QUASIMOLECULAR MODELLING
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QUASIMOLECULAR MODELLING
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world scientific Lecture Notes in Physics-vol. 44
QUASIMOLECULAR MODELLING Donald Greenspan Department of Mathematics The University of Texas at Arlington
V|fe World Scientific «•
SinaaDore New Jersey Jersey •• London L Singapore •• New • Hong Kong
Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH
Library of Congress Cataloging-in-Publication Data Greenspan, Donald. Quasimolecular modelling / Donald Greenspan. p. cm. - (World scientific lecture notes in physics : vol. 44) Includes bibliographical references and index. ISBN 9810207190 1. Quasimolecules—Mathematical models. 2. Nonlinear theories. I. Title. II. Series. QC173.4.Q37G74 1991 539'.6-dc20 91-35232 CIP
Copyright © 1991 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.
Printed in Singapore by JBW Printers and Binders Pte. Ltd.
Preface "If, in some cataclysm, all of scientific knowledge were to be destroyed, a n d only one sentence passed on t o the next generations of creatures, w h a t s t a t e m e n t would contain the most information in the fewest words? I believe it is the atomic hypothesis (or the atomic fact, or whatever you wish t o call it) t h a t all things are made of atoms — little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In t h a t one sentence, you will see, there is an enormous amount of information a b o u t the world, if j u s t a little imagination and thinking are applied." Richard Feynman Lectures on Physics In this monograph we have tried to apply "a little imagination and thinking" to modelling dynamical phenomena from a classical atomic and molecular point of view. Nonlinearity is emphasized, as are p h e n o m e n a which are elusive from the continuum mechanics point of view. F O R T R A N programs are provided in the Appendices. Throughout, the spirit is t h a t in the quotation cited above. D. Arlington, Texas 1990
Greenspan
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Contents
Preface
Part I Chapter
INTRODUCTION 1
Quasimolecular Modelling: What It Is What It Is Not 1.1. Introduction 1.2. Classical Molecular Forces 1.3. General Modelling Principles 1.4. Numerical Solution Exercises
Part II Chapter
QUANTITATIVE MODELLING 2
Falling Water Drops 2.1. Introduction
vii
Contents
Chapter
Chapter
Chapter
Chapter
3
4
5
6
2.2. Mathematical, Physical, and Modelling Preliminaries
19
2.3. Parameter Selection for Water Drops
21
2.4. Dynamical Equations
26
2.5. Examples
27
Exercises
31
Colliding Microdrops of Water 3.1. Introduction
33
3.2. Mathematical and Physical Considerations for Water Molecule Interaction
33
3.3. Examples
35
Exercises
41
Crack Development in a Stressed Copper Plate 4.1. Introduction
43
4.2. Formula Derivation
43
4.3. Examples
47
Exercises
52
Stress Wave Propagation in Slender Bars 5.1. Introduction
55
5.2. Development of the Force Formulae
55
5.3. Particle Modelling of a Slender Bar
58
5.4. Examples
58
Exercises
62
Melting Points of Atomic Solids 6.1. Introduction
63
6.2. Mathematical Preliminaries
63
6.3. Model Formulation, Implementation and Results
65
Exercises
67
Contents
Part III
QUALITATIVE MODELLING
Chapter
7
Chapter
Chapter
8
9
Chapter 10
Biological Self Reorganization 7.1. Introduction
71
7.2. Cellular Self-Sorting
72
7.3. Dynamical Formulation
73
7.4. Examples
73
7.5. A Parameter Selection Process
77
Exercises
79
Cavity Flow 8.1. Introduction
81
8.2. Example
81
8.3. Additional Examples
93
Exercises
93
Turbulent and Nonturbulent Vortices 9.1. Introduction
95
9.2. Basic Definitions
96
9.3. Examples
99
9.4. Remark
105
Exercises
105
Vortex Street Modelling 10.1. Introduction
107
10.2. Vortex Street Development without Fixed Boundaries
107
10.3. Vortex Development in an Open Ended Channel
110
Exercises
116
Contents
C h a p t e r 11
C h a p t e r 12
Part IV
Porous Flow 11.1. Introduction
117
11.2. Model Formulation
117
11.3. Examples
122
Exercises
122
Q Modelling Combustion 12.1. Introduction
125
12.2. Model Formulation
125
12.3. Examples
127
12.4. Remarks
130
Exercises
130
CONSERVATIVE A N D COVARIANT
Chapter 13
Chapter 14
MODELLING
Conservative Q Modelling 13.1. Introduction
133
13.2. T h e Collisionless TV-Body Problem
134
13.3. Conservative Numerical Solution of the Three-Body Problem
135
13.4. T h e Oscillatory Nature of Planetary Perihelion Motion
143
13.5. Remarks
146
Exercises
147
Relativistic Motion 14.1. Introduction
149
14.2. T h e Concept of Simultaneity in Relativity
149
14.3. T h e Lorentz Transformation
150
14.4. Covariance
152
14.5. Relativistic Oscillation
154
14.6. Numerical Methodology
155
Contents
xi
14.7. A Relativistic Harmonic Oscillator
156
Exercises
159
A P P E N D I C E S - F O R T R A N Programs A. DROP.FOR
161
B. CRACK.FOR
164
C. CONSERVE.FOR
172
References and Sources for Further Reading
177
Subject Index
193
Chapter 1 Quasimolecular Modelling: What It Is and What It Is Not 1.1. I n t r o d u c t i o n Science is the study of N a t u r e . We study Nature not only because we are curious, b u t because we would like to control its very powerful forces. Understanding the ways in which N a t u r e works might enable us to grow more food, to prevent normal cells from becoming cancerous, and to develop relatively inexpensive sources of energy. In cases where control may not b e possible, we would like to be able t o predict what will happen. T h u s , being able to predict when and where an earthquake will strike might save lives, even though, at present, we have no expectation of being able to prevent a quake itself. T h e discovery of knowledge by scientific means is carried out in the following way. First, there are experimental scientists who, as meticulously as possible, reach conclusions from experiments and observations. Since experimental conditions can never b e reproduced exactly, and since n o one is perfect, not even a scientist, all experimental conclusions have some degree of error. Hopefully, the error will be small. T h e n there are the theoretical scientists, who create models from which conclusions are reached, often using m a t h e m a t i c a l m e t h o d s . Experimental scientists are constantly checking these models by planning and carrying out new experiments. Theoreticians 3
4
Quasimolecular
Modelling
are constantly refining their models by incorporating new experimental results. The two groups work in a constant check-and-balance refinement process to create knowledge. And only after extensive experimental verification and widespread professional agreement is a scientific conclusion accepted as valid. Our concern in this book is with a new area of theoretical modelling which is called quasimolecular modelling, or more succinctly, Q modelling, or, less precisely, particle modelling. Though specifics will follow in later sections, we observe now, for the purpose of providing an overview, that quasimolecular modelling is the study of the dynamical behavior of solids and fluids in response to external forces, the solids and fluids being modelled as systems of molecules or molecular aggregates, which interact in a fashion entirely analogous to classical Newtonian molecular interaction. The dynamical equations of Q modelling are large systems of second order, nonlinear, ordinary differential equations. Note that for linguistic simplicity, the term molecule will be used throughout as a generic term which includes both atom and molecule. The primary differences between quasimolecular modelling and molecular mechanics modelling (Alder and Wainwright (1960); Hoover (1984)) can be describe as follows. The field of statistical mechanics combines the rules of statistics with the laws of Newtonian mechanics to describe quantitative, large scale properties of continuous solids and fluids from the most probable behavior of constituent molecules. Primary goals of statistical mechanics are the derivation of macroscopic thermodynamic properties relating to such quantities as temperature, stress, internal energy, and heat flow, and the derivation of equations of state which relate pressure, energy, volume and temperature. Molecular mechanics modelling is a computer approach applied directly to a small molecular subset of a given substance with the objective of confirming or modifying large scale statistical mechanics properties or equations. The major results in molecular mechanics have been equilibrium, that is, steady state, results. Q modelling, on the other hand, is concerned, primarily, with nonsteady state phenomena and with variations in dynamical response due to variation of system parameters. In addition, Q modelling applies both to sets of molecules and to sets of molecules which have been aggregated into larger units called quasimolecules. It is through quasimolecular systems that Q modelling can be made to simulate exorbitantly large systems of molecules. For linguistic ease, we will often use the term particle rather than quasi-
Quasimolecular
Modelling:
What
It Is and
What
It Is
5
Not
molecule. However, it must be noted that this usage of the term particle is different from the usage of others. Buneman et al. (1980) and Hockney and Eastwood (1981) use the term particle to represent an ion in a plasma. Amsden (1966) and Harlow and Sanmann (1965) use the term to represent a fluid point of positive mass which moves in accordance with mass, energy and momentum conservation properties which are incorporated in a system of partial differential equations in two space dimensions. In the present context, the term particle will always mean an aggregate of molecules. 1.2. Classical Molecular Forces From the classical, Newtonian point of view, both atoms and molecules exhibit the following behavior. Two molecules, for example, interact only locally, that is, when they are in close proximity to each other. Qualitatively, this interaction is of the following character (Feynman, Leighton and Sands (1963)). If pushed together, the molecules repel; if pulled apart they attract; and the repulsive force is of a greater order of magnitude than is the attractive one. A mathematical formulation of the behavior can be given as follows (Hirshfelder, Curtiss and Bird(1954)). o r « P
•
^ X
l F i g . 1.1.
Consider two molecules Pi and P2 on an X-axis, as shown in Fig. 1.1. Let Pi be at the origin and let P2 be at a positive distance r from Pi. Let the force F which Pi exerts on Pi have magnitude F given by P = - - + - , rP
(1.1)
r«
where G,H,p,q are positive constants with q > p. Consider, for example, G — H — l,p = 6,q = 12, which are good approximations for a variety of experimental results (Hirschfelder, Curtiss and Bird (1954)). Then F
= ~r\
+^ -
(1-2)
6
Quasimolecular
Modelling
If, in (1.2), r = 1, then F = 0, so that P x exerts no force on P 2 . In this case, one says that the molecules are in equilibrium. If r > 1, say r — 2, then ^ = - ^ + ^12'
(1-3)
which is negative, so that Pi exerts an attractive force on P 2 . If, on the other hand, 0 < r < 1, say, r — 0.1, then F = _
(0l)6
+
((lip >
( L4 )
which is positive, so that Pi exerts a repulsive force on P 2 . As r approaches zero, the force F in (1.2) becomes unbounded in magnitude. Mathematically, r is not allowed to be zero because, if it were, F in (1.2) would be undefined. Physically, r is not allowed to be zero because one assumes conservation of mass, so that the same position cannot be occupied simultaneously by different physical entities. If one sets F = 0 in (1.1), then, using the reasoning above for (1.2), one finds that equilibrium results if
r=y
,
(1.5)
with an attractive force resulting for larger values of r and a repulsive force for smaller ones. It is important to observe that even though the gross motion of, for example, a fluid may be physically stable, the motion between two neighboring molecules of the fluid, in accordance with (1.1), may be highly volatile. This volatility, however, is strictly local. 1.3. General Modelling Principles To simulate the dynamical response of a solid or fluid to external forces, we will proceed in general as follows. First, we will group the large number of molecules physically present into a smaller number of subunits called quasimolecules or particles. In the case of fluids, for example, this aggregation process is exactly the same as that utilized by both Boussinesq (1913) and Prandtl (1925). Assume, then, that the number of particles which results is N. Denote these by P;,i = 1,2,... ,N and let the mass of P; be m,-. From given initial data, the motion of each P,- is then prescribed by the coupled system of ordinary differential equations: Fi = miT,
i = l,2,...,N,
(1.6)
Quasimolecular
Modelling:
What It Is and What It Is Not
7
in which F,- is the force on P,-, r; is the position vector of P,-, and differentiation is with respect to time. In (1.6), we assume t h a t F , = FJ* + F* ,
(1.7)
where F** is an external or long range force, which, like gravity, can act on all the particles uniformly or, like a driving force, can act on a particular subset of the particles; and F* is the local, short range force on P,- due to molecular type interaction with its immediate neighbors. In practice, a positive parameter D, called the distance of local interaction parameter, will often b e associated with F * . It will assure t h a t F* is, in fact, local by only allowing particles whose distance to P,- is less t h a n D to have a nonzero effect on P,-. Hence, D can be viewed as a switching parameter which turns off F? for all paricles except those close to P,-. Observe t h a t when a substance under consideration has, approximately, 10 2 2 molecules, then a choice of N in the range 10 2 < N < 10 4 yields a relatively small number of particles. For such choices of N, parameter choices like p = 6, q = 12, G = H — 1 in (1.1), which are realistic for systems of molecules, are not realistic for systems of particles, since local volatility will then also yield system volatility. In order to insure the physical stability of a system of particles, it will be necessary then to decrease the exponents p and q appropriately. Thus, molecular type attraction and repulsion will be incorporated in Q modelling, b u t with decreased local volatility in order to assure physical stability of the system. 1.4. N u m e r i c a l S o l u t i o n In general, system (1.6) cannot be solved analytically from given initial d a t a and must be solved numerically. T h e choice of a numerical method is simplified by the fact t h a t the physics of Q modelling demands small time steps. T h e reason is t h a t only with small time steps can the repulsive component H/r9 in (1.1) be treated accurately for small r, since H/rq is unbounded as r goes to zero. T h u s , the advantages in using high-order numerical methods, which allow the choice of large time steps in obtaining high-order accuracy, are not applicable in Q modelling. Hence, for economy, simplicity, and relative numerical stability, we will utilize the leap-frog formulae (Greenspan (1980a)), which are described as follows. For positive time step At, let tk = kAt,k — 0 , 1 , 2 , . . . . For i = 1 , 2 , . . . ,N, let Pi have mass m* and at tk let Pi be located at r ^ , have
8
Quasimolecv.hr Modelling
velocity vik and have acceleration a, *. T h e n the leap-frog formulae, which relate position, velocity and acceleration are
(At)
Vi.l/2 = V,-,0 + —^ai,0
tarter formula)
,
v,-,fc+i/2 = v .\Jb-i/2 + (At)a,- iib r,,it+i = ritk + {At)vik+i/2
,
(1.8)
Jb = 1 , 2 , 3 , . . .
(1.9)
Jb = 0 , l , 2 , . . . .
(1.10)
T h e name "leap-frog" derives from the way position and velocity are denned at alternate, sequential time points. Note also t h a t if (1.9) is solved for a t| jt and (1.10) is solved for Vitk+i/2, then the resulting formulae are central difference, 0((At)2) approximation formulae. If at time tk one rewrites (1.6) and (1.7), respectively, as Fi.jt = rrii&i^,
i = 1 , 2 , . . -,N
(1.11)
F
«" = 1 , 2 , . ..,N
(1.12)
>,fc -
F
F
i,fc + i,fci
then (1.8)—(1.12) determine the positions and velocities of all N particles recursively and explicitly from given initial data. E x a m p l e . To illustrate the numerical procedure to be followed, consider the following simple example in only one space dimension. On an X-axis, let P\ and P 2 , with masses m i = 2 , m 2 = 1, be located initially at x\to = 0,^2,0 = 1 and have initial velocities t ^ o = —1,^2,0 = 3. Let the distance of local interaction be D — 1.5 and set At = 0.1. Let the forces on P i and P2 at tk be given by F**k = - 9 8 0 , (1.13)
F
b = ( - 1 — ' — ^ + i — 2 — i * ) T^ir-i V
\x2,k-Xitk\
\x2,k -Xltk\
J
(1.15) •
(116)
T h e n , from (1.8)-(1.11), one finds for P i t h a t ^1,1/2 = v i,o + (0.05)01,0 , i>i,fc+i/2 = n,k-i/2 xi,k+i
\X2,k-xl,k\
n% = 0 , F*2,k = ~Kk
-
(Starter formula)
+ (0.1)ai ifc ,
k = 1,2, 3 , . . .
= *i,fc + (O.l)vi,k+1/2 ,
k = 0,1, 2 , . . .
Quasimolecular Modelling: What It Is and What It Is Not
9
or, equivalently, t h a t wi.i/3 = - 1 + (0.05)(Fi, 0 /2) = - 1 4 - ^ ( 0 . 0 2 5 ) ^ , 0 ,
(1.17)
"i,*+i/2 = "i.fc-i/2 + ( 0 . 1 ) ( F M / 2 ) = wi,fc_i/2 + ( 0 . 0 5 ) F l i t , (1.18) *l,Jfc+l = xl,k + (0.1)wi,fc + i/ 2 •
(1-19)
Since |x 2 ,o - *i,o| < 1.5 = D, it follows from (1.12)-(1.14) t h a t •Fi.o = F**o + ^i*o
= -980+ ( - ,
L _ +
1*2,0 - * l , o | 3
2
^ «M-«».o
\X2,0 - Xl,o\6 J 1*2,0 - X i , 0 |
= -980 + (l)(-l) = -981 . T h u s , from (1.17), wi.i/2 = - 1 + (0.025)(-981) = - 2 5 . 5 2 5 .
(1.20)
One finds in a n analogus fashion t h a t "2,1/2 = "2,0 + (0.05)a 2 ,o , "2,fc + l/2 = "2,ib-l/2 + (0.1)a2,jb , X2,k+1 - *2,k + (0.1)t>2,Jfe+l/2 '
"2,1/2 = 3 4- (0.05)(F 2 , 0 /1) = 3 4- (0.05)F 2 , 0 , "2,fc+l/2 = "2,fc-l/2 4" (0.1)(F 2 ,fc/l) = "2,*-l/2 4- ( 0 . 1 ) F 2 i t , Z2,i + 1 = X2,k + (0.1)"2,fc + l/2 •
(1.21) (1.22) (1-23)
Since ^2,0 = ^2*0 + ^2*0 '
it follows from (1.13), (1.15) and (1.16) t h a t ^2,0 = 0 - Fl0
= 1.
T h e n , from (1.21), "2,1/2 = 3.05 .
(1.24)
Quasimolecular
10
Modelling
Thus, the velocities "1,1/2 and "2,1/2 °f ^1 a n ( l -^2 a t the time t = 1/2 have now been determined and are given by (1.20) and ( 1 2 4 ) . T h e formulae (1.19) and (1-23) with k = 0 now yield the new positions i n and £2,1 of P i and P2, as follows: *i,i = *i,o + (O.IK,1/2 = ° + (0.1)(-25.525) = - 2 . 5 5 2 5 ,
(1.25)
*2,i = z 2 ,o + (0.1)^,1/2 = 1 + (0.1)(3.05) = 1.305 .
(1.26)
T h e process now continues to determine next "1,3/2, "2,3/2- But since the distance |x2,i — # i , i | = 3.8575 > 1.5 = D, the switch is applied so t h a t Ki
= F 2,i = 0 •
(1.27)
Observe also t h a t the notation in (1-27) should always remain clear if one remembers t h a t the first subscript is always the particle number and the second is always the time step. Once formulae (1.17) and (1.21) have been used to determine "1,1/2 a n d "2,1/2 > they are no longer used. All the remaining trajectory calculations are done with (1.18), (1.19), (1-22) and (1.23). Hence, the counter is now set to k = 1. From (1.18) and (1.22), then, "1 3/2 = "1 1/2 + (0.05)Fi** = - 2 5 . 5 2 5 + (0.05)(-980) = - 7 4 . 5 2 5 , (1.28) "2,3/2 = "2,1/2 - ( 0 . 0 1 ) P 2 1 = 3.05 - (0.01)(0) = 3.05 .
(1.29)
Now, having the velocities of Pi and P 2 at £ = 3/2, we find their new positions from (1.19) and (1.23) to be 112 = 111 + (0.1)ui 3/2 = - 2 . 5 5 2 5 +• (0.1)(-74.525) = - 1 0 . 0 0 5 , (1.30) *2,2 = x2,i + (0.1)^,3/2 = 1-305 +• (0.1)(3.05) = 1.610 .
(1.31)
T h e counter is then increased to k = 2 and the iteration continues in the indicated fashion. W i t h regard to the leap-frog formulae and their application, several relevant observations must now be made. First, note t h a t (1.8)—(1.12) have been given in vector form, so t h a t they can be applied in 1, 2, or 3 space
Quasimolecular
Modelling:
What It Is and What It Is Not
11
dimensions, as needed. Of course, in two dimensions, one would, in general, have r.',fc = (ri,k,x,ri,k,y)
—
V:,fc = (Vitk,x,
Vi,k,y),
a
a
a
j',fc — \ i,k,x>
(xi,k,yi,k)
i,k,y)t
F«,ib = (Fi,k,x,Fitk,y)
,
while in three dimensions one need only append a 2-component to the above formulae. Next, note t h a t typical F O R T R A N programs for IBM, D E C , VAX, and CRAY mainframe computers are provided in the Appendices. Finally, note t h a t for N relatively large, t h a t is, N ~ 5000, the determination of the nearest neighbor for each particle of a system will usually be the most time consuming part of any simulation. This is particularly valid in simulations of fluids. Indeed, when one simulates a solid, the near neighbors of any P, can often be given uniquely and explicitly for all time. But when one simulates a fluid, this is not the case. For this reason, there have been a variety of "economical near-neighbor" algorithms developed recently for simulations of fluids(Boris (1986)). However, in each case, one either does not include all the neighbors or else one is forced to alter the particle ordering. In the latter case, one cannot follow the trajectory of any particular particle from an initial to a later time, and this capability is desirable for our purposes, since we may wish to explore, for example, the motion of individual particles at the onset of turbulence. Thus, we will not take advantage of "near-neighbor" algorithms, which, at present, have their primary value in molecular mechanics modelling. Exercises 1.1 Argue for or against any one of the following: (a) Mathematics is a science. (b) Physics, chemistry and biology are sciences. (c) Astronomy is not a science because it has no experimental component. (d) All things change with time, including science. (e) Astrology has aspects of science. (f) Economics, sociology, and psychology are sciences.
12
Quasimolecular
Modelling
1.2 Find the equilibrium distance for each of the following: , ^ r,
1-2
(b)F = - ^ . , „
1-2 ^ .
+
1.2
1.0
1.3 Find the equilibrium distance for each of the following:
wj? = - ^ + ^ • „
N
„
1
1-21
(fe)^ = - ^ + — 1-21
•
1
1.4 Consider two particles Pi and Pi in motion in an XY-plane. Let Pi have mass rn\ — 2 and initial data x\to = 0,yi,o = 10,«i,o,r = 0,vio,j, = —15. Let Pi have mass mi = 1 and initial data £2,0 — 10,2/2,0 = 0,112,0,1 = — 10,i>2,o,y = —4. Let the local distance of interaction be D = 5. Using the leap-frog formulae with At = 0.01, determine the motion of Pi and P2 through iioo for the force formulae
F f t = (*!,*,«. *i,*,v) = (°>- 9 - 8 )> 'r2i,* 1,4
[
r
ia,k
r
UkJ
2,k — * l , J b 1 * 2 , * -
- J !
Vw,fc l,k
1.5 Repeat Exercise 1.4 but use the force formulae
V r !2,*
'
171* +
F* *
2,k -
F *2,fc
~*l,fc > -p*
-
"*!,* '
r
?2,*y r12,t
Quasimolecular
Modelling:
What It la and What It Is Not
13
r
ij,k = \]{xi,k ~ xj,k)2 + (Vi.k ~ Vj,k)2 •
1.6 Repeat Exercise 1.4 but use the force formulae
Fi; t =0, • **
F 2,ifc
r
r12tk
\
r
12,k
T +* t
- ~ l,k
'
2,k — ~tl,k
>
12,kJ
r
12,k
o\* = v(*»".* - xj*f + (&-* _ w.*)2 •
1.7 Consider two particles Pi and P2 in motion in three dimensions. Let Pi have mass mi = 2 and P2 have mass m.2 = 1. Let the initial data be «i,o = -1,2/1,0 = 0, z1>0 = 0, ui)0)ar = 0, i>i|0|y = 0, u li0|Z = - 1 , ^2,0 = 1, 2/2,0 = 0, Z2>0 = 0, V2,0,x = 0, T>2,0,y = 5, V2,0,z = 1 •
Let the local distance of interaction be D = 5. Using the leap-frog formulae with At — 0.01, determine the motion of Pi and P 2 through tfioo for the force formulae ST* = (0,0,-9.8), F
l,fc \
J r i2,fc
| r21,t ' _4 1 »"l2,t r i2,fc;
*2,fc - *l,fc ' * 2 , i - —* l.Jfc ' r
O\* = Vfa*",*~ X J,*) 2 + (j/«,* - yj.kf + (z',k ~ Zj.kf
1.8 Repeat Exercise 1.7 but use the force formulae
F
i,* -
- 3 — + 3— (irrr) > \
r
12,*
r
l2,*/
\r12,t/
Quasimolecular
14 2,k *
F 2,* r
t
l,k
Modelling
'
Tp*
-
— *l,fc
.
o\* = v(x>,fc - x:,k)2 + (yi,k - yj,kf + (zi,k - zj,k)2
1.9 Repeat Exercise 1.7 but use the force formulae r *» F
M =
f_ \
1° \ r2i,fc Ir2 — r7 " ^ ' i 2 , * / 12,*
Fft = 0 ,
r12>k < D ,
F^/^ '
r
,
+^ W r
V
i2,*
2,ifc - ~tl,k
F
^ n i2,k>D
r
'-12,*/ r i2,*
'
IP*
*
2,k -
-1!
l,fc >
'J,* = V (*«",* - xJ,k)2 + (2/«,* - 2/;,it)2 + (*,-,* - Zjik)2
.
>
1.10 Consider 3 particles Pi,P2,^ 3 of respective masses mi = 1000, m 2 — 20, m 3 = 1. The given initial data are *i,o = 0. 2/i,o = 0, zi,o = 0,
VII0IX
= 0, i>i,o,y = 0, vifiiZ = 0,
*2,0 = 10, 2/2,0 = 0, 2 2 ,0 = 0, V2,0,x = 0, V2>0,y = 10, l>2,0,z = 0, «3,0 = - 1 0 ,
J/3,0 = 0,
Z3.0 = 0, 03,0,1; = 0, V3,0,y = 0,
v 3 ,o,, = 20 .
Let no distance of local interaction be prescribed. Using the leap-frog formulae with At = 0.01, determine the motion from the force formulae „•„ _ (
mim2\
r 2 i,t
(
mim 3 A r 3 i , t
„„,
mim 2 \ r 12|i;
/
m2m3 ^\ r32,fc
i2,it ) i2,fc
V
r
mim3\ n3,t
/
m2m3\
V
r
F
_ /
2,k
~ V
„
_ (
^
~ V
r
r
r
r
13,* / 13.»
23,fc ) r23,k ' r23,fc r
23,* / 23,* '
Fi >t = r;,fc = Fs >t = o, r
«j,t = v (^i,* - xi,*)2 + (*/»,* - yj,*)2 + (*«,* -
z
i*f
Quasimolecular
Modelling:
What It 1$ and What It Is Not
15
1.11 Consider 3 particles P\,P2,P3 which are in motion in the X Y - p l a n e . Assume t h a t mi = m 2 = m3 = 1 and t h a t the initial d a t a are: xifi = 0.50, yifi = 0,vi ) O ) r = -0.5,1)1,0^ = 0 , *2,0 = -0.50,2/2,0 = 0,^2,0,1 = 0.5,V2,0,y = 0.01 , ^3,0 = 0,1/3,0 = 0.87, V 3 | 0 | I = 0.01,V 3|0 ,y = - 0 . 9 . Let all long range forces be 0 and, for D acting on P i at time tk be given by (1)
FlikiT
mim.2 3~~ 12,it
+ (2)
F;i.*,y
m\mi r
mi?7i3 ' 13,k
m\rri2 5 12, k
m\ni3
mims
'13,*
Xl,k - XZ,k ri3,k
'13,* 2/1,* -
? '12,it
+
2,k
ri2,k
m\mi r
x
Xl,k -
5 12,*
+
2.5, let the local force
2/2,*
ri2,* miffl3
2/1,* -
'13,*
2/3,*
7*13,*
(3) rijik = \J(xitk - xjik)2 + (yi>h - yjtkf Let the local acting on P 2 be given by ( l ) - ( 2 ) with the numbers 1 and 2 interchanged, while t h a t on P3 is given by ( l ) - ( 2 ) with 1 and 3 interchanged. T h e n determine the motion of the system through t = 2 using the leap-frog formulae with At = 0.0001. 1.12 (a) Discuss the following free translation of Zeno's "Achilles and the Tortoise"" paradox. A fast runner and a slow tortoise are to have a race. Because of the runner's superior speed, the tortoise is allowed to begin the race at a positive distance d ahead of the runner (see the figure). Let the runner's initial point be P and t h a t of the tortoise be Q. After the race has begun, the runner must reach the point Q, which takes time, during which the tortoise moves ahead to a new point Q\. The runner must then reach the point Q l f which takes time, during which the tortoise moves ahead to a new point Q2- T h e runner must then reach the point Q2, which takes time, during which the tortoise moves ahead to a new point Q3, and so forth. T h u s the runner must always reach a
16
Quasimolecular
Modelling
point where the tortoise has already been, from which it follows that the runner, no matter what his speed, can never overtake the tortoise, (b) Show that if one takes a molecular viewpoint, then no paradox exists. \
,
—i
[•
d
1
•
P
Q
Qx
•—< Q2
.
Q3
1.13 Show how to derive the Navier-Stokes equations from a molecular model of a fluid. 1.14 From a molecular point of view, what is fluid surface tension?
Chapter 2 Falling Water Drops 2.1. Introduction T h e study of fluid drops has long been of interest to mathematicians, scientists and engineers (Bond (1927), Boussinesq (1913), Finn (1986), Harlow and Shannon (1967), P r a n d t l (1914), Schlichting (1960), and Simpson (1923)). In the spirit described in C h a p . 1, we explore in this chapter a Q model approach to falling water drops. In the present discussion, attention will be limited to two space dimensions. In this and in all succeeding chapters, cgs units will be the initial choice. A major difficulty in modelling liquid drops in due to the large gradients which result from surface tension. And, since surface tension is not a consequence of the Navier-Stokes equations, these important partial differential equations are not directly applicable to the formulation of viable continuum models (Daly (1969)). 2.2. M a t h e m a t i c a l , Physical, and Modelling Preliminaries T h e gross physical response of a fluid to external forces is, primarily, the result of forces due to gravity and due to molecular interaction. Gravity acts uniformly on all molecules in a fluid. Molecular interaction forces have components of b o t h attraction and repulsion. Classically, these forces have magnitude F given by 19
20
Quasimolecular F
= ~^
G > 0
+ ^'
H>0
>
<
3>P>7,
Modelling
(2.1)
where r is t h e distance from molecule P to a neighboring molecule. Because of the singularity in Eq. (2.1) at r — 0, the motion of an individual molecule can be relatively volatile locally, even though the gross motion of the fluid is physically stable. To simulate fluid motion, we proceed as follows. First, we group t h e large number of fluid molecules which are physically present into a relatively small number of quasimolecules. Consider then, N quasimolecules P,-, i — 1 , 2 , . . . , N. For A t > 0, let tk = kAt, k = 0 , 1 , 2 , . . . . For each z, let m,denote t h e m a s s of P ; a n d let Pi at tk be located at r , * = (xi.fc, y,-,*), have velocity v,-^ = ( f t | j ; , r , Vi,k,y) and have acceleration a,jt = (a,-*^, 0,-^y ). At tk, let the force acting on P,- be F , ^ = (FiklX,Fi,k,y)We relate force a n d acceleration by t h e dynamical equation F,-,* = m,-a j i t .
(2.2)
T h e motion of each P,- will be determined explicitly and recursively by t h e leap-frog formulae from given initial d a t a once the force F,-^ is prescribed, and this is done as follows. First, fix positive p a r a m e t e r D, the distance parameter. We do not exclude the possibility t h a t D is infinite. Any particle Pj, different from P,-, which lies within a circle of radius D a n d center P,- is called a neighbor of P j . If Pj is a neighbor of P,-, let r,jk b e t h e vector from Pi t o Pj at time tk, so t h a t r ^ * = ||rt|fc — r ^ H is t h e distance between t h e two particles. T h e n t h e force F*- k on P,- due to Pj at time tk is defined by H
F
7j,fc =
.
(rii,k)P
+ (rij,k)<>
(2.3) m.k
T h e force F* k on Pj at tk is defined by
F
U = Y,Fh,x.
(2-4)
i=i jjti
where the summation is taken over all neighbors of P j . Finally, t h e total force Fi}k on P,- at tk is defined by Fi,k,x = Fi,k,x>
F
i,k,y = Fi,k,y -
m
i9
where g is the constant of acceleration due to gravity.
>
(2-5)
Falling Water Drops
21
2 . 3 . P a r a m e t e r S e l e c t i o n for W a t e r D r o p s For the simulation of water drops by quasimolecules, the choices of m, G, H, and D will be dominated by physical considerations. T h e choice of N will be dominated by computational and budgetary considerations. T h e choices of p and q will be dominated by b o t h computational and physical considerations. To begin, set p = 3,5 = 5. Such a choice of exponents will result in a physically stable system. In addition, the calculation of force components in the coordinate directions will result in divisions by r 4 and r 6 , thus avoiding time consuming and expensive square root processes. In addition, a relatively large time step At can be employed in the numerical routine. Next, we will choose N to be approximately 1000 and proceed as follows. Consider a rectangular basin of width 2 cm, as shown in Fig. 2.1. Into this basin set N — 1111 quasimolecules P, at the respective points (xi,yi) determined by Xl
= - 1 , 2/1 = 0; X52 = - 0 . 9 8 , 2/52 = 0.034641016 ,
xi+i
= 0.04 +xit
xi+i
- 0.04-t-x,-, j/j+i = 2/52, t = 5 2 , 5 3 , . . . ,100 ,
Xi = xi-i01,
yi+i
= yi,
i = 1 , 2 , . . . ,50 ,
y,-= 0.069282032+ 2/i_ioi, i = 1 0 2 , 1 0 3 , . . . . 1111 .
T h e resulting arrangement is shown in Fig. 2.2. T h e (x,-,2/i) are vertices of a regular triangular mosaic in which the distance from any P,- to an immediate neighbor is 0.04 cm. T h e height of the system is 0.72746133 cm and there are 22 rows of particles which contain, alternately, 51 and 50 particles. Next, the mass parameter m of each quasimolecule will be determined by mass conservation. For this purpose, suppose t h a t the region filled by quasimolecules, as shown in Fig. 2.2, were t o be filled by molecules. Now, for two water molecules, a simplistic potential function <j)(r) is given by (Hirshfelder, Curtiss and Bird (1954)) <^(r) = Ac
<26)
(T-(v)V
where the distance r between the molecules is in angstroms and a = 2.725 A, e = 707 cal/mol. Conversion to ergs yields -13 <j>{r) = 1.9646383 x 10
2.725\12
^2.725
erg .
(2.7)
22
Quasimolecular
Modelling
Fig. 2.1. The basin.
-. X
Fig. 2.2. The initial configuration of quasimolecules
Hence F(r) = 1.9646383 x 1(T 5
1 2 (2.275)
12
6 (2.275)
,13
6
dynes .
(2.8)
Now, when <j> is a minimun, then F — 0, which yields 12(2.725)12 _ _13
~ °
(2.275)6 _7
— " l
the solution of which is r = 3.058709 A .
(2.9)
23
Falling Water Drops
We now fill the basin up to the height 0.72746133 cm with molecules which are also arranged on a regular triangular mosaic, b u t so t h a t the distance from any molecule to a nearest neighbor is 3.058709 A . T h e generating regular triangle for such a mosaic has edge length 3.058709 A and height 2.6489196 A. Since the basin area to be covered is 2 cm by 0.72746133 cm, it follows t h a t the number N* of molecules which fill the area is, approximately, N
.
=
_2_xT0*_ 3.058709
x
0-72746133 x 10*
=
2.6489196
v
'
Moreover, since the mass of an individual water molecule is 30.103 x l 0 _ 2 4 g , the total molecular mass M is approximately, M = 5.4055854 x 1 0 _ 8 g .
(2.11)
This total mass is now distributed over the 1111 quasimolecules by the choice 5.4055854 x 10~ 8 = 4.8655134 x H T n g . (2.12) 1111 To approximate G and H, we will impose energy conservation as follows. Assume first t h a t the total energy E of the molecular configuration described above is totally potential. Now, the energy determined by one molecule and an immediate neighbor is, from (2.7), (/.(3.058709) = -4.9115957 x 1 0 ~ 1 4 erg .
(2.13)
Since the m i n i m u m potential results for two such adjacent molecules, we can estimate E as follows. Suppose one traverses the rows of molecules from left to right, starting at the top, and suppose one comes to a particular molecule Po, as shown in Fig. 2.3. Except at boundary points, PQ will have exactly six immediate molecular neighbors, which are shown as PI,P2,PZ,PA,PS and PQ in Fig. 2.3. However, the potential energies of the pairs P0P2, P0P3, P0P4 have already been determined by the m e t h o d prescribed for traversing the rows. T h u s , at Po, only three contributions are made to the energy calculation, and these are for the pairs P0Pi, P0P5, PoP6- Hence, from (2.10) and (2.13) the total energy is approximately,
24
Quasimolecular
Modelling
E = 3(1.7956966)1015 x (-4.9115957)l(r 1 4 erg or, equivalently, E = -2.6459207 x 102 erg .
(2.14)
Fig. 2.3. A quasimolecule and its six nearest neighbors.
Now, for the quasimolecular system, we have thus far G
H
,
H 4^
e i g
(2.15)
with R measured in centimeters. Hence ^
=
G - 2 ^
+
(2.16)
Let us assume that all the energy of the quasimolecular system is also potential. Thus, for R = 0.04 cm, it follows that G (0.04) 3
+
H (0.04) 5
0
(2.17)
while conservation of energy implies, from (2.14), the approximation 3(1111) ( -
G H 2 + 2(0.04) 4(0.04)
= -2.6459207 x 102 .
(2.18)
25
Falling Water Drops
The solution of system (2.17), (2.18) is G = 5.0806758 x 10~ 4 , H = 8.1290812 x 10~ 7 , which yields for the quasimolecules the force magnitude 5.0806758 x 10~ 4 8.1290812 x lO" 7 # + & •
„,„, *"<*) =
,0 i n , (219)
Unfortunately, force formula (2.19) is, itself, unrealistic and requires normalization. This can be seen dynamically by examining force components in the Y direction only. For, if R = (X, Y), the motion of quasimolecule P would require that d2Y
v ^ 17 5.0806758 x 10~ 4
= -980m + X; (
jp
+
8.1290812 x 1 0 - 7 \ Y
#
JS (2.20)
where the summation is taken over those quasimolecules which interact with P. However, from (2.12) it then follows that
w ="
980+
£
1.0442219 x 107 R3
+
1.6707755 x 1 0 4 \ Y_ R5 J R
, (2.21)
in which the magnitude of the force interaction between quasimolecules is so great that gravity may have no effect at all on any resulting motion. Thus, normalization is required and is accomplished appropriately as follows. In place of (2.21), consider d2Y dP
v ^ 17 10442219 x 107
1.6707755 x 1 0 4 \ Y
(2.22) where a is a normalization constant. Guidance for the choice of a is now derived from actual molecular interaction. In the molecular case, formula (2.8) is local in the sense that molecules more than five equilibrium distances away from a given molecule have, relative to gravity, a negligible effect on that molecule. We extend this result by assuming that any quasimolecule more than five equilibrium distances away from a given quasimolecule has, relative to gravity, a negligible effect on that quasimolecule.
26
Quasimolecular
Modelling
Now, for quasimolecules, the equilibrium distance .Ris, from (2.19), approximately, R — 0.04. Hence, 6R = 0.24 and we will determine a in accordance with the inequality 1.0442219 x 107 (0.24) 3
+
1.6707755 x 104 < (1%)(980) , (0.24)5
(2.23)
from which it follows that a ~ 1.3344478x 1 0 - 8 . However, this quantity can be approximated reasonably in terms of the constants already developed. Indeed, let Q be the number of molecules which are aggregated into a quasimolecule, so that Q = JV"7llll. Then, Q = 1.6162885 x 1012 and 1/VQ = 7.8657574 x 1 0 - 7 . Hence, for simplicity, we set a = 1/y/Q = A / ^ ^ = 7.8657574 x 1 0 - 7
(2.24)
and the magnitude F in (2.19) is replaced finally by F = 7.86575574 x 10" 7 / _ 5-0806758 x 1 0 ^ R3
8.1290812 x 10~ 7 R5
or more concisely, F =
^_3.99g3541()_10
+
6.3941366 1Q _ 13 ^
(JJ
^
Finally, since force interactions greater than six times the equilibrium distance are relatively small, we now set the distance parameter D by D = 0.2. 2.4. Dynamical Equations For the motion of the system of quasimolecules from initial data, it follows from the discussion in Sec. 2.3 that the dynamical system to be solved is d2R
«
non
«- V^ 17
3.9963354 3.9963354l 1A f t __110 0 ^ 6.3941366 R5
13\
R,, ) R(j j (2.26)
27
Falling Water Drops
in which 6 = (0,1) and the summation is taken over all neighbors within a radius of 0.20 cm. However, using (2.12) and the transformation R : = 0.074330624 R* ,
(2.27)
t = 0.027263643 T ,
(2.28)
the equation transforms into
£f-* + £
V
200
58 ^
(2.29)
which is less sensitive to the choice of time step than is (2.26). It is (2.29) which is utilized in the examples to be described next. 2.5. Examples From (2.29), it follows that, in the new coordinates R* = (X*,Y*), the equilibrium distance is now 0.54 and the coordinates of the 1111 quasimolecules are X{ = -13.75, x;+1
Yj* = 0,
= o.55 + x;,
X*+l = 0.55 + X*, X* = Xi-101,
x;2 = -13.475, Y5*2 = 0.47631397 ,
¥;+! = ¥{,
.- = 1,2,... ,50,
Y/ +1 = Y*3,
i = 5 2 , 5 3 , . . . , 100 ,
V = 0.95262794 + Y/_ 101 ,
i = 102,... , 1111 .
In addition, we will avoid complete symmetry by prescribing small initial velocities, which keep the kinetic energy close to zero, by V? = (-0.00001, 0.0),
i = 1,2,... , 101 ,
V ; = (0.0, -0.00001),
i = 102,103,... , 201 ,
v ; = (o.ooooi, -o.ooooi),
% = 202,203,..., 1111.
We now set A T = 0.0002, Tk = kAT, k = 0 , 1 , . . . . The system of 1111 particles is then allowed to interact in accordance with (2.29) and the numerical solution is generated by the leap-frog formulae. If any particle collides with a wall of the basin, it is reflected symmetrically, but its velocity is damped by the factor 0.8. The interaction distance used is D = 2.7. The system is allowed to interact through Tgoooo,
Quasimolecular
(a)T0
•->; - « : ^.
(b) T 6 0 0 0
(c) T12ooo
(d) Ta,
Fig. 2.4. Pendent drop formation.
Modelling
29
Falling Water Drops
at which time the fluid particles are reflected about t h e X-axis, the walls are deleted, and a ceiling of 201 additional particles is added by the rule x ; = - 2 5 . 0 + (i-llll)(0.25),
Yi*=0.25,
t = 1111,1112
1312 .
This result is shown in Fig. 2.4(a). T h e quasimolecules of the ceiling are fixed, t h a t is, they are not allowed to move, and are called the solid particles. However, they are allowed to interact with the fluid particles with a force effect given by
F
= ~W? + Wf-
<230)
One should observe now t h a t the motion of real fluid drops can take relatively long periods of time (Adamson (I960)). To avoid such a situation, we shall choose G and H in (2.30) so t h a t fluid adhesion at the ceiling is relatively strong and drop formation relatively rapid. For this reason, we set G = 250 and H = 75. T h e entire system is now allowed to interact for 6000 time steps, with no imposed damping, b u t with the gravity force decreased from g = 9.8 to g — 0.98 to prevent the immediate fall of the fluid from the ceiling. T h e counter was reset to T — 0 and the resulting formation is shown in Fig. 2.4(b) at Tgooo- T h e system was then allowed to continue its interaction for 6000 more time steps to T^oooj but with g reset correctly to 9.8. T h e result is shown in Fig. 2.4(c). Finally, the system is allowed to interact for an additional 14000 time steps t o T26000, but with velocities damped initially and every 2000 steps thereafter by the factor 0.2. T h e result is shown in Fig. 2.4(d). T h e effect of the damping is to decrease the total kinetic energy, or internal heat. T h e result shown in Fig. 2.4(d) is in complete agreement with experimental shapes of pendent water drops (Finn (1986)). Moreover, it is worth noting from Fig. 2.4(d) t h a t the fluid quasimolecules at the free surface show a greater separation t h a n do the interior fluid quasimolecules, which is consistent with the theory t h a t it is this relatively large separation in the free surface which manifests itself as surface tension. Next, let us begin with the d a t a for the pendent drop shown in Fig. 2.4(d) and use t h e m as initial d a t a to study the fall of a pendent drop. We again reset the timer. Now, the drop shown in Fig. 2.4(d) will not fall of its own accord. T h u s , it must be dislodged by a suitable force. This can b e done in a variety of ways. A computationally convenient way
30
Quasimolecular
Modelling
(a) T 1 0
(b)Ts
(0T11JO
(c) T,, (g) J1200
(d) T 9 5
(h) T „
0) T , 5 0 (e) T , ,
(J) T l t F i g . 2.5. P e n d e n t drop fall.
to do this would be simply to increase g, because once the drop has been dislodged, its shape will depend only on internal forces, though, of course, the distance it has fallen will not. Since our primary interest is in the shape of the falling drop, and since, again, we desire a relatively rapid reaction, we merely take the initial data as that provided by the output for Fig. 2.4(d), eliminate all damping, and set g to 98.0. The resulting fall of the drop is
Falling Water Drops
31
shown in Fig. 2.5(a)-(j), at the respective times shown. Note finally t h a t the computer program which was adapted to the simulation is given in Appendix A. Exercises 2.1 Reproduce Fig. 2.5. 2.2 Show t h a t after Xi6ooo> the falling drop in Fig. 2.5 becomes more circular with time. 2.3 Simulate a falling drop with each of the following changes from those assumed for Fig. 2.5: (a)V;* = 0, i = 1 , 2 , . . . , 1 1 1 . (b) 1% is replaced by 0 . 1 % in (2.23). (c) G = 225, # = 65 in (2.30). (d) Dislodge the pendent drop differently. 2.4 Simulate a falling drop of oil. 2.5 Simulate a drop of water which forms on top of a flat carbon surface. 2.6 Simulate the rise of water in a glass capillary t u b e . 2.7 Simulate the fall of a drop of water from the t a p of a sink. 2.8 Simulate the fall of a drop of water into a basin of water.
Chapter 3 Colliding Microdrops of Water 3.1. Introduction Collisions of microdrops are important in microwave, chemical nucleation and raindrop studies (Adam, Lindblad and Hendricks (1968), Peterson (1985), Simpson and Haller (1988)). In this chapter we shall show how to simulate collisions of microdrops of water. However, since attention will be restricted to the micro level only, we shall deal directly with molecules rather than with quasimolecules. The general dynamical formulation is then given completely in Sec. 2.2 by Eqs. (2.1)-(2.5). Further, since the interaction during collisions will be independent of gravity, we will set g — 0 in (2.5). 3.2. Mathematical and Physical Considerations for Water Molecule Interaction Recall first the elementary water molecule potential (2.7), that is, 12 2.725 2.725 (3.1) erg <j>{r) = 1.9646833 x 10"
33
Quasimolecular
34
Modelling
in which r is measured in angstroms. From (3.1) the force F, in dynes, between two molecules r A apart has magnitude F given by 2.725
F(r) = 4.325809 x 10"
13
2.725
(3.2)
In considering how to proceed, let us recall first that all mathematical models are only approximations of the real thing. Second, note that one of our primary objectives is to explore different aspects of, and approaches to, Q modelling. For these reasons, we will not use (3.2) directly in the present chapter, but will develop a modified, simpler approach. The direct use of a molecular formula will be explored in another context later. Consider then a least square fit of (3.2) by the function F*(r) = 4.325809 x 10 - 5
G
(3.3)
For the fit one can determine as many data points (r,F(r)) as one desires from (3.2). For simplicity, let us consider only the five values r = 2.5,2.75,3.0,3.25,3.5, which contain the equilibrium point r = 2.75. Then, from (3.2) F(3.5) = -0.09616,
F(3.25) = -0.08888,
F(2.75) = 0.83084,
F(3.0) = 0.06291,
F(2.5) = 4.30357
(3.4)
from which it follows that the least square fit is F*(r) = 4.325809 x 10"
115
1104
(3.5)
Since F is in dynes and since the mass of water molecule is 30.103 x 10~ 24 g, a dynamical equation which describes the motion of one water molecule which interacts with only one other water molecule r A away is 4.325809 x 1 0 - 5
115
1104
•^-+—5-
= 30.103 x 10 _ 2 4 a
(3.6)
where a is measured in cm/s 2 . Changing to A/s 2 and also introducing the computationally convenient time transformation T2 = 10nt2 yields the dynamical equation d2^__16J5 158.6 • (3J) dT2 - - r 3 + rs
35
Colliding Microdrops of Water
For a system of N water molecules Pi, P21 • • • 1 P/v > it follows from Eq. (3.7) that, from given initial data, the motion of each P,- can be determined by solving the system of second order, nonlinear, ordinary differential equations d*r, A / 16.5 158.6 "j r,,.
^ SrT+^rJ^'
<38>
-
in which r,- is the position vector of P,-, r,-,- is the vector from Pj to P,- and rtj is the length of r,,-. 3.3. Examples Before studying the interaction of two water drops, it is necessary to generate a single drop, which is done as follows. Since <^(2.725) = 0, consider a regular triangular mosaic of points (a;j,t/,), i = 1,2,... ,9000, given by: xi = -68.125, 1/1 = -58.997975, xi+i = 2.725 +a;,-,
y,-+1 = yi,
xi+i = 2.725 + xit
yi+i-y52,
*,- = *i-ioi,
x52 = -66.7625, y52 = -56.638056,
i = 1,2,... ,50, i = 52,53,... ,100,
» = 4.7198384+ y,-_ioi,
t'= 102,203,... ,9000 .
From these we choose only those which satisfy xf + yf < 2320, thus yielding 1128 points which lie in a relatively circular pattern. At each such point (xi>yi) w e place a water molecule P,-. Each of the 1128 water molecules P,- is now allowed to interact with all other molecules in accordance with Eq. (3.8). For simplicity, we assume first that all initial velocities are zero. The differential system is then solved numerically with AT = 0.0002 until T = 11.2. At this time the system has contracted maximally, so that its energy should be almost all potential. Thus, at T =11.2 all velocities are reset to zero and the system is allowed to interact until T=14.0, at which time all velocities are again reset to zero. Thereafter, the molecules are allowed to interact without further damping. The resulting system configurations are shown at T = 14.0, 16.8, 19.6, 22.4, 25.2, 28.0, 30.8 and 33.6 in Figs. 3.1(a)-(h), respectively. These figures show the presence of surface waves, which, in fact, are due to the system's contractions and expansions with time. Note also that the density at any time is always greater in the interior of the system than at the boundary,
36
Quasimolecular
Modelling
Fig. 3.1. T=14.0(a), 16.8(b), 19.6(c), 22.4(d), 25.2(e), 28.0(f), 30.8(g), 33.6(h).
Fig. 3.2. Two microdrops of water 3 A apart.
which is consistent with the surface tension theory which holds that the surface molecules are in an attraction mode. To simulate the interaction of two drops, we proceed as follows. The drop in Fig. 3.1(b) was reflected symmetrically about the Y-axis so that
37
Colliding Microdrops of Water
(I,
(t)
U)
(P)
Fig. 3.3. Oscillating oblateness mode.
Quasimolecular
38
Modelling
•\2 (al
(d)
(d)
(e)
(f)
Fig. 3.4. "Raindrop" and "dumbell" modes resulting from variation of the second component of v*.
39
Colliding Microdrops of Water
-0.2
(c)
(d)"
Ce) ""'""'
(f)
Fig. 3.5. A brush type collision.
the minimal distance between the resulting two drops was 3.0 A. To avoid complete symmetry, however, velocity components were taken t o be the same for any particle and its image. T h e arrangement of the drops is shown in Fig. 3.2. Thereafter, each molecule in the left drop in Fig. 3.2 h a d its velocity components increased by v*, while each molecule in the right drop h a d its components decreased by v*. Various choices for v* yielded the following examples. For t h e first example, let v* = (0, 0), so t h a t the two drops are allowed to interact with no changes in their velocity components. System (3.8) consists now of 2256 coupled equations. Figures 3.3(a)-(p) show, through T — 85, the resulting formation into a single drop which then oscillates with alternating vertical and horizontal oblateness. For the next two examples, we considered, in order, v* = (0.2,1.5) and v* = (0.2,2.2). These vectors differ only in their second components. T h e results are shown simultaneously in Fig. 3.4 through T — 60. T h e mode shown on the left, corresponding to v* = (0.2,1.5), is typical of a
40
Quasimolecular
Modelling
"raindrop" mode; while the mode shown on the right, corresponding to v* = (0.2,2.2), is typical of a "dumbell" mode (Peterson (1986)).
-3.0
3.0
-<».o
• (a)
• (c)
k.O
(d)-vSi
Fig. 3.6. Direct collision.
Colliding Microdrops of Water
41
Finally, let us give two examples of collision without total adhesion. Such interactions are not usually observable experimentally. In the spirit of the last two examples, let us first increase v* to (0.2, 2.5), thereby changing again only the second velocity component of v*. Figures 3.5(a)-(f) show the resulting interaction through T = 60. During this brush type collision only a small amount of mass is transferred between the drops and only a small amount of mass is lost, as is seen in Fig. 3.5(f). If we next increase v* to (4, 3), then a direct and violent collision occurs, as is seen in Figs. 3.6(a)(i) through T — 55. Approximately half of each emerging drop shown in Fig. 3.6(i) comes from both the left and right drops shown in Fig. 3.6(a). In addition, the amount of mass lost, as seen in Fig. 3.6(i), is now more extensive. Moreover, the density patterns in Figs. 3.6(d)-(f) indicate that the system is also expanding and contracting with time. Finally, it should be noted that the results of the simulation are entirely in agreement with experimental results (Adam, Lindblad and Hendricks (1968)). Exercises 3.1 Reproduce Fig. 3.6 using the color green for each molecule in the left drop in Fig. 3.6(a) and the color red for each molecule in the right drop. 3.2 Simulate the collision of two microdrops of water using (3.2) rather than (3.5). 3.3 Simulate the collision of two microdrops of water using a potential which allows for dipole moments. 3.4 Simulate the collision of two microdrops of water in three dimensions. 3.5 Simulate the collision of two microdrops of honey.
Chapter 4 Crack Development in a Stressed Copper Plate 4.1. Introduction Thus far, attention has been directed to fluid behavior. We turn next, then, to Q modelling of solids. The problem we will consider is that of crack development in a stressed plate. The ability to predict where a crack will develop in a stressed plate is of fundamental importance in the design of buildings, aircraft, and nuclear reactors. Specifically, we will simulate crack and fracture development in a stressed, slotted copper plate. 4.2. Formula Derivation An approximate potential function for the interaction of two copper atoms r A apart is (Greenspan (1989)) 1.398068 x 10- 1 0 4>(r) =
1.55104 x l O - 8 +
-JJ
„ erg .
, (4.1)
From (4.1) it follows that the magnitude F of the force F, in dynes, between two copper atoms r A apart is 8.388408 x 10~ 2 F
(r) =
+
77
1.861248 x 10 jsj •
, x (4.2)
The minimum occurs when F(r) = 0, that is, at r = 2.46 A and yields ^6(2.46) = -3.15045 x 10~ 13 erg . 43
(4.3)
44
QuasimoIecuIaT
Modelling
With these observations made, let us then consider a rectangular copper plate which is approximately 8 cm x 11.4 cm. To simulate the plate, let the points Pi with respective coordinates (x,-, y,), i = 1, 2 , . . . , 2713, be defined by x(l) = - 3 . 9 ,
2/(1) = -5.71576764 ,
x(41) = - 4 . 0 ,
y(41) = -5.54256256 ,
x(i + 1) = x(i) + 0.2,
y(i + l) = y(l),
x(i + 1) = i(t) + 0.2,
y(i + l) = y(41),
x(i) = x(i - 81),
i = 1,2,... ,39 i = 41,42,... ,80
y(f) = y(i - 81) + 2(0.17320508),
i = 8 2 , 8 3 , . . . ,2713 . The resulting arrangement is shown in Fig. 4.1. The (x,-,y;) are vertices of a regular triangular mosaic in which the distance from any Pi to an immediate neighbor is 0.2 cm. The P,- are assumed to represent quasimolecules of an 8 cm x 11.43 cm rectangular copper plate. The neighbors of any Pi are those particles which are 0.2 cm from Pi. The neighbors of any P,- are defined to be the neighbors of P,- for all time. In order to determine a mass m for each Pj, we use total mass conservation. Suppose the rectangular plate were to be filled with copper atoms using, again, a regular triangular mosaic, but in which the distance between two immediate neighbors is 2.46 A. Then the number N* of atoms in the plate is approximately, Ar„ N
8 x10s = ^46-
X
11.43xl08 2.13
=
__A_ 1ftl7 1 J 4 5 X 10
•
t t t . (44)
Since the mass of a copper atom is 1.0542 x 1 0 - 2 2 g, the total mass M of these copper atoms is then M = 1.840 x 1 0 - 5 g. Distributing this mass over the 2713 quasimolecules yields a quasimolecular mass m given by m = 6.782 x 1 0 - 9 g .
(4.5)
To determine computationally convenient force and potential formulae, we utilize energy conservation. Since the minimum potential between two copper atoms is given by Eq. (4.3), it follows, as in Sec. 2.3, under the assumption of zero kinetic energy, that the total energy E* of the system of atoms is, approximately, E* = 3(1.745)10 17 (-3.15045)10 -13 = -1.6493 x 105 erg .
(4.6)
Crack Development
in a Stressed Copper Plate
45
Y P
2674
I
2713
41 40 Fig. 4.1. The initial configuration.
Let us assume now that the force F, in dynes, between two quasimolecules has magnitude F given by (4.7)
in which R is measured in centimeters. Hence ,/m 0.5G 0.25H (4.8) «*) = - _ + — e r g . Assuming cj>{R) is minimal for R = 0.2, so that f (0.2) = 0, implies G H 0 . (4 9) (0.2) 3 +' (0.2) 5 ' Approximating the total energy E of the quasimolecular system in the fashion used to obtain E* yields E = 3(2713)
G
H
2(0.2) 2
4(0.2) 4
+
erg
(4.10)
Quasimolecular
46
Modelling
Equating E and E* implies
-2(blF + 4 W = - 2 0 - 2 6 4 -
<4'U>
T h e solution of Eqs. (4.9) and (4.11) is G = 3.24224, H = 0.12969, so t h a t Eq. (4.7) takes the specific form „.„.
-3.24224
F
0.12969
+
.„ , „ ,
(412)
w = -w- ^?--
Again, as in Sec. 2.3, the force between two quasimolecules must be local in the presence of gravity. T h u s , we will introduce a normalizing constant a such t h a t at a distance 0.4 cm the force between two quasimolecules is small relative to gravity. This is essential because we have assumed t h a t forces act only between neighbors, and initially the distance from any Pi to a neighbor is 0.2 cm. If we now define "small relative to gravity" to mean 0 . 1 % of the effect of gravity, then we must have a\ - 3.24224/(0.4) 3 + 0.12969/(0.4) 5 | < (0.001)980 m
(4.13)
which results in our choice a = 1.25 x 1 0 - 1 0 . Since the effect of gravity will b e relatively insignificant in the problems to be considered, the dynamical equation for the motion of each quasimolecule is then d2Ki
1.25 x l O -
1 0
^
(
-3.24224 3
(Ri:)
0.12969\ R , , +
jRiJWJ'P~
(4.14)
in which R j , is the vector from Pj to P,- and summation is taken over the neighbors of P,-. From Eq. (4.5) and introducing the computationally convenient transformations R* = 47?, T 2 = 10t2, Eq. (4.14) reduces to 1.52981 2
dT
^
' (^o)
3 +
0.97908\ R?,
TO5 J Rij
(4.15)
For the distance D* at which a quasimolecular bond breaks we choose, as recommended by Ashurst and Hoover (1976), the value R* at which dF/dR* first becomes negative. From Eq. (4.15), then, D* = 1.033.
Crack Development
in a Stressed Copper Plate
Fig. 4.2. The slotted plate.
4.3. E x a m p l e s As a first example, consider a slotted plate, t h a t is, as shown in Fig. 4.2, a plate in which the 15 particles P,-, i = 1070 +41k,k = 0 , 1 , 2 , . . . ,14, have been removed. At each time step the particles in t h e b o t t o m row are relocated so t h a t their Y coordinates are decreased by 0.00002 units. T h e particles in top row have their Y coordinates increased by 0.00002 units per time step. Thereby, the plate is stretched. Solving system (4.15) numerically with time step AT = 0.0001 yields the following results. Figures 4.3-4.6 show the developing force field throughout the b o t t o m half of the plate at the times T = 2.0,6.0,1Q.0 and 12.0. In these figures the force field is represented by vectors emanating from the centers of the quasimolecules. Figures 4.3-4.6 reveal the stress effect being t r a n s m i t t e d to the interior of the plate. T h e fact t h a t this transmission is not instantaneous serves t o differentiate w h a t a non-impulsive force from an impulsive one,
6uiii9-poy\[
•09=x -vv -3rd
/ < . i » t t l i i i i
i i i i i i i ,
l / / , , * # * » # t i i # i i t i i t i t i i » t i i t t » * » * . . % l l I I / # # * / I I I I I I I I I I I I I I. I I I I t I I I I I I I * I I i 1 I I I I / I * I I I I I I I I I I I I I I I I I I I I I t I I I I 1 I I I » » I I i I i i i i i i i I I I I I I I I I I I I / I I I i i i i i » i • • i i > > i l / / / i i i i i I I I I I I I I I I I I I I I t i i i i i i I i I I i i i i \ i i i i i i i i I I I I I I I I I I I I i I I I i i t i I I i I I t i it » i i / / i i j t I i I I I I I I I I I I I I I I I i I I i I i i I I I I i i i i > i t * i i i f i i r i i i i i i i i i i i i I i i i t I I I i i i i i i i > • «
•OZ = X -£f
'StJ
/ 1 i i 11 t i i i i i i i i i i i i i t i i i i i i i i i i i i i i i 11 • * • i ; i * t i i i i • I i i • i i i i i i i i i i i i i i i i i t i i i
dVjTioQioiuisvnfo
Crack Development
in a Stressed Copper Plate
•
» 1
I *
• »
'
• I
»
> I
* I
< I
•
• I
•
i I
*
*
i »
»
i I
'
• •
I I I
*
i I
I
»
i I
I
*
» t
I
»
» I
t
-
* I
I
| t
*
|
« .
t II
i
| (
i I
i i
i i
i i
i i
* i
i
i I
i
i
*
i i
,
t
i i
,
i i
,
,
» \ I I I I » I I # I I I » t • • • t I I I I I I 1 I • I I ' I I I I • * • I • I I I t I I • I • I t t I t f t
I I I
Fig. 4.5. T=10.0.
Fig. 4.6. T=12.0.
i
i i
,
i
i
i
i
,
i
t
i
i
i
,
i
,
,
,
,
,
i
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/
i
,
i
,
i I
I
/
/
,
I I
I
t I
I
i I
I
I
QuasimoU'ecular Modelling
: - ; - : • : • : - / . / 1 •
1 •
1 •
1
1
•
<
* •
* /
. - > » » : - : • : • : • : • ; • : • : .
.
.
.
<
1
1
1
1
1
1
1 1 . ' . '
I . , . . , , , , , , ,
Fig. 4.7. T = 12.8.
Fig. 4.8. T=13.8.
Fig. 4.9. T=15.8.
which would simply tear the top and bottom off the plate. Figures 4.5 and 4.6 reveal large forces on the left and right sides of the slot, which yield
Crack Development
in a Stressed Copper Plate
51
Fig. 4.10. T=18.8.
Fig. 4.11. Row dislocation at T=16.8.
a widening of the slot. Figure 4.7, at T — 12.8, shows clearly from the force field t h a t the first crack occurs at the lower left of the slot, and hence by symmetry, simultaneously at the upper right. Figures 4.8-4.10 show the gross effect on the plate at the respective times T = 13.8,15.8,18.8. Figure 4.11 shows the associated row dislocation at T = 16.8.
52
Quasimolecular
Modelling
Figures 4.12 and 4.13 show the fracture of a full plate under shear at the times T = 20.0 and T = 30.0. In this case, the top and bottom rows were again moved 0.00002 units per time step in the Y direction, while they were moved simultaneously, and again symmetrically with respect to the origin, 0.000005 unit in the X direction per time step. Note finally that the FORTRAN program adapted to the examples of this section is available in Appendix B.
.•;.•:•• . ; • • ; . — •• ..••."•••••
v.v
-'&• ..•:
'&?'•&'•££.£©:•• 'M
Fig. 4.12. T=20.0.
Fig. 4.13. T=30.0.
Exercises 4.1 Reproduce Fig. 4.7. 4.2 In the slotted copper plate shown in Fig. 4.2, apply an impulsive force which will tear off the top and the bottom of the plate. Show the development of the resulting force field in the plate. 4.3 Using a Q model, analyze crack development in a stressed copper plate which has a wedge cut out on one side.
Crack Development
in a Stressed Copper Plate
53
4.4 Consider the copper plate shown in Fig. 4.2. Suppose four holes are punched near the corners for the insertion of bolts. If the top edge is stressed upwards and the bottom edge downwards, where will cracks develop first. 4.5 How can one differentiate between hard and soft solids, like steel and butter, in quasimolecular modelling?
Chapter 5 Stress Wave Propagation in Slender Bars 5.1. Introduction With the availability of today's advanced technology, experimental data is often available in various types and forms. It is important to examine as many sources of such data as is reasonable in formulating a Q model. In this chapter we will show how to incorporate available stress and strain measurements for slender aluminum bars in order to simulate stress wave propagation in such bars. As an important byproduct of the development, we will see that the number of particles N in a Q model need not always be large in order to achieve excellent quantitative results. Indeed, we will require only N = 20. In addition, since N is relatively small, there is no substantial disadvantage in allowing square root routines into the leap-frog formulae, so that we will, for variety, set the exponent parameters p and q to p — 2,q — 4.
5.2. Development of the Force Formulae The basic ideas are summarized as follows. Only a one-dimensional array of particles will be considered. Each particle will interact locally only with its immediate neighbors. Experimental results will be incorporated into the local interaction formula, and the leap-frog formulae will be applied with an exceptionally small time step, that is, At is chosen to be one half a microsecond, so that, At = 0.5 x 1 0 - 6 . 55
Quasimolecular Modelling
56
In order to allow for nonuniform mass distributions, we assume the local interaction formula G
mirrijH
(5.1)
rij,k
In the present chapter, because we will be guided by experimental data, we will, of necessity, have to deviate from cgs units. Thus, the units for (5.1) will not be prescribed until particular examples are discussed in Sec. 5.3. However, we will simplify (5.1) under the following assumptions. All N particles will be ordered linearly on an X-axis so that the particle numbers increase from left to right. Any particle P,- will be acted upon only by its adjacent particles. Thus, Pi will be acted upon only by P2,PN will be acted upon only by P J V - I , and for i = 2 , 3 , . . . , iV — 1, P,- will be acted upon by both P,_i and Pj+i. Let us then consider the most complex case immediately, which occurs when P,- is an interior particle. Let Pi be located at x,- *.. Assume first that Pj = P;+i, which is located at Xi+x *. Then the force on P,- due to P,+i is rrti rrij G • ij,k
p
(rij,k)
rrii rrij H Xj ~ '
(r 0 - l t )«
Xi+i
Xi+i - Xi
(5.2)
Thus, Fii.h ijtk =
rriimjG l>.-j,*) p
rrumjH (rijtk)o \
j = i+ 1 •
(5.3)
Of course, the equilibrium distance ro for (5.3) satisfies _P-«
_
' n
—
G H
(5.4)
Relation (5.4) establishes one constraint on the four parameters p, q, G, H. We next establish a second condition by introducing Young's modulus E. The strain Cijtk on P,- due to P, + i is denned by
«M =
^
r0
j=i+1 •
(5.5)
ro
The stress on P,- due to Pi+i is defined as Fij^/A, where j — i + 1 and A is the area over which the force acts, i.e., the cross-sectional area defined
Stress Wave Propagation
in Slender
Bars
57
for P{. The modulus of elasticity E is defined as the derivative of the stress with respect to the strain at the zero strain point. Hence, _ E =
d(Fijik/A) drij^
d(Fijih/A) deij,k
f
drijik dcijtk
e=0
ij,*=°
d(Fijtk/A) drijik
/deiiik drij>k
/
(5.6)
But, d(Fijik/A) drijik
_ mjmj ~ A
qH
-PG
(r,-;,*)
p+1
+ (r,'i,*)<+1
(5.7)
n
3 I
Fig. 5.1. Lumping of a distributed mass.
and dejjik
_
drijik
1_
(5.8)
r0
But (5.4)-(5.8) imply that G =
EAri m.imj(q-p)
EArl mimj(q-p)
(5.9)
_£_ |Y ro V _ / _ r o _ V q-p[\rij,k) \rij,kj
(5.10)
H =
'
so that substitution into (5.3) implies Fi %]L A
=
Thus, the force acting on Pj due to P,+i is expressed in (5.10) as a function of »"y,Jti the constants E, A, TQ; and the parameters p and q. The relationship (5.10) represents the stress as a function of the strain through jy,-^ and is called a stress-strain function. Now consider the case where Pj — P,_i, that is, Pj is the point to the left of P,-. In this case (5.3) is replaced by Fijik
=
m,mjG p
(r0-,fc)
mimjH (r,;,*)» J '
1 .
58
QiLa.simolecv.laT Modelling
But, the strain on P,- due to P,_i is defined by Uj.k =
ro -
r
ij,k —
ro
,
. . , j = t - l .
Hence, the derivation from (5.6) onward is the same because of the two sign changes, so that (5.10) is also valid in this case. Of course, we are assuming that the force effect on P,- due to any point different from Pj+i and P,_i is zero. Now we set p = 2, q = 4 throughout this chapter. 5.3. Particle Modelling of a Slender Bar A single chain of particles, as shown in Fig. 5.1, each linked with its immediate neighbors, has proved a satisfactory representation for the slender bars analyzed in this chapter. The particles were initially spaced at their equilibrium distance to avoid start-up transients. The mass associated with each element represents the distributed mass of the length of the bar represented by the elements. For simplicity, the stress and strain at a fixed right end were assumed to have the same values as the element to the left. Since gravity and bar support forces do not affect the axial stress wave propagation, they were neglected. An impulse force was applied to the first element at the left end of the bar as a compressive force along the axis. Except for bars with fixed end conditions, no axial restraint was considered in the analysis. 5.4. Examples We now consider a variety of types of bars which are of engineering interest. Example 1. Uniform Bar with Free-Free Ends. The simplest case for study of stress wave propagation is the constant cross-section bar with uniform, homogeneous density and elasticity. This case was analyzed for a one-half inch diameter aluminum bar. A half sine wave shaped force pulse of thirtyfive microseconds duration and 5000 pounds peak magnitude was applied to one end of the bar. Both ends were otherwise unrestrained. The bar was ten inches in length and was simulated by sub-division into twenty equal particles, each representing a one-half inch segment. Bar characteristics were chosen to facilitate comparison with results of a similar study in the
Stress Wave Propagation in Slender
Bars
59
literature (Sandlin (1970)). The mass of each segment is 0.2541 x l O - 4 lb.sec 2 /inch. The bar strain response to the impulse is shown in Fig. 5.2 for several points along the span. The wave shapes and magnitudes are identical with those reported by Sandlin (1970).
Fig. 5.2. Uniform bar strain history. Free-free end conditions.
In the case of a bar with a free end condition, a compression wave is reflected at the free end as a tension wave traveling in the opposite direction. According to one-dimensional wave theory the time between successive compression peaks is given by: 21 y/E/P 2 x 10 v/10.6 x 106(386.4/0.1) = 98.8 microseconds
Quasimolecular
60
Modelling
Fig. 5.3. Uniform bar strain history. Free-fixed end conditions.
and the peak strain magnitude is _ 5000 epeak-
A E
5000 ~ ( T T / 4 ) ( 1 / 2 ) 2 x 10.6 x 106
= 2.402 x 10~ 3 inches/inch . Examining the strain history of point 6 in Fig. 5.2, we see that the two compression peaks are approximately 97 microseconds apart and that their strain magnitudes are near 2.40 x l O - 3 inches per inch. Thus, the numerical results show good agreement with the theoretical predictions.
61
Stress Wave Propagation in Slender Bars
Example 2. Uniform Bar with Free-Fixed Ends. The uniform cylindrical bar was also analyzed for the case where the end opposite the applied impulse was fixed. This case was simulated mathematically by specifying the position of the end particle to remain fixed at its initial position. Otherwise, properties of the bar and the applied impulse were the same as for the free-free case.
—1 F(M-
KTv -*s
lis
5"
-30
5
-2 0
5 A
-
15
-1 0
y
r 1 0-
15
2 0
5
30
4 0
I 20
I
40
I
1
60 80 T. (US)
1
100
i
120
140
Fig. 5.4. Stepped area bar strain history. Free-free end conditions.
Response of the free-fixed bar is shown in Fig. 5.3. The compression wave in this case is reflected at the fixed end as another compression wave traveling in the opposite direction. Again, the results of the analysis agree identically with those in the literature (Sandlin (1970)).
62
Quasimolecular
Modelling
Example 3. Free-Free Bar with Stepped Area Change. A free-free cylindrical bar with a stepped area change at its midspan was analyzed to compare the predicted wave propagation characteristics with those described elsewhere by Sandlin (1970). Bars of this type are commonly used as shafting in machinery with gears, pulleys or sprockets. T h e pulse input and the bar dimensions of the left half of the bar are the same as those of the uniform bar discussed in the previous examples. T h e cross-section area of the right half of the bar was twice t h a t of the left half. T h e strain time history for the stepped bar is shown in Fig. 5.4. T h e results agree closely with those found by others and the "bump" noted by Sandler (1970), produced by reflection from the step, is evident. Exercises 5.1 Duplicate Fig. 5.2. 5.2 Extend the quantitative approach of this chapter to rectangular plates. 5.3 Simulate the flow of heat in a slender aluminum bar which is heated at one end.
Chapter 6 Melting Points of Atomic Solids 6.1. Introduction In C h a p . 3, the dynamical interaction between molecules was simulated in two space dimensions using a simplified least square fit of a given intermolecular force. In this chapter, we shall show how to deal directly in three space dimensions with such given intermolecular potentials. For variety, we shall direct attention to a problem of wide chemical interest, the determination of the melting point of a given solid. Melting points are critical to the study of material changes of state. T h e present chapter will illustrate also how Q modelling can lead t o a new type of model which is substantially different from t h a t in common practice. In addition, t h e model will yield a simple formula for experimenters to study and test. 6.2. M a t h e m a t i c a l Preliminaries T h e melting point of a solid is usually defined in terms of the average kinetic energy of a large ensemble of atoms or molecules (Cotterill, Kristensen, and Jensen (1974)). We now explore a model of the melting point which uses only four atoms or molecules. Consider, for example, four identical atoms P\,Pi,Ps,P±, each of mass m. Let <(>{r) be a related classical interatomic potential and let F be the interatomic force defined by . Let F be zero when r — r*, the equilibrium 63
64
Quasimolecular
Modelling
distance. Though r and r* will be given in angstroms, all other quantities will be given in cgs units. Next, set P,-, i = 1,2,3,4, initially at the respective points (x^, y,-, 2,), which are vertices of a regular tetrahedron of edge length r*, as shown in the figure below, in which, for convenience,
z i
P
P
4 /
l
/ - -
\
/
I
/
/
\ / \ /
/
/
'*
s ^
^
Y
(*i,tfi,*i) = (0, 0, [ ( r * ) 2 - ( § r ' s i n 6 0 ° ) 2 ] * ) , (xa,ya,2a) = (0, | r * s i n 6 0 ° , 0 ) , (*3,2/3,Z3) = ( K , - ± r * s i n 6 0 ° , 0 ) ,
{xA,yA,zA) = {-\r\
-±r'sin60°, 0) .
For this arrangement, Pi,Pz and PA are in the XF-plane and are equidistant from the origin, while Pi lies on the Z-axis.
Melting Points of Atomic
Solids
65
6.3. Model Formulation, Implementation and Results We begin by studying, in particular, copper. For this purpose, recall the potential (4.1), that is,
*W=-(i^)lO-°+(^i)lO-!et8. (6.1) In dynes, the force F between two copper atoms has magnitude F given by F =
_(8.38M08)10_a+lg:^48>
^
from which it follows that r* = 2.460 A, and the initial positions of Pi, P21 Ps,P\ are determined. The initial velocities are chosen to be V i = (0,0, Vz), V2 = V3 = V4 = 0, in which V2 is a parameter. We will proceed to determine numerically the minimum value of Vz for which P\ passes through the plane of P2,P3,Pi- Intuitively, such behavoir isfluid-likeand should enable one to characterize the transition to melting. With r in angstrom units, m,- = 1.0542 x 1 0 - 2 2 g, and t in seconds, the dynamical system to be solved is
(1.0542).0-(10.)^ = ± f-i2JS»I0-> + i»*»») ffi. jjti
i= 1,2,3,4, or, equivalently
jjti
In molecular mechanics, it is common practice to solve (6.3) numerically with exceptionally small time steps, like At = 1 0 - 1 4 . However, for present purposes, it will be adequate to proceed in the following fashion. To solve this system from the initial data, we will apply the time transformation T = 107t
(6.4)
Quasimolecular
66
Modelling
so that (6.3) reduces to d2Tt
^
/
0.0795713
17.65555 \ r,,-
,
,
The time step to be used in solving (6.5) by the leap-frog formulae is AT =
io-6. The minimum Vz for which Pi passes through the plane of P2P3P4 is now determined by bisection (Young and Gregory (1972)) as follows. First note that dz dz dT *, dz •, * = * = iff • A" = 1 0 7 f f = 1 0 V " <6-6> in which Vz = dz/dT. We now consider a variety of choices Vz in the range 0.01-1.00. For each choice, (6.5) is solved numerically. It is found that for Vz < 0.20, Pi does not pass through the plane of P2P3P4 while for Vz > 0.21 it does. Using bisection and continuing in this fashion, one finds, to five decimal places, that the minimum value of Vz which is desired is Vz = 0.20356. Hence, Vz = 0.20356 x 10- 7 A/sec, or Vz = 0.20356 x lO" 1 5 cm/sec. If vi is the initial speed of Pi relative to the center of mass of the system, so that v\ = \VZ, then v\ = 0.15267 x 10~ 15 cm/sec. We now define the melting point To of copper in K by the weighted kinetic energy formula To = c(^mvfj , (6.7) in which C is a constant which is determined as follows. Since the mass of a copper atom is 1.0542xl0 - 2 2 gr and its melting point is To = 1357 K, formula (6.7) implies C = 1.10453 x 10 57 . But, to 0.03%, which is less than the error in the experimental value 1357 K, one finds
C=(|)lO«. in which h is Planck's constant 6.6251 x l O - 2 7 . From the result above, we propose now that the melting point for any system Pi, P 2 , P3, PA is T
o = hr
1 0r.84
o"™i
(6-8)
Melting Points of Atomic
Solids
67
and proceed to examine the applicability of this formula to other atomic species. In the Table, we have listed the parameter values of the Noble gases Ne, Ar, Kr and Xe (Hirshfelder, Curtiss and Bird (1954)) for the potentials (r) = 4e
O'+G)
12
(6.9)
These potentials were derived using second-virial coefficients. Also listed are the respective masses, equilibrium distances r*, and numerically determined minimum values Vz. The experimental melting points and the theoretical values determined by formula (6.8) are given in the final two columns. The agreement is quite good.
Table Noble Gas
c (mult, by 1 0 - " for ergs)
r*
(A)
(A)
mass (mult, by 1 0 " 2 4 for grams)
V, (mult, by 1 0 - " for cm/sec)
Melting point
To
(°K)
(°K)
Ne
48.1837
2.78
3.12044
33.49754
0.04763
24
24
A
168.436
3.40
3.81637
66.31447
0.06292
84
82
Kr
236.086
3.60
4.04086
139.10966
0.05251
116
119
Xe
299.595
3.963
4.44832
217.96061
0.04872
161
161
Exercises 6.1 Verify the Vz entries in the Table for Ne, A, Kr and Xe. 6.2 If PI,P2,P3,PA are N2 molecules, show that formula (6.8) yields the melting point 64 K, which agrees with the experimentally determined melting point. 6.3 Calculate the melting point of a carbon rod using (6.8) and compare your result with the experimentally determined melting point. 6.4 Calculate the melting point of a diamond using (6.8) and compare your result with the experimentally determined melting point.
Chapter 7 Biological Self Reorganization 7.1. Introduction In Chaps. 2-6 we have presented approaches and examples of quantitative Q modelling. Unfortunately, though desirable, such quantitative modelling may not always be possible because experimental results are not always available for proper guidance. Thus, for example, when two molecules are different, one, for example, being water and the other being silicate glass, formulae for intermolecular potentials are almost nonexistent. In addition, even if two molecules are of the same kind, experimental formulae, of type (2.6) and of other important types, may disagree to the point that no quantitative inferences can be made with reasonable assurity (Hirschfelder, Curtiss and Bird (1954)). For the above reasons, we turn now to qualitative Q modelling. And, though there are those few who consistently experience near apoplexy at the mention of "qualitative modelling," such models can provide advantages which would not otherwise be available. For example, qualitative Q modelling can provide insights into mechanisms of complex physical behavior, like turbulence, which have been elusive to all types of modelling. Also, such modelling often allows one to determine which parameters are significant and which are relatively insignificant in a phenomenon of interest. And, finally, qualitative Q modelling can provide physical intuition, 71
72
Quasimolecular
Modelling
that most valuable precursor to quantitative modelling. For our first study, we will explore a fundamental capability of Q modelling, not mentioned thus far, that is, the ability to simulate self reorganization. Indeed, self reorganization is important in evolutionary processes in physics, it is important in many reaction processes in chemistry, and it is often important in growth and developmental processes in biology. We will begin with a self reorganization process in biology called cell sorting. 7.2. Cellular Self-Sorting In general, a sorting problem is one in which one has to reorganize a given set of objects into disjoint subsets, each of which has a unique set of defining characteristics. In a self-sorting problem, the objects have to accomplish this reorganization by themselves. One motivation for the mathematical study of self-sorting comes from the area of experimental biology. Holtfreter, for example, began with a mixture of individual mesoderm, endoderm, and ectoderm cells and was able to induce self reorganization into a normal tissue in which the mesoderm cells were interior to the tissue, the endoderm were at the interior periphery and the ectoderm were at the exterior periphery. The prototype mathematical problem of cellular self-sorting was posed by Steinberg (1963) and can be stated as follows. Consider a collection of objects which consists of two or three different types. For illlustrative purposes, consider a set with two types of objects, say, type A and type B. Let the initial positions of the objects be given at random. Then, using local interaction rules only, induce the type A cells to organize into a central core and the type B cells to organize into a layer around the core. Steinberg, himself, approached this problem by developing a concept of differential adhesion which was constrained so as to minimize free energy. Computer implementations by others (Antonelli, Rogers and Willard (1973), Goel and Rogers (1978), Gordon, Goel, Steinberg and Wiseman (1972)) have explored diverse dynamical possibilities, including special motility rules, extended zone effects, hexagonal tessellation, viscosity, topological exchange and interface tension. In all cases, simulation is from the physics point of view. Our lack of knowledge of biological mechanisms precludes more sophisticated modelling. The requirement by Steinberg that cellular self reorganization be accomplished using local rules of interaction only is immediately suggestive
Biological Self
73
Reorganization
of the applicability of Q modelling, and it is to a related model that we turn next. 7.3. Dynamical Formulation Consider N particles Pi, i — 1,2,... ,N and let Mt be an "adhesive measure" associated with P,. The motion of the system from given initial positions and velocities is assumed to be determined by the dynamical equations F,- = M i a i , i = l , 2 , . . . ,N .
(7.1)
The force F,j exerted on Pi by Pj is assumed to be of the form
*V =(-%- +??)
Tji
>
(72)
in which Gij and Hij are functions of the particular pair under consideration. The total force F,- on P,- due to particles different from P,- will be the sum of the resulting forces due only to those particles Pj within a prescribed local interaction distance D. The formulation, though now complete, will, as usual, require the numerical solution of system (7.1), which will be accomplished by the leap-frog formulae with At = 0.0001. 7.4. Examples In this section we will describe computer examples of cell sorting. Though the parameter selections may seem to be arbitrary, their choices result from methodologies quite different from those used in quantitative modelling. In order not to confuse the presentation of the examples, the discussion of parameter selection has been deferred until Sec. 7.5, where it will be given in detail. Set p-
3,
g= 5 ,
(7.3)
Gij = H^ = hMiMj ,
(7.4)
D = 2.1 .
(7.5)
Note that (7.2)-(7.5) imply that F,j = 0 if r,-;- = 1.0.
Quasimolecular
74
Modelling
Consider now a square region in the XY-plane whose vertices are (16,16), (—16,16),(—16,—16), (16,—16). In this region, construct a triangular mosaic of 1072 grid points in the following fashion, which, incidentally, is sufficiently general to allow the construction of both larger and smaller sets of such grid points. Set *i = -25.0,
yi = 25.0,
x 52 = -24.5,
y52 = 24.0,
xi+1 = l + Xi,
j/.+i = 25.0;
t'= 1,2,... ,50,
xi+1 = 1 + x,-,
j/,+1 = 24.0;
i = 52,53,... , 100,
*.• = *.--ioi,
W = - l + W-ioi;
• = 102,103,... ,2576.
The resulting 2576 points P,-, with respective coordinates (x,-,y,), are the vertices of a triangular mosaic which fills the square whose vertices are (25,25), (-25,25), ( - 2 5 , - 2 5 ) , (25,-25). To determine the 1072 such points which lie within and on the square whose vertices are (16,16), (—16,16), (—16,-16), (16,-16), we merely exclude those of the 2576 points which satisfy any one of x,- > 16, x,- < —16, y,- > 16, y,- < —16. Next, we fix the set A to have 304 particles, each with Mj = 10,000, and the set B to have 768 particles, each with M,- = 2000. Each particle is set at a distinct grid point as shown in Fig. 7.1, where the particles of set A are represented by circles, while those of set B are represented by triangles. The particles of set A have been distributed widely throughout the square. Next, a velocity is assigned to each particle, by a random process, in one of the four directions N, S, E, W. For each particle in B, the speed is 150. For each particle in A, the speed is either 50 or 80, determined at random. The velocity of each particle is shown in Fig. 7.1 as a vector emanating from each particle's center. All initial data are now assigned. During the numerical calculation, in order to allow for a small amount of energy dissipation and to keep the particles within the square, the following reflection rules will be applied: (a) (b) (c) (d)
i f x < > 1 6 , reset x,- —> 32 — Xj, vXti —* — 0.99vX]J-, vVii —* 0.99uy_,-, if x,- < —16, reset x,- —* —32 — x,-, vX|1- —• —0.99^^, i^,- —* 0.99vyi,-, if yi > 16, reset yt —> 32 - y,-, vXii —• 0.99^,,-, vy,j —• 0.99u yi , if yi < - 1 6 , reset y{ -* 32 - yt-, vXii —» 0.99t^)t-, vVii —> —0.99vy),-.
Figures 7.2-7.5 show the self reorganization process of the A particles at the respective times T = 0.5,1.5,2.4,4.0. They reveal that these particles first form into small subsets, which then form into larger subsets, and finally
Biological Self
75
Reorganization
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fc-ft»If © * ft. If © *ft>If © *ft»B»©*© If © * l!» If © * IS> If © * ft. If © i»i>-*© if >©© if i y * © if i y * o if i>*© if &-*© if > * © B> o-e>© i > ft. if © * ft. if © * ft. if © * 6> if ©©ft.o> © *ft-if © *ft»if © *ft>if © * i •-*© if © * © if > * © if > * © if o-e>© if > * © if >oo if E>-e>© if i • if © * ft. if © * ft. if © * ft. if © *ft>if © * ft. if © *ft*if © * ft. if © * ft> H »©© > * © if t>-*© if &-e>© if > * © if > * © if © * o if > * © if I H > © * ft. if © * ft. if © * ft.©©* ft. if © *ft>if © * ft-if © *ft>if © *ft>if O If &-*© If !>*© If > * © If > * © If !>*>©© t>-e>© If B^»© If >*(> 'l> ftjlfOI!>l!>lfO{!>OlfOC>ft'lf OI>ft*I f © *ft»IfOI»ft*If i y *ft*If y Fig. 7.1. Initial data.
form a central core. Figure 7.6, at the time T = 24.0, shows the self reorganization of the set A into a relatively circular core. At this point, it was observed that the formation of an outer layer by the B particles was a relatively slow process. Hence, as an economy move, we reduced the damping factor in rules (a)-(d) to 0.9 after T = 24.0. Figure 7.7 then shows, at T = 31.5, the circular central core of A particles and the layer of B particles around the core. From other examples run, in which various parameters were selected differently, the following observations resulted. If damping is eliminated, calculation can become unstable unless one chooses At < 0.0001. In the choices M,- = 9500, Mj = 10000, self reorganization was very slow. The choice of the damping factor 0.9 from the start in rules (a)-(d) yielded particle trapping, that is, several B particles were found to be always interior to the A set, the reason being that the loss of energy was sufficiently
Quasimolecular
76
h2&»
oo
,a&?
o°
Fig. 7.2. T=0.5.
Fig. 7.3. T=1.5.
Fig. 7.5. T=4.
Fig. 7.4. T=2.5.
Fig. 7.6. T=24.
Modelling
Biological Self
77
Reorganization
Fig. 7.7. T=31.5.
excessive to yield premature solidification. This last observation implies the reasonable conclusion that cell sorting can only occur above well defined cell temperatures, which are characteristic of the cells under consideration. 7.5. A Parameter Selection Process Because parameter selection for qualitative models is difficult, we will discuss in this section the general strategy used and indicate how it was applied to the example in Sec. 7.4. The very same ideas and techniques apply to all qualitative models yet to be discussed, and hence will not be elucidated in detail in the future. In general, one proceeds in four steps, as follows: Step 1. Fix those parameters which will make the calculation simple. In cell sorting, then, we set p = 3, q = 5, M,- = 10000 for A type
78
Quasimolecular
Modelling
particles, M; = 2000 for B type particles, and Gij = Htj. The choice of exponents follows from our experience with quantitative models. The choice of adhesion constants should promote sufficiently large attraction between the A particles to induce the formation of a central core. The choice Gij = Hij implies that the equilibrium distance between any two particles is unity. Step 2. Choose a large range of choices for all other parameters. For cell sorting we considered the parameter choices dj
= O.lMtMj, MiMj, bMiMj,
D=
25MiMj
1.2,1.5,2.0,2.5,3.0,3.5.
For the initial speeds v(A) for type A particles and v(B) for type B particles, we chose v(A) = 1,5,25,100,250,500, v(B) = 0.5,2.5,10.0,20.0,50.0,125.0,200.0 . For the wall damping factor 6, we set 6 - 1.0,0.99,0.975,0.96,0.925,0.90,0.85,0.75,0.5,0.25 .
Step 3. Run numerical simulations with sampled parameter sets from Step 2. For cell sorting, sixty sampled parameter sets were selected and the numerical simulations yielded five in which the A particles seemed to be forming into a central core. The most promising case was then selected for further improvement. Step 4- Using the result of Step 3, continue the simulations by varying only one parameter at a time, until the best value of each parameter has resulted. If the resulting model is still not completely acceptable, repeat the individual parameter modification process until an acceptable result follows. For cell sorting, beginning with the result after Step 3, simulations using all the possible choices for Gij were studied and the best result was
Biological Self
Reorganization
79
chosen. With this result, simulation using all the possible values of D were then studied, and the best result of these was chosen. With this result, simulation using all the choices for v(A) were studied, and the best one of these was chosen. The simulations were continued in this fashion until an acceptable model resulted. It should be observed that use of Steps 1-4, above, can yield several acceptable parameter sets. Also, note that additional knowledge should always be incorporated into the procedure whenever possible. Thus, for example, once the adhesive measures are fixed, one can readily determine an appropriate range for v(A) and v(B) which assures the fluid nature of the configuration (Greenspan (1980)). Exercises 7.1 Reproduce Fig. 7.7. 7.2 Consider 1072 particles, arranged as in Fig. 7.1. At random, let 38 of these constitute set A, 266 constitute set B, and 768 constitute set C. Let the adhesive measures of sets A,B,C be 10000, 4000, and 2000, respectively. At random, assign each particle a speed in one of the four directions N, S, E, W, with the A particles having speeds 60 while the B and C particles have speed 150. Assuming all other conditions to be the same in the example in Sec. 7.4, show that the particles self reorganize into a core consisting of the A particles, a layer around the core consisting of B particles, and finally a layer of C particles around the B particles. 7.3 Simulate cell sorting in three dimensions. 7.4 Simulate cell sorting in three dimensions and include the effects of gravity. 7.5 In the XF-plane, let three particles Pi, P2, P3 be located at (0,0.8660), (—0.5,0), (0.5,0), respectively. Assign to each particle a zero initial velocity and let the distance of interaction be D = 2.1. Determine the minimum velocity (0, v), v > 0, for Pi such that PiP2 = P1P3 = 2.11. For this value of v, Pi no longer interacts with Pi and P3, and hence behaves like a gas particle. Thus, for v < v,Pi behaves either like a liquid or a solid particle relative to P2 and P3. If G,j = Hy — 5MiMj, show that v ~ 170 if Mi = M 2 = M 3 = 2000, while v ~ 75 if Mi = Mi = M 3 = 10000. Verify that, for the two cases, the quantities | M , v 2 are approximately equal.
80
7.6 Simulate the evolution of a planet. 7.7 Simulate the inversion of volvox.
Quasimolecular
Modelling
Chapter 8 Cavity Flow 8.1. Introduction Perhaps the simplest type of nontrivial, fluid dynamical problem which is used to test various types of models and numerical methods is the twodimensional cavity flow problem. This problem is formulated as follows: Determine the motion of fluid which fills a square basin, or cavity, when the upper side, or lid, is in uniform horizontal motion. In this chapter we will formulate and explore a Q model of the cavity problem. 8.2. Example Let us begin directly by describing in detail a specific example. For this purpose, consider a square ABCD as shown in Fig. 8.1. The coordinates of the vertices are ^(-6.25,6.25), 5(6.25,6.25), C(6.25,-6.25), D{—6.25,— 6.25). The inside of the square is called the basin and the top side AB is called the lid of the basin. We now construct a triangular mosaic of 2576 points within and on ABCD, as shown in Fig. 8.1. The coordinates of the points are given by (*«i2/i)> where xi = -6.25 ,
yi = 6.25 ,
xi+1 = x{ + 0.25 ,
x52 = -6.125 ,
yi+1 = 6.25 ; 81
y52 = 6.0 ,
i = 1,2,... ,50 ,
°*
Quasimolecular
Modelling
xi+1 = Xi + 0.25 , yi+1 = 6.0 ; i = 5 2 , 5 3 , . . . , 100 , * t = Z i - i o i , Vi = Vi-wi ~ 0.5 ; i = 102,103,... ,2576 . This point set is symmetrical about both axes and the origin. The 51 rows contain, alternately, 51 and 50 points. In each row the distance between two adjacent points is 0.25. The distance between two consecutive rows is also 0.25.
Y A
B
-~ X
D
'
'C
Fig. 8.1. The square cavity region.
Each point will represent a particle and the mass of each particle is taken to be unity. The particle with coordinates (xt,yi) is denoted by P,. Thus, the subscripts of the P, increase from left to right on any row and the numbering begins on the top row and proceeds from any row to the next lower row. The long range force on each particle Pi is taken to be gravity, for which we choose g = 980.0. Since all masses are unit masses, we take, as
Cavity Flow
83
usual, the local force on P,- due to neighbor Pj to be H • 0\k
(rij,k)p
+
(rijtk)i
(8.1)
r
ij,k
Now, since gravity will be a dominating force, local repulsion must be significant to keep all the particles from falling to the bottom of the basin. Hence, we now choose G = 0, H = 100, p — 3, q = 5. Assume, also, that local force interactions are restricted to pairs of particles whose distance of separation is less than D = 0.35. Each particle is now assigned a small, randomly generated velocity vector whose speed is less than 0.002, so that all initial data are determined. The resulting system of 2576 second order, dynamical equations was then solved numerically by the leap-frog formulae with At = 0.0001. Whenever a particle crossed a side of the square, it was reflected back symmetrically across that side with a velocity damping factor of 0.9. However, when a particle had moved across the top side AB of the square, that is, whenever the particle had collided with the lid of the square, the particle was reflected as indicated and then the constant V was added to its x-component of velocity. For the present, let V = —10.0. In the usual notation t^ = kAt, fc = 0 , 1 , 2 , . . . , the system was solved numerically to <250000- The results are described as follows. Figure 8.2 shows the velocity field after 6000 time steps, that is, at ^6000- The figure is relatively meaningless because the field is dominated by Brownian type motions, which reflect the strong molecular type interactions. A smoothing or filtering process is therefore required to clarify the gross fluid motions, and this is implemented as follows. Definition 8.1. For J a positive integer, let particle Pi be at (xik,yi,k) at time tk and at (ar^jt-j, 2/>,jfc-j) at time tk-j. Then Pi's average velocity v i,Jfc,.7 at time tk is defined by , Xi,k - Xik-j V.'fc.J = ' —
JAt
Vi,k ~ ^ ~
Vi,k-J
J At
In this chapter, we choose J = 1500, which was decided upon after several comparison runs with other values of J. In the remainder of the discussion, all velocity fields are average velocity fields.
Quasimolecular
Modelling
Fig. 8.2. Instantaneous velocity field at t6ooo.
Figures 8.3-8.9 show the motion of the fluid by displaying the velocity fields at times in the range tisoo ~ ^200000- The development of a primary vortex is clear as is the motion of its core from the upper right to a more central location. These results are in agreement with experimental results (Pan and Acrivos (1967)). Most important, however, is the observation that the model reveals the mechanisms of vortex development. Indeed, Figs. 8.3-8.5 reveal that there is compression in the upper left corner and partial vacuum in the upper right corner. The compression yields large repulsive forces, which result in motion downward. The repulsive forces between particles just below the partial vacuum result in upward motion to fill the void. Thus, rotational motion begins. The continued driving force of the lid results in an increase in the size of the vortex. However, when the vortex has reached the size shown in Fig. 8.6 at tisooo, it changes relatively slowly thereafter. This is consistent with the usual assumption of a steady state (Anderson, Tannehill and Pletcher (1984), Pan and Acrivos (1967)), which results when the energy being added to the system is dissipated at the same rate.
Cavity Flow • —••»•»• i - a ^ - - o - p '
I ' 4 '
mi ;; .^v,C.;v«v:<:^v.;.::^A . :V.
Fig. 8.3. Average velocity field at ti 5 0 o. *•—
i.'i\ \ y :•••!'.
•- \ w
• -
1/ . . . , , • • . .
: \ \ \
\-.:; •
,y.ii..; ( y;};y,:
Fig. 8.4. Average velocity field at (30oo.
i
r <
Quasimolecular
. m'VV
^
" '-"
^
X
[
l
I . '
L
%£WS'& U.>,>Nv>^i.,.,v::::-i:::^.-^^=rr
Fig. 8.5. Average velocity field at t 6DO o.
Fig. 8.6. Average velocity field at ti 50 oo-
'/ • '
J
J
Modelling
Cavity Flow
Fig. 8.7. Average velocity field at t35oo0.
,1.11 •„• 1 \ \ V \ i \ ! i\'\ <. w o"-'—'
Fig. 8.8. Average velocity field at tl0oooo-
Quasimolecular
Modelling
Fig. 8.9. Average velocity field at (200000•
The figures reveal also other interesting aspects of the flow. Figure 8.5 accentuates a downward flow near the left wall. Figures 8.4 to 8.9 show, with clarity, the development of "arms" at the bottom of the primary vortex which penetrate into the relatively quiescent fluid area below the vortex. Figures 8.7-8.9 accentuate the development of a relative dead zone near the right wall. Since there is interest in the motion in the upper corners, we have shown in Figs. 8.10-8.13 at every 500 times steps from to to £3000 the dispersive mixing of various upper left corner particles. Figure 8.10 shows the motion of P 1 - P 4 , Fig. 8.11 shows P52-P55, Fig. 8.12 shows P102-P105, and Fig. 8.13 shows P153 —Pi56- Figures 8.10-8.13 reveal that Pi,Ps2,Pio2, and P153 exhibit small oscillations during this initial period, but remain relatively stationary. Figure 8.10 reveals an erratic motion for P2, while Fig. 8.11 reveals large motions toward the interior for P53 — P55. Figure 8.12 shows motion along the wall for Pio3- Figure 8.13 reveals, initially, strong
Cavity Flow
Fig. 8.10. Initial motion of P 1 - P 4 .
Fig. 8.11. Initial motion of P52— P s 5 .
Quasimolecular
90
Fig. 8.12. Initial motion of P102— -Pios6
3
Fig. 8.13. Initial motion of
Piss-Piss-
Modelling
Cavity Flow
91
Fig. 8.14. Lower left corner secondary vortex.
Fig. 8.15. Lower right corner secondary vortex.
Quasimolecular
Fig. 8.16. Velocity field for
.',•"•'•-'•'.V-v«
v
V-~7.
•V^'---.\--.;rJ'.•'•••::'•'••: ••'••' ••'•'•'
Fig. 8.17. Velocity field for V = - 1 3 at tisooo-
Modelling
Cavity Flow
93
backward motions for P154 and P155. It may be noted again, as at the end of Chap. 1, that if we had utilized a near neighbor algorithm like that of Boris (1986), we could not have described the individual motions of these particles. Secondary vortices were not identified easily, primarily because a moving vortex is not readily recognizable and because small motions may require a different value of J for proper display. However, two examples of secondary vortex development are given in Figs. 8.14 and 8.15, one in the lower left corner at £150000 and one in the lower right corner at 1200000For display purposes, the velocity vectors required magnification. If CTV represents the magnification factor, then CTV = 25 was used for Fig. 8.14 while CTV = 15 was used for Fig. 8.15. 8.3. Additional Examples The number and variety of examples which can be explored are, of course, unlimited. We therefore describe only three of these studies in which V was varied. For V = —2.5, no vortex ever develops. Instead, there is an undulation through the fluid which resembles a compression wave. For the cases V = —7, V = —13, the results as tisooo are shown in Figs. 8.16 and 8.17, respectively. Relative to the results for V = —10 shown in Fig. 8.6, Fig. 8.16 shows a smaller primary vortex while Fig. 8.17 shows a larger one, which are not unexpected. Exercises 8.1 Reproduce Fig. 8.6. 8.2 Consider a fluid in a square cavity. Let all the sides be stationary. Describe the fluid motion if various portions of the boundary are heated. 8.3 Consider a fluid in a square cavity which is open at the top, that is, it has only three solid walls. Describe the fluid motion if a wind blows across the top.
Chapter 9 Turbulent and Nonturbulent Vortices 9.1. Introduction Since turbulence is the most commonly observed form of fluid behavior, interest has been exceptional in the modelling of related phenomena. Beginning with G. I. Taylor's seminal paper (Taylor (1921)), one school of study (Favre (1964)) has emphsized the statistical approach. A second major school of thought, following fundamental papers of Landau (1944), Hopf (1984), Lorenz (1963), and Ruelle and Takens (1971) uses Galerkin approximations to simplify the Naviern-Stokes equations and bifurcation theory to analyze the resulting ordinary differential system (Barenblatt, Looss and Joseph (1983)). A more recent, computer-oriented approach is to solve the full Navier-Stokes equations numerically, assuming, of course, t h a t these equations represent turbulent flow in some average sense (Markatos (1986)). In addition, there exists a variety of n o less interesting, b u t less studied, approaches, as, for example, the thermodynamic model of Malkus (1960). Unfortunately, i m p o r t a n t and realistic aspects of turbulent motions have defied inclusion in all the models described above (Favre (1964), Barenblatt, Looss, and Joseph (1983), Markatos (1986), Saffman (1968)). T h u s , for example, whereas homogeneous turbulence has received intensive theoretical study, it is not known to exist anywhere in N a t u r e .
95
96
Quasimolecular
Modelling
Our purpose in this chapter is to initiate a Q modelling approach to the study of turbulence. In a n a t u r a l way, turbulent behavior will be induced by conditions which permit the large, repulsive effects of intermolecular forces to prevail. T h e particle equations used will, in fact, constitute a primitive set of Navier-Stokes equations in the sense t h a t they approximate a molecular formulation from which the continuous Navier-Stokes equations can be suitably derived (Hirschfelder, Curtiss and Bird (1954)). Note, finally, t h a t , for simplicity, attention will be restricted only t o vortex motion. T h e kind of turbulent behavior which can result in the upper left corner in the cavity problem of Chap. 8 will not be discussed. 9.2. Basic Definitions Let us begin, rather specifically, by considering the 609 particle arrangement shown in Fig. 9.1. (A more general approach follows in a n a t u r a l way from t h e physical assumptions to be made.) T h e enclosed rectangle is bounded by the lines whose equations are y = 9.526, y = —9.526, x = 13.0, x = —13.0, and the sides of the rectangle are called the walls. T h e particles are arranged on and within the rectangle in such a fashion t h a t the distance between any particle and its immediate neighbors is unity. T h e position (x;, yi) of each P; is given by Xl
= -12.5 ,
x27 = - 1 3 . 0 ,
xi+i
= 1.0 + xi ,
yi+i = yi ;
xi+i
= 1.0 + Xi ,
y,-+i = 7/27 ;
xi = x,_53 ,
z/i = 9.526 ,
j / 2 7 = 8.660 ,
i = 1 , 2 , . . . , 25 , i = 2 7 , 2 8 , . . . ,52
Vi = - 1 - 7 3 2 + J/.--53 ;
i = 5 4 , 5 5 , . . . , 609 .
By construction, each particle in the interior of the rectangle has exactly six immediate neighbors, each a unit away and on the vertices of a regular hexagon. Next, as shown in Fig. 9.2, the seven particles P26C-P261, ^286,-^287, ^288, P313, ^314, are assumed to form a solid hexagon, which will be allowed to move as a rigid body at a fixed velocity V = (VX, VY) per unit time. Our problem will be to describe the motion of the remaining particles, called the fluid particles, as the hexagon proceeds on its fixed, linear trajectory. For intuition, one can think in terms of a planar suspension of particles in a gas as the hexagon moves through it (van Dyke (1982)). Long range forces will be considered to be negligible.
Turbulent and Nonturbulent
97
Vortices
,r
x
Fig. 9.1.
o:-:-:-:
Fig. 9.2.
Quasimolecular
98
Modelling
Now, as the hexagon moves, the fluid particles will experience displacement. Relative to these displacements, the following definitions are given for the fluid particles. Illustrative examples will be given later. D e f i n i t i o n 9 . 1 . For a given time step At, let P,- b e located at (x,-jt, Vi,k) at time tk — kAt, Jb = 0 , 1 , 2 , . . . . T h e n P,'s instantaneous velocity v,-^ at Ik is defined by _ Vi fc
(xj,k
' " V
- g»,*-i
At
Ui,k -
yi,k-\
At
Definition 9.1 uses backward differences because these are consistent with whatever measurements an observer of the hexagon can have made at time tk. D e f i n i t i o n 9 . 2 . For an integer J > 1, let P; be at (x,-^, y,tk) at time tj. and at (xjk-j, j/,-,t_j) at time tk — J At. T h e n P,'s average velocity v,-,fcj at time tk is defined by _ I xi:k - xitk-j
yj,k - Ui,k-J
Definition 9.2 enables us to describe gross particle motions over extended periods by filtering the Brownian-type local motions, as was demonstrated in C h a p . 8. D e f i n i t i o n 9 . 3 . If, relative to a particle P,-, the six particles closest to Pt have instantaneous velocities, relative to P,-, which are all either clockwise or counterclockwise, then the resulting seven-particle configuration is called an instantaneous vortex. D e f i n i t i o n 9.4. If, relative to a particle P,-, the six particles closest to Pi have average velocities, relative to P,-, which are all either clockwise or counterclockwise, then the resulting seven-particle configuration is called an average vortex. Definitions 9.3 and 9.4 prescribe the minimum size of the vortex t h a t we will try to observe. Vortices of all magnitudes exist, from the microscopic to the macroscopic, b u t they can be recognized only within the limits of
Turbulent and Nonturbulent
Vortices
99
accuracy of one's observational instruments. Definitions 9.3 and 9.4 assume that we cannot observe vortices that have only five or fewer particles in clockwise or counterclockwise rotation relative to P,-. Definition 9.5. Relative to P,- and / , a vortex is said to be turbulent at time tk if either of the following is valid: (a) It is average vortex at tk but not an instantaneous vortex at either tk or tk-j • (b) It is an instantaneous vortex at tk but not an instantaneous vortex at tk-j, not an average vortex at tk, and not an instantaneous vortex at tk+j. Definition 9.5, part (a), allows local motions to be of Brownian type while average motions are not, whereas part (b) implies that for relatively small J the vortex appears and disappears quickly. It should now be readily apparent that the choice of J in Definitions 9.2-9.5 is critical to an appropriate simulation of turbulencec. A proper physical choice of J may, indeed, depend on position, time, V, and/or other physical factors. Note that a vortex that is not turbulent is called, quite naturally, a nonturbulent vortex. The forces in our equations of motion are, again, for any two particles, G Ftj.fc
H
=
(r>j,k)p
(ry,*)«
njik < D
(9.1)
in which D is the distance of local interaction. 9.3. E x a m p l e s Throughout this section, set At = 0.0001 and, for reasons to be discussed later, set J — 600. Example 9.1. Consider first the solid-fluid arrangement in Fig. 9.2. Fix m,- = 1, i — 1,2,3,... ,609. Assume that the initial velocities of all the fluid particles are relatively negligible, that is, are much smaller than unity. For simplicity, then, set these to zero. Let the solid particles P260, -P261, ^286,^287, ^288,^313, .P314, each have V = ( - 7 . 5 , - 1 . 0 ) , so that the hexagon moves linearly with this velocity through the fluid. This is implemented mechanically at each time step tt by discarding the numerical calculations for the positions and velocities of the solid particles, decreasing their x and y coordinates by 0.00075 and 0.0001, respectively, and resetting
Quasimolecular
100
\
.
-
-
\
\
- \ X
~
i
v
, -.
.
..
I -
'
. «.
V
,
s
>
,
.
/
I
1
'
'
'
> v. ' - N
/ I .
•
'
' '
Fig. 9.3. Instantaneous velocity field at t 96 oo.
w,\ * -
N ' «
\ \
M i
. \ '
, • ^
\ \ ^
'
,
"
- <
• - - - ' - ' : / -
:~:'fs\
- . » v~~'
'i
, z'
/ V ,
/
/• /
i
/ ' / ' /' /
Fig. 9.4. Instantaneous velocity field at ti02oo-
.
Modelling
Turbulent and Nonturbulent
101
Vortices
. \
_
i
.
,
,
,
N
\V*V. -
N
- \ - - vV
-
\
'
-
/ /
<•
•
Fig. 9.5. Instantaneous velocity field at tioso
N
-
-
-
-
-
-
,
'
•
N
N
,
N
/
-\
\
•
/>
"' -
f
'
'
i
'
<
,
/ -.v. Fig. 9.6. Average velocity field at tio2oo-
;
102
Quasimolecular
Modelling
their velocities to (—7.5,-1.0). Let the parameters of local interaction b e G = 30, H = 50, p = 1, q = 3, and D = 1.5, which allow the fluid particles to move relatively freely. Reflection from the walls is implemented by resetting particles t h a t have moved out of the rectangle back to the nearest wall, in a perpendicular direction, with zero velocity. T h e numerical results for the resulting fluid motion was studied every 600 time steps in the range £2400 to £12600 from the points of view of b o t h instantaneous velocities and average velocities with / = 600. T h e value / = 600 was chosen, after several values of J were studied, because it proved to reveal vortex motion readily. Figures 9.3-9.5 show the instantaneous velocity fields at the respective times <96oo, £10200, £iosoo- Figure 9.6 shows the average velocity field at £10200, and Fig. 9.7 shows the average velocity field at £10200 relative to P258, revealing t h a t P258 is the center of a moving vortex. Particle P258 is marked crossed in Fig. 9.7. Moreover, though six points near P258 in b o t h Figs. 9.3 and 9.4 are moving counterclockwise relative to P258, the closest six points are not in such motion. T h u s , the vortex shown in Fig. 9.7 is a turbulent vortex of the type (a) described in Definition 9.5. Moreover, Fig. 9.8, which shows the instantaneous velocity field at £10200 relative to ^365, reveals a second vortex which is an instantaneous vortex. As shown in Fig. 9.9, this vortex is not an average vortex at £10200- Moreover, it is not an instantaneous vortex at either £9600 or £iosoo> so it is a turbulent vortex of the type (b) described in Definition 9.5. Indeed, the vortex in Fig. 9.8 has resulted at £10200 from extensive counterclockwise motion just above it, and it then dissipates quickly with the counter-rotating vortex it adjoins.
Example 2. Example 1 was repeated with the single change V = (—5.0, —0.5), thus decreasing the speed and altering the direction of the hexagon. No vortices ever developed.
Example 3. Example 1 was repeated with the change p — 2,q = 5, thus making the fluid more cohesive. The velocity fields at £5400 and £6000 indicated the existence of a n instantaneous frontal vortex between these time steps, which additional computation and printout confirmed. Figure 9.10 shows the instantaneous velocity field at £5700 relative to P362, and reveals t h a t P362 is the center of an instantaneous vortex. However, Fig. 9.11, six hundred time steps later, shows t h e average velocity field at £6300 and
Turbulent and Noniurbulent
103
Vortices
•I
_ - " " " - N * . . - - * ' / V ^ N ^ .
N
.
\
-
N N-
-'..'A/.V^V-^ ^ < /
/
,
/ / \ -V - v
- * > • . t. \ ^ .
'
/ ^
~ - - -"
-
.
.
.V.'V.NV.;-.— '. \ -
s
- ^
Fig. 9.7. Type (a) turbulent vortex relative to P258 at ti02oo-
, '
- ,
- N ^
\
N
\
- : > - v ^ -^, - / i - • , . N.
'J,
\^-
..' • J/ s
- _ - . < '-J
-.-.--
Fig. 9.8. Type (b) turbulent vortex relative t o P365 at iiosoo-
Quasimolecular
104
J /
..v^iV-^y.^-^
Fig. 9.9. Average velocity field relative to P365 at ti 0 2oo. I I I I I I I l\ I i I I I I / I I 1 1 1 1 1 i 1 1 V\ 1 I 1 1 1 i 1 i 1 I I I I I I I » ' I I I I J J I I I I I I I I I 1 \ \ [.l, ' I I J J I I I I I I I I \ \ \'ll. J < / / / , , ,
1
' WViY/Ti 1"'
'" '
I I l 1 1 l \ ' l \ f y /S * I
^ ' <^/A
. 1
' KA
l
'v-^iS-'Ji'Y.v I
»
V V _
1 i 1
K__;
--
• • • • . /
' —
/
,
'/ , i
/
/
I
'
1
j ~Z. 1 1 I ' ' '
'
•
'
'
'
'
' ' < / / » 1 1 1 / s t 1 1 i I I I ^ 1 , 1 / 1 1 1 1 1 1 1 1 i , / 1 1 J / / 1 1 1 1 1 1 1 1 ( 1 1 / 1 1 1 1 1 1 1 1 1 1 1 1 1 1 / 1 1 1 1 i 1 1 1 j Fig. 9.10. Intantaneous velocity field relative to P362 at t57oo-
Modelling
Turbulent
and Nonturbulent
Vortices
105
I I I I I I i I I I I 1 I II 1 I 1 I I I I I I I I ( I I I I 7 I 1 I 1 I I I I I V I I I I / 1 1 I 1 I I I I I II I I I I I I i I I \ V , I ' ' ' / I I I I I I i I i I I I 1 I t I I I V \i ' l. I i i I i i i i I i i t i i i i i i i i i i \ \ V ' i , / ' / « i t i i i I I I I I l l V I I t \ \ \ \ ,M ' ' j / / 1 l 4 v ! , I I I 1 I I l l V 1 l l V \ \ J , \ ,' y , S - I _ v V i I I I I I i l \ V I I I V^ \ \ , • ' ' \ v i ' i i i i i i V \ . N » i / i v. ^ O V V
'
i i i i i i iv.v.\ i , V ^ J O V / l l l l l l l » . l l \ . \ .
N
> - ^
W
'
s
\ V \
'
i i
vv/
.V ,
I
V
1
' /
• I • I • I I I I V V V v. v. v ' < 0 ) / ' I 1 \ V I I I I I I I I I I I I I V ^ ,. s. . - ' ~ ' , " i I I I I I I I I I I I I I I I I I V v. >. I ' ' y" i | I I I I i i i i i i i i i i i i i <. r • , , ;~ / i i i i i i i i i i i . i i i i i i,.,. , i i i i i / ; i i i I I I I I I I I 1 I I I 1 I v. . k , I , I I 1 I I I I I I / / I I I I t I I I i I - I I I I I I 1 I I I I I I I I / I I 1 I I I 1 I I I I I I I I I I I I I I I I I I I I I I I I i I I I I I I I I 1 I I I I I I I F i g . 9.11. n o n t u r b u l e n t vortex relative t o P362 a t t 6 3oo-
reveals t h a t P362 is the center of an average vortex. T h u s , the average vortex shown in Fig. 9.11 does not satisfy the conditions of Definition 9.5, so t h a t it is a nonturbulent vortex. 9.4. R e m a r k It is interesting to observe, from the examples of Sec. 9.4 and from additional examples, t h a t nonturbulent vortices were relatively circular and the speeds of all six rotating particles, relative to the central particle, were about the same size. Turbulent vortices showed no such regularity. Exercises 9.1 Duplicate Fig. 9.11. 9.2 Discuss the development of vortices if the hexagon in Fig. 9.2 is replaced by a circle. 9.3 Develop definitions of turbulent and nonturbulent vortices for a threedimensional fluid. Produce computer examples of such vortices. 9.4 Simulate the turbulent motion which develops under a Boeing 747 in flight.
Chapter 10 Vortex Street Modelling 10.1. Introduction In Chap. 9, we introduced the study of vortex development when a solid object moves through a fluid. The emphasis was on the nature of a particular vortex, that is, whether it was turbulent or nonturbulent. In this chapter, we again study vortex development in a similar physical setting. However, our emphasis will be somewhat different. We will be interested more in multiple sets of vortices and the patterns in which they may develop. We will not be concerned with questions related to turbulent or nonturbulent structure. Again we consider only the local force effect
on Pi due to Pj. Long range effects will be considered to be negligible. 10.2. Vortex Street Development without Fixed Boundaries The developemnt of a vortex street, sometimes called a Karman vortex sheet, has received wide attention both experimentally and theoretically (see, e.g., Davis and Moore (1982), Perry, Chong and Lim (1982), Shapiro (1972), van Dyke (1982), von Karman (1962) and the numerous additional 107
108
Quasimolecular
Modelling
references contained therein). To simulate vortex street generation let us set p = 1, q = 3, G = 99, H = 100 in local force formula (10.1). In addition, no boundaries will restrict the fluid motion, so that all surfaces will be free. Next, let us arrange and number 998 particles, each of unit mass, in the following way. For each Pi, the initial position (ar,-, t/,),i = 1,2,... ,998 is determined by x i = - 1 9 . 0 , x i + i = x, + 1.0,
» = 1 , 2 , . . . 38
j / ! = 10.392, yi+i
= yi,
i = 1 , 2 , . . . 38
— Xi + 1.0,
1 = 4 0 , 4 1 , . • ,77
x 4 0 = - 1 8 . 5 , xi+i
2/40 = 9.562, yi+1 = y40,
i = 40,41,. -,77
* i = X,_78,
1 = 7 9 , 8 0 , . . ,974
Vi = yt-78 - 1-732,
i = 7 9 , 8 0 , . . ,974
The resulting particles form the triangular mosaic of points shown in Fig. 10.1. This set will be called the fluid particles. The distance between each fluid particle and any immediate neighbor is unity. Twenty four additional particles P975-P99S are determined by 0t = (0.261799)t, xi+974
t = 1 , 2 , . . . . 24
= 20 + (0.2) cos 0,-, i = l , 2 , . . . , 24
y i + 9 7 4 = (O.2)sin0 i ,
i = 1,2,... ,24 ,
and are positioned uniformly on the circle whose center is (20, 0) and which has radius 0.2. These particles are called the solid particles because they will be moved rigidly through the fluid. Each solid particle is assigned the initial velocity Vo = (—0.0006, 0.0). Each fluid particle is assigned the initial velocity vo = (0.0, 0.0001). Thus, all initial data are now available. Assuming that the local force interactions are restricted to any two particles whose distance of separation D is less than 1.5, the motion of the fluid particles is determined numerically by the leap-frog formulae with At = 0.0001. At each time step tk = kAt, the solid particles are moved rigidly by using the velocity V = (—0.0006, 0.0). Before describing the computational results, however, two essential definitions must be recalled. The first definition is that of average velocity.
Vortex Street
109
Modelling
D e f i n i t i o n 1 0 . 1 . For J a positive integer, let particle P,- be at (xiik, j/,-^) at time tk and at (x t | fc_j, yiik-j) at time tk_j. Then P.'s average velocity v , - 1 ; at time tk is defined by
V,
'' M
_ fxitk
-xi:k-j
_
JM
V
'
yj,k - Vi,k-J
\
J At
J '
In the present section, let = 5000. Other values of / will be considered later. T h e present choice is a relatively large one which will reveal sustained motions and was decided upon after several comparison runs. In the remainder of this chapter, all velocity fields will be average velocity fields. At any time step tk, a vortex is defined only in terms of average velocities as follows. D e f i n i t i o n 1 0 . 2 . If, relative to fluid particle Pi, the six particles closest to Pi are fluid particles and are either in clockwise or in counterclockwise motion, then the seven-particle configuration is called a vortex. Of course, the vortex defined above is, in fact, an average vortex. However, no distinction as t o vortex type is required in the present chapter. We are now ready to discuss the computational results. Figure 10.2 shows the velocity field at £30000- T h e two vortices shown are moving vortices. To show t h a t the conditions of Definition 10.2 are valid, Fig. 10.3 displays the system also at £30000, but the velocity field is given relative to the center particle of the upper right vortex. T h e figure reveals clockwise motion most clearly. Figure 10.4 shows the existence of three vortices at time <60000, the upper right vortex in Fig. 10.2 having dissipated. Each of these vortices is located near a section of s t r e a m flow which has a sharp change in direction, as is indicated in Fig. 10.5. Note t h a t most of the vortices yet to be discussed will be moving vortices. These are often difficult to locate. But, once located, each can be shown to satisfy the conditions of Definition 10.2 by as appropriate linear transformation of the velocity field. A large number of other cases were studied. All results were consistent with experimental results and are summarized as follows. Decreasing the speed of the solid yields a more laminar type flow with fewer vortices, while increasing this speed yields irregular vortex p a t t e r n s and increased
110
Quasimolecular
Modelling
centrifugal effects. Decreasing the interaction distance D to 1.4 also yields an increase in centrifugal effects. Finally, allowing the initial direction field of the fluid particles to be determined by a r a n d o m process, b u t one in which no speed is greater t h a n 0.0001, yields results entirely analogous to those shown in Figs. 10.2 and 10.4. Y
1
•—* x
I
Fig. 10.1. The initial configuration.
10.3. Vortex D e v e l o p m e n t in a n O p e n E n d e d Channel We proceed now in the same spirit as in Sec. 10.2, but incorporate b o u n d a r y constraints. We will show t h a t the resulting vortex p a t t e r n s can be much more complex t h a n those described in Sec. 10.2. Indeed, wall effects are responsible for t h a t very important component of fluid flow called the b o u n d a r y layer. In addition, for variety and to indicate the broad applicability of Q modelling, we will alter certain of the previous procedures, vary certain parameters, and increase the number of particles significantly. In the plane, consider a triangular mosaic of 5126 particles arranged within and on the rectangle whose vertices are (—50.0,21.65), (—50.0,21.65), (—50.0,-21.65), ( 5 0 . 0 , - 2 1 . 6 5 ) . T h e coordinates of each P; are given by (x,-,2/ t ), where
Vortex Street
111
Modelling
•\'W •• -
»'
- •'
_. • < * % V \ 1 ) - - ' - V * - /
•• •- -
*•• V o \\
- *~
' '~ - - - / ' , /
; l \ " I " ^ \ N < N ^ / i ! - 's^ . .' '"<• i -"
ww.wwv/.ww*/^<£> «"-x '"^-""-Cz-v- :::::.. .\'/S « ' , ' " 0 - . «-'w • ••//.
'
'
'
/
(
/
/
•
•
l
s
l
>
Fig. 10.2. Velocity field at t3oooo.
Fig. 10.3. Relative velocity field at t 30 ooo.
'
^
-
"V
.
Quasimolecular
112
Fig. 10.4. Velocity field at t 60000*
Fig. 10.5. Stream flow near vortices at *6oooo.
Modelling
Vortex Street
113
Modelling
*i = - 5 0 , 2/1 = 21.65, x102 = -49.5, y102 = 20.784 , xi+1 = Xi + 1.0, yi+1 = 21.65, i = 1,2,... ,100 *i+i = a;. + 1-0, 2/,-+i = 20.784, t = 102,103,... , 200 Xi = Xi-i0i, w = W-aoi - 1-732; t = 202,203,... ,5126 . Again, let each point have unit mass. The seven particles P2510, -P2511, -P2610, -P26II, ^2612, -P2711, ^2712 are now removed. In their places are inserted 24 "solid" particles, situated uniformly on the circle whose equation is (x — 47.5) 2 + y2 = 1. The total number of particles is now 5143 and the initial configuration is shown in Fig. 10.6. Each solid particle is assigned an initial velocity V 0 = (—0.0015, 0.0). Each fluid particle is assigned an initial velocity vo = (0.0, 0.00025). Thus, all initial data are known. This time, let p = 3 and q = 5, with G = H = 1000 for the local force between two fluid particles and with G = H = 200 for the local force between a fluid and a solid particle. Gravity is neglected and the distance D of local interaction is taken to be 1.25. Y
k
Fig. 10.6. The initial configuration of 5126 particles.
Next, assume that the fluid flows in an open ended channel. To this end, the two lines whose equations are y = ±21.65 are taken as fixed boundaries, or walls, so that the channel is open at the left and right ends. Whenever a computation reveals that a particle has crossed a boundary line, the particle will be reflected back symmetrically across that boundary
Quasimolecular
114
Modelling
and then assigned a zero velocity, thus imposing a strong frictional effect along the walls. T h e fluid motion is generated again by moving the solid particles rigidly and uniformly t o the left at each time step. T h u s , each solid particle is assumed to have a uniform velocity V = (—0.0015, 0.0) at each time step fj. = jfe(O.OOOl). To determine the average velocity of each fluid particle, we assume this time t h a t / = 3500, thereby including vortices whose life spans are shorter t h a n those considered in Sec. 10.2. Figures 10.7 and 10.8 show the fluid motion at t50Q0 and £15,000Observe first t h a t , whereas the velocity field in the first figure is outward, the flow behind the solid in t h e second figure is inward, indicating t h a t wall reflection is significant. In addition, Fig. 10.8 reveals quite clearly, j u s t below the solid, a K a r m a n type vortex in development. Figure 10.9 shows at £74,000 five K a r m a n type vortices, spaced approximately where one would expect to find them. Nevertheless, Fig. 10.10 shows the existence of many additional vortices at £74,000- Most, b u t not all, of these appear near the walls and indicate how complex motion near the walls can b e .
V.\\\I.V(Utl,l I
III.
:
":::fttfmh//mffmw-
•"""•••'
fff////l!\'\\f{
•••:--:::fJ/jfff/ff:'&\\!.\W&
Fig. 10.7. Velocity field at
15000-
A large number of related computations were carried out, the results of which are summarized as follows. Completely analogous results followed when the radius of t h e solid was decreased t o 0.5. For the solid velocity V = (—0.001, 0.0), and for smaller ones, many vortices dissipated
Vortex Sired
Modelling
115
Fig. 10.8. Velocity field at tisooo-
^i^^ffl^^^^^^^^BP Fig. 10.9. Velocity field at *74ooo-
Fig. 10.10. Additional vortices at t740oo-
116
Qv.asimolecv.lar
Modelling
before the solid had traversed the full length of the rectangle. For V = (—0.002, 0.0), the motion increased to the point that vortices were difficult to identify. Introduction of the parameter changes d > 0.5, D > 1.5 were both counterproductive to vortex generation.
Exercises 10.1 Reproduce Fig. 10.5. 10.2 Reproduce Fig. 10.8 and graph five streamlines. 10.3 Simulate vortex development in a two-dimensional channel which is closed on all four sides. 10.4 Simulate an ocean wave generated by a suboceanic earthquake. 10.5 Simulate the generation of Taylor vortices. 10.6 Simulate the generation of a tornado. 10.7 Simulate the breaking of wave on a sloping beach. 10.8 Simulate the development galaxy arms.
Chapter 11 Porous Flow 11.1. Introduction Another class of problems of wide interest and exceptional difficulty is the class of interface problems. Stefan problems, for example, are problems in describing the changing shape of the boundary, or interface, between the liquid and solid portion of a melting solid or of a crystallizing liquid (Crank (1957), Douglas and Gallie (1955), Ehrlich (1985), Greenspan (1978c), Jamet and Bonnerot (1975), Osterby (1974), Stefan (1889)). In Stefan problems, the interface is between a solid and a liquid. Interface problems which involve two different liquids also occur naturally (Bulgarelli, Casulli and Greenspan (1984), Vargas (1986)). In the study of porous flow, for example, if water is injected below oil which is immersed in porous ground, with the idea of floating the oil out above the water, the boundary between the oil and the water changes with the volume of injected water. In this chapter we will develop a Q model of the oil-water problem described above. The methodology is that developed first by Vargas (1986). The discussion will reveal that interfaces are not as clearly demarked as one often assumes. 11.2. Model Formulation The problem is described physically as follows. Consider a region R 117
118
Quasimolecular
Modelling
which is a porous medium, that is, an area of fixed rock formations with open spaces of separation. In the open region between the rocks, we assume there is a fluid, which will be called oil. From an opening in the base of R, we inject a second, heavier fluid, which will be called water. We wish to describe the way the water forces the oil to rise against gravity, and, in particular, we wish to explore the interface boundary between the two fluids. For simplicity, then, let R be the square with vertices (0, 0), (4, 0), (0, 4), (4, 4). Using Ax = Ay = 0.5, construct 81 grid points within and on the boundary of R. At these 81 grid points, we will set 81 particles. For the purpose, let the particles Pi,P^, ... ,P-25 represent rocks; particles P26, • • • ,-Pgi represent oil; and particles P$2,... ,Pi6i represent the incoming water. The rock particles are set at the 25 points with integer coordinates, as shown in Fig. 11.1. The oil particles, shown as the smaller darkened circles are set at the remaining grid points, as shown in Fig. 11.2. It is also assumed that R is bounded by impermeable walls, so no fluid can enter or leave the region without further assumptions. For purpose of injection of water and production of oil, two wells are now opened, one in the bottom left corner of R for injection, and another in the diagonally opposite corner for production, as shown in Fig. 11.2. The diameter of the production well is taken to be 0.5 and that of the injection well 0.25. The rock particles Pi and P25 have been removed to allow for injection and production. To simplify the study, the particles P2-P24 will be kept fixed and only the liquid particles will be allowed to move. However, all three types of particles will be allowed to interact. Now, the mass of each oil particle is taken as unity, the mass of each water particle as 2.5, and the mass of each rock particle as 69.22. (For the motivations of all parameter choices, see Vargas (1986)). Let the distance of local interaction between Pi and Pj be £>,j. We assume that Dij — Dji and that
( 1.3; Di
i; 1.15; 13/36;
I i/n/W;
,3 t,j = = i=
= 26,27, = 82,83, 26,27,.. 82,83,.. 26,27,..
..,81
(oil-oil)
..,161
(water-water)
,81, j = 8 2 , 8 3 , . . . , 161 (oil-water) ,161, j = 2 , 3 , . . . , 24 (water-rock) ,81, j = 2 , 3 , . . . ,24. (oil-rock)
119
Porous Flow
T h e force of local interaction on P; due to Pj is taken to b e m i rrij G
m,- m j H
r
r
ij,k
ij,k
(11.1)
nj,*
T h e parameters G a n d H in (11-1) are chosen as follows for t h e moving particles:
G = 3,
H = 1;
i = 26,.. . , 8 1 ,
i = 2,.. , 24
(oil-rock)
G = l,
# = 1;
i = 26,.. .,81,
(oil-oil)
G = 0,
# = 1;
i = 26,.. -,81,
3 = 26,. . , 81 J = 82,. . , 161
G = 0,
# = 1;
i = 82,.. .,161,
(water-water)
G = 0,
# = 1.5; i = 82,.. . , 1 6 1 ,
3 = 82,. . , 1 6 1 3 = 21, • . , 24.
(oil-water) (water-rock)
For the long range force, we choose gravity a n d let g = 98. T h e time step chosen is At = 1 0 - 4 . T h e injection procedure into R will be simplified in the following way. Assume t h a t t h e water particles enter singly at the point (0, 0) through a t u b e extending from (—1, —1) t o (0,0). In this portion of the t u b e , t h e particles always have a separation distance of unity a n d a speed V towards the opening into R. T h e water particles will be shown as undarkened circles. Finally, consider wall reflection in R. This is done by introducing t h e following parameters: Si = 0 . 4 for i = 2 6 , . . . ,81 6i = 0.8 for i = 8 2 , . . . , 161 . In implementing such reflections, we use t h e following rules, in which t h e parameter choices allow oil to attach itself to the walls of R more readily t h a n does water. (a)
If xiik
(6)
< 0, then
xi>k -> -xiik,
viik}X -> -£,•«,-,*,„.,
If xi>k > 4, then
xi>k —• 8.0 - xiik,
vi:kiX —>
-SiVitk>x,
(c)
If yi,k < 0, then
yi>k —• - j / , - , * ,
viik>y —>
-SiViikiy,
(d)
If yi>k > 4, and
xi>k < 3.5 then
y,->fc - ^ 8.0 Vi,k,y —*
yi>kt SiViky.
120
Quasimolecular
Modelling
PRODUCTION WELL
Fig. 11.1. Array of rocks.
Fig. 11.2. Region R with production and injection wells.
Fig. 11.3. Oil out = 3 ,
Fig. 11.4. Oil out = 5,
water in = 7, i=4500.
water in = 12,fc= 7800.
121
Porous Flow
Fig. 11.5. Oil out = 6,
Fig. 11.6. Oil out = 8,
water in — 14, fc=9000.
water in = 16, i = 10500.
Fig. 11.7. Oil out = 18,
Fig. 11.8. Oil out = 27,
water in = 23, jt = 15000.
water in = 32, water out = 1 , £ = 21000.
122
11.3.
Quasimolecular
Modelling
Examples
Let us now set V = 15. Then Figs. 11.3-11.8 give the evolution of the system. At iterationn k = 4500, i.e., at T = 0.45, shown Fig. 11.3, the water has displaced most of the resident fluid in the neighborhood of the entrance. Also we see t h a t gravity forces a cluster of oil in the lower side, opposite to the injection point. For k = 7800, Fig. 11.4 shows t h a t the water finds less resistance to its motion above t h a n to the right. T h e same observation remains valid for Fig. 11.5. In all three of these first figures, with the exception of oil particles which cling to the walls, the water particles advance in a homogeneous fashion. By the time shown in Fig. 11.5, six oil particles have left R through the product well. T h o u g h the water particles consistently find less resistance from above t h a n to the right, Fig. 11.6 at k = 10500 shows a new phenomenon. W h a t appear t o be isolated water particles are beginning to penetrate into the relatively dense oil area in the lower right region of R. At this time, 8 oil particles have left R through the production well and 16 water particles have entered R. For k — 15,000, shown in Fig. 11.7, the water front continues its movement upward and to the right. At this time, 18 oil particles have been retrieved and 23 water particles have entered R. Most interestingly, the figure reveals t h a t oil particles are now being t r a p p e d within the water region. By the time shown in Fig. 11.8, when k = 21,000, the front is relatively high in R and the number of trapped oil particles has increased. At this time, 27 oil particles have left R and, for the first time, a single water particle has also emerged through the production well. T h e example indicates t h a t , under appropriate conditions, the interface between the oil and water is not an explicit, clearly demarked curve. Indeed, oil can be t r a p p e d within the water flow as the water volume increases. Variation of the parameters in this example also yield the process of "fingering." This is especially apparent as V is increased (Vargas (1986)). It is also possible t h a t a Q model with many more particles would reveal t h a t the results shown in Fig. 11.6 are, indeed, the results of fingering. Exercises 11.1 Duplicate Fig. 11.5. 11.2 Using the same parameters as those for the results in Fig. 11.5, repeat the calculation but for a region whose boundary is a rectangle whose vertices are (0,0), (5, 0), (5, 3), (0, 3).
Porous Flow
123
11.3 Simulate the melting of a snowball. 11.4 Simulate the flow of a liquid through a porous dam. 11.5 Provide an example to show that as a liquid flows into porous ground under the force of gravity, the fluid particles follow paths of least resistance.
Chapter 12 Q Modelling Combustion 12.1. Introduction Interest in combustion is as old as man's acquaintance with fire. However, it is only of more recent origin that chemists, physicists, mathematicians, and engineers have made extensive analytical and experimental studies of the subject. Of late, environmentalists have become additional interested participants. Current theories of combustion are primarily continuum theories, even though it is understood clearly that combustion is a noncontinuum, molecular phenomenon (see, e.g., Hirschfelder, Curtiss, and Bird (1954), Kanury (1975), Zeldovich et al. (1985), and the numerous references therein). The equations studied are relatively sma//systems of differential equations which relate the reaction of a physical system to the release of chemical energy. Major areas of inquiry relate to fuels, ignition, heat transfer, mass transfer, flames, explosions, equations of state, enthalpy, equilibrium, turbulence and conservation laws. In this chapter, we demonstrate the feasibility of simulating combustion phenomena by means of Q modelling. 12.2. Model Formulation Consider 1080 particles P<, 1,2,... , 1080, each of unit mass, arranged 125
Quasimolecular
126
in nine rows of 120 particles each, with respective coordinates given by
Modelling
(x(i),y(i))
x(l) = 0.0, y(l) = 0.0, s(121) = 0.5, t/(121) = 0.866, z(t + l) = 1.0 + z(i), y ( » + l ) = y(l), t = 1,2,... ,119 x(i+l)
= 1.0 + x(i), y(i+l)
= 2/(121), i = 121,122,... ,239
x(i) = x ( i - 2 4 0 ) , 2/(i) = 1.732 + j/(i - 240), t = 241,242,... ,1080. These positions are the vertices of a regular triangular grid whose basic triangular unit has edge length unity. The particles, shown in Fig. 12.1, are bounded by a rectangular region which is 120 units long and 6.928 units high.
y^AvAVAyAv.v.v.v.'Sav.v.v.v.v.T.v^v.v;vS 120
Fig. 12.1. The initial particle configuration.
Fig. 12.2. The initial ignition area at the left end.
Initial particles velocities (vx(i), vy(i)) are assigned in a relatively random fashion in terms of parameters v and V as follows: vx(i) = vy(i) — v,
i = 1,2,... , 119
vx(i)-vy(i)
-v,
i= 121,122,... ,239
= -v,
i = 241,242,... , 1080
vx(i) = -vy(i) which are then modified by
vx(bi) — —vy(4:i) = vx{Ai) = —vy(bi) — v,
i = 1, 2 , . . . , 200 ,
Q Modelling
127
Combustion
and which are further modified by vx(i) = V,
vy{i) = 0, if x{i) < 2.1 .
T h e choice of v will be much smaller than t h a t of V. T h e particle velocities in the range x(i) < 2.1, which is the shaded region in Fig. 12.2, will be chosen to simulate ignition. We assume t h a t any particle P; is acted upon only by those particles Pj which are within 2 units of P,-. T h e force of interaction will be of local molecular type, with magnitude F given by
W h e n v and V have been prescribed, the motion of the system will b e determined numerically by the leap-frog formulas with At = 0.0001. During the simulation, the particles will fall into three distinct categories: combustion, fuel, and burned particles. Prior to ignition, all 1080 particles are assumed to be fuel particles. If and when a fuel particle's speed exceeds 12 units, it will be considered as a combustion particle. At the time of ignition, the parameters v and V will b e prescribed so t h a t particles in the range 0 < x < 2.1 will be combustion particles while all others are fuel particles. A combustion particle will have its velocity increased by the factor C every T time steps until its speed exceeds 300, at which time its velocity will be reset to zero and it will be considered to be a b u r n e d particle. Finally, we will allow symmetric particle reflection from the walls. However, the velocity of each reflected particle will be multiplied by the positive parameter d. A choice d < 1 will imply transmission of heat to the walls. We consider next examples for various choices of the parameters v,V, C, T and d. 12.3.
Examples
We consider now several sets of parameter choices which show the diversity of results which can be achieved. In the figures, the combustion, fuel and burned particles are represented, respectively, by the colors red, yellow and gray. Case 1. Consider the parameter choices v = 3, V = 60, C = 1.75, T = 50, d = 0.90. Figures 12.3(a)-(h) show the particles every 1000 time steps
Quasimolecular
128
M
Modelling
K
Fig. 12.3. u = 3, V = 60, C = 1.75, T=50, cf=0.9.
through tgooo. at which time only one combustion particle remains. T h e combustion is relatively efficient in t h a t only 10 fuel particles remained at b u r n o u t . T h e combustion particle configuration at each time step compares favorably with experimental observations of turbulent flames (Kanury (1975)).
Fig. 12.4. t/=3, V = 60, C=1.75, T=50, d=1.0.
Case 2. In this case the parameter choices are the same as in Case 1 with the single exception d = 1.0. T h u s , the significance of the material structure of the wall is under consideration. Figures 12.4(a)-(c) show the results every 1000 time steps through <3000- T h e combustion in this case is relatively explosive and most efficient, with only 2 fuel particles remaining at b u r n o u t .
Q Modelling
Combustion
129
£fJF
Fig. 12.5. 0=1, V=S0, C=1.75, T=70, d=0.9.
Fig. 12.6. »=1, V=S0, C=1.75, T=70, rf=1.0.
Case 3. This time the parameters are u = 1, V = 80, C = 1.75, T = 70, d = 0.90. T h e particle configurations are shown every 1000 time steps through i6ooo m Figs. 12.5(a)-(f). T h e combustion was rapid and relatively efficient. However, interest in this case increased with the observation of the centrally located combustion particle in Fig. 12.5(a) and led to the next example.
Case 4- T h e parameters choices in this case are the same as in Case 3 with the single exception d = 1. T h u s , as in Case 2, the walls are again completely insulated. T h e results are remarkably different from those of Case 3. Figures 12.6(a)-(d) show the particle configurations every 1000 time steps through <4ooo, at which time there is b u r n o u t . Figure 12.6(a) reveals combustion particles t o the far right as early as
130
Quasimolecular
Modelling
t h a t the combustion particles have developed as two disjoint sets, separated by fuel particles which are centrally located. T h e final combustion stage in this case thus occurs at the center of the t u b e rather t h a n at the right end, as in the previous cases. T h e energizing of the right end of the t u b e in t h e present example is the result of a compression wave induced at the left by the rapid combustion rate created by the choices T = 30 and the relatively large ignition velocity V = 80. 12.4. Remarks A very large number of additional examples were run in b o t h two and three dimensions. All yielded expected variations from the examples discussed in the last section. T h e parameter choices considered were v = 1,2,3; V = 60,70,80; T = 30,50,70; C = 1.5,1.6,1.75; and d = 0.90,0.95,1.0. Cases with relatively small combustion regions, as when one sets v = 1, V = 60, T = 30, d = 0.90, were relatively inefficient in t h a t a disproportionate number of fuel particles remained after b u r n o u t . It should be observed t h a t the approach developed can be applied to solids, liquids and gases. Our examples, however, were motivated by the wide interest in cylindrical combustion engines. Mathematically, t h e related vapor combustion is exceptionally difficult to model because it is of a turbulent n a t u r e and is usually not accessible through continuum models. Exercises 12.1 Reproduce Fig. 12.4. 12.2 Simulate combustion in a bunsen burner. 12.3 Simulate the burning of a wooden log. 12.4 Simulate quantitatively the burning of methane gas. 12.5 Show t h a t if one does not d a m p the velocities of the burned particles, then Figs. 12.3-12.6 correspond to flame generation when the left end of the tube is open.
Chapter 13 Conservative Q Modelling 13.1. I n t r o d u c t i o n When the Newtonian dynamical equations of a system of particles are not dissipative, the system has three fundamental invariants, that is, quantities, which do not change with time. These quantities are energy, linear momentum and angular momentum. Such systems are called conservative. The solar system, for example, is considered to be conservative system because the hydrogen in interstellar space is so dilute as to make frictional effects negligible as the planets traverse their orbits. The solar system can be modelled reasonably as a system of particles because the radii of the sun and the planets are relatively small compared to the distances between these bodies. Unfortunately, the leap-frog formulae and other traditional numerical methodology, like Runge-Kutta, Taylor series, and predictor-corrector methods, fail to preserve the system invariants of conservative systems, which induces corresponding physical errors in related quantities of interest, like particle trajectories and periods of oscillation. In this chapter, then, we will develop numerical methodology which, when applied to conservative particle systems, will conserve the very same invariants as those of the given system. For variety, the discussion will be applied to planetary type motion. However, it will be shown later how the methodology can be extended to all the models described in Chaps. 2-12. 133
Quasimolecular
134
Modelling
For consistency with the terminology in astronomy, in this chapter, particles will be called bodies. 13.2. The Collisionless TV-Body Problem Given the initial positions r,- and velocities v,- of TV bodies Pi,P2, . . . , PN of respective masses mi, m^, . . . , mj\r, the TV-body problem is that of determining the motion of the system if each body is under the gravitational influence of all the other bodies. We assume, implicitly, that collisions are not allowed. The equations of motion of the TV-body problem are:
"*< = £(-!%?%)•
' = >•'
»• <'">
in which G is a positive constant. The most complete existence and uniqueness theorem for the TV-body problem is that of Wintner (1947). However, actually solving the problem in closed form without the need for numerical approximation has defied all efforts for TV > 3. The two-body problem can be solved analytically, the three-body problem can be "solved" in terms of integrals, and all efforts for solving the four-body problem resulted in failure (Pollard (1976)). For clarity, we shall direct attention to the numerical solution of the three-body problem. The methodology will conserve exactly the same energy, linear momentum, and angular momentum as the system defined by (13.1) with TV = 3. All the ideas and theorems extend completely to the full TV-body problem, but with some additionally required mathematical sophistication (Greenspan (1974c)). Further, for simplicity only, attention will be directed to planar motion, which is not unreasonable physically since the solar system is nearly planar.
Conservative
Q Modelling
135
1 3 . 3 . C o n s e r v a t i v e N u m e r i c a l S o l u t i o n of t h e T h r e e - B o d y Problem For t > 0 and i = 1,2, 3, let particle P,- of mass m,- be at (x,-, y,), and have velocity (i»,)X, f»,y)- From (13.1), the motion of the system from given initial d a t a is determined by Gmim2 2
m\x\
xi-x2 r
i2
r
2/i - y2
Gm1m3
r(2
Gm1m2 f"i2/i =
=
m3x3
=
^2 x2 - « i
'
r12 y2 - J/i
2
r|3 Gm2m3 2 r\z Gm2m3 2 r^ 3 Gm2m3
5
.
'
V1"5-4)
>
(13.5)
'
(13.6)
r23 y2 - y3 r23 x3 - x2 r23 y3 - y2
2
r(3
rl3
(.la.3)
r13
2
r\2 r12 Gmim3 x3 - x i 5 r(3 r13 G m i m 3 y3 - j/i
m3j/3 =
i3
j/i - y3
*i3 Gm2m3x2-x3
2
=
r
,,,., (li.2)
'
2
r(2 Gm1m2 m2y2
xi - x3
i3
2
rf2 Gm1m2 m2x2
Gmim3 2
, '
r<3
.
(13-<)
r23
in which
r?- = [ ( x i - x i ) 2 + ( y , - - J / i ) 2 ] . To approximate the solution of (13.2)-(13.7), we proceed as follows. For At > 0 and tfc = fcAf, k = 0 , 1 , . . . , let P,- be at (x,-jjfc, r/,-^) and have velocity (wi,fc,x, v»,fc,y) at £&. For i = 1,2,3, define Xi,k+1 - Xi,k _ Vi,k + l,x + Vi,k,x
At
-
Vi.k+l - Vi,k _
At
v
i,k+l,y
~
«i,t + l,g — i,k,x
m—
' v
+ i,k,y
2
v
rm
, 1 Q Qx
2
_
(13 8)
'
/,,„>
'
(
"'9)
• •. , At
—
= FiikiX
,
(13.10)
-~
—
- Fitk,y
,
(13.11)
136
Quasimolecular
Modelling
in which, motivated by the right-hand sides of (13.2)-(13.7) and the value of averaging procedures (Greenspan (1980)), we choose G m i m 2 [(zi,jt+i + x1>k) - {x2,k+i + fi,k,x
=
x2ik)]
F ; i »"l2,fc»"l2,* + i r i 2 , f c + r12,Jfc + lJ
_ Gm1m3[(xi>k+1
+ xlik)
- (x3ik+1
+
x3:k)]
ri3,*r13,fc + l[r13,lfc + r i 3 | f c + 1 ] Gmlm2 f\,k,y
'
Q 3 22)
[(yi,*+i + 2/i,jb) - (j/2,ifc+i + 2/2,*)]
—
F
\
1
ri2,fcri2,;fc+i|ri2,fc + ^i2,fc+ij _ G m i m 3 [ ( t ; i i H i + yr>k) - (y3,k+i ri3,kri3,k+i[ri3ik Gmim2
J4
2,k,x
—
f ; '*12,Jfc'"l2,ifc+l|.ri2,fc +
'
r
Gm1m2[(y2:k+1
+ l[r23,k
+ y2ik)
i^)
+
x3,k)]
r23,fc+l]
(13.14)
- (yi,t+i + Vi.k)]
,
ri2,ifl2,fc+l[' i2,ib + ri2,*+l] _ Gm2m3[(y2,k+i
+ V2,k) ~ (?/3,fc+i + y3>k)]
r23,kr23,k + l[r23lk + 7"23,*:+l] G m i m 3 [(x3,ib+i + x3>k) - (x1
(\^
^ 12,fc+lJ
+ x2>k) - (a?3,fc+i +
r23,kr23tk
—
+ ri3ik+i]
[(z2,it+i + x2,k) - («i,*+i + *i,Jfc)]
Gm2m3[(x2}k+1
F,2,k,y
+ y3,k)]
— ri3,kri3,k
_ Gm2m3[(x3ik
F + l[ri3,k
Q g jg-j xlik+1)]
; i + ri3,ifc + l j
+ x 3| fc +1 ) - (x2>k +
x2ik+i)]
r23,jfc**23,ifc + l[r23,jfc + r23,Jfe+l]
Qg
^g\
G , mim 3 [(j/3,t + 2/3,*+i) - (j/i,* + 2/i,ifc+i)] -T3,ib,y
=
F r
;
T r
»'l3,Jt''l3,fc+lL 13,*: + 13,Jk+lJ _ Gm2m3[(y3>k
+ j / 3 , t + i ) - (ife.fc + 3/2,fc+i)] r
'*23,Jfc 23,fc+l[»"23,fc + r23,fc + l]
(^3 J J \
Hj.m = K*t,m - *J,m) 2 + (j/.\m ~ yj',m)2] , m = fc, fc + 1 .
(13.18)
in which
Substitution of (13.12)-(13.17) into (13.10), (13.11) implies t h a t (13.8)(13.11) are then twelve equations for x^jt+i, 2/,-,*+i, vi:k+1>x, Vitk+i>y, i =
Conservative
137
Q Modelling
1,2,3, in terms of xi>k, yiik, viiktX,viikiy, i = 1,2,3. Then, from given initial data Xi o, j/,,o> Vi o x, vi,o,y> ' = 1> 2,3, the positions and velocities for the P,are defined implicitly in terms of the positions and velocities at the previous time step. The solution of the resulting nonlinear system is readily available by Newton's Method (Greenspan (1980a)) and the numerical method has now been described. In order to prove that the above numerical method yields, independently of At, the same energy, linear momentum, and angular momentum as does the solution of system (13.2)—(13.7) from given initial data, the following definitions will be essential. Definition 13.1. The kinetic energy Kitk of P,- at tk is defined by Ki,k = \mi{vh,*
+ vi*,y)-
(13-19)
Definition 13.2. The kinetic energy Kk of the three-body system at tk is defined by 3
( 13 - 2 °)
Kk = 53(#i lJb ) •
Definition 13.3. The potential energy Vijik, i ^ j , determined by the pair of particles P,- and Pj at tk is defined by Vi.^_Grn1rnL.^. r
^
^
ij,4
Definition 13.4. The potential energy Vk of the three-body system at tk is defined by Gmim2 Gmim3 Gm2m3 ,.„.., Vk = . (13.22) ?"i2,fc
ri3ik
r23,k
Observe that Definitions 13.1-13.4 are identical to those of Newtonian mechanics, but are given only at the discrete time steps tk. Theorem 13.1. (Conservation of Energy) Independently of At, Kn + Vn = Ko + V0,
n = 1,2,... .
(13.23)
138
Quasimolecular
Modelling
P r o o f . Let r»-l w
i,n
= ^2[(xi,k+i
~ *i,k)Fitk,x
+ (yi,k+i ~ yi,k)Fiikiy]
,
i = 1, 2, 3 .
fc=0
(13.24) For i = 1, it follows from (13.10)-(13.11) and (13.24) t h a t n-i r, x Wi,„ = m i ^ (^l.fc+l - Sl.fc), -(«l,fc + l , r - t>l,*,*) At i =0 (yi,fc+i - yi.k) (Vl,k + l,y — At
v
l,k,y)
which, with the aid of (13.8)—(13.9), implies n-l
W
l,» = -£51 [iVlk + l,r ~ "l,*,J + («l,t + l,» - «?.*,»)] • fc = 0
However, the last s u m is telescopic, which implies W l . n = ~(vl,n,x
+ v\ny
- v\fi>x
- v\fiiy)
,
or by (13.19), Wi„
= A'i,„ - A'i, 0
Similarly, W2,n = I<2,n - I<2,0, W3,n = K3,n ~ #3,0 • from which it follows t h a t W
n = tl
W
i,n = Kn ~ #,0
(13.25)
•
1= 1
However, direct substitution of (13.12)—(13.17) into (13.24) and algebraic manipulation yields Wn =Wi,n
+ Wiin + W3in
r,
= — umim,2
ST^
V12,fc + 1
fc=o n-l
— Gm\m3
'12, k)
y .ri2 ,kri2,k + l(ri2,k + rn,k+ l ) ( r 13,fc+l
— r
(r23,t + l
— r
13,k)
\ , , k=o ri3 ,Jfl3,it+l(' 13,fc + ri3tk+ l ) n-l
Gm2m3 ] P Jfc = 0
r23,kr23,k + l(r23,k
23,it) +
r23,k+l)
Conservative
Q
139
Modelling
which, by factoring and cancelling, yields n-l
/ 1 W„ = - Gmim 2 V ] f^0\ri2,k n_1
- Gm2m3 k22 =0
1 \ / 1 I - Gm!m3 V I r12tk+ij f^0 \ri3,k
/ 1
1
V23 k
^
1 \ 1 rl3:k+1)
r
'
23,/fc+l
However, each summation in the last expression is telescopic, which yields Wn — - Gm1m2 ( Gm2m3
)-
r
^"12,0
Gmim3
l2,nJ
1
\ r 13,0
^13,1
1
,»"23,o
r
2 3 n
so that, . Gmim2 Wn = (
Gm\m3 1-
r\2,n
Gmim.2 ri2}o
h
ri3|„
Gm\m3 ri3i0
Gm2m3\ r23}H
J
Gm2m3 r23,o
From (13.22), then Wn = ~Vn + V0 .
(13.26)
Hence, elimination of Wn between (13.25) and (13.26) yields (13.23) and the theorem is proved. Note finally that A'o + Vo depends only on the initial data, so that the numerical method conserves exactly the same energy as does the differential system (13.2)-(13.7). Let us turn then to concepts of momentum by considering first linear momentum. Definition 13.5. The linear momentum M,(ijt) = M,-.* of P, at tk is defined to be the vector M , ] J t = {miVitktX,miVi>kiy)
.
Definition 13.6. The linear momentum Mjt of the three-body system at time tk is defined to be the vector 3
Mi = ]TMi,,t . !=1
140
Quasimolecular
Modelling
Theorem 13.2. (Conservation of System Linear Momentum) Independently of At: Mn = M 0 , n = 1,2,3,... . (13.27) Proof. It follows directly from (13.12), (13.14) and (13.16) that ^1,*,* + F2,k,r + F3,kiX = 0 ,
4 = 0,1,2,3,... .
(13.28)
Thus, from (13.10) mi
"l,fc + l,x — vl,k,x
1
—•
•
,
V2 ,k+l,x — V2,k,x J
h Tdn—!
—
A*
-—
At V3,k + l,x — V3k,x
+ m3—'•
'—
n
— = 0,
,
n 1 o o
4 = 0,1,2,3,.
or equivalently, rrai(vi,jfc+i,x — vi,k,x) + rri2(v2,k+i,x - "2,*,*) + rn3(v3lk+i,x - v3ik,x) = 0,
k = 0,1,... . (13.29)
Summing both sides of (13.29) from 4 = 0 t o 4 = n— 1 implies mi(i>i,„,x - vi,o,r) + mi{v2,n,x - vi,o,x) + rn3(v3}„iX - v3)0tX) = 0 . (13.30) Thus, " l l U l . n , * + rri2V2tn,x + ™3t>3,n,i: = C\,
Tl > 1 ,
where mi«i,o,i + m2V2,o,x + m3v3io,x = C\. In a similar fashion, miV\tnty
+ m2V2,n,y
+ ™3V3,n,5, = C 2 ,
n > 1 ,
where »™l«l,0,y + "l2«2,0,!/ + W3«3,0,y =
C2.
Thus, 3
3 S
M n = ^ M i , n = ^2,{miViintX,miVitnty) t= l
i=l
and the theorem is proved.
= (Ci, C 2 ) = M 0 , n > 1 ,
Conservative
Q Modelling
141
Since linear momentum has been defined exactly as in Newtonian mechanics, it follows that conservation principle (13.27) is the same as that for the differential system (13.2)-(13.7). Because the definition of angular momentum is in terms of a cross product vector, and because the proof of angular momentum conservation is truly unwieldy using anything but vector notation and methodology, we now proceed as follows. For * = 1,2,3 and k — 0,1, 2 , . . . , let ri.it = (xi,k,yi,k)
,
Vi.jt = (Vi.jfe.x.Vi.Jfc.y) . Fi.fc = (i ? i,it,r,.F'i,ib,j,) ,
where the xiik,yiik,
vitkiX,vi}kiy,
Fiikix,Fi,k,y
satisfy (13.8)-(13.18).
Definition 13.7. The angular momentum Li t of P,- at tk is defined to be the vector U,k = mifa.k x v.-,*,) . (13.31) Definition 13.8. The angular momentum hk of the three-body system at tk is defined by the vector 3
Lk = Y^Litk.
(13.32)
i=i
T h e o r e m 13.3. (Conservation of System Angular Momentum) Independently of At: L n = L 0 , n = 1,2,3,... . (13.33)
142
Quasimolecular
Modelling
Proof. Consider Li^+x — ^>i,k- Then L»,fc+i
-
L
«,fc =ra.'(r>,ifc+i x vi
x v i | f c + i) + -(r.-^+i x v i|fc )
1
1
- ^(^.fc+i
x v
«'.*) + 2^i,k
X v,
1
''* ;+1 ^
_
2^i,k
X V,,fc
^
= m; r(r,-,fc+i - rj.fc) x (vi,jt+i + Vj|fc) + —(ri.jt+i + riik) x (v i|fc+ i - vi>k)
= rrii
(r,-,fc+i - r,-|fc) x (r i|Jfe+ i -r,- |fc )/(A*)
+ 2(r»',*+i + r«'.*) x (v«',*+i ~ = -At(riik+i
v
a)
+ i\,k) x Fj ifc
Now, set T,-lfc
1 = r ( r i > i + 1 +r i > J : ) x F ^ . T * =
3
Y,Ti.k i=i
Then, L,,jt+i — L^fc = AtTitk , from which it follows that Lfc+i - L* = AtTfc But, for the three-body problem, direct calculation yields T^ = 0, k — 0 , 1 , 2 , 3 , . . . . Thus, Lfc+i = Lfc for all k and the theorem follows readily. Again, note that (13.33) is the same system invariant as for differential system (13.2)-(13.7).
Conservative
Q Modelling
143
13.4. The Oscillatory Nature of Planetary Perihelion Motion In this section we will apply the conservative methodology developed in Sec. 13.3 in order to predict a phenomenon present in planetary motion which is not usually reported. To do this, we will study several examples of what is known as perihelion motion. In each example which follows the time step is At = 0.001 and cgs units are used, so that G = 6.67 x 10~ 8 .
Fig. 13.1.
E x a m p l e 1. Consider the three-body problem for particles Pi,P2 and P3 with the following initial data: mi =(6.67)- 1 10 8 , x1>0 = 0,
m2 = (6.67)-1106, x2io = 0.5,
m3 = (6.67)-H05, x3fi = - 1 ,
2/i,o = 0,
2/2,0 = 0,
2/3,0 = 8,
w
«2,0,x = 0,
l,0,r = 0,
t>l,0,!/ = 0>
"2,0,5, =
163
V3fi>x = 0, >
v
3,0,j, = - 3 . 7 5 .
In the absence of P 3 , the motion of P2 relative to Pi is the periodic orbit shown in Fig. 13.1, for which the period is r = 3.901. If the major axis of the motion is the line of greatest distances between any two points of an orbit, and if the length of the major axis is defined to be 2a, then the
144
Quasimolecular
Modelling
Fig. 13.2.
Fig. 13.3.
major axis of P 2 's motion relative to P x lies on the X-axis and a = 0.730. Incidentally, this orbit was constructed by solving the three-body problem w i t h 7TI3 = 0.
Conservative
145
Q Modelling
Fig. 13.4.
T h e initial d a t a for P 3 were chosen so t h a t this particle begins its motion at a relatively large distance from b o t h P i and P 2 , arrives in the vicinity of (—1,0) almost simultaneously with P2, and proceeds past (—1,0) at a relatively high speed, assuring only a short period of strong gravitational attraction. Particles Pi and P3 come closest in the third quadrant a t <2i25, when P2 is at ( - 0 . 9 2 9 6 , - 0 . 1 1 0 8 ) a n d P3 is at ( - 0 . 9 3 2 5 , - 0 . 1 0 1 2 ) . T h e effect of the interaction is to deflect Pi outward, as is seen clearly in Fig. 13.2, where the motion of Pi, relative to P i , has been plotted from £0 to £5000, with the integer labels n = 0 , 1 , 2 , 3 , 4 , 5 , marking the positions tiooon- After having been deflected, Pi goes into the new orbit about P i which is shown in Fig. 13.3. T h e end points of the new major axis are (0.4943, 0.1664) and ( - 0 . 9 1 0 5 , - 0 . 3 0 7 5 ) , so t h a t a = 0.74135. T h e new period is r = 3.9905. Now, the perihelion point of any of Pi about P i is its position which is closest to P i during t h a t orbit. Since Pi has been deflected into a new orbit, its perihelion point has moved. T h e perihelion motion is measured by the angle of inclination 6 of the new major axis with the X-axis, and is given by t a n 0 = 0.34. Note t h a t the perihelion motion of this example is positive.
146
Quasimolecular
Modelling
E x a m p l e 2 . T h e d a t a of Example 1 were changed only for P3 by setting x 3 j 0 = - 0 . 5 , t / 3 ) 0 = 8.0,V3I0IX = -0.25,i>3 i0i j, = - 4 . 0 0 . This time the strongest gravitational effect between P2 and P3 occurs in the second quadrant at t 1 9 6 6 when P2 is at (—0.94582,0.01950) a n d P3 is at (—0.94418,0.01796), and P2 is p e r t u r b e d into the new orbit shown in Fig. 13.4. T h e end points of the new major axis are (0.50724,-0.18349) a n d (-0.92692, 0.33474), so t h a t a = 0.76246, a n d t h e new period is r = 4.162. T h e resulting perihelion motion is now negative, since the angle 6 of t h e new major axis with t h e X - a x i s is given by t a n # = —0.36. From the above a n d similar examples, it follows t h a t the major axis of P2 is deflected in t h e same direction as is P2. In actual planetary motions, as, for example, in a Sun-Mercury-Venus system, where the mass of t h e sun is distinctly dominant, it can b e concluded t h a t when Mercury a n d Venus are relatively close in the first or in t h e third quadrants, the perihelion motion of Mercury must be p e r t u r b e d a very small amount in the positive angular direction, while relative closeness in the second or in the fourth quadrants must result in a very small negative angular perturbation. All such possibilities can occur for t h e motion of Mercury and Venus. T h u s t h e perihelion motion of Mercury is a complex, nonlinear, oscillatory motion. These conclusions were verified on t h e computer with t e n full orbits of Mercury. Most astronomy books give the incorrect impression t h a t t h e perihelion motion of Mercury is uniform a n d always positive, which is valid only in an average sense. 13.5. Remarks Observe first t h a t all the definitions, methods, and theorems given for the three-body problem extend t o t h e TV-body problem by changing t h e indices i = 1,2, 3 t o i = 1 , 2 , . . . ,N. Further, t o apply the methodology to N bodies which interact like molecules in a local fashion, t h a t is, a t t r a c t like -y and repel like ^ , the corresponding conservative numerical force formula is (Greenspan (1974c)):
F
v^ f
°o- E r i e [(nJ-,Qm(ri,,Hirm-2]
, hj E S £ o [ ( ^ n r . - ^ + i r " - 2 ] 1 rjitk+1 + rjiik (rij,k)q~1(rij,k+i)q~1
J T^ij.k+i + rijik
(13.34)
Conservative
Q Modelling
147
Finally, note t h a t for the interested reader a F O R T R A N program for conservative modelling is given in Appendix C. Exercises 13.1 Prove t h a t for the three-body problem: Tk = 0,
k = 0,1,2,... .
13.2 Consider the conservative, planar motion of a single particle P of mass m under a central Newtonian potential V(r). Classically, one assumes
<1335»
* = -(£);• In particle modelling, at each tk assume (Vk+1 n+i
- Vk) ,rk ^ rk+1 - nt (rk+i + rk)
,
(13.36)
where r\ = x\ + y\, k — 0 , 1 , . . . . Then, show t h a t lim Ffc = F At^O
and, for V =
F
_
, show t h a t formula (13.36) implies Gmlm2{xk+l rkrk+1(rk
+xk) + rk+i)
_ '
'
y
Gm1m2(yk+1 rkrk+i(rk
+ yk) +
rk+1)
13.3 Are there arithmetic formulae, other than those discussed in this chapter, which also conserve the system invariants of the three-body problem? 13.4 Model a computer whose logic is conservative. 13.5 (a) (b)
Develop from the Develop from the
a conservative, quasimolecular approach to modelling Lagrangian point of view, a conservative, quasimolecular approach to modelling Hamiltonian point of view.
148
Quaaimolecular
Modelling
13.6 Let particle P be in motion in an XY-plane. At time tk let vjfc = (vk,x,Vh,y), fc = 0 , 1 , 2 , . . . ,n «>+!,« + P t | , _ Xk+1
2 Vk+l,y +Vk,y
2
- Xk
At
'
_ Vk+1 ~ Vk
" ~ A l ' F* = (FktX,FkiV),
. _ „ . „
,
*-u>i>^--->n
1
L_ n 1 O
1
„
*-0.1-2.--."i = 0)l,2,...,n.
1
If Ffc = FJ., show that under each of the transformations (a) x' = x — a, y1 = y — b, (a,b constants) (b) x' = x cos 0 + y sin 0 y' = y cos 0 — x sin 0, (c) x' = x —ctjt, Jb = 0 , l , . . . y' — y — ctk, k = 0 , 1 , . . . , (c constant) the dynamical equation F* = m transforms into F'k = m
is covariant, that is, it
'k+i
At 13.7 Duplicate Fig. 13.1 and show that the numerical results conserve the energy of the system. 13.8 Duplicate Fig. 13.2 and show all system invariants are conserved. 13.9 Given the initial data of Example 1 in Sec. 13.4, how can one change the velocity of P3 so that it pulls P2 out of orbit? 13.10 Simulate three-body gravitational interaction which allows for collision and/or capture.
C h a p t e r 14 Relativistic M o t i o n 14.1. Introduction When the time it takes for the light to travel from an observed event or object to the eye of the observer is significant, Newtonian mechanics is no longer applicable. In Newtonian mechanics, one assumes that the speed of light to the eye of the observer is infinite. Thus, in observing the motion of a distant galaxy or in modelling the motion of an electron which oscillates at exceptionally high speeds, classical physics has to be modified. For such problems, the Special Theory of Relativity provides a means for correct modelling and analysis. In this chapter, by limiting attention only to certain particular rudiments of Special Relativity, we will explore the need for computers in related problems and the special numerical formulae which preserve the physics of a particle in motion. 14.2. The Concept of Simultaneity in Relativity At the outset, we must note that the types of problems discussed in Chap. 13 cannot be studied in the context of Special Relativity. We will show this quite simply by reproducing the following simple example first given by Einstein, which assumes subtly that the speed of light is the same to any observer. 149
Quasimoltcular
150
Modelling
Three men A, O, B are riding in a fixed direction on a train whose speed relative t o the ground is constant. A is in front, O in the middle, and B is in the rear. A fourth man, O', is standing beside the rails. At the instant t h a t O passes O', both receive light signals from A and B. T h e question to be decided is which of A and B sent the signal first. Now, O is in the same reference system as A and B, t h a t is, A and B are at rest relative to O. T h e y are equidistant from O, and this can be verified by measuring. Hence, flashes from A and B require equal time to reach O, who then concludes t h a t A and B emitted the flashes at the same time. But, O' is standing at the rails and reasons as follows. T h e two flashes arrived when the middle of the train passed O'. However, it took time for the light to travel and both flashes were emitted before the middle of the train reached O'. But, at t h a t time, A was nearer to O' t h a n was B, so t h a t B h a d to emit his signal first. Thus, taking into account the speed of light, the concept of simultaneity becomes a relative one, not an absolute one, and the usual concept of absolute simultaneity is meaningless in the context of Special Relativity. Note, for example, t h a t the three-body problem studied in C h a p . 13 is then not a proper problem in Special Relativity, because in it one assumes t h a t the gravitational forces between the three bodies all act simultaneously. 14.3. T h e Lorentz
Transformation
Let XYZ and X'Y'Z' be two rectangular, cartesian coordinate systems which at some initial time t — 0 coincide. At the origins of each coordinate system, let there be an observer. Let the observers have identical synchronized clocks. Assume now t h a t the X'Y'Z' system is in uniform motion relative t o the XYZ system with constant speed u. For simplicity, assume t h a t the motion is in the X direction only. XYZ is called the lab frame and X'Y'Z' the rocket frame. T h e name rocket frame is derived from the assumption t h a t the speed u is relatively large. Suppose next t h a t at some time b o t h P, the observer in the lab frame, and P', the observer in the rocket frame, observe an exploding star. Assume t h a t P records the s t a r ' s position as {x, y, z) in his coordinates and the time of the explosion as t on his clock. Assume t h a t P' records the star's position as (x',y',z') in his coordinates system and the time t' of the explosion on his clock. T h e quadruples (x,y,z,t), (x', y', z',t') are called events. Assuming t h a t the times taken by the light to travel to each observer's eye is significant, the precise mathematical formulae relating events were
Relativistic
Motion
151
developed first by Lorentz (Bergmann (1942)), and are given as follows. , -'-
c(x — ut) 2
2
(c — u )i
. y' = y,
c2t — ux t'=— - r , c(c 2 —u 2 ) =
z'=z,
, (14.1)
or, equivalently, by c(x' + ut') Z
x=-— —r,
.
.
y = y,
z = z',
c2t' + ux'
t=
—r.
.
(14.2)
(c* — u1) 2 eye1 — uz)2 In (14.1) and (14.2), c is the speed of light, and it is assumed t h a t \u\ < c. Note, also, t h a t when one considers motion in the x direction only, the y1 and z' equations of the transformation (14.1) are the identity transformations, so t h a t , in the discussion which follows, these will be neglected. For later consideration, let us define first continuum concepts of velocity and acceleration. These are taken to be the usual ones, t h a t is, in the lab dx dv v =
a
Tt>
<14-3)
=-dl'
but we assume t h a t \v\ < c, while in the rocket ,
dx'
dv1
t
~ ~dF'
~ ~dU
(14.4)
assuming t h a t \v'\ < c. To relate v and v', we have from (14.1) ,
that
dx1
c~(dx — udt)
dV
c2dt — udx
c2{v — u) v = c2 — uv i
(14.5)
Equivalently, from (14.2), one has c2(v' + u) v =
—7,
c- + uv
— .
(14.6)
Similarly, the relationship between a and a' is found to be ,
c3(c2 2 z
u2)~
(c — uv)A
a ,
(14.7)
152
Qv.asimohcv.lar
Modelling
or, equivalently, c3(c2 - u 2 )t a = —2\ J-a'
(c + ««')
14.4.
.
(14.8)
Covariance
W h e n one develops a physical theory, not only are conservation laws i m p o r t a n t , b u t covariance, or symmetry, is also i m p o r t a n t . By covariance, one means t h a t the structure of the dynamical equations associated with the physical formulation is invariant under fundamental coordinate transformations. These transformations include not only translation and rotation, b u t also the transformation associated with the uniform motion of one coordinate system relative t o another. I n the formulation of Special Relativity, t h e n , this means under the Lorentz transformation. Around 1900 it was shown t h a t Newton's equation was not covariant under the Lorentz transformation, whereas Maxwell's equations, t h a t is, t h e equations of electromagnetic, were. T h e question arose, then, as t o what dynamical equation for particle motion was covariant under the Lorentz transformation, and Einstein showed t h a t if we assume t h a t mass varies with speed, then with only a slight modification of Newton's equation, t h e result was a covariant dynamical equation. Let us prove this result first, since it is essential for an understanding of later discussions. T h e o r e m 1 4 . 1 . Let a particle P be in motion along the X-axis in the lab frame and along the X'-axis in the rocket frame. In the lab frame, let the mass m of P be given by (
(14.10)
where mo is the same constant as in (14.9) and v' is the speed of P in the rocket. Let a force F b e applied t o P in the lab. In rocket coordinates, denote the force by F', so t h a t F = F'.
Relativistic
Motion
153
Then, if in t h e lab the equation of motion of P is given by F = -(mv)
,
(14.11)
it follows t h a t the equation of motion in the rocket is (
IT"'
F
I
l\
(14.12)
=-(mv).
Proof. From (14.11) and (14.9) ,_, dm dv F = v——h n i di dt (-i)(cm0) ( c 2 _ „2)3/2
{-2va)
+ ma
'ma -—r + ma , so t h a t F =
(14.13)
From (14.10) and (14.12), then, we must have .2
F'
2
c -
(14.14)
(v1)2
Since F — F', the proof will follow if we can establish the identity
ma
(V>y
or, equivalently, c" -
(v1)2
(14.15)
However, substitution of (14.5), (14.7), and (14.10) into the right-hand side of (14.15) yields, remarkably, t h a t the identity is valid and the theorem is proved.
154
Quasimohcular
Modelling
14.5. Relativistic Oscillation Let us consider now a particle P which oscillates on the X-axis in the lab frame. Assume also t h a t the force F on P is one whose magnitude depends only on the x coordinate of P. T h e n , let F = f(x)
.
(14.16)
Assume t h a t initially, i.e., at time t = 0, P is at x0 and has speed VQ. T h e n the equation of motion of P in the lab frame is ^-(mv)
dt
= f(x)
.
(14.17)
ma = f(x)
(14.18)
From (14.13), (c2-v2) or c2mx = f(x)(c2
- x2) .
From (14.9), this can be reduced to c3£m0 = / ( x ) ( c 2 - f 2 ) 3 / 2 , so t h a t , finally, x
c 3 mo
(c2 -
x2)3/2 =
0
( U
l g )
is the differential equation one has to solve in the lab frame, given the initial data x(0) = x0, x(0) = v0 . (14.20) In general, (14.19) cannot be solved in closed form, so t h a t the observer in the lab frame must now introduce a computer to approximate the solution. However, the observer in rocket frame also observes the motion of P, but in his coordinate system. His equation and initial conditions are found by applying (14.2), (14.6) and (14.8) to (14.9) and (14.20). T h u s , he too will be confronted with a differential equations problem which requires computational methodology, so a computer identical to t h a t in the lab is introduced also into the rocket. A fundamental problem in preserving the physics of Special Relativity t h e n arises in how the two observers should approximate the solutions of
Relativistic
155
Motion
their initial value problems. Their differential equations are covariant under the Lorentz transformation. To preserve the physics, if they use difference approximations, then, the difference approximations of these differential equations should also be covariant under the Lorentz transformation. We will show next how this can be accomplished. 14.6. Numerical M e t h o d o l o g y In the lab, let At > 0 and tk = kAt. At time tk, let P be at xk in the lab. T h e n , in the rocket P will be at x'k at time t'k, where c2tk - uxk
c(xk - utk) X
* -
(C2 _ „ 2 ) l / 2 '
<* -
C(C2
a
_ „2)l/2
^-Zl>
>
or equivalently,
T h e formulae (14.21) and (14.22) are valid because they are merely special cases of (14.1) and (14.2), t h a t is, they result from the particular choices x — Xk and t = tk. T h e concepts of velocity and acceleration are now approximated by the following formulae. At tk in the lab, let Axk * = ITT
v
=
^•tk
xk+i-xk Z
T
Avk vk+1-vk * - ~TT~ ~ Z T
a
'
ffc+l — tk
&tk
tk
+
,.,„„, (14.23)
•
i — tk
At t'k in the rocket, let ,,, _ Ax' t _ x'k+i ~ x'k Vi — . .. — —. ~r~ , * A/' /' - /' ' L L
^k
l
k+l
, _ *
0,1. —
l
k
Av'k A/'
/'
^lk
'ifc + l
(14.24)
— i' l
k
Then, corresponding to (14.5)-(14.8), one has by direct substitution t h a t
c^)_
Vk =
c1 — uvk
c3(c2 - u2fl2 (c 2 - uvk)2(c2
- uffc+i)
^k±At
c3(c2 -
_ a
<"
'
a k
=
(14.25)
cl + uvk
t„22
(c +, ....l\2,„2 uv'k)2(c2
u2f +,
..../
uv'k+1)\
a
k
(14.26)
156
Quasimolecular
Modelling
Of course, in the limit, (14.25) and (14.26) converge to (14.5)-(14.8). Our problem now is one of choosing an approximation to F = -(mv) ,
(14.27)
in the lab which will transform covariantly into the rocket. The clue for this choice comes from (14.13), which is equivalent to (14.27). What we choose at tk is the approximation F
* =m — ^ 2
) c
\
m/2a*'
"»(**) = - 7 = = f
•
(14-28)
Note first that, again, in the limit, (14.28) converges to (14.13). What we must prove is that if Fk = F'k, and if in the rocket m'(t'k) =
CT /2
"°
,
(14.29)
then, in the rocket m'(t'k)c2 b
k ~ —7 1..,\7\(»1 [(C2 _ K )2)(C2 _ ~ K +, i
) 2\2Yll/2 ) ] 1 / 5
a
* •
(14.30)
Theorem 14.2. If Fk = F'k, then (14.28) and (14.29) imply (14.30). Proof. The proof is entirely analogous to that of Theorem 14.1, but uses (14.21)-(14.26) instead of (14.1)-(14.8). We will show now how, in the lab, (14.28) is applied to generate the numerical solution of a particular oscillator. 14.7. A Relativistic Harmonic Oscillator For the usual definition of a harmonic oscillator in Newtonian mechanics, one assumes that f(x) in (14.16) is given by f(x) = —K2x, where K is a nonzero constant. We will assume that this same choice of f{x) in (14.17) defines a relativistic harmonic oscillator and examine its motion in the lab frame from given initial data.
Relativistic
Motion
157
Fig. 14.1.
Fig. 14.2.
158
Quasimolecular
Modelling
For simplicity, we normalize the equation of motion by choosing m 0 = c = K = 1. Then, (14.19) and (14.20) reduce to i-x(l-x2fl2
= 0,
x(Q) = x0,
x(0) = v0 .
(14.31)
To generate a numerical solution, we have first from (14.23) xk+1=xk
+ (At)vk,
k = 0,l,....
(14.32)
Next, from (14.28), the equation (14.31) is approximated by _
_
1
Vk+i - Vk
Xk
~(i-vi)(i-vi+1y/i
At
'
the solution of which for Vk+i is _ v Vk+1
~
k
- (At)xk(l
- vpl
[l + xl(Atni
l + x2(A02(l-^2)2
-
yp]' '
fc 0 1,2
- ' "--
• (14.33) Formulae (14.32), (14.33) are explicit and yield the desired numerical solution. Let us assume now t h a t z(0) = XQ = 0 and examine the results for various values of vo in the range 0 < VQ < 1. In particular, this is done for 30000 time steps with At = 0.0001 for each of the cases v0= 0.001, 0.01, 0.05, 0.1, 0.3, 0.5, 0.7, 0.9. T h e n Fig. 14.1 shows the amplitude and period of the first complete oscillation for the case VQ = 0.001. For such a relatively low velocity, the oscillator should behave like a Newtonian oscillator and, indeed, this is the case, with the amplitude being 0.001 and, to two decimal places, the period being 6.28 ( ~ 2ir). Subsequent motion of this oscillator continues to show almost no change in amplitude or period. At the other extreme, Fig. 14.2 shows the motion for t^o = 0.9, which is relatively close to the speed of light. To two decimal places, the amplitude of the first oscillation is 1.61 while the period is 8.88. These results are distinctly non-Newtonian, and, to thirty thousand time steps, these results remain constant to two decimal places but do show small increments in the third decimal place. Finally, Fig. 14.3 shows how the amplitude of the relativistic harmonic oscillator deviates from that of the Newtonian harmonic oscillator with increasing VQ.
Relativistic
159
Motion
Fig. 14.3.
Exercises 14.1 Reproduce Fig. 14.2. 14.2 Describe the motion of a relativistic oscillator for which f(x) — —x2, x(0) = 0, x(0) = c/2. 14.3 Show that Newton's dynamicaj equation is not covariant under the Lorentz transformation. 14.4 Show that Maxwell's equations are covariant under the Lorentz transformation.
Appendices - FORTRAN Programs Appendix A - FORTRAN Program Drop.For c c c c c
THIS PROGRAM TESTS PENDENT DROPS. THE SOLID PARTICLES ARE 2501:2703. THE FLUID PARTICLES ARE 1:2500. THE TOTAL NUMBER OF PARTICLES IS 2703. P AND Q ARE CHOSEN TO AVOID SQUARE ROOTS. DIMENSION X0(2703),Y0f 2703),VX0(2703),VYO(2703), 1X12703,31,1(2703.3),VXI2703,2),VYI2703,21, 1ACXI2703),ACY(2703),E2t2703),FX(2703),FY(2703),F(2703) 1,PMASS(2703),XG(2703),XH(2703) OPEN IUMT=21,file='drop.dat , ,status='old') OPEN (UNIT:31.file:'drop.out',status:'new') OPEN (UNIT=41,fiie='drop.xke',status:'new') c THE XG AND XH PARAMETERS ARE LOCAL FORCE PARAMETERS. K=i KPRINT:1000 DO 4 1:1,2703 XGII):75.0 XH(I)=30.0 X(I,1)=0. 711,11=0. VXII,i|:0. 161
162
4
515 10 265
30 65
70
701 3456
Quasimolecular
VY(I,i)=0. X(I,2)=0. Y|I,2)=0. VX{1,2)=0. VY(I,2)=0. ACX(I)=0. ACY(I)=0. PMASS(I ) = 1.0 CONTINUE DO 515 1=2501,2703 XG(I} = 300. XH(I)=120, PMASS(I } = 1.0 CONTINUE READ ( 2 1 , 1 0 ) 1 X 0 ( 1 ) , Y 0 ( I I , V X 0 ( I l f V Y 0 ( I ) , 1 = 1 , 2 7 0 3 1 FORMAT I4F16.10) FORMAT (I12.F20.5) DO 30 1=1,2703 X(I,1)=X0(I) Y|I,1)=Y0II) VX(I,l)=YXO|I) VY|I,l)=VYO(I) CONTINUE GO TO 3456 DO 70 1=1,2703 X(I,1)=X(I,2) Y(I,1)=Y(I,2) VX|I,1)=VX(I,2) VY(I,1)=VY(I,2) CONTINUE DO 701 1=1,2703 ACXII)=0. ACY(I)=0. F(I)=0. CONTINUE DO 78 1=1,2500 PMASSI=PHASS( I) IP1=I*L DO 76 J = I P 1 , 2 7 0 3 82IJ) = (X(I,l)-X(J,l))**2 + I Y ( I , n - Y U , 1 ) 1 * * 2
Modelling
Appendices
76
CONTINUE DO 77 J=IP1,2703 IF (R2IJI.LT.16.0) GO TO 7924 C R=4.0. BUT WE CALCULATE WITH SQUARES TO AVOID ROOTS. F{J)=0.0 GO TO 727 7924 FIJ)=PMASS(J)*PMASSI*i-XG(J)+XH(J)/R2(J))/ l(R2!J)*R2(Ji) 727 FX|J) = F(J)'(X(I,n-XIJ,l)) FY ( J ) = F ( j ) * ( y ( r , i } - Y ( J , i ) > 77 CONTINUE ACXI=ACX(I) ACYI=ACY(I) DO 777 J=IP1,2703 ACXI=ACXI+FXIJj ACX(J)=ACXIJ)-FX|J) ACYI=ACYI+FY(J) ACY|J)=ACY(Ji-FY(J) 777 CONTINUE ACX(I>=ACXI ACY(I)=ACYI 78 CONTINUE C NOW INSERT GRAVITY. DO 8406 1=1,2500 ACY(I) =ACY(13+0.98 8406 CONTINUE C FORCES HAVE JUST BEEN ACCUMULATED. BUT F=A SINCE MASS=1. DO 7509 1=2501,2703 ACX(I)=0.0 ACY(I) = 0.0 7509 CONTINUE DO 7123 1=1,2703 VX(I,2)=VX(I,1)+0.0002*ACXJI) VY(I,2)=VY(I,1)+0.0002*ACY(I) X(I,2)=X(I,1)+0.0002*VX(I,2) Y(I,2)=Y(I,1)+OJ002*VY(I,2! 7123 CONTINUE C WE NOW KEEP THE SOLID PARTICLES FIXED. DO 9917 1=2501,2703 X(I,2j=X!I,l)
Quasimolecular
164
YII.2)=Y(I,1)
m i , 2)=o.o VY11,2) =0.0 9917 CONTINUE C NO WALL DAMPING IS IMPOSED IN THIS PROGRAM. K=K + 1 IF |MOD(K,KPRINT).GT.O) GO TO 82 2014 WRITE (31,10) (X(I,2!,Y(1,2),VX(1,2),VY|1,2) ,1 = 1,2703) XKE=0.0 DO 6555 1=1,2500 XKE=XKE+.5*|VX(I,2)*VX(I,2)+VY|I,2)*YYII,2)) VX(I,2)=VX|I,2I*0.2 VY!I,2)=YY(I,2)*0.2 6555 CONTINUE WRITE 141,265) K,XKE 82 IF ( L L T . 4 0 0 1 ) GO TO 65 STOP END
Appendix B - FORTRAN Program Crack.For c c c c c c c c c
This is the 2-d fracture prograi for a copper plate which is approxiiiately 8 ca by 11.4 ca. The given plate has 2713 quasiaolecules. The force foriula has magnitude -1.5260/r**3 + 0.97665/r**5. The particles which form the slot are given 0 Bass so that they can be included in the calculations. These particles foru a 60 degree angle with the horizontal. Note that the file for this prograa had been erased and that the present progran is a retyping froa hard copy, diaension xol2713),yo(2713),vxo(2713),vyo(2713), lxl2713,2),y|2713,2),vx(2713,2),vy(2713,2), lacx(2713),acy(2713),paass(2713) OPEN (UNIT=21,£iie='crack.dat',status='old') OPEN (UNIT=31,file='crack.kir',status:'new') OPEN (UNIT=41,£ile='crack.kxya',status = 'new'I OPEN |UNIT=51,file='crack.out',status='new') c Unit 21 is the input file, c Unit 31 signals the fracture -gives k,i,r * * 2,
Modelling
Appendices
c Unit 41 is the kprint file-gives k,x,y,acx,acy, c Unit 51 is the output file. 10 fornat (4f16.10) 265 foriat (il2,4f20.5) 81 foriat (2il2,fl6.10) kprint=20000 kk=2000 k=l do 4 i=l,2713 x|i,l)=0.0 y (i, 1) = 0.0 vx(i,l)=0.0 vy(i,l)=0.0 x(i,2)=0.0 y(i,2)=0.0 vx(i,2)=0.0 vy(i,2)=0.0 acx(i) = 0.0 acy(i) =0.0 piass(i)=1.0 4 continue c Ho» fix the initial slot. p»a3S(1070)=0.0 do 5 i = l, 14 p«ass(1070+41*i)=0.0 5 continue read (21,10) (xo(i),yo(i),vxo(i),vyo(i),i=l,2713) do 30 i=l,2713 x(i r l)=xo(i) yli,l)=yo(i) vx(i,l)=vxo(i) vy|i,l)=vyo|i) 30 continue go to 71 65 do 70 i=l,2713 x(i,l)=x(i,2) y(i,l)=y(i,2) vx(i,l)=u(i,2) vyfi,l)=vy(i,2) 70 continue do 701 i=l,2713
166
Quasimohcular
acx11} = 0.0 acy(i) =0.0 701 continue c We utilize the syaaetry of the probles. 71 do 77 i = l,1377 c Force calculations are distributed in terns of a c particle's neighbor configuration. if 1».40 go to 7001 if eq.41) go to 7005 if It. 81) go to 7003 if eq.81) go to 7006 if i.eq.82) go to 7002 if i.It.121 go to 7003 if i.eq.121 go to 7004 if iodli-41,81 eq.0) go to 7005 if «od(i-82,81 .eq. go to 7002 if iodfi-40,81 ,eq. go to 7004 if uodli-81,81 ,eq. go to 7006 go to 7003 7001 acx i)=0.0 acy i)=0.0 go to 77 7002 rll=(x(i,l •x|i-41,l **2+(y(i,l)-y i-41, 1))**2 r22=(x(i,l -xli-40,1 **2+(yfi,l)-y i-40, 1))**2 r33=|x(i,l -xfitl,l))"2+(y(i,l)-y i+1,1) )**2 r44=(x(i,l -x(i+40,l))**2+(y(i,l)-y i + 40,1))*»2 r55=(x(i,l)-x(iMl,l))**2+(y(i,l)-y i + 41,1)1**2 £l=pna3s(i)*piias8(i-41)*(-1.5260 l+0.97665/(rll))/(rll**2) if (rll.lt.1.0667) go to 5000 c If rll is greater than 1.0667, then c a crack has developed. f1=0.0 if |»od(k,kk).gt.0) go to 5000 write (31,81) k,i,rll 5000 f2=piass(i)*pnass(i-40)*(-1.5260 l+0.97665/fr22))/(r22**2) if (r22.lt.1.0667) go to 5001 £2=0.0 if (aod(k,kk).gt.0) go to 5001 write (31,81) k,i,r22
Modelling
Appendices
5001
5002
5003
5004
7003
f3=pnass(i)*piiass(i + l)*(-1.5260 l+0.97665/(r33))/(r33**2) if (r33.lt. 1.0667) go to 5002 {3=0.0 if («od(k,kk).gt.O) go to 5002 write 131,81) k,i,r33 f4=p«ass(i)*pnass|i+40)*f-1.5260 H0.97665/(r44))/lr44**2j if (r44.lt.1.0667) go to 5003 f4=0.0 if (nod(k,kk).gt.0) go to 5003 write (31,81) k,i,r44 f5=piBss(i)*pias8(i+41)*(-i.5260 l + 0.97665/(r55))/U55**2) if (r55.lt,1.0667) go to 5004 £5=0.0 if (Bod(k,kk).gt.0) go to 5004 write (31,81) k,i,r55 fxl=fl*(x(i,l)-x(i-41,li) fyl=fl*(y(i,l)-yli-41,l)) fx2=f2*(x(i,li-x(i-40,D) fy2=f2*(y(i,l)-y(i-40,l)) fx3=f3*(x(i,l)-x(i+l,l)) fy3 = f3*(y|i,l)-y(iH,l)) fx4=f4*fx(i,l)-x(i+40,lj) £y4=f4*(yli,l)-y(i+40,l)) fx5=f5*(x(i,l)-x(i+41,l)) fy5=f5*(yli,l)-yli+41,l)) acx(i)=fxl+fx2+fx3+fx4+fx5 acy(i)=fyl+fy2+fy3+fy4+fy5 go to 77 rll=(x(i,l)-x(i-41 f l))**2+(y(i,l)-y(i-41,l))**2 r22=(x(i,l)-x(i-40,l))'*2+(y(i,l)-y(i-40,l))**2 r33 = (x(i,l)-x(i-l,l))**2 + (y(i,l)-y(i-l,D)**2 r44=(x(i,l)-x(i+l,l))**2+(y(i,D-y(i+l,l})**2 r55=(x(i,l)-x(i+40,l))**2+(y(i,l)-y(i+40,l))**2 r66=(x(i,l)-x(i+41,l))**2+(y(i,D-y(i+41,l))**2 fl=pnass(i)*piiass(i-41)*(-1.5260 l+0.97665/(rll))/(rll**2) if (rll.lt.1.0667) go to 5010 £1=0.0
Quasimolecv.hr
168
5010
5011
5012
5013
5014
5015
if (lodlk.kk) . gt. 0) go to 5010 trite (31,81) k,i, rll f2=pEass(i)*pnass i-40)*(-1.526C 1 0.97665/(r22))/(r 22**2) f (r22.lt.1.0667) go to 5011 2=0.0 f («od(k,kk).gt.O ) go to 5011 rite 131,81) k,i, r22 f3=ptass(i)*p«ass i-1) *(-1.5260 1 0.97665/(r33))/(r 33**2) f (r33.lt.1.0667) go to 5012 £3=0 - 0 f («od(k,kk).gt.O ) go to 5012 rite (31,81) k,i, r33 4=pnass(iI*pnass( i + 1)*(-1.5260 0.97665/(r44))/(r 44**2) f (r44.lt.1.0667) go to 5013 4=0.0 f (Bod(k,kk).gt.O ) go to 5013 rite (31,81) k,i, r44 5=pBass(l)*pcass( i+4O)*(-1.5260 0.97665/(r55))/(r 55**2) £ (r55.lt.1.0667) go to 5014 5=0.0 f (Bod(k,kk).gt.O) go to 5014 rite (31,81) k,i, r55 6=pBass(i)*pBass( i + 41) *(-1.526C 0.97665/(r66))/(r 66**2) £ (r66.lt.1.0667) go to 5015 6=0.0 £ (B0d(k,kk) .gt.O ) go to 5015 r66 rite (31 81) £xl=£l*(x i,l -41,1)) fyl=fl* yli.l l-x(i -41,1)) i-7li fx2=£2* x(i,l )-x(i -40,1)) £y2=£2*(yy(i,l i-y(i -40,1)1 £x3=£3*(xx(i,l i-x(i -1,D) fy3=f3*(yy(i,l i-y(i -1,1!) £x4=£4*(x i,l i-x(i +1,1)) fy4=f4*(y i.l -y(i +1,1)) £x5=£5*(x(i,l i-x(i •40,1))
U,
Modelling
Appendices
7004
5020
5021
5022
5023
fy5=f5*(y(i,1 -y(i + 40,1)) x6=£6*|x(i,l -x(i+41,l)) fy6=f6*(y(i,l)-y(i+41,l)) cx(i)=£xl+£x2+fx3 +fx4+£x5+fx6 cy(i)=fyl+£y2tfy3 +fy4+fy5+fy6 go to 77 ll=(x(i,l)-x(i-41 , l ) ) * * 2 M y ( i , D - y (i-41, 1))**2 22=(x(i,l)-xfi-40 ,l))**2+(y(i,l)-y (i-40, 1))**2 33=(x|i,l)-x(i-l, l))**2+(y(i,D-y( i-1,1) )**2 44=(x(i,l)-x(i+40 ,l))**2+(y|i,l)-y (i+40, 1))**2 55=(x(i,l)-xfi+41 ,l))**2+(y(i,l)-y (i+41. 1))**2 l=pnass(i)*piiass( i-41)*(-1.5260 1 0.97665/(rll))/(r 11**2) £ (rll.lt.1.0667) go to 5020 1=0.0 f («od(k,kk).gt.0 ) go to 5020 rite (31,81) k,i, rll 2=p»ass(i)*pnas8( i-40) *(-1.5260 0.97665/ (r22)) / (r22**2) £ (r22.lt.1.0667) go to 5021 2=0.0 £ (iod(k,kk).gt.O ) go to 5021 rite (31,81) k,i, r22 3=p»ass(i)*piiass( i-1 )*(-!.5260 0.97665/(r33))/(r 33**2) £ (r33.lt.1.0667) go to 5022 3 = 0.0 £ |iod(k,kk).gt.O) go to 5022 rite (31,81) k,i, r33 4=pnass(i)*piiass( i + 40)*(-1 - 5260 0.97665/(r44))/(r 44**2) £ (r44.lt.1.0667) go to 5023 4=0.0 £ (iod(k,kk).gt.O ) go to 5023 rite (31,81) k,i, r44 5=pnass(i)*piass( i+41)*(-1.5260 0.97665/(r55))/(r 55**2) £ (r55.lt.1.0667) go to 5024 5=0.0 £ (Bod(k,kk).gt.O ) go to 5024 rite (31,81) k,i, r55
169
170
5024
7005
5030
5031
5032
Quasimolecular
fxl=£l*(x(i,l)-x|i-41,l fyl=fl*(y(i,l)-y(i-4l,l fx2=£2*(x(i,l)-x(i-40,l fy2=£2*(y(i. i-y(i-40,l fx3=£3*(x(i, i-xli-1,1) fy3=f3*(y(i, i -y(i-1,1) fx4=f4*(x(i, l-xli+40,1 fy4=f4*(y(i, i-y|i+40,l fx5=f5*(xfi, i-x(i+41,l fy5=£5*(y(i, ... . .'-yli+41,1.. acx(i)=fxl+fx2+£x3+fx4+£x5 acy(i)=fyl+fy2+fy3+fy4+fy5 go to 77 rll=|x(i,l)-x(i-40,l))**2Hy(i ( l)-y(i-40, 1))**2 r22=|x|i,l)-xli+l,l))"2+{yli f l)-yli+l,l) )**2 r33 = (x(i,l)-x(i + 41,l))**2 + ( y ( U ) - y ( i + 41,1)1**2 fl=pia88!i)*piass|i-40)M-1.5260 l+0.97665/(rll))/(rll**2) if (rll.lt.1.0667) go to 5030 f1=0.0 if lnod(k,kk).gt.O) go to 5030 write (31,81) k,i,rll f2=pnass(i)*pnass(i+1)*(-1.5260 l+0.97665/(r22))/(r22**2) if (r22.lt.1.0667) go to 5031 £2=0.0 if (nod(k,kk).gt.0) go to 5031 write (31,81) k,i,r22 f3=pnas8(i)*pias8(it41)*(-1.5260 l+0.97665/(r33))/(r33**2) if (r33.lt.1.0667) go to 5032 f3 = 0.0 if («odfk,kk).gt.O) go to 5032 '" k,i,r33 write "" (31,811 fxl=fl*(x(i,] £xl=fl*(xli,l)-xfi-40,l)) £yl=fl*(y(i,l -yli-40,1)) fx2=f2*(x(i,l -xli+1,1)) fy2=f2*(y(i,] -y(i+l,l)) £x3=£3*(x(i,] -x!i+41,l|) fy3=f3*(y(i,l i-yli+41,1)) acx(i) = fxl + fx2 + fx3
Modelling
Appendices
acy{i)=fyl+fy2+£y3 go to 77 7006 rll=|x(i,l)-x(i-41,l))* *2+(y(i )-yd -41,1 > ) " 2 r22=(xfi,l)-x(i-l,l))** 2+(y(i, -y(i- 1,11! **2 r33=(x(i,l)-x(i+40,l))* *2+(y(i )-y(i + 40,1 fL=pmasa(i}*pnass(i-41) *(-1.52 l+0.97665/(rll))/(rll**2 ) if (rll.lt.1.0667) go t o 5040 f 1 = 0.0 if (nod(k,kk).gt.0) go to 5040 write (31,81) k,i,rll 5041 £2=pnass(i}*pnass(i-1) * (-1.526 l+0.97665/|r22))/(r22**2 ) if (r22.lt.1.0667) go t o 5041 f2=0.0 if (nod(k,kk).gt.0) go to 5041 write (31,81) k,i,r22 5041 f3=pnass(i)*pnass( i + 40)'(-1.52 60 l+0.97665/(r33))/(r33**2 ) if (r33.lt.1.0667) go t o 5042 f 3 = 0.0 if l»od(k,kk).gt.0) go to 5042 write (31,81) k,i,r33 50 42 fxl=fl*(x(i,l)-x(i-41,l fyl=fl*|y(i,l)-yli-41,l fx2=£2*(x(i,l)-x(i-l,l) fy2=f2*(y(i, 1) -y(i-1,1) fx3=£3*[x(i,l)-x(i+40,l fy3 = f3*(y(i, 1) -y(i + 40,1 acx(i)=fxl+fx2+fx3 acy(i) = fyl + fy2 + fy3 continue 77 c Ca lculations of acceleratio n of s y n e t r i c particles. do 7019 i=1378,2713 acx(i)=-acx(2714-i) acy(i)=-acy(2714-i) 7019 continue do 7123 i = l r 2713 c Ca lculation of new position s and accelerations for if (i.le.40) go to 7985 if (i.le.1377) go to 79 86
172
Quasi-molecular
Modelling
if (i.gt. 1377) go to 7988 c noving particles: 7986 vx(i,2)=vxfi,l)+0.0001*acx(i) vy(i,2)=vy|i,l)+0,0001*acy(I) x (i, 2)= x( i,1)+0.0001*vx(i,2} y|i,2)= y(i,l)+0.0001*vy(i,2) go to 7123 c stretched particles: 7985 x(i,2)=x(i,l) y(i,2)=y(i,l)-0JOQ2 vx(i, 2) = 0.0 v y (i, 2) = 0.0 go to 7123 7988 xfi,2)=-x(2714-i,2) y!i,2)--y[2714-i,2) vxii,2)=-vx(2714-i,2) vy(i,2)=-vy(2714-i,2) 7123 continue 8043 k=k+l if (nod(k,kprint).gt.0) go to 82 do 810 i=l,1377 write (41,265) k,x(i,2) ,y(i,2) ,acx(i) ,acy(i) 810 continue 82 if (k.It.20001) go to 65 102 write (51,10) (x(i,2) ,y(i,2),vx(i,2) ,vy(i,2), li=l,2713) stop end
Appendix C - FORTRAN Program Conserve.For c c c c c c c c c c
In this progran is shown the iterative technique to conserve energy when the potential is given. The potential chosen is a classical nolecular potential, the Morse potential M ( x ) , given in the f o m : Mfx)=74.855925*(-(dexp|-3.8919124*x))/(3.8919124) +(dexp(-7.7838246*x))/(l.839145)) This is used for the nunerical simulation of the E a tons in a hydrogen nolecule. The potential gives special difficulty because the singularity in the nunerical fornulation does not cancel out, as is the case for 1/r type potentials, for exanple.
Appendices
c c c c c c c
For this reason special care oust be taken and double precision is inposed, He assune the initial data is so chosen that the notion is along the X axis and is synnetrical with respect to the origin. In this way, we need Eollow the notion of only one aton to have the notion of the systen. The progran is written to assure that all constants nust be double precision constants. double precision xl,x2,x3,vl,v2,v3,xnin,xnax,xo, lvo,delt,dif£v,xx,vx,energy I,cl,c2,c3,c4,c5,c6,c7,c8,c9,cl0,cll,cl2,cl3 open Iunit=21,file='conserve.dat',status='old') open (unit=31,file='conserve.out',status='new') open (unit=41,file='conserve.xke',status='new') cl=0.5 c2=3.8919124 c3=7.7838246 c4=74.855925 c c5 and c6 are convergence tolerances. c5=0.000000001 c6=0.0000000001 c7=0.16733 c8=0.760429 c9=8.4646357 cl0=4.2323178 cll=4.35912 cl2=l.839145 cl3 = 17.912514 KPRINT=100 K=l c xiin and xnax are nin and nax distances to the origin, c These values enable one to deternine the dianeter of the c nolecule. delt is the tine step. xnin=1000. xnax=0.0 delt=.00001 C BEAD IN THE INITIAL DATA. READ (21,10) xO,vo 10 FOEMAT(fl6.U,fl6.12) C SET THE IHPUT DATA. xl=X0 vl=vO
173
174
Quasimolecular
GO TO 100 xl=x2 vl=v2 C FIX THE FIRST GUESS x2 OF THE ITERATION AND ITERATE. 100 x2=xl v2=vl KK=1 go to 110 105 x2=x3 v2=v3 110 x3=xl+cl*delt*lv2+vl) diffv=(-dexp(-c2*x3)+dexp(-c2*xl)) l/c2 diffv=diffv-(-dexp(-c3*x3)+dexp(-c3*xl)) 1/C12 diffv=c4*diffv di£fv=diffv/(x3-xl) v3=vl-delt*diffv KK=EK+1 c Fix the laxinun nunber of allovable Newtonian iterations. IF (KK.GT.1000) GO TO 9876 xx=abs(x3-x2) vx=abs(v3-v2) index=l c Test if the convergences tolerances are satisfied. if Ixx.gt.c5) index=-l if (vx.gt.c6) index=-l if (index.eq.-l) go to 105 x2=x3 v2=v3 if (x2.lt.xnin) x«in~x2 if (x2.gt.xnax) xnax=x2 9876 k=k+l 1999 if (HODIK,KPRINT).GT.0) GO TO 82 c Calculate and print energy, c The energy is (10** 11)*E8EEGY energy=c7*(v2*v2)+c8*(-c9* Ildexp(-c2*x2))tcl3*(dexp(-c3*x2))) 1-cll write (41,8112) x2,v2,k,energy,xnin,xiax 8112 foruat (2fl5.10,il0,fl5.11,2fl5.10) 65
Modelling
Appendices
8113 fonat (£25.15) c Go to the aext tine step. 82 if (K.le.lOOODO) GO TO 65 WRITE (31,10) x2,v2 83 STOP END
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References and Sources for Further Reading A d a m , J. R., L i n d b l a d , N. R. a n d Hendricks, C. D . (1968), The collision, cence, and disruption of water droplets, J. Appl. Phys. 3 9 , p . 5173. A d a m , N . K. (1930), The Physics Oxford).
and Chemistry
A d a m s o n , A. W . (1960), Physical
Chemistry
of Surfaces
of Surfaces
coales-
( C l a r e n d o n Press,
(Interscience, N. Y . ) .
A g u i r e - P u e n t e , J., a n d F r e m o n d , M . (1976), Frost propagation in wet porous media, in P r o c e e d i n g s of J o i n t I U T A M / I M T J S y m p o s i u m on Application of M e t h ods of F u n c t i o n a l Analysis t o P r o b l e m s of Mechanics, Marseille, 1975 (Springer, Berlin). Akiyoshi, T . (1978), Compatible Eng. Mech. Div., p . 1253.
viscous
boundary
for discrete
models, A S C E , J.
A l b r y c h t , J., a n d Marciniak, A. (1981), Orbit calculations nearby the points by a discrete mechanics method, Cel. Mech. 2 4 , p . 391. Alder, B . J., a n d Alley, W . E . (1984), Generalized J a n u a r y , p . 56.
hydrodynamics,
equilibrium
Physics
Today,
Alder, B . J., a n d W a i n w r i g h t , T . E . (1960), Studies in molecular dynamics Behavior of a small number of elastic spheres, J. Chem. Phys. 3 3 , p . 1439.
II.
A m s d e n , A. A. (1966), The particle-in-cell method for the calculation of the dynamics of compressible fluids, LA-3466, Los A l a m o s Sci. Lab., Los A l a m o s , N. M. Antonelli, P., Rogers, T . D., a n d Willard, M. (1973), Cell aggregation Theor. Biol. 4 1 , p . 1. 177
kinetics,
J.
178
Arzelies, H. (1966), Relativistic
Quasimolecular Kinematics
Modelling
( P e r g a m o n , N . Y.).
A s h u r s t , W . T . , and Hoover, W . G. (1976), Microscopic fracture two-dimensional triangular lattice, Phys. Rev. B 1 4 , p . 1465.
studies
in the
A u r e t , F . D . , and S n y m a n , J. A . (1978), Numerical study of linear and nonlinear string vibrations by means of physical discretization, Appl. Math. Modelling 2, p . 7. B a r e n b l a t t , G. I., Looss, G., and Joseph, D . D. (1983), Nonlinear Turbulence ( P i t m a n , B o s t o n ) . Barker, J. A., a n d Henderson, D . (1976), What states of matter, Rev. Mod. Phys. 4 8 , p . 587. B a r t o , A. G. (1975), Cellular Automata Thesis, Univ. Michigan, A n n A r b o r . B e r g m a n n , P. G. (1942), Introduction Englewood Cliffs, N. J.).
is 'liquid'?
as Models
to the Theory
Dynamics
and
Understanding
the
of Natural
Systems,
of Relativity
P h . D.
(Prentice-Hall,
Birkhoff, G., and Lynch, R. E. (1961), Lagrangian hydrodynamic computations and molecular models of matter, LA-2618, Los A l a m o s Sci. Lab., Los A l a m o s , N. M. Bombolakis, E. G. (1968), Photoelastic study of initial stages of brittle fracture compression, Int. J. Tectonophys. 6, p . 4 6 1 . B o n d , W . N . (1927), Bubbles
and drops and Stokes'law,
Boris, J. (1986), A vectorized 'near-neighbor' tonic logical grid, J. Comp. Phys. 6 6 , p . 1.
algorithm
in
Phil. Mag. 4 , p . 889. of order N using a
mono-
Boussinesq, J. (1913), Contribution to the theory of capillary action with an extension of viscous forces to the surface layers of liquids and an application notably to the slow uniform motion of a spherical fluid drop, Comp. Rend. 1 5 6 , p . 1124. Broberg, K. B . (1971), Crack-growth Mech. Phys. Solids 1 9 , p . 407.
criteria
in non-linear
fracture
mechanics,
Bulgarelli, U., Casulli, V., a n d G r e e n s p a n , D . (1984), Pressure Methods Numerical Solution of Free Surface Fluid Flows (Pineridge, Swansea).
for
J.
the
B u n e m a n , O., B a r n e s , C. W., Green, J. C., a n d Nielsen, D. E. (1980), Principles and capabilities of 3-D, E-M particle simulations, J. Comp. Phys. 3 8 , p . 1. Burridge, R. (ed.), (1978) Fracture Cadzow, J. A . (1970), Discrete
Mechanics,
calculus
Am. Math. Soc, Providence, R. I.
of variations,
Int. I. Control 1 1 , p . 393.
Cauchy, A. L. (1828), Sur I'equilibre et le mouvement d'un system de points materiels sollicites par des forces d'attraction ou de repulsion mutuelles, Exerc. de Math. 3 .
179
References
Ciavaldini, J. F. (1975), Analyse numerique d'un probleme de Stefan, SIAM J. Numer. Anal. 12, p. 464. Coppin, C , and Greenspan, D. (1983), Discrete modelling of minimal Appl. Math. Comp., p. 17.
surfaces,
Coppin, C , and Greenspan, D. (1988), A contribution to the particle modelling of soap films, Appl. Math. Comp., to appear. Costabel, P. (1973), Leibnitz and Dynamics (Cornell University Press, N. Y.). Cotterell, B. (1972), Brittle fracture in compression, Int. J. Fracture Mech. 8, p. 195. Cotterill, R. M. J., Kristensen, W. D., and Jensen, E. J. (1974), Molecular dynamics studies of melting. III. Spontaneous dislocation generation and the dynamics of melting, Phil. Mag. 30, p. 245. Crank, J. (1957), Two methods for the numerical solution of moving-boundary problems in diffusion and heat flow, Quart. I. Mech. Appl. Math. 10, p. 220. Cryer, C. W. (1969), Stability analysis in discrete mechanics, TR # 6 7 , Computer Sciences Department, University of Wisconsin, Madison, Wisconsin. Daly, B. J. (1969), A technique for including surface tensin effects in hydrodynamic calculations, J. Comput. Phys. 4, p. 97. Daly, B. J., Harlow, F. H., and Welch, J. E. (1965), Numerical fluid dynamics using the particle-and-force method, LA-3144, Part I, Los Alamos Sci. Lab., Los Alamos, N. M. Das, S., and Aki, K. (1977), Fault plane with barriers: A versatile earthquake model, I. Geophys. Res. 82, p. 5658. Davis, R. W. and Moore, E. F. (1982), A numerical study of vortex shedding from rectangles, J. Fluid Mech. 116, p. 475. de Celis, B., Argon, A. S., and Yip, S. (1983), Molecular dynamics simulation of crack tip processes in alpha-iron and copper, J. Appl. Phys. 54, p. 48. Denavit, J. (1974), Discrete particle effects in whistler simulation, J. Comp. Phys. 15, p. 449. Dezin, A. A. (1984), Discrete models in mathematical physics, in Current Problems in Mathematical Physics and Numerical Mathematics, "Nauka", Moscow, p. 75. Dienes, G. J., and Paskin, A. (1983), Computer modelling of cracks, in Atomistics of Fracture (Plenum, N. Y.), p. 671. Dryden, H. L., Murnaghan, F. D., and Bateman, H. (1956), (Dover, N. Y.).
Hydrodynamics
Quasimolecular
180 Dukawicz, J. K. (1980), A particle-fluid Comp. Phys. 3 5 , p . 229.
numerical
Ehrlich, L. W . (1958), A numerical method moving boundary, J. ACM 5, p . 161. Erickson, R. O. (1973), Tubular Science 1 8 1 , p . 705.
packing
model for liquid sprays,
of solving
of spheres
Modelling
a heat flow problem
in biological
fine
J.
with
structure,
E r i n g e n , A . C. (1968), Mechanics of micromorphic continua; mechanics of generalized continua, in P r o c e e d i n g s I U T A M S y m p o s i u m (Springer, Berlin), p . 18. E v a n s , D . J., a n d Hoover, W . G. (1986), Flows far from equilibrium dynamics, Ann. Rev. Fluid Mech. 1 8 , p . 243. Favre, A . (ed.)(1964), The Mechanics
of Turbulence
Fermi, E., P a s t a , J., a n d U l a m , S. (1955), Studies 1940, Los A l a m o s Sci. L a b . , Los A l a m o s , N. M.
via
molecular
( G o r d o n and Breach, N. Y . ) . of nonlinear
problems-I,
F e y n m a n , R. P., Leighton, R. B . , a n d S a n d s , M. (1963), The Feynman on Physics (Addison-Wesley, R e a d i n g , M A . ) . F i n n , R. (1986), Equilibrium
Capillary
Surfaces
Fredkin, E., and Toffoli, T . (1982), Conservative p . 219. F r e u n d , L. B . (1978), Stress intensity factor integral, Int. J. Solids Struct. 1 4 , p . 241.
Lectures
(Springer-Verlag, N. Y . ) . logic, Int.
calculations
J. Theor.
based on a
F r o d a , A. (1977), La finitude en mecanique classique, ses axiomes cations, S y m p o s i u m on t h e A x i o m a t i c M e t h o d , p . 238. Gentil, S. (1979), A discrete 3 , p . 193.
LA-
Phys. 2 1 ,
conservative
es leurs
model for the study of a lake, Appl. Math.
impli-
Modelling
Goel, N . S., C a m p b e l l , R. D., G o r d o n , R. D., Rosen, R., M a r t i n e z , H., Yeas, M. (1970), Self-sorting of isotropic cells, J. Theor. Biol. 2 8 , p . 423. Goel, N . S., Leith, A. G. (1970), Self-sorting 2 8 , p . 469.
of anisotropic
cells, J. Theor.
Goel, N . S., Rogers, G. (1978), Computer simulation of engulfment movements of embryonic tissues, J. Theor. Biol. 7 1 , p . 103.
and
Biol.
other
G o r d o n , R., Goel, N . S., Steinberg, M. S., W i s e m a n , L. L. (1972), A rheological mechanism sufficient to explain the kinetics of cell sorting, J. Theor. Biol. 3 7 , p . 43. G o t t l i e b , M. (1977), Application of computer simulation molecular theories, Comput. and Chem. 1, p . 155.
techniques
to
macro-
References
181
G o t u s s o , L. (1985), On the energy theorem for the Lagrange discrete case, Appl. Math, and Comp. 1 7 , p . 129. G r e e n s p a n , D . (1968a), Introduction G r e e n s p a n , D. (1968b), Discrete Univ. Wisconsin, Madison. G r e e n s p a n , D. (1970), Discrete, p . 195.
to Calculus mechanics,
nonlinear
G r e e n s p a n , D . (1971a), Computer 1 1 , p . 399.
equations
in the
( H a r p e r and Row, N . Y . ) .
Tech. R p t . 49, D e p t . C o m p . Sci.,
string vibrations,
simulation
of transverse
The Computer
J. 1 3 ,
string vibrations,
BIT
G r e e n s p a n , D . (1971b), Computer power and its impact on applied mathematics, in Studies in Mathematics, ed. A. H. T a u b (Prentice-Hall, Englewood Cliffs, N. J.), Vol. 7, p . 65. G r e e n s p a n , D . (1972a), Discrete Utilitas Math. 2, p . 105.
Newtonian
gravitation
and the N-body
problem,
G r e e n s p a n , D . (1972b), An energy conserving, stable discretization of the monic oscillator, Bull. Poly. Inst. Iasi., X V I I ( X X I I ) , Section 1, p . 205. G r e e n s p a n , D . (1972c), Discrete versity of Wisconsin, Madison.
solitary
waves, T R 167, D e p t . C o m p . Sci., Uni-
G r e e n s p a n , D . (1972d), A discrete numerical ceedings of I F I P S , 71 ( N o r t h - H o l l a n d ) . G r e e n s p a n , D . (1972e), New forms
G r e e n s p a n , D. (1973a), Discrete
approach
of discrete
G r e e n s p a n , D. (1972f), A new explicit Franklin Inst. 2 9 4 , p . 231.
har-
to fluid dynamics,
mechanics,
discrete
Kybernetes
mechanics
in P r o -
1, p . 87.
with applications,
J.
Models (Addison-Wesley, R e a d i n g , M A . ) .
G r e e n s p a n , D . (1973b), Symmetry 3 , p . 247.
in discrete
G r e e n s p a n , D . (1973c), A finite difference 8 0 , p . 289.
mechanics,
Foundations
proof that E= mc2,
Am.
of
Phys.
Math.
Mon.
G r e e n s p a n , D. (1973d), An algebraic, energy conserving formulation of molecular and Newtonian N-body interaction, Bull. AMS 7 9 , p . 423. G r e e n s p a n , D . (1974a), Discrete Newtonian lem, Foundations of Phys. 4 , p . 299.
gravitation
G r e e n s p a n , D . (1974b), Discrete bars, conductive Computers and Structures 4 , p . 243. G r e e n s p a n , D. (1974c), A physically Math. Soc. 8 0 , p . 553.
consistent,
and the three-body
heat transfer,
discrete
classical
and
prob-
elasticity,
N-body model, Bull.
Am.
182
Quasimolecular
G r e e n s p a n , D. (1974d), An arithmetic, Math. Appl. Mech. Eng. 3 , p. 293. G r e e n s p a n , D. ( I 9 7 4 e ) , Discrete (Academic Press, N . Y . ) .
particle
Numerical
theory of fluid
Methods
Modelling
dynamics,
in Physics
and
Corap.
Engineering
G r e e n s p a n , D. (1975), Computer Newtonian and special relativistic mechanics, in Proceedings of t h e Second U S A - J a p a n C o m p u t e r Conference, A m . Fed. Inf. P r o c . S o c , Montvale, N. J., p . 88. G r e e n s p a n , D . (1976), The arithmetic Phys. 1 5 , p . 557.
basic of special
relativity,
Int.
J.
Theor.
G r e e n s p a n , D. (1977a), Conservative discrete models with computer examples nonlinear phenomena in solids and fluids, Am. J. Phys. 4 5 , p . 740. G r e e n s p a n , D. (1977b), Computer studies of interaction masses, J. Comp. Appl. Math. 3 , p. 145. G r e e n s p a n , D . (1977c), On the arithmetic Polit. Din. IASI, X X I I I ( X X V I I ) . G r e e n s p a n , D. (1977d), Arithmetic Appl. 3 , p . 253.
of particles
basic of special
applied mathematics,
G r e e n s p a n , D . (1978a), Computer and Comp. 4 , p . 15.
relativity,
Comp.
G r e e n s p a n , D . (1977e), Computer studies of interactions masses, J. Comp. Appl. Math. 3 , p . 145. studies of a von Neuman
and Math,
with
with
differing
of classical
model of the Stefan problem,
G r e e n s p a n , D . (1980a), Arithmetic
Applied
Mathematics
Comp.
Math,
and spe-
Meth.
in
( P e r g a m o n , Oxford). free surface
G r e e n s p a n , D . (1980c), New mathematical Modelling 4 , p . 95.
models
of porous
G r e e n s p a n , D . (1980d), Discrete modelling cosm, Int. J. Gen. Syst. 6, p . 25.
in the microcosm
G r e e n s p a n , D . (1981a), Computer-Oriented Oxford).
Mathematical
G r e e n s p a n , D. (1981b), A classical molecular approach biological sorting, J. Math. Biology 1 2 , p . 227.
Inst.
type fluid, Appl.
G r e e n s p a n , D. (1978c), A particle Appl. Mech and Eng. 1 3 , p . 95.
modelling of nonlinear, XXII, p . 200.
differing
Bui.
of particles
G r e e n s p a n , D. (1978b), A completely arithmetic formulation cial relativistic mechanics, Int. J. Gen. Syst. 4 , p . 105.
G r e e n s p a n , D . (1980b), N-body Math, and Comp. in Simulation,
with
of
flow,
liquid
flow,
Appl.
Math.
and in the
macro-
Physics
to computer
(Pergamon,
simulation
of
References
183
G r e e n s p a n , D . (1982a), Qualitative and quantitative particle modelling with applications to wave generation, vibration, and biomathematics, in Dynamical Systems / / ( A c a d e m i c P r e s s , N. Y.), p . 7 1 . G r e e n s p a n , D . (1982b), Deterministic p . 505.
computer
physics,
Int. J. Theor. Phys. 2 1 ,
G r e e n s p a n , D . (1982c), Direct computer simulation of nonlinear waves in solids, liquids and gases, in Nonlinear Phenomena in Mathematical Sciences (Academic P r e s s , N. Y . ) , p . 4 7 1 . G r e e n s p a n , D . (1982d), Computer modelling of double-layer circularization and gasticulation, in Discrete Simulation and Related Fields ( N o r t h - H o l l a n d ) , p . 153. G r e e n s p a n , D. (1983a), A new computer approach to the modelling stars, Astrophys. and Space Sci. 9 3 , p . 3 5 1 . G r e e n s p a n , D . (1983b), Computer-oriented, faces, Appl. Math. Modelling 7, p . 423.
N-body
modelling
of close of minimal
G r e e n s p a n , D . (1983c), Direct computer modelling, in Mathematical Science and Technology ( P e r g a m o n , N. Y.), p . 46. G r e e n s p a n , D . (1983d), An arithmetic Models (Springer, N. Y . ) , p. 46. G r e e n s p a n , D. (1984a), Computer ena, Math. Modelling 6, p . 273.
in particle
modelling
of
sur-
Modelling
theory of gravity, in Discrete
studies
binary
and fluid
in
System phenom-
G r e e n s p a n , D . (1984b), A new mathematical approach to biological cell rearrangement with application to the inversion of volvox, in Syst. Anal. Model. Simul. 1, p . 5. G r e e n s p a n , D. (1984c), Conservative Phys. 5 6 , p . 28.
numerical
models for x= f(x),
J.
Comp.
G r e e n s p a n , D. (1985), Discrete mathematical physics and particle modelling, Adv. in Electronics and Electron Phys. (Academic P r e s s , N . Y . ) , p . 189. G r e e n s p a n , D . (1986a), Quasimolecular particle fracture, Comp. and Structures 2 2 , p . 1055. G r e e n s p a n , D. (1986b), Particle Comp. in Simul. 2 8 , p . 13.
simulation
modelling
of crack generation
of compression
waves,
G r e e n s p a n , D. (1987a), Discrete arithmetic based simulation modelling in Encyclopedia of Systems and Control ( P e r g a m o n , Oxford), p . 4345. G r e e n s p a n , D . (1987b), Quasimolecular modelling vortices, Appl. Math. Modelling 1 1 , p . 465.
of turbulent
and
Math,
in and and
formalism, nonturbulent
G r e e n s p a n , D. (1987c), Particle modelling by systems of nonlinear ordinary differential equations, in Nonlinear Analysis and Applications (Dekker, N. Y.), p . 203.
184
G r e e n s p a n , D. (1987d), Particle elling 9, p . 785.
Quasimolecular simulation
of spiral galaxy evolution,
G r e e n s p a n , D. (1988a), Quasimolecular channel supercomputer, Comp. Math. Appl. 1 5 , p . 141. G r e e n s p a n , D. (1988b), Mechanisms Comp. Math. Appl. 1 6 , p . 331.
Modelling
Math.
Mod-
and vortex street modelling
on a
of capillarity
G r e e n s p a n , D . (1988c), Supercomputer simulation Math. Computer Modelling 1 0 , p . 8 7 1 . G r e e n s p a n , D. (1988d), Quasimolecular modelling tor computer, Appl. Math. Modelling 1 2 , p . 305.
via supercomputer
of sessile
simulation,
and pendent
of the cavity problem
on a vec-
G r e e n s p a n , D . (1988e), Particle modelling of cavity flow on a vector Comp. Meths. in Appl. Math, and Eng. 6 6 , p . 291. G r e e n s p a n , D . (1989a), Quasimolecular D.: Appl. Phys. 2 2 , p . 1415.
simulation
simulation
G r e e n s p a n , D . a n d Casulli, V. (1985), Particle Math. Modelling 9, p . 215.
computer,
of large liquid drops, J.
G r e e n s p a n , D. (1989b), Supercomputer simulation of cracks quasimolecular dynamics, J. Phys. Chem. Solids 5 0 , p . 1245. G r e e n s p a n , D . (1990), Supercomputer Computer Math. Appl. 1 9 , p . 9 1 .
drops,
and fractures
of colliding microdrops
modelling
Phys.
by
of
water,
off an elastic arch,
Appl.
G r e e n s p a n , D . and Collier, J. (1978a), Computer studies of swirling particle and the evolution of planetary-type bodies, JIMA 2 2 , p . 235.
fluids
G r e e n s p a n , D . a n d Collier, J. (1978b), Computer tion, J. Comp. and Appl. Math. 4 , p . 235.
evolu-
studies
of planetary-type
G r e e n s p a n , D., C r a n m e r , M., a n d Collier, J. (1976), A particle model of ocean waves generated by earthquakes, Tech. R p t . 277, D e p t . C o m p . Sci., Univ. Wisconsin, Madison. G r e e n s p a n , D . a n d H o u g u m , C. (1978), New investigations of von Newmann fluids, T R 323, D e p t . C o m p . Sci., Univ. Wisconsin, Madison. G r e e n s p a n , D., and R o s a t i , M. (1978), Computer Computers and Structures 8, p . 107.
generation
Halicioglu, T . a n d C o o p e r , D . M. (1984), An atomistic Materials Sci. and Eng. 6 2 , p . 121.
of particle
model of slip
type
solids,
formation,
Harlow, F . H. a n d S a n m a n n , E. E . (1965), Numerical fluid dynamics using the particle-and-force method, LA-3144, Los A l a m o s Sci. Lab., Los Alamos, N. M.
References
185
Harlow, F . H. a n d S h a n n o n , J. P. (1967), The splash Phys. 3 8 , p . 3855. H e r t z b e r g , R. W . (1976), Deformation Materials (Wiley, N. Y . ) .
and Fracture
of a liquid drop, J.
Mechanics
of
Hess, S. (1985), Dynamics of dense systems of spherical particles Mec. Theor. et Appliq., Special N u m b e r , p . 1.
Appl.
Engineering
under shear,
Hida, K. a n d Nakanishi, T . (1970), The shape of a bubble or a drop attached flat plate, J. Phys. Soc. 2 8 , J a p a n , p . 1336. Hirschfelder, J. O., C u r t i s s , C. F . , a n d Bird, R. B . (1954), Molecular Gases and Liquids (Wiley, N . Y . ) . Hockney, R. W . a n d E a s t w o o d , J. W . (1981), Computer cles (McGraw-Hill, N . Y . ) .
Simulation
Holtfreter, J. (1943), A study of the mechanics p. 261.
of gastrulation,
Hoover, W . G. (1984), Computer Today, January, p . 44.
of many-body
Hopf, E . (1948), A mathematical Pure. Appl. Math. 1, p . 303.
simulation
example
H u a n g , K. (1950), On the atomic A 2 0 3 , p . 178.
theory
displaying
features
of elasticity,
to a
Theory
Using
J.
of
Parti-
J. Exp. Biol. 9 4 ,
dynamics,
Physics
of turbulence,
Comm.
Proc. Roy.
Soc.
London
H u d s o n , J. A., Hardy, M. P., a n d F a i r h u r s t , C. (1973), The failure of rock Part I-Theoretical studies, Int. J. Rock. Mech. Min. Sci. 1 0 , p . 69.
beams:
Ikeda, T . (1986), A discrete model for spatially Math. Appl. 1 8 ( N o r t h - H o l l a n d ) , p . 385.
Studies
Jackson, J. C. (1977), A quantization
aggregating phenomena,
of time, J. Phys. A: Math.
in
Gen. 1 0 , p . 2115.
J a m e t , P. a n d B o n n e r o t , R. (1975), Numerical computation of the free boundary for the two-dimensional Stefan problem by finite elements, C e n t r e d ' E t u d e s de Limeil, Service M. A., B . P. 27, 94190 Villeneuve-St. George, F r a n c e . K a n a t a n i , K. (1984), The accuracy and the preservation mechanics, J. Comp. Phys. 5 3 , p . 181.
property
of the
discrete
K a n n i n e n , M. F . (1978), A critical appraisal of solution techniques in dynamic fracture mechanics, in Numerical Methods in Fracture Mechanics, Univ. Coll. of Swansea, Swansea, U K . K a n u r y , A . M. (1975), Introduction Breach, N. Y . ) .
to Combustion
Phenomena
(Gordon and
Quasimolecular
186
K a r d e s t u n c e r , H. (1975), Discrete N. Y . ) . Killand, J. L. (1964), Inversion Kittel, C. (1971), Introduction
Mechanics-A
Unified Approach
Modelling
(Springer-Verlag,
of Volvox, Univ. Microfilms, A n n A r b o r , M I . to Solid State Physics
(Wiley, N. Y . ) , 4 t h edition.
Kobayashi, A. S., W a d e , B . G., and M a i d e n , D. E. (1972), Photoelastic gation on the crack-arrest capability of a hole, Exp. Mech. 1 2 , p . 32. Kopal, Z. (1978), Dynamics drecht, Holland).
of Close Binary
L a B u d d e , R. A. (1980), Discrete 6, p . 3.
Systems
Hamiltonian
investi-
(D. Reidel P u b l . Co., Dor-
mechanics,
Int. J. General
Systems
L a B u d d e , R. A. a n d G r e e n s p a n , D . (1974a), Discrete mechanics for nonseparable potentials with application for the LEPS form, T R 210. (Dept. C o m p . Sci., Univ. Wisconsin, M a d i s o n ) L a B u d d e , R. A . a n d G r e e n s p a n , D. (1974b), Discrete ment, J. Comp. Phys. 1 5 , p . 134.
mechanics
- a general
treat-
L a B u d d e , R. A. a n d G r e e n s p a n , D. (1976a), Energy and momentum methods of arbitrary order for the numerical integration of equations I, Numerische Math. 2 5 , p . 323.
conserving of motion -
L a B u d d e , R. A . and G r e e n s p a n , D . (1976b), Energy and momentum methods of arbitrary order for the numerical integration of equations II, Numerische Math. 2 6 , p . 1.
conserving of motion -
L a B u d d e , R. A. a n d G r e e n s p a n , D . (1978), Discrete potentials, Virginia I. Science 2 9 , p . 18.
mechanics
for
anisotropic
L a B u d d e , R. A. and G r e e n s p a n , D. (1987), An energy conserving modification of numerical methods for the integration of equations of motion, Int. J. Math, and Math. Sci. 1 0 , p . 173. L a n d a u , L. D. (1944), On the problem p . 311.
of turbulence,
L a n g d o n , A. B . (1973), 'Energy-Conserving' Comp. Phys. 1 2 , p . 247. Lax, M . (1965), The relation elasticity, in Lattice Dynamics
plasma
Dokl. Akad. Nauk
simulation
algorithms,
between microscopic and macroscopic ( P e r g a m o n , Oxford), p . 583.
Leith, A. G. and Goel, N. S. (1971), Simulation sorting, J. Theor. Biol. 3 3 , p . 171.
of movement
USSR 4 4 ,
theories
of cells during
J.
of
self
Leontovich, A. M., P y a t e t s k i i - S h a p i r o , I. I., and Stavskaya, O. N. (1971), The problem of circularization in mathematical modelling of morphogenesis, Avtomatika i Telemekhanika 2, p . 100.
187
References
Liebowitz, H. (ed.) (1968), Fracture, an Advanced Treatise (Academic, New York). Lorente, M. (1974), Bases for a discrete special relativity, Publ. #437 (Center for Theoretical Physics, MIT, Cambridge, MA.) MacPherson, A. K. (1971), The formulation of shock waves in a dense gas using a molecular dynamics type technique, J. Fluid Mech. 45, p. 601. Madariago, R. (1976), Dynamics of an expanding circular fault, Bull. Seism. Soc. Amer. 65, p. 163. Maeda, S. (1979), On quadratic invariants in a discrete model of mechanical systems, Math. Japonica 23, p. 587. Mahar, T. J. (1982a), Discrete conservative oscillators: periodic and asymptotically periodic solutions, SIAM J. Numer. Anal. 19. Mahar, T. J. (1982b), Discrete almost-linear oscillators, SIAM J. Numer. 19, p. 237.
Anal.
Malkus, W. V. R. (1960), Summer Program Notes, Woods Hole Ocean. Inst., Woods Hole, MA. Malone, G. H., Hutchinson, T. E., and Prager, S. (1974), Molecular models for permeation through thin membranes: the effect of hydrodynamic interaction on permeability, J. Fluid Mech. 65, p. 753. Marciniak, A. (1985), Numerical Solutions of the N-Body Problem (D. Reidel, Dordrecht). Markatos, N. C. (1986), The mathematical Math. Modelling 10, p. 190.
modelling of turbulent flows,
Appl.
Matela, R. J., and Fletterick, R. J. (1980), Computer simulation of cellular selfsorting, J. Theor. Biol. 84, p. 673. May, R. M. (1975), Biological populations obeying difference equations: stable points, stable cycles and chaos, J. Theor. Biol. 5 1 , p. 511. Meyer, G. H. (1973), Multidimensional 10, p. 522.
Stefan problems, SIAM J. Numer.
Anal.
Meyer, R. (1971), Introduction to Mathematical Fluid Dynamics (Wiley, N. Y.). Miller, R. H. and Alton, N. (1968), Three dimensional n-body calculations, ICR Quart. Rpt. # 1 8 , Univ. Chicago. Mostow, G. D. (ed.) (1975), Mathematical Models for Cell Rearrangement Univ. Press, New Haven, CT.).
(Yale
Muetterties, E. L. (1977), Molecular metal clusters, Science 196, p. 839. Murdoch, A. I. (1985), A corpuscular approach to continuum mechanics: Basic considerations, Arch. Rat. Mech. Anal. 88, p. 291.
188
QuasimoleculaT
Modelling
Neumann, C. P. and Tourassis, V. D. (1985), Discrete dynamics robot models, IEEE Trans, on Systems, Man and Cybernetics, Vol. SMC 15, # 2 , Maich/April, p. 193. Nichols, B. D. and Hirt, C. W. (1971), Improved free surface boundary conditions for numerical incompressible flow calculations, J. Comp. Phys. 8, p. 434. Pan, F. and Acrivos, A. (1967), Steady flows in rectangular cavities, J. Fluid Mech. 28, p. 643. Pasta, J. R. and Ulam, S. (1957), Heuristic numerical work in some problems of hydrodynamics, MTAC 13, p. 1. Pavlovic, M. N. (1986), A simple model for thin shell theory-Part 2: Discretized surface, bending theory, and membrane hypothesis, Int. J. Mech. Eng. Ed. 13, p. 199. Perrone, N. and Alturi, S. N. (eds.) (1979), Nonlinear and Dynamic Mechnics (ASME, N. Y.)
Fracture
Perry, A. E., Chong, M. S. and Lim, T. T. (1982), The vortex-shedding process behind two-dimensional bluff bodies, J. Fluid Mech. 116, p. 77. Petersen, R. A. and Uccellini, L. (1979), The computation of isentropic atmospheric trajectories using a discrete model, Monthly Weather Rev. 107, p. 566. Peterson, I. (1985), Raindrop oscillation, Sci. News 2, p. 136. Phan-Thien, N. and Karihalov, B. L. (1982), Effective moduli of particulate solids, ZAMM 62, p. 183. Piest, J. (1974), Molecular fluid dynamics and theory of turbulent motion, Physica 73, p. 474. Pollard, H. (1976), Celestial Mechanics, Cams Monograph # 1 8 (Math. Assoc. Am., Washington. D. C.) Popov, Yu. P. and Samarskii, A. A. (1970a), Completely conservative difference schemes for the equations of gas dynamics in Euler's variables, Zh. vychisl. Mat. mat. Fiz. 10, p. 773. Popov, Yu. P. and Samarskii, A. A. (1970b), Completely conservative difference schemes for magnetohydrodynamic equations, U.S.S.R. Comput. Math, and Math. Phys. 10, p. 233. Potter, D. (1973), Computational Physics (Wiley, N. Y.). Prandtl, L. (1925), On the development of turbulence, ZAMM 5, p. 136. Preisendorfer, R. W. (1965), Radiative Transfer in Discrete Spaces (Pergamon, N. Y.).
Rtjtrtncts
189
R a v i a r t , P. - A . (1985), An analysis of particle methods, ematics (Springer, N. Y . ) , Vol. 1127, p . 243. Rawlinson, J. S. and W i d o m , B . (1982), Molecular d o n Press, Oxford).
in Lecture Notes in
Theory
of Capillarity
Math-
(Claren-
Reeves, W . R. a n d G r e e n s p a n , D . (1982), An analysis of stress wave propagation in slender bars using a discrete particle approach, Appl. Math. Modelling 6, p . 185. Roger, G., Goel, N . S. (1978), Computer simulation of cellular movements: sorting cellular migration through a mass of cells and contact inhibition, J. Biol. 7 1 , p . 141. R o s e n b a u m , J. S. (1976), Conservation ods for systems of ordinary differential
properties equations,
of numerical integration methJ. Comp. Phys. 2 0 , p . 259.
Ruelle, D . a n d Takens, F . (1971), On the nature Phys. 2 0 , p . 167. Saffman, P. G. (1968), Lectures on homogeneous ear Physics (Springer-Verlag, N. Y . ) , p . 485. Schlichting, H. (1960), Boundary
Layer
of turbulence,
turbulence,
Comm.
Math.
in Topics in
Nonlin-
Theory (McGraw-Hill, N. Y . ) .
Schubert, A . B . a n d G r e e n s p a n , D . (1972), Numerical studies of discrete strings, T R 158, D e p t . C o m p . Sci., Univ. Wisconsin, Madison. Schwartz, H. M. (1968), Introduction
to Special Relativity
Shapiro, A . H. (ed.) (1972), Illustrated Press, Cambridge, MA.). Sih, G. C. (ed.) (1973), Dynamic Simpson, G . C. (1923), Water
CellTheor.
Experiments
Crack Propagation
in the Atmosphere,
vibrating
(McGraw-Hill, N. Y . ) .
in Fluid
Mechanics
(MIT
(Noordhoff, Leiden.) Nature
1 1 1 , p . 520.
Simpson, S. F . a n d Haller, F . J. (1988), Effects of experimental variables on of solutions by collisions of microdroplets, Analyt. Chem. 6 0 , p . 2483.
mixing
Sneddon, I. N . a n d Lowengrub, M. (1969), Crack Problems of Elasticity (Wiley, N. Y . ) .
Theory
in the Classical
S n y m a n , J. A. (1979), Continuous and discontinuous numerical Troesch problem, J. Comp. and Appl. Math. 5 , p . 171. S n y m a n , J. A. a n d S n y m a n , H. C. (1981), Computed tures, Surface Science 1 0 5 , p . 357. Soos, E . (1973), Discrete 2 5 , p . 687.
and continuous
models
epitaxial
solutions
to the
monolayer
struc-
of solids III, Stud.
Cere.
Mat.
Soules, T . F . a n d Bushey, R. F . (1983), The rheological properties and fracture of a molecular dynamic simulation of sodium silicate glass, J. Chem. Phys. 7 8 , p . 6307.
Quasimolecular
190
Modelling
Stefan, J. (1889), Uber die Theorie der Eisbildung, insbesondere uber die Eisbildung im Polarmeere, Sitz. Akad. Wiss. Wien, Mat.-Nat. Classe 9 8 , p . 965. Steinberg, M . S. (1963), Reconstructing 141, p. 401.
of tissues
Su, C. H. a n d Mirie, R. M. (1980), On head-on waves, J. Fluid Mech. 9 8 , p . 509. Synge, J. L. (1965), Relativity: Taylor, G. I. (1921), Diffusion Soc. A 2 0 , p . 196.
The Special
collision
by continuous
movements,
Teodorescu, P. P. a n d Soos, E. (1973), Discrete models of elastic solids, ZAMM 5 3 , p . T 3 3 .
and computations,
Trefethen, L. (1972), in Illustrated Cambridge, MA.).
Science
two
solitary
between
Physics
Proc. London
attraction,
Math.
(Freeman, San Fran-
quasi-continuous
T h o m s o n , W . (Lord Kelvin) (1886), Capillary
T o o m r e , A. and T o o m r e , J. (1973), Violent
cells,
Theory (North-Holland, A m s t e r d a m ) .
Taylor, E . F . a n d Wheeler, J. A . (1966), Spacetime cisco.)
Toffoli, T . (1982), Physics
by dissociated
and
continuous
Nature 3 4 , p . 270.
Int. J. Theor. Phys. 2 1 , p . 165. tides between galaxies,
Experiments
Sci. Am.
in Fluid Mechanics
38.
( M I T Press,
Uccellini, L. W . a n d P e t e r s e n , R. A. (1980), Applying discrete model concepts the computation of atmospheric trajectories, Int. J. Gen. Syst., 6, p . 13. van Dyke, M. (1982), An Album
of Fluid Motion
(Parabolic Press, Stanford, CA.)
Vargas, C. (1986), A discrete model for the recovery of oil from a reservoir, Math, and Comp. 1 8 , p . 93. von K a r m a n , T . (1963), Aerodynamics
to
Appl.
(McGraw-Hill, N. Y . ) .
von N e u m a n n , J. (1963) Proposal and analysis of a new numerical method for the treatment of hydrodynamical shock problems, in The Collected Works of John von Neumann, ( P e r g a m o n , N. Y.) Vol. 6, No.27. Vul, E . B . a n d P y a t i t s k i i - S h a p i r o , I. I. (1971), A model of inversion Problemi Peredachi Informatsii 7, No.4, p . 9 1 .
in
volvox,
W a d i a , A . R. a n d G r e e n s p a n , D . (1980), An arithmetic approach to gas dynamical modelling, in Innovative Numerical Analysis for the Engineering Sciences, Univ. Press of Virginia, p . 272. Welch, J. E., Harlow, F . H., S h a n n o n , J. P., a n d Daly, B . J. (1966), The method, T R LA-3425, Los A l a m o s Sci. L a b . , Los A l a m o s , N. M.
MAC
References
191
Wente, H. C. (1980), The symmetry of sessile and pendent drops, Pacific J. Math. 88, p.387. Whitrow, G. J. (1961), The Natural Philosophy of Time (Harper's, N. Y.). Wintner, A. (1947), The Analytical Foundations of Celestial Mechanics (Princeton Univ. Press, Princeton, N. J.). Young, D. M. and Gregory, R. T. (1972), A Survey of Numerical (Addison-Wesley, Reading, MA.).
Mathematics
Zeigler, B. P. (1976), Theory of Modeling and Simulation (Wiley, N. Y.). Zeldovich, Y. B., Barenblatt, G. I., Lebrovich, V. B., and Makhviladze, G. M. (1985), The Mathematical Theory of Combustion and Explosions (Consultants Bureau, N. Y.).
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Subject Index accuracy 7, 99 adhesion 29, 72, 78 aggregates 4 aluminum 55, 58 amplitude 158 angular 133, 134, 137, 141, 146 approximation 5, 8, 24, 34, 134, 155, 156 argon 67 atom 4 , 5 , 4 3 , 4 4 , 6 3 , 6 5 , 6 6 attraction 7, 19, 36, 78, 145 average 63, 83, 96, 102, 105, 109, 114, 146 axial 58 bar 55, 58, 59, 61, 62 basin 21, 23, 27, 31, 81, 83 bifurcation 95 bodies 133, 134, 146, 150 boundary 23, 35, 93, 110, 113, 117, 118 boundary layer 110 Brownian 83, 98, 99 burnout 128-130 cartesian 150 cavity 81, 93, 96 cell sorting 72, 73, 77-79 193
194
Subject
Index
centrifugal 110 changes of state 63 channel 110, 113, 116 chemical 33, 63, 125 clock 150 collision 33, 41, 134, 148, 158 collisionless 134 combustion 125, 127-130 compressive 58 conservation 5, 6, 21, 23, 24, 44, 125, 137, 140, 141, 152 conservative 133, 135, 143, 146, 147 constant 4, 5, 25, 26, 66, 78, 83, 134, 148, 150, 152, 156 contractions 35 copper 43, 44, 65, 66 core 72, 75, 78, 79, 84 covariance 148, 152 crack 43, 51 damping 29, 30, 35, 75, 78, 83 data 6-8, 20, 26, 29, 30, 34, 35, 55, 56, 65, 74, 83, 113, 135-137, 139, 143, 145, 148, 154, 156 density 35, 41, 58 difference 4, 8, 98, 155 difference equation 8, 135, 155 differential equation 4-6, 19, 35, 125, 154, 155 dislocation 51 dissipative 133 distributed mass 58 drop 19, 21, 29-31, 35, 36, 39, 41 dumbell 40 dynamical 4, 6, 20, 26, 33, 34, 46, 63, 65, 72, 73, 81, 83, 133, 148, 152, 159 elasticity 57, 58 energy 3-5, 23, 24, 44, 45, 72, 74, 75, 84, 125, 134, 137, 139, 148 engines 130, 133 equilibrium 4, 25, 27, 56, 58, 63, 67, 78, 125 events 149, 150 expansions 35 filtering
83, 98
Subject
Index
195
fingering 122 flames 125, 128 fluid 11, 19, 20, 29, 43, 65, 79, 81, 83, 84, 88, 95, 98, 99, 102, 105, 107, 109, 110, 113, 114, 118, 122 force 3-8, 19-21, 25, 26, 29, 30, 34, 43, 44, 46, 47, 50, 51, 56-58, 63, 65, 73, 82-84, 96, 99, 107, 108, 113, 118, 119, 122, 127, 146, 150, 152, 154 FORTRAN 11, 52, 147, 161 fracture 43, 52 frame 150, 152, 154, 156 free energy 72 fuel 125, 127, 128, 130 function 21, 34, 43, 57, 73 gas 67, 79, 130 gradients 19 gravitational 134, 145, 146, 148, 150 gravity 7, 19, 20, 25, 29, 33, 46, 58, 79, 82, 83, 113, 118, 119, 122 harmonic 156 heat 4, 29, 125, 127 homogeneous 58, 95, 122 ignition 125, 127, 130 impulsive 47 injection 118, 119, 122 instantaneous 47, 98, 99, 102 insulated 129 interaction 4, 5, 7, 8, 19, 25-27, 29, 33, 35, 36, 41, 43, 55, 56, 63, 72, 73, 79, 83, 99, 102, 110, 113, 118, 119, 127, 145, 148 interface 72, 117, 118, 122 invariants 133, 142, 147, 148, 152 iteration 10, 122 Karman vortex sheet 107 kinetic energy 27, 29, 44, 63, 66, 137 knowledge 3, 4, 72, 79 krypton 67
196
Subjeci
Index
lab 150-156 laminar flow 109 layer 72, 75, 79 leap-frog 7, 8, 10, 20, 27, 55, 66, 73, 83, 108, 127, 133 least 34, 63 lid 81, 83 light 141-151, 158 linear momentum 133, 134, 137, 139, 140, 141 local 7, 8, 25, 55, 56, 72, 73, 99, 102, 107, 108, 113, 118, 119, 127, 146 long range 7, 15, 82, 96, 107, 119 Lorentz transformation 150, 152, 155, 159 mass
5-7, 20, 21, 23, 34, 41, 44, 56, 58, 63, 66, 82, 113, 118, 125, 134, 146, 147, 152 melting point 63, 66, 67 Mercury 146 microdrop 33, 41 microsecond 55, 58, 60 midspan 62 modulus 57 molecule 4-7, 19-21, 23, 25, 26, 33-36, 39, 41, 63, 71, 146 molecular mechanics 4, 11 momentum 5, 133, 134, 137, 139, 141 mosaic 21, 23, 35, 44, 74, 81, 108, 110 JV-body problem 134, 146 Navier-Stokes equations 16, 19, 95, 96 neighbor 7, 11, 20, 21, 23, 27, 44, 46, 55, 93, 108 neon 67 Newtonian mechanics 4, 133, 137, 141, 149, 156 nonlinear 4, 35, 137, 146 nonuniform 56 normalizing constant 46 observer 98, 149, 150, 154 oil 31, 117-119, 122 orbit 133, 143-146, 148 oscillation 88, 133, 158
Subject
Index
197
paradox 16 parameters 4, 7, 20, 21, 26, 55-57, 65, 67, 71, 73, 75, 77-79, 102, 110, 116, 118, 119, 122, 126-130 particle 4-8, 10, 11, 20, 21, 27, 29, 39, 44, 47, 55, 56, 58, 61, 73-75, 78 79, 82-84, 88, 93, 96, 98, 99, 102, 105, 109, 113, 114, 118, 119, 122, 125-130, 133, 134, 137, 143, 145, 147-149, 152, 154 particle modelling 58 peak 58-60 perihelion 143, 145, 146 period 29, 133, 143, 145, 146, 158 periphery 72 Planck's constant 66 planetary motion 143, 146 plasma 4 plate 43, 44, 47, 50-52 porous flow 117 potential 21, 23, 24, 33, 35, 41, 43, 63, 65, 67, 71, 147 potential energy 137 production 118, 122 propagation 55, 58, 62 pulse 58, 62 Q modelling 4, 7, 34, 43, 63, 71-73, 96, 110, 125 qualitative 71, 77 quantitative 4, 55, 71-73, 78 quasimolecule 6, 20, 21, 23, 25-27, 29, 33, 44-47 quasimolecular modelling 4 raindrop 33, 40 Relativistic Mechanics Relativity 149 reorganization 72 repulsion 7, 19, 83 rest mass 152 restraint 58 rock 118, 119 rocket 150-156 Runge-Kutta 133
149
198
Subject
Index
science 11 self reorganization 72, 74, 75 shafting 62 shear 52 sheet 107 simultaneity 150 slender 55, 58 slotted 43, 47 smoothing 83 solid 4, 6, 11, 29, 43, 63, 79, 93, 96, 99, 107-109, 113, 114, 116, 117, 130 sorting 72 Special Relativity 149 square 21, 34, 55, 63, 74, 81, 83, 93, 118 stable 6, 20, 21 statistical mechanics 4 steady state 4, 84 Stefan problem 117 strain 55-60, 62 stress 4, 47, 55-58 stree-strain 57 subscript 10, 82 sun 133, 146 surface waves 35 symmetry 27, 39, 51, 152 temperature 4, 77 tension 16, 19, 29, 36, 59 tortoise 15, 16 trajectory 10, 11, 96, 133 transformation 34, 46, 65, 148, 151 transients 58 turbulence 11, 71, 95, 96, 99, 125 undulation
93
vapor 130 velocity 8, 20, 27, 39, 41, 74, 79, 83, 84, 93, 96, 98, 99, 102, 108, 109, 113, 114, 127, 130, 135, 137, 148, 151, 155, 158 Venus 146 vertices 21, 44, 64, 74, 81, 110, 118, 126
Subject
Index
199
vortex 84, 88, 93, 96, 98, 99, 102, 105, 107-110, 114, 116 vortices 93, 98, 99, 102, 105, 107, 109, 114, 116 water 19, 21, 23, 29, 31, 33-35, 41, 71, 117, 119, 122 waves 118 well 118, 122 xenon
67
Young's modulus
56