 2N

Introducing the same notation as the one implied by Eq. (3.3), we obtain the following: PðX ; t; X1 Þ ¼

1 2ðpDtÞ1=2

( )!  X2 ð2X1 X Þ2 þ exp  : exp  4Dt 4Dt

ð3:4Þ

Note that the distribution (3.4) obeys the well-known boundary condition ðqP=qXÞX ¼X1 that is responsible for the vanishing of the flux at the plane wall. For an absorbing boundary, the probability that after N steps the particle will be found at a point m < m1 is equal to Pðm; N; m1 Þ ¼ Pðm; NÞPð2m1 m; NÞ: The minus sign before the second term expresses the fact that the absorbing boundary prohibits the particle to get into a mirror-reflected point. The limiting case

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112

N  1 and m  N results in the formula PðX ; t; X1 Þ ¼

1 2ðpDtÞ1=2

( )!  X2 ð2X1 X Þ2 exp  : exp  4Dt 4Dt

ð3:5Þ

The distribution (3.5) satisfies the condition P(X1, t; X1) ¼ 0, which corresponds to a well-known boundary condition at the completely absorbing wall. The derived expressions make it possible to determine the probabilistic flux of particles deposited on the wall. In order to do so, it is necessary to find the probability that after N steps the particle will arrive at the point m1 and that while performing these N steps, it will never cross or touch the boundary, that is, the point m ¼ m1. This probability is A(m1, N) ¼ m1/NP(m1, N). In the limiting case of large N it is equal to  X1 1 X2 DX ; ð3:6Þ exp  AðX1 ; tÞ ¼ Nt ðpDtÞ1=2 4Dt where X1 ¼ m1l; N ¼ nt; D ¼ nl2/2; l is the step length; n is the number of displacements in a unit time. If the initial number of particles (t ¼ 0, X ¼ 0) is known, we can find the rate of deposition particles on the absorbing wall. At the moment t, the ratio of the number of deposited particles per unit time to the initial number of particles will be equal to qðX1 ; tÞ ¼

 X1 1 X12 exp  : t 2ðpDtÞ1=2 4Dt

ð3:7Þ

The distribution (3.7) obeys the condition   qP ; qðX1 ; tÞ ¼ D qX X ¼X1

ð3:8Þ

where P is determined by the relation (3.5). If P stands for the concentration of a matter, then Eq. (3.8) can be interpreted as a probability flux or, from the physical viewpoint, as a diffusion flux of matter. Consider now the general random walk problem. The particle accomplishes N displacements, whose positions are determined by radius vectors r1, r2, . . . , rN, so P after N displacements the particle will be found at the point R ¼ N i¼1 r i . Each displacement is assigned its own probability density ti(ri). Thus ti(ri)dri is the probability for i-th displacement vector to belong to the range (ri, ri þ dri). In the coordinate form, this probability is written as ti ðr i Þdr i ¼ ti ðXi ; Yi ; Zi ÞdXi dYi dZi ;

ði ¼ 1; 2; . . .; NÞ:

The problem is to find the probability pN(R)dR that after N displacements the particle will be found in the interval (Ri, Ri þ dRi). The solution of this problem obtained by

3.1 Random Walk of an Isolated Particle

j113

Markov’s method has the following form: 1 pN ðRÞ ¼ 3 8p AN ðrÞ ¼

N Y j¼1

Z1 AN ðrÞexpðirRÞdr; 1 Z1

ð3:9Þ

t j ðr j Þexpðirr j Þdr j ;

1

where r is an N-dimensional vector and AN(r) is the Fourier transform of the Q function Nj¼1 t j ðr j Þ. Hence, the solution of the general random walk problem depends on the form of probability density ti(ri) for each displacement. Let us take a look at some distributions commonly encountered in applications. 3.1.1 Isotropic Distribution

A distribution is isotropic if it does not depend on the direction of the displacement vector ri, and depends only on its length (or, which is the same, on the square of length jr 2i j ¼ r 2 if all displacements have the same length). Clearly, such a distribution possesses a spherical symmetry. That is why such distributions are sometimes called spherical distributions of displacement directions. Also, distributions of displacements must to be identical for all displacements, so that we can drop the index after t, writing t j ðr j Þ ¼ tðr 2 Þ:

ð3:10Þ

Then the second relation (3.9) yields 81 9N <ð = tðr 2 ÞexpðirrÞdr : AN ðrÞ ¼ : ; 1

Switching to a spherical coordinate system and integrating, we get: 8 1 1 2p 9N 8 1 9N <ð ð ð = < ð sinðjrjrÞ = AN ðrÞ ¼ r 2 tðr 2 ÞexpðijrjrtÞdw dt dr ¼ 4p r 2 tðr 2 Þdr : ; : ; jrjr 10 0 811 9N  < ð 1 = ¼ 4p 1 jrj2 r 2 þ . . . r 2 tðr 2 Þdr : : ; 6 0

Tending N to 1, we obtain the following expression:  Njr2 jhr 2 i ; AN ðrÞ ¼ exp  6 where hr2i is the root-mean-square displacement.

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A substitution of this expression into the first relation (3.9) and a subsequent integration results in )  3=2 ( 3 3jRj2 : ð3:11Þ exp  pðRÞ ¼ pN ! 1 ðRÞ ¼ 2Nhr 2 i 2pNhr 2 i Suppose the particle accomplishes n displacements per second. If particle makes N displacements during the time t, then N ¼ vt. Designating D ¼ vhr2i/6, we can rewrite Eq. (3.11) as ( ) 1 jRj2 : ð3:12Þ exp  pðRÞ ¼ 4Dt ð4pDtÞ3=2 So, after time t the particle will be found in spherical shell volume element between R and R þ dR with probability p(R)dR, where p(R) is given by Eq. (3.12). 3.1.2 Gaussian Distribution

Let the displacements have a probability density described by a Gaussian distribution:

t j ðr j Þ ¼

!3=2 ( ) 3jr j j2 3 exp  ; 2hr 2j i 2phr 2j i

ð3:13Þ

where hr 2j i is the mean square of j-th displacement. Since the distribution (3.13) depends on |rj|, the direction of each displacement vector can be arbitrary. A substitution of Eq. (3.13) into the second relation (3.9) leads to

AN ðrÞ ¼

N Y j¼1

3 2phr 2j i

!3=2 1 ð 1 ð 1 ð expðiðr1 X j þ r2 Y j þ r3 Z j Þ 111 Z2j Þ=2hr 2j iÞdX j

þ dY j þ dZ j 3ðX 2j þ Y 2j þ N



 Y ¼ exp ðr21 þ r22 þ r23 Þhr 2j i=6 ¼ exp jrj2 Nhr 2j i=6 ; j¼1

P where hr 2 i ¼ N1 Nj¼1 hr 2j i: Now it follows from the first relation (3.9) that  pN ðRÞ ¼

3 2pNhr 2 i

3=2

(

) 3jRj2 exp  : 2Nhr 2 i

ð3:14Þ

Hence, the assumption of a Gaussian distribution of probability density of displacements allows to obtain an exact solution of the general random walk problem for N  1.

3.1 Random Walk of an Isolated Particle

j115

3.1.3 An Arbitrary Distribution t(r) in the Limiting Case N  1

We assume that all tj(rj) are identical and equal to t(r). In this case one can get an exact expression for the probability density pN(R) at N ! 1. Eq. (3.9) gives us 81 1 1 9N <ð ð ð = AN ðrÞ ¼ tðX ; Y; ZÞexpðiðr1 X þ r2 Y þ r3 ZÞdX dY dZ : : ; 111

Expanding the exponent as a Taylor series and recalling the expressions for moments of the distribution t(r), we get: 81 1 1 <ð ð ð 1 ð1 þ iðr1 X þ r2 Y þ r3 ZÞ ðr21 X 2 þ r22 Y 2 þ r23 Z2 AN ðrÞ ¼ : 2 111

þ2r1 r2 XY þ 2r2 r3 YZ þ 2r3 r1 ZX Þ þ . . .ÞtðX ; Y; ZÞdX dY dZgN 1 ¼ 1 þ iðr1 hX i þ r2 hYi þ r3 hZiÞ ðr21 hX 2 i þ r22 hY 2 i þ r23 hZ2 i 2 þ2r1 r2 hXYi þ 2r2 r3 hYZi þ 2r3 r1 hZX iÞ þ . . .ÞN : At N ! 1, the derived expression tends to an exponential expression, so that  1 AN ðrÞ ¼ exp iNðr1 hX i þ r2 hYi þ r3 hZiÞ NQðrÞ ; 2 where Q(r) denotes the following quadratic form: QðrÞ ¼ r21 hX 2 i þ r22 hY 2 i þ r23 hZ2 i þ 2r1 r2 hXYi þ2r2 r3 hYZi þ 2r3 r1 hZX i: Let us choose a coordinate system whose axes coincide with the principal directions of the quadratic form (x, Z, z). Then Q(r) assumes the canonical form QðrÞ ¼ r2c hc2 i þ r2h hh2 i þ r2z hz2 i; where hx2i, hh2i, and hz2i are the eigenvalues of the symmetric matrix of the quadratic form. In the new coordinate system, the vector R has components (Rx, Rh, Rz). Then Eq. (3.9) gives the following expression for the case N ! 1: 1 pðRÞ ¼ 3 8p

1 ð 1 ð 1 ð

 1 exp  Nðr2x hx2 i þ r2h hh2 i þ r2z hz2 iÞ 2

111 o iðrx ðRx NhxiÞ þ rh ðRh NhhiÞ þ rz ðRz NhziÞÞ drx drh drz ( ) Rx Nhxi Rh Nhhi Rz Nhzi 1 :  ¼ exp   2Nhh2 i 2Nhz2 i 2Nhx2 i ð8p3 N 3 hx2 ihh2 ihz2 iÞ1=2

The obtained distribution p(R) shows that the particle experiences a systematical drift Nhxi, NhZi, Nhzi along the principal axes, onto which fluctuations Nhx2i, NhZ2i, Nhz2i get superimposed.

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3.2 Random Walk of an Ensemble of Particles

The previous section dealt with the motion of an isolated particle. Since interactions of the particle with the surrounding fluid and with the other particles were not taken into account, we may as well consider an ensemble of particles that were all initially located at one and the same point and then started a random walk (we again disregard the influence of the surrounding fluid and interparticle interactions). The results of the previous section can be applied directly to this case, but their interpretation should be changed. For instance, the expression (3.9) and all subsequent relations have the meaning of fraction of the total number of particles that will be found at time t in the spatial interval (R, R þ dR) given that all particles were initially located at the point R = 0. In this formulation of the problem the total number of particles is conserved and is equal to the initial value. We are mostly interested in the limiting case of an extremely large number of particles N ! 1. Then particle positions can be regarded as continuous random variables, and a continuous probability distribution can be introduced. Besides, since the random walk is a Markowian process, the results of Sections 1.11 and 1.12 can be used, in particular the differential Fokker–Planck equation for the PDF of a random variable in a Markowian process. For an isotropic distribution t(r2) considered in the previous section, the Fokker–Planck equation assumes the form  2 qp q p q2 p q2 p þ þ ¼D ¼ DD p; qt qX 2 qY 2 qZ2

ð3:15Þ

because, due to the symmetry of the distribution t(r2), the drift A is zero and the diffusion tensor is reduced to D ¼ DI, where I is a unit tensor with components d ij. In the more general case 3 (see Section 3.1), the distribution t(r) that controls individual particle displacements does not possess any special symmetry properties. Therefore moments of the first order that determine the drift vector are nonzero, and moments of the second order that determine the symmetric diffusion tensor cannot be thought of as components of a diagonal matrix. But when the coordinate system is chosen in such a way that its axes coincide with principal axes of the diffusion tensor, the diffusion tensor reduces to D ¼ jjDi j di j jj and the Fokker– Planck equation to qp qp qp qp q2 p q2 p q2 p ¼ A1 þ A2 þ A3 þ D11 2 þ D22 2 þ D33 2 ; qt qX qY qZ qX qY qZ

ð3:16Þ

where A1 ¼ nhXi, A2 ¼ nhYi, A3 ¼ nhZi, D11 ¼ nhX2i/2, D11 ¼ nhY2i/2, D33 ¼ nhZ2i/2, and n is the number of displacements per unit time. Eq. (3.16) can be rewritten as a continuity equation: qp þ rj ¼ 0; qt

3.3 Brownian Motion of a Free Particle in a Quiescent Fluid

j117

where j is probability flux with components  jX ¼

A1 PD11

qp ; qX

 jY ¼

A2 PD22

qp ; qY

 jZ ¼

A3 PD33

qp : qZ

These fluxes have the meaning of probable numbers of particles crossing the unit elements of surfaces perpendicular to axes X, Y and Z in a unit time. If that all particles are initially concentrated at the point (X0, Y0, Z0), the solution of Eq. (3.17) is pðX; Y; ZÞ ¼

1 3=2 ðD11 D22 D33 Þ1=2 8ðptÞ( ) ðX X0 þ A1 tÞ2 ðYY0 þ A2 tÞ2 ðZZ0 þ A3 tÞ2 exp    4D11 t 4D22 t 4D33 t

3.3 Brownian Motion of a Free Particle in a Quiescent Fluid

Brownian motion of a free particle is defined as the particle’s motion in the fluid under the action of two forces: the random force caused by collisions between the particle and molecules of the fluid, and the systematic force of hydrodynamic resistance. Assume that the fluid is at rest, the external force field is absent, and interparticle interactions are negligibly small. The last assumption means that the suspension is infinitely dilute. The modern theory of Brownian motion is based on the Langevin stochastic equation (see Section 1.13): mu ¼ hu þ F fl ðtÞ;

u ¼ X_ ðtÞ:

ð3:17Þ

Here X is the particle’s position in some fixed coordinate system; u is the particle’s velocity; m – the particle’s mass; h ¼ 6pme – the coefficient of hydrodynamic resistance; F fl – the random force; me – the coefficient of dynamic viscosity; a – the radius of the particle (the particle is assumed to be spherical). As to random force, it is assumed that: F fl varies much faster than u, therefore it is independent of u; the average value is hF fl i¼ 0; the correlation is hF fl (t) F fl (t þ t)i ¼ Fd(t). The last two conditions follow from Eqs. (1.14) and (1.15). First, we should note that the characteristic time of velocity change that happens due to the action of a systematic force on the particle has the order of m/h. Since m  a3 and h  a, we have m/h  a2. The particle size is small, so m/h is small as well. At the same time it is much longer than the characteristic time of change of the

j 3 Brownian Motion of Particles

118

random force. Hence, for t  m/h, the inertia term in the Langevin equation can be dropped, and Eq. (3.11) reduces to 0 ¼ hu þ F fl ðtÞ: The parameter tv ¼ m/h is called the characteristic time of dynamic (viscous) relaxation of the particle. If the process is taking place on a time scale much longer then the characteristic time of random force fluctuation and yet shorter than tv, the particle’s velocity varies with time, and the corresponding PDF should be a function of u. For t  tv, the PDF is a function of the particle’s position r and the initial velocity u0, while being independent of u. Further on, our main attention will be focused on these two limiting cases. Consider the motion of a particle during the time interval Dt ¼ t  t0 long enough compared to the period of random force fluctuation (and thereby to the period of random acceleration of the particle), yet short enough compared to the time of change of the particle’s velocity. Since the Langevin equation is the equation for velocity u in a Markowian process, one has to specify the velocity u0 at the previous instant of time and then, using the properties of F fl, determine statistical properties of the distribution p. In this case one should consider the dependence of p not on particle position X but on particles velocity u, in other words, p ¼ p(u, t|u0, t0). Then p(u, t|u0, t0)du gives the probability that the particle’s velocity at time t will belong to the interval (u, u þ du) given that at the previous (initial) moment t0 ¼ t  Dt it was equal to u0, u  Du. The PDF defined in this way should satisfy the initial condition pðu; t0 ju0 ; t0 Þ ¼ dðuu0 Þ ¼ dðuX uX 0 ÞdðuY uY0 ÞdðuZ uZ0 Þ: Consider now the condition that the distribution p(u, t|u0, t0) must satisfy at t ! 1. Look at the Fokker–Planck equation. We already wrote it for the coordinate distribution p(Z, t|Y, t0 ) in Section 1.12. A transition to the velocity distribution leads to a similar equation, but with different drift and diffusion coefficients. This is selfevident, because the term ‘‘coordinates’’ usually implies generalized coordinates, which includes velocities. The one-dimensional Fokker–Planck equation is qp qðAu pÞ q2 ðDu pÞ : ¼ þ qt qu qu2

ð3:18Þ

The coefficient Au stands for the particle’s acceleration, therefore, according to Eq. (3.17), it is equal to Au ¼  fu;

f ¼ h=m ¼ 6pme a=m:

Now Eq. (3.18) for Du ¼ const takes the form qp qð fu pÞ q2 p ¼ þ Du 2 ; qt qu qu

ð3:19Þ

This form allows us to find statistical moments hui and hu2i without actually solving the equation. To this end, we multiply both sides successively by u and

3.3 Brownian Motion of a Free Particle in a Quiescent Fluid

j119

u2 and integrate the result over u from 1 up to þ 1, assuming that at u ! ± 1, p ! 0 faster than the function un. The results are hui ¼ u0 e ft ;

hðuhuiÞ2 i ¼ Du ð1e ft Þ= f :

For t  tv ¼ 1/f ¼ m/6pmea, we have hui ¼ 0;

hu2 i ¼ Du = f :

The characteristic time of viscous relaxation for the particle is tv ¼ 2a2rp/9rene, where rp is the particle’s density and ne ¼ me/re is the coefficient of kinematic viscosity. Smallness of the particle’s size results in a short viscous relaxation time and, consequently, very short transient period for the particle’s velocity. The energy of thermal motion of the particle is mhu2i/2 ¼ kW/2, so hu2i ¼ kW/m ¼ Du/f and Du ¼ f kW/m. Introduce new variables n ¼ ueft, F ¼ p3ft, and dt ¼ Due2ftdt. In these variables, Eq. (3.19) transforms to a standard diffusion equation qF q2 F ¼ 2; qt qv whose solution in the original variables is n o f ðuu0 e ft Þ2 exp  2D 2 ft Þ ð1e u pðu; tju0 ; t0 Þ ¼

1=2 : 2pDu 2 ft Þ ð1e f In the limiting case of t  1/f, we have  pðu; tju0 ; t0 Þ ¼

f 2pDu

1=2

 

fu2 m 1=2 mu2 : exp  exp  ¼ 2Du 2kJ 2pkJ ð3:20Þ

So, after a sufficiently long time a Maxwellian velocity distribution will be established. Consider now the PDF of displacements p(r, t|r0, u0, t0). Recall that p(r, t|r0, u0, t0)dr is the probability for the particle’s displacement at time t to be in the interval (r, r þ dr) given that r0 was the initial displacement and u0 – the initial velocity. In the one-dimensional case, this problem boils down to the Fokker–Planck equation qp qðA pÞ q2 ðD pÞ ; ¼ þ qt qX qX 2

ð3:21Þ

j 3 Brownian Motion of Particles

120

where A ¼ u. Solving this equation is tricky because p now depends on X. Therefore, in order to get p(r, t|r0, u0, t0), we proceed as follows. Since r_ ¼ u, we can write ðt rr 0 ¼ uðtÞdt:

ð3:22Þ

0

It follows from the Langevin equation (3.17) that uu0 e

 ft

¼e

 ft

ðt e f x F fl ðxÞdx:

ð3:23Þ

0

At t  1/f, the left-hand side of Eq. (3.23) is equal to u. Since statistical properties should be the same for both sides of Eq. (3.23), then, as we showed earlier, at t  1/f (or, which is the same, at t ! 1) the distribution e

 ft

ðt e f x F fl ðxÞdx: 0

should be Maxwellian, in other words, 8 < lim

t ! 1:

 ft

ðt

( )

m 3=2 mjuj2 e F ðxÞdx ¼ exp  : ; 2kJ 2pkJ fx

e

0

9 =

fl

ð3:24Þ

The difference between the distributions (3.24) and (3.20), namely, the different exponents in the first term of the product, is explained by different dimensionalities of these distributions. From Eq. (3.22) and Eq. (3.23), we find that ðt ðh 1  ft fh rr 0 ¼ u0 ð1e Þ þ e dh þ e f x F fl ðxÞdx: f 0

0

Integration by parts gives ðt 1  ft rr 0  u0 ð1e Þ ¼ yðxÞF fl ðxÞdx; f

ð3:25Þ

0

where yðxÞ ¼

1 ð1e f ðxtÞ Þ: f

ð3:26Þ

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j121

Now, consider the random variable ðt R ¼ yðxÞF fl ðxÞdx: 0

The random function F fl(x) varies extremely rapidly as compared to any other quantity. The period of fluctuation of F fl(x) is of the same order as the characteristic time of interaction between the particle and molecules of the surrounding fluid. Therefore a Maxwellian distribution of the random variable R !3=2

1

pðRÞ ¼

Ðt 4pDu 0 y2 ðxÞdx

( exp 

jRj2

)

; Ðt 4Du 0 y2 ðxÞdx

ð3:27Þ

is established very quickly in the course of Brownian motion. Substituting Eq. (3.26) into Eq. (3.27), we get  3=2 mf2 pðr; tjr 0 ; u0 ; t0 Þ ¼ 2pkJ½2 ft3 þ 4e ft e)2 ft ( 2 m f jrr 0 u0 ½1e ft = f j2 exp  : 2pkJ½2 ft3 þ 4e ft e2 ft

ð3:28Þ

It is readily seen that at t  1/f, this equation can be simplified: (

) jrr 0 j2 ; exp  pðr; tjr 0 ; u0 ; t0 Þ ¼ 4Dt ð4pDtÞ3=2 1

ð3:29Þ

where D ¼ kq/mf ¼ kq/6pame. Eq. (3.29) gives us the particle’s root-mean-square displacement in the X-direction: 1 ð 1 2 hðX X0 Þ i ¼ hðjrr 0 jÞ i ¼ jrr 0 j2 pðr; tjr 0 ; u0 ; t0 Þdr ¼ 2Dbr t; 3 2

ð3:30Þ

0

where Dbr is called the coefficient of Brownian diffusion. The relation (3.30) was first obtained by Einstein. It is valid only for t  1/f. The expression for root-mean-square displacement for an arbitrary time t can be derived from (3.28): jrr 0 j2 ¼

ju0 j2 3kJ ð1e ft Þ2 þ ð2 ft3 þ 4e ft e2 ft Þ: f2 mf2

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122

Averaging this relation over u0 and taking into account the fact that h|u0|2i ¼ 3kq/m, we get hjrr 0 j2 i ¼

6kJ ð ft1 þ e ft Þ: mf2

ð3:31Þ

The relation (3.30) follows from Eq. (3.31) at t ! 1. An equation describing the initial stage of Brownian motion can be obtained from (3.31) by setting t ! 0: hjrr 0 j2 i ¼

3kJ 2 t ¼ hju0 j2 it2 : m

ð3:32Þ

3.4 Brownian Motion of a Particle in an External Force Field

Consider Brownian motion of a particle under the action of two systematic forces: the drag force hu and the external force Fe (gravity force, electric force), and one random force F fl(t). We assume that the characteristic time of random force fluctuations is much shorter then the characteristic times of variations of systematic forces. Let us examine the particle’s motion in the time interval Dt ¼ t  t0 that is long enough as compared to the period of random force fluctuations (i.e., to the duration of the accelerating force), but short enough in comparison with the time it takes for physical parameters, such as the particle’s velocity, to change by a noticeable amount. The particle’s motion is described by the Langevin equation: mu_ ¼ hu þ F e þ F fl ðtÞ;

u ¼ X_ ðtÞ:

ð3:33Þ

The Fokker–Planck equation for the PDF p(r, u, t|r0, u0, t0) in this case reduces to qp þ urr p þ ru ð f u þ F e Þ ¼ Dr2r p þ Du r2u p: qt

ð3:34Þ

The subscripts r and u in the operator r mean that the derivatives are taken with respect to ri and ui, respectively. Since during the time Dt the particle’s position changes by Dr ¼ uDt, the meansquare displacement is h(Dr)2i|u|2(Dt)2 and (see Eq. 1.123) D¼

1 hðDrÞ2 i lim ¼ 0: 2 Dt ! 0 Dt

The obtained estimate allows to ignore the first summand on the right-hand side of equation (3.34):   qp Fe ¼ Du r2u p: þ urr p þ ru  p  f u þ m qt

ð3:35Þ

3.4 Brownian Motion of a Particle in an External Force Field

j123

It is convenient to rewrite this equation as qp Fe þ urr p þ ru p ¼  f ru ð puÞ þ Du r2u p; m qt

ð3:36Þ

so that the left-hand side represents the total (substantial) derivative: D pðr; u; tjr 0 ; u0 ; t0 Þ q p ¼ þ urr p þ F e ru p: Dt qt In the one-dimensional case, Eq. (3.35) has the form qp qp q þu þ qt qX qu

   Fe q2 p ¼ Du 2 : p  fu þ m qu

ð3:37Þ

An attempt to solve Eq. (3.37) for the general case run into difficulties. Therefore we consider simple case when the external force is absent: Fe ¼ 0. Then Eq. (3.37) reduces to qp þ urr p f uru p ¼ 3 f p þ Du r2u p; qt

ð3:38Þ

which can be solved by the method of characteristics. We write the two characteristic vector equations for Eq. (3.38): qu ¼  f u; qt

qr ¼ u: qt

The two families of solutions are: ue ft ¼ I 1 ;

rþ

u ¼ I2 f

where different I1 and I2 are different vectors. The general solution of Eq. (3.38) can be written as p ¼ p(I1, I2). In new variables n ¼ ueft, R ¼ r þ u/f, and F ¼ pe3ft, Eq. (3.38) assumes the form   qF 2 1 ¼ Du e2 ft r2v F þ e ft ðrv rR ÞF þ 2 r2R F qt f f or, in the coordinate form ( ) 3 qF 2 ft X q2 F 1 2 ft þ 2 DX F ; ¼ D u e Dx F þ e qt f qxi qXi f i¼1

ð3:39Þ

j 3 Brownian Motion of Particles

124

where the vector v has coordinates (x1, x2, x3) and the vector R has coordinates (X1, X2, X3). We look for the solution in the form F¼

3 Y Fi ðxi ; Xi Þ: i¼1

By substituting it into Eq. (3.39), it is easy to convince ourselves that Fi obeys the equation qFi q2 Fi q 2 Fi q2 Fi þ y2 ðtÞ ; ¼ j2 ðtÞ 2 þ 2jðtÞyðtÞ qt qxi qXi qXi2 qxi 1=2

ð3:40Þ

1=2

where jðtÞ ¼ Du e ft ; yðtÞ ¼ Du = f . The solution of this equation with the source-type initial condition, at the point x = X = 0, is ( ) 1 ax2i þ 2bxi Xi þ gXi2 Fi ðxi ; Xi Þ ¼ ; exp  2D 2pD1=2 where ðt 2

D ¼ agb ;

ðt

a ¼ 2 y ðtÞdt; 2

b ¼ 2 jðtÞyðtÞdt;

0

ðt g ¼ 2 j2 ðtÞdt:

0

0

After some math, one finally obtains: ( ) e3 ft ½ajvv0 j2 þ 2bðvv0 ÞðRR 0 Þ þ gjRR 0 j2 pðr; u; tjr 0 ; u0 ; t0 Þ ¼ exp  ; 2D 8p3 D3=2 R  R0 ¼ r þ u/f  r0  u0  u0/f; where v  n0 ¼ e ftu  u0; b ¼ 2Du(e ft  1)/f 2, g ¼ Du(e2ft  1)/f.

a ¼ 2Dut/f 2,

3.5 The Smoluchowski Equation

Consider Brownian motion of a free particle when t  1/f. As was shown above, this is the time it takes for the particle velocity to reach its equilibrium value (another term for t is ‘‘dynamic relaxation time’’). The particle’s motion is considered inertialess, and the probability density of particle transition from r to r þ dr in time Dt  1/f is given by Eq. (3.29), namely, ) jDrj2 tðDrÞ ¼ exp  ; 4DDt ð4pDDtÞ3=2 1

where D ¼ kq/mf.

(

ð3:41Þ

3.5 The Smoluchowski Equation

j125

Then the problem of Brownian motion of a free particle reduces to the random walk problem (see Section 3.1). In this case the particle’s motion can be characterized by a PDF of displacements p(r, t|r0, t0) and the corresponding Fokker–Planck equation is considered in the displacement space r rather than in the phase space (r, u). If the external force is absent, the Fokker–Planck equation reduces to the ordinary diffusion equation qp ¼ Dr2r p: qt

ð3:42Þ

The presence of external force field means that the particle’s velocity will be influenced by an external systematic force Fe in accordance with the Langevin equation (3.33). If the force Fe does not change noticeably on distances of the order (D/f)1/2, then any initial velocity distribution has enough time to turn into a Maxwellian distribution, and there is no significant change of r : since |Dr| ¼ |Du|/f(D/ f )1/2. Then the PDF p(r, u, t|r0, u0, t0) can be written as ( ) n m o3=2 mjuj2 pðr; u; tjr 0 ; u0 ; t0 Þ ¼  pðr; tjr 0 ; t0 Þ: exp  2kJ 2pkJ As a result, the particle acquires the average drift velocity along the direction of the external force Fe. Since particle inertia is neglected, this drift velocity is equal to Fe/h. Then on the left-hand side of Eq. (3.42) there appears an additional convective term r(Fe/h), and the equation for the PDF p(r, t|r0, t0) becomes   qp Fe p : ¼ rr  Drr p mf qt

ð3:43Þ

Equations (3.42) and (3.43) are called the Smoluchowski equations. By analogy with Eq. (1.153), Eq. (3.43) can be rewritten in the divergent form: qp þ rr j ¼ 0; qt

j ¼ Drr p þ

Fe p; mf

ð3:44Þ

where j has the meaning of diffusion flux. Let L be the characteristic linear size of the region where the Brownian motion is considered. Then the time it takes for a stationary distribution p to establish is estimated by tsL2/D ¼ 6pameL2/kT. For t  t