group expansions for the bulk properties of a statistically homogeneous, random suspension

Group Expansions for the Bulk Properties of a Statistically Homogeneous, Random SuspensionAuthor(s): D. J. JeffreySource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 338, No. 1615 (Jul. 16, 1974), pp. 503-516Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/78595 .

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Proc. R. Soc. Lond. A. 338, 503-516 (1974) Printed in Great Britain

Group expansions for the bulk properties of a statistically homogeneous, random suspension

BY D. J. JEFFREY

Department of Applied Mathematics and Theoretical Physics, University of Cambridge

(Communicated by G. K. Batchelor, F.R.S. - Received 14 November 1973)

The problem considered is the general one of calculating the bulk properties of a random, statistically homogeneous suspension. The bulk properties are expressed first as ensemble averages and then as expansions (derived from the ensemble averages) whose successive terms involve averages over successively larger groups of particles. These expansions correspond to the expression of the bulk properties as series in the volume fraction of the particles. The derivation is applicable to any shape of particle but for simplicity the description of the method is mainly confined to spherical particles. The form of the expansion depends upon the strength of the interactions between the particles, and the forms for two cases are given. The conditions for the existence of the expansions are discussed and an example is given of interactions which are too strong to be handled by this method.

1. INTRODUCTION

In the study of the bulk properties of suspensions and composite materials, those

suspensions in which the concentration of the particles is small have been of long- standing theoretical interest. A description of some of the properties of these

suspensions and a summary of known results can be found in a recent review by Batchelor (1974). One problem which is discussed in that review, and which is the subject of this paper, is the possibility of obtaining expressions for bulk properties as series in powers of the volume fraction of the particles, in particular as series in

integral powers of the volume fraction (denoted by c). Belief in the possibility sprang from the early successes of Maxwell (I873) and Einstein (i906) in obtaining expressions linear in c for the effective conductivity and viscosity of a suspension of

spheres and it was strengthened by the success of the virial expansion in the theory of gases. Recently, the results of Einstein and Maxwell have been extended to the term in c2 by Batchelor & Green (1972) and Jeffrey (1973) respectively, thus pro- viding further evidence for the existence of a series in integral powers of c. At the same time other work (Childress I972; Saffman 1973; Howells 1974) has found that some problems have series solutions which contain terms such as ci, c2 and clogc. This paper shows that the c2 calculations referred to above have used the first two terms of an expansion which is given below. With certain restrictions on the

[ 503 ]



504 D. J. Jeffrey

statistics of the array of particles this expansion corresponds to an integral power series in c, thus confirming the early belief, but without these restrictions the

expansion can produce non-integral powers of c. The expansion is not applicable to all situations (e.g. flow past a fixed array of particles) but it has the advantage that, for those situations to which it does apply, it can be written down explicitly.

One of the principal difficulties in studies of the sort undertaken here is the

appearance of integrals which are not absolutely convergent. Although there has been a vague awareness of this difficulty for some time, ways of dealing with it have been slow to develop. One possible reason is that the non-convergence comes in the form of conditional convergence and in spite of the potential divergence, finite

(but not unique) values can be found for the integrals. This led people to look for a 'correct' way to evaluate the integrals instead of a way to avoid them. Einstein

(1906) evaluated conditionally convergent integrals for both energy dissipation and mean rate of dilatation and then combined them together in a way which cancelled out the contributions from the non-convergent parts of the two integrals. Failure to

appreciate this aspect of Einstein's method caused some confusion in later work. The aim of the work described here is to derive an expansion which involves only absolutely convergent integrals. The method used is an extension of that used by Batchelor (I972) who found the leading term of the expansion which will be given here in full. Readers are reminded that a sufficient condition for an integral to be

absolutely convergent is that the integrand goes down faster than r-3 at large distances r from the origin (Smirnov I964, ? 86).

The work below is mostly concerned with the statistics of the suspension and the

particular physical context is important only for certain general properties it must

possess, thus several physical problems can be treated simultaneously by using an abstract formulation of the problem. This, however, will be hard to follow for those not familiar with the subject, so the heat conduction problem will be sketched first and used during the development as a specific example. In this context one seeks the average heat flux F as a function of the average temperature gradient G, the conductivities of the matrix (A1) and particles (A2) and the statistics of the suspension. F and G are stationary random functions of position so F and G can be defined by the volume averages

(~ = lim r1 r G- lim V GdV, (1.1)

tnd Ff = lim l- 1 r and F= lim v- AGdV, (1.2) v-+Oo v

and combining these gives

FA= G+ lim (A2-A) GdV (1.3)

where KT is the volume occupied by particles. For each particle define

S = (A2-A1) GdV, (1.4) volume of particle



Bulk properties of a statistically homogeneous, random suspension 505

then (1.3) becomes F = AG+nS, (1.5)

where n is the number density of the particles and S is the average of S over the

particles. The general statistical problem can be stated thus: a heterogeneous material is

in the form of a suspension of particles randomly and homogeneously dispersed in a continuous matrix. With each particle is associated a quantity S (a vector or

tensor) whose value depends upon the distribution of all the other particles and

linearly upon the known average value of a field G (likewise vector or tensor) which is defined everywhere in the suspension. An expansion for the average of S over all

particles in the suspension is to be calculated when the concentration of the particles is small. It is expected from experience in other branches of statistical physics that successive terms in the expansion will be averages of the interactions within succes-

sively larger groups of particles. The problem is not purely statistical because the form of the expansion is governed by some general properties of the interactions. The expansion takes its simplest form when the interactions fall off so quickly (faster than (particle separation)-3) that they can be considered to be short-range interactions. This will be called the first group expansion. Its form is similar to that of the virial expansion and has been so called by some authors, but respect for

etymology should prevent this extension. The main interest of the paper is in the second group expansion given below, which is for those interactions which fall off more slowly and cause definite long-range effects to enter the expansion without

completely upsetting its form. Still stronger interactions cannot be treated by this method.

2. FORMULATION AND FIRST GROUP EXPANSION

The analysis proceeds by expressing the averages of S and G as ensemble averages over all configurations of the suspension. So that the principles involved do not become obscured by unnecessary detail, in this and the following two sections the

suspension will be supposed to consist of identical spherical particles of radius a and the extension to more general collections of particles (which requires no new

principles) will be outlined in ? 5. The reference particle for the ensemble averaging is taken as the origin of a set of coordinates and the configuration of the suspension is described by the set of the position vectors of the centres of the spherical particles, denoted by {= {rl, r2, r3, ...}. A subset of ' containing k members is denoted Wk

and the set of all such subsets is denoted tk('). Two probability density functions are associated with each Ck:P(W[lo) and P(Ck), being respectively for the con-

figuration k given that the reference sphere is in its place at the origin and for % when the reference sphere is absent: from the system. Conditional probability densities when a group W? is fixed ar P(klo, ~ ) and P((W,l'); the rules for their use are discussed later.

The normalization conditions for the probability density functions pose a

problem well known in statistical physics. If the suspension occupies an infinite



volume and contains an infinite number of particles (and this is what is wanted) then it is unavoidable that infinities turn up in the normalization conditions. The standard way of dealing with this is to consider a large finite volume V containing N spheres and then to take an appropriate limit when the final expansion has been obtained. This is followed here but with the difference that by a particular choice of normalization conditions it is possible to write almost all equations in a form which does not involve the finite volume explicitly, thus allowing the limit process to be anticipated in the notation. The normalization conditions which will be used (for a volume V containing N particles) are

fP(k|Io, W)dk = fP(|kle)d - ( ki), (2.1)

where d == drdr2 ... drk. This is the number of groups W contained in N-i

particles. Two other common choices for the right-hand side of (2.1) are

(N- i) !l(N- - k) !

and Vk. The former was used by Batchelor (1972) and the latter by Fisher (I964). The form some of the equations take with these choices instead of (2.1) will be noted at the appropriate places. The choice (2.1) differs in an important way from the other two. It implies that each 'k is only counted once instead of c! times as is more usual. This means that a little extra care is needed with conditional probabilities because P('k) -P(C~)P(WC6,_j|). The actual rule is given after (2.11). Also a counting convention is needed so that in later work each 6k is only considered once. The one used here is that P(V) is zero unless the members of k are numbered in order of increasing distance from the origin (reference particle). Thus P(rj, r2) = 0 for r1 > r2.

The normalizations of P(W) and P(|lo) are found from the limit of those for P(WN) and P(|NIo) as N and V become infinite with N/V = n. From (2.1)

P(Wlo)dW= lim fP(coA o)dv= 1 (2.2) N, V-> oo

and fP(W)d'= lim fP(CN)d N= . (2.3) ,, ~J S?N, V-oo

Two other properties of the functions which can be noted immediately are (remem- bering the homogeneity of the statistics)

P(i) = P(r) = n, (2.4) and when all members of k are far from the origin then

P(Ck |Io) IlP(ek) (2.5)

because there is no long-range order in the suspension (the strength of this assumption is discussed in ? 4).

To complete the formulation in terms of ensemble averages, let S((k; G) or S(r r2 ... rk; G) be the value of S for the reference particle when it is surrounded by

506 D. J. Jeffrey




a configuration lk and the whole group is placed in a field which far from the

configuration is uniform and equal to G. Similarly let G(W,; G') be the field at the

origin (in the absence of the reference particle) produced by a configuration ', placed in a uniform field G'. Then the ensemble average expressions for the averages of S and G (denoted S, G) are

S -= S('; G)P(i|o)dr, (2.6)

G = fG(W; G) P(W) d, (2.7)

where (2.2) and (2.3) have been used. This completes the formulation of the problem. The first group expansion will now be derived. To this end a set of quantities Sk

must be defined that divide up the contributions to S between the various subgroups of W. Such a set is built up by the following scheme:

S,(G) = S(G), S )(r; G) = S(r1; G)- S(G),

S2(rlr2; G) = S(rlr2; G)- S(r,; G)-S(r2; G)- So(G),

and so on. In general k-i

Sk(k; G)- S(k; G)- E E S (; G). (2.8) i= =0 %eq/i(Wk)

Similarly for G: Go = G, k-i

and Gk(lc; G) = G(k; G)- E - Gi( W; G) (2.9) i=o tEqti('k)

Since Go and G are the same, Go will be used for both from now on. Note for later use that Sk and Gk are both linear in the argument G.

As a result of the definition (2.8), S is divided between the Sk according to

S(W; Go)= E S,( s; Go), (2.10) k=0 'keWk*(e)

and substituting this into (2.6) gives ao

S = SE Sk(W; Go)P(tlo) dW. (2.11)

To understand the next step it is easiest to return to explicit use of the limit process used in (2.2). Using (2.1) to integrate P([NI o) and P(%k|o) P(N-k IO, %k) shows that

P(CN o) dxN = f P( | eNo)P(-k l o,X ek) dN-kddk, (2.12)

so E Sk(e( ; Go)P(eNlo)dN

=-N Jf Z Sk (; GO) P( Io)P( N-kIo, k) d-k dk- (2.13)

k'J StVWf



D. J. Jeffrey

Now there are (k) sets in 2/k('N)) and after integration with respect to ^ each set

will give the same result. So the summation over the Wk. in (2.13) can be dropped and

the integral multiplied by ( ). Thus takin tthe V ->oo limit as before, (2.13)

becomes

T S, E ,; Go) P( Jo) d6

= lim ff S k(c; Go)P( ko) P(C,_1|o, k) d -k d .? (2.14) N, V--> oo

The integration with respect to the Wv-k can be performed with the aid of (2.1) and so (2.11) becomes

S= ( Go)fP(,1o) dk. (2.15) k==O

Similarly GO= 2 fGk(k; Go)P() . (2.16) k=0=

These expansions are the first group expansions. If the alternative normalizations for the probability density functions are used, then the integrals in (2.15) and (2.16) are multiplied by 1/k! or nk/k! respectively for the Batchelor and Fisher choices.

Because of the analogy between (2.15) and the virial expansion it is necessary to comment on one aspect of previous treatments of this expansion (see, for example, Landau & Lifschitz I959, ? 71). It is the question of whether the W% in (2.15) are the k particles nearest the reference sphere, or whether they are any k particles. This

obviously affects the form of P(klo) and the convergence of the integrals. From

(2.10) it is clear that S has been broken up into contributions coming from groups of

particles scattered over all space and similarly in the steps leading to (2.15) the

average over k is taken regardless of particles other than those belonging to Ck. So the 6k are definitely any k particles. One must not be confused by the way in which Landau & Lifschitz assume that for rapidly vanishing integrands only the close

particles contribute to the integrals. This is possible for the problems of interest to them, but if the interactions fall off more slowly such an approach does not show that a modification of (2.15) is needed. It is needed because, in many problems of interest, the integrals in (2.15) are not absolutely convergent and as explained in ? 1 cannot be evaluated.

3. THE SECOND GROUP EXPANSION

The heat conduction problem can again be used to motivate what follows. It is shown elsewhere (Jeffrey 1973) that for two equal spheres, radius a

S(r; Go) = -4a3.3A 2A G +2A (G -3 2 r) +0(r-6). (3.1)

So S,(r; G0) = -4wa3AA (1 2- 1 ( (Go- r) +O(r-6). (3.2) 1 A2 + 2/1,) r

508




Thus if (3.2) were used in the k = I integral of (2.15) the integral would be conditionally convergent with all the consequences already described. Now (3.2) can be rewritten SI(r; Go) = SO(Gl(r; Go))+O(r-6). (3.3)

This says that with an error O(r-6), the reference sphere reacts as though it were alone in a uniform temperature gradient G1. It will be noticed that if the term in SO is subtracted from the S1 term, the difference is O(r-6) and so the difference is an integrable quantity. It is a repetition of these observations in a general framework that leads to the second group expansion.

Let the new expansion be denoted by

S= E Sk (fk; Go)dek. (3.4) k=0

The search for the form of (3.4) is based on the hypothesis that it is a modification of (2.15) of the type

Sk = Sk(k; Go)P(kl\o) - (terms which ensure convergence).

A study of the asymptotic behaviour of the first few integrands is made and from this a trial form for SI* is induced which is then verified. In what follows the tilde, ~-, is used in the sense that A - B implies that A - B is absolutely convergent for the mode of integration under consideration. The first few integrands are:

(i) k 0, S (Go) = S0(Go).

(ii) k = 1. When the particle at rx is far from the reference sphere, the 'method of reflexions' can be used to obtain the asymptotic form of S1. The particle at rx produces a field G(rl; Go) at the origin. The reference particle then takes up a value of S appropriate to an isolated particle in a uniform field. So

S(r; Go) / SO(G(rl; Go)), therefore SI(rl; Go) % So(G(rl; Go)-Go) =- S(G1(rl; Go)).

This is only the first step of the method but it is all that is needed here as it is now assumed (as exampled in (3.3)) that

Sl(rl; Go) - SO(GI(r1; Go)).

This assumption is not always correct and the several references which have already been made to interactions too strong for the approach described in this paper refer to those interactions for which several steps of the method of reflexions must be made to capture the asymptotic behaviour of S. Combined with P(rlIo) P(rl), the asymptotic relation is

SI(r; GO)P(ro) SO(GL(rl; Go))P(rl). So one conjectures

SL(rl; Go) = Sl(r; Go)P(r,o)-So(Gl(rl; Go))P(rl)

Vol. 338. A. 32



510 D. J. Jeffrey

(iii) k = 2. There are now particles at r, and r2 and two cases must be checked: r -> oo while r1 remains finite, and r1 -> oo and r2 - oo while I rl - r\ remains finite. The third case of r1 -> oo while r2 remains finite can be ignored because of the convention adopted in line with (2.1). Proceeding as before,

(a) for r2 -> oo while r, remains finite,

S2(rlr2; G0)P(r,r21o) Sl(rl; Gl(r2; Go))P(r,1o)P(r2).

Now if r1 -> oo also

Sl(r,; G1(r2; Go))P(r,|o)P(r2) SO(G_(rL; G1(r2; Go)))P(r,)P(r2).

(b) Now for r1 -> oo, r2 -> oo while ] rl - r2 remains finite,

S2(r1 r; Go)P(r, r2 o) SO(G2(r, r2; Go)) P(r1 r2),

and for I rl- rl -> oo also,

SO(G2(r,r2; Go))P(r,r2) SO(Gl(rl; Gl(r2; Go)))P(r1)P(r2).

A form which combines all these cases is

S*(r r2; Go) = S2(r r2; GO)P(r r21o) - S(r; Gl(r2; Go))P(r,[o)P(r)

-So(G2(rl r2; Go)) P(r1 r2)+ So(GI(r1; G1(r2; Go))) P(r) P(r2).

For case (a) the first two terms are asymptotically equal as are the last two. For case

(b) the first and third terms are asymptotically equal as are the second and fourth. The expression for S* can be rewritten

S*(r1 r2; G) = S2(rl r2; G) P(r1 r2lo)-S (r1; G(r2; Go)) P(r2)

-S*(G2(r,r2; Go))P(r r2).

It is now possible to jump to a form for the general term and it is

k-i

Sk('k; GO) = Sk( (k; G)oP(W|o)- " S(; Gkc_ k Go))P(e6-,), (3.5) i=0

where it is understood that Ti = {rl, r2, r, ..., r} and W_i = {r.+ ... rk}. If either of the alternative probability density functions mentioned in ? 2 is used, a sum over &i(6k) is needed in the last term of (3.5). The structure of (3.5) can be understood from a consideration of the general term in the summation. Now by definition of S*, integration with respect to a member of ci is convergent and so non-convergence can only come from the Gk_i term. For each i, this non-convergence cancels a similar one in Sk. This is the basis of the proof by induction that (3.5) is convergent for all k. That (3.5) is convergent has been shown to be true for k = 0, 1, 2 so now it is taken as true for all instances up to k - 1 and it must be shown that this implies it is true for k. The equations in the proof become unwieldy if all the notation in (3.5) is retained so the arguments will be suppressed when it should be clear what they are. Thus S*( i; Gk-_(W7_-; Go)) becomes Si*(Gk-). It must be shown that Sk




is convergent for integration over any subset of k. Let % divide into two groups T

and _k- and let _k- go to infinity as a group. (3.5) can be rewritten

j-1 k-i

Sk = SkPP(( ) o) - S(Gk_) )Pk_k)-S~(Gkj)P(K_)k- S (Gk_i)P(k_i), i-O i=j+l

and then used again to substitute for S* j-1

Sk = SkP(|ko)- Sj(Gk_) P(lo) P(k_4) + E S (GJ_(Gk)) P(_kJ) P({-') i=0

j-1 k-1 - S*(Gk-i) P(t-i)- E Si* (Gk-)P(_-). i= O i=j+1

The first two terms are asymptotically equal as before, the next two summations are asymptotically equal because

Gk-iP(%_i) - Gj_i(Gk_j) P('k-J) P(W-i)

and all the terms in the last sum are convergent by previous construction. This shows that (3.5) is a possible way of obtaining convergent integrals (provided

the assumptions about asymptotic behaviour are true) and it must now be proved that the sum of the integrals is S. Integrating and summing both sides of (3.5) gives

c0 co o k-1 r E sk:d = E SkP(klo) d z E Sti(Gk-iP(Wi))d

k==O k=O k=0 i=O

00 co 00

k=OJ i=O k=i-lJ

= SkP(|klo)d~k-- S H S (GP(Wj)) dejdi. (3.6) k=OJ ij=Oj.-1

At this point it is necessary to recall the remarks made before (2.1) that the notation used anticipates the taking of the limit N, V -> co. Thus (2.15) and (2.16) can be used to reduce the right-hand side of (3.6) to S provided that the substitutions are understood to take place before the limit is taken. By the linearity of Si with respect to G, the integration with respect toCj in the last term can be taken into the argument of S* as can the sum over j,

d k = Sf SP((|lo) dK - E S* GjP(j) d dWi. (3.7) k=OJ k=O i==tO \j=G i1

The first term on the right-hand side becomes S by (2.15) and the second is zero by (2.16) (note that the sum over j starts at j = 1) and so (3.4) is recovered. (3.4) together with (3.5) will be called the second group expansion. Some further insight into (3.7) can be gained by reviewing the steps from (2.6) to (2.15) to (3.4). The first

group expansion of (2.6) was (2.15) and in its derivation no use was made of (2.7). This is in line with its equivalence to the virial expansion, because for problems to which the virial expansion applies there is no equivalent of (2.7). For problems involving suspensions the integrals in (2.15) depend upon the way the N, V -> co

32-2



512 D. J. Jeffrey

limit is taken (but their sum does not). The second group expansion uses (2.7) to find an equivalent to (2.15) in which the integrals are independent of the limiting process. In (3.7) it can be clearly seen that the two expansions are equivalent because of

(2.7). The second group expansion was contained in the work of Finkel'berg (1964) but, perhaps because of the difficulty of following the terse treatment, it remained unnoticed until after the present work was completed. Finkel'berg also lacked the

physical background we now have, and in particular seems to have been unaware of situations in which the expansion breaks down and of the influence of the structure of the probability density functions on the form of the final series.

4. APPLICATIONS AND PROPERTIES OF THE EXPANSION

The main interest in this section is the connexion between (3.4) and the series in c

(the volume fraction of the particles) mentioned in the introduction. Before this can be investigated, the conditions under which the integrals in (3.4) converge must be discussed. S* can be written

S (r; Go) = {S(r; Go)-S S(GI(r; G0)))P(r[o) + SO(Gr(r; Go)) {P(r o)-P(r)}. (4.1)

For the integral of S* to be convergent each term must go down faster than r-3 at

large r. The first term on the right-hand side of (4.1) shows that there is a restriction on the

type of interaction which the method can handle. Suppose S1- = S(G1) + O(r-m), then the interactions must be such that m > 3. For the problem of thermal conduction and its mathematical equivalents (Jeffrey 1973) and the problem of stress in an elastic solid or a fluid suspension (Batchelor & Green 1972) it is found that m = 6, while for the problem of sedimentation of a suspension (Batchelor 1972) m=4. (It is worth mentioning that for this last problem G cannot be identified with a single physical quantity but is the sum of u and (a2/6/) V. d, the fluid velocity and the

divergence of the deviatoric stress tensor.) One situation to which the method cannot be applied is the flow past an array of fixed particles, since for this m = 2. Childress (1972) solved this problem by two methods, the first of which can be

regarded as an extension of the method used here. He was forced to introduce an elaborate classification of possible interactions and the complexity of the calculation suggests that this approach is unsatisfactory for this problem when compared with his second method which is the basis of more recent solutions of the problem (Saffman 1973; Howells I974).

The second term on the right-hand side of (4.1) shows that there is a restriction on the statistics of the array of particles. This condition is strongest for the sedimentation problem, being

P(r[o) = P(r)+O(r-2-)

for some e > 0. All forms of P(rlo) which have so far occurred in specific calculations have easily met this requirement. Any limitations the constraint places upon the




statistical structure of the suspension are as yet unknown (apart, of course, from the initial assumptions that it must be random and have no 'long-range' order) and

attempts to show that the restriction is contained in the concept of no long-range order have failed.

The discussion will now be confined to those problems for which the integrals in

(3.4) converge and the question of the dependence on c taken up. If the probability density functions contain no length scale other than a (the sphere radius) then the

expansion corresponds to a series in integral powers of c. This is so because the

integrals can be non-dimensionalized with respect to a. Thus if r = ar' then

dk = a3kd,. Further, P(kJ]o) -= np(), where n is the number density of the

particles and p is some function which is 0(1). Finally, since Sk and Gk are functions only of Wk, each term in the expression for Sk, (3.5), is nk times a function of le which is 0(1). Thus

S: dWk = a3knkO(1) - O(ck).

The calculation of S to O(ck) requires the solution of a problem involving k + 1 particles and a knowledge of P(Wklo). Examples of probability functions which contain only the length scale a are those used in Batchelor (1972) and Batchelor & Green (I972). In fluid suspensions in which the particle distribution is produced by hydrodynamic interactions, with or without the additional effects of Brownian

motion, a is certain to be the relevant length scale. If other length scales exist in the probability density functions (as, for example,

with the 'well-separated array' mentioned in Batchelor & Green (I972) and Jeffrey (i973)) then non-integral powers of c can arise. This was suggested by Batchelor

(I972) and shown explicitly by Saffman (1973) for the sedimentation problem. Saffman's results appear very different in form from the results found here and in Batchelor (I972) and it is worthwhile showing that they are equivalent. Saffman works with the Fourier transform of the velocity field and replaces the particles by point forces. His expression for the sedimentation velocity (his equation (4.10)) is in his notation

Uo- u = - k-2F.(I--2kk) - (k)dk.

Parseval's theorem (together with the Fourier transform results given in Saffman's

paper) allows this to be rewritten

Uo- U= n 8rF. (I + r-2rr)G(r)dr.

This is equivalent to Batchelor's equations (3.9) and (5.1) (within the approximation of replacing a particle by a point force) as can be seen by noting that nG(r) in Saffman's notation is P(r) -P(rlo) in Batchelor's and F is 67a/uUo0 to first order.

There are interesting differences between the ways a length scale in the probability density functions affects the different physical situations. Consider a probability



distribution function which contains a length scale acq (e.g. the well-separated array mentioned above). Then

P(rlo)-P(r) np(a ),

where p is some function which is 0(1). The dominant term is the second one on the

right-hand side of (4.1), i.e. So(G1), and if this is proportional to (a/r)s, then the orders of magnitude of interest are

f(?)np (ar)dr=fnc( acI) CI(S )c1-d r_ na3l-a3 cis (l) ac d- r csac- a-

= O(CiS).

For the sedimentation problem s = 1 and the ci dependence is recovered as shown

by Saffman (1973). For the heat conduction and stress problems, s = 3 and the

leading term is still proportional to c as was shown for the well-separated array by Batchelor & Green (I972) and Jeffrey (1973). In all cases, fractional powers will enter at the higher orders. The difference in first-order behaviour between the two

types of problems is present also in periodic arrays. Length scales other than c-+ in the probability distribution are at present hard to imagine, but if they were present they would contribute their own set of non-integral powers to the series in c.

The physical significance of the extra terms introduced in the transition from (2.15) to (3.4) can be seen from the 'excluded volume effects'. Since the spherical particles cannot overlap,

P(rjo) =0 for r < 2a. (4.2)

The centre of each sphere is surrounded by an 'excluded volume' of radius 2a. This allows the k = 1 integral in (3.4) to be written

/*~oo / r \ rPr2a S$(r; Go)dr -So( G1(r; Go)P(r)dr + S(r; Go)dr. (4.3)

J=r=O \Jr-O J r=2a

The first term on the right-hand side involves only the solution for an isolated particle and does not depend upon the two-particle interaction. It corresponds to the reference particle 'seeing' an effective G which differs from G0. This can be seen another way by dividing the sample (volume V) up into V', the excluded volumes, and V".Then

Go = - Gd V J GdV. (4.4)

If interactions between particles are ignored then the average of G outside the excluded volume is

1 G V=G+8eon2a

f GdV Go+8cGo-n G(r; Go) dro(c) 2a

= Go- G1(r; Go)P(r)dr+o(c), (4.5) Jr=0

where P(r) = n has been used.

514 D. J. Jeffrey




It is now argued that the reference particle reacts to the effective field of (4.5) as an isolated particle. Thus it takes up a value of S equal to

So (f GdV) = S(Go)-o (S G,dr).

The first term on the right-hand side corresponds to the =k 0 integral and the second term is the' excluded volume' effect. Thus the excluded volume effects display the effects of the interactions which involve the suspension as a whole and show that the need to modify (2.15) has a physical basis and is not a mere mathematical

quibble. It should be mentioned at this point that the solution to O(c2) given by Walpole (I972) for the stress in an elastic suspension is the excluded volume effect

only, this is why the solution is independent of the two-particle distribution. The same effects are present in the higher orders, the k = 2 integral is

f0: X oo

rid r=o 2a {f S'dr dr- Sf 2'dridr2-f (r,;* FG(r2; Go)P(r2)dr) dr JO JO J 2a 2a J2 \ 2O d

oo 2a

+ SO {Gl(r; G1(r2; Go))P(rl)P(r2)

-G2(r1r2; Go) P(r1 r2)}drdr2).

Note that in the second integral on the right-hand side, r. > r2. The restriction

r2 > r1 mentioned in ? 2 only applies to joint probabilities and does not apply when the two are treated independently.

5. NON-SPHERICAL PARTICLES OR MIXTURES OF PARTICLES

The extension to more general situations is, at the formal level, largely a matter of

finding an appropriate representation for the configuration of the suspension. The actual evaluation of the integrals would involve many difficult problems. Each

particle is assigned a generalized coordinate q which consists of two parts r, the

position vector of some point in the particle (e.g. the centroid) and a vector t (in some appropriate space) whose components record the type of particle and its orientation. The configuration of the suspension is now defined by

= {ql, q2, q3, *,

the qi for particles other than the reference particle, and to the value of t for the reference particle. As before there are the two probability density functions for fi denoted P(ilto) and P(i) and an additional one for t: g(t), for which

Tg(t) dt= 1. (5.1)

The normalization is the same as in (2.1), namely (for a volume containing N

particles) N P(k) dVk = P(kt0) d%c = ( (5.2)



516 D. J. Jeffrey

The alternatives to (2.1) mentioned earlier are more difficult to generalize because of the complexity introduced by repeated counting of identical configurations. The equations and first group expansions for S and G become

S-= S(to, ) P( jto)g(to)dto dV= E| Sk(to 'e) P(eklto)g(to)dto dk7 (5.3) k=O

and = G fG()P( dW)d - = Gk() P(%)d%. (5.4) J k^oj

The asymptotic considerations are not altered by the generalization and the second

group expansion is

S- E Sk (to, Ik) g(to) dtodWke (5.5) k=OJ

k-I with S(to, k) = Sk(to Wk)P(k|to)- ] S*(to, i; Gk_(-k _)) P k-). (5.6)

i'=0

The first term in (5.5) has been extensively studied (reviewed in Batchelor 1974) and the second term was given for the case of unequal spheres, for which to is simply the radius of the sphere, by Batchelor & Green (1972) and Jeffrey (I973) using the notation

P(WQ I to) P(r, b o, a)g(b). (5.7)

I have been supported during this work by a Gonville Research Studentship for which I gratefully thank Gonville and Caius College. This paper was greatly improved by discussions with Professor G. K. Batchelor, F.R.S.

REFERENCES

Batchelor, G.K. 1972 J. Fluid Mech. 52, 245. Batchelor, G. . . 974 Ann. Rev. Fluid Mech. 6, 227 Batchelor, G. K. & Green, J. T. 197z J. Fluid Mech. 56, 401. Childress, S. 1972 J. Chem. Phys. 56, 2527. Einstein, A. 1906 Annln Phys. 19, 289 (and 34, 591). Finkel'berg, V. M. 1964 J.E.T.P. 19, 494 (Russian original 46, 725). Fisher, I. Z. 964 Statistical theory of liquids. University Chicago Press. Howells, I. D. 1974 J. Fluid Mech. (In the press.) Jeffrey, D. J. 1973 Proc. R. Soc. Lond. A 335, 355. Landau, L. D. & Lifshitz, E. M. 1959 Statistical physics. London: Pergamon Press. Maxwell, J. C. 1873 Electricity and magnetism. Clarendon Press. Saffman, P. G. 1973 Studies in applied maths, vol. LII, 115. Smirnov, V. I. I964 Course in higher mathematics, vol. i. London: Pergamon Press. Walpole, L. J. 1972 Q. Jl mech. appl. Math. 25, 153.



group expansions for the bulk properties of a statistically homogeneous, random suspension

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