conduction through a random suspension of spheres

Conduction Through a Random Suspension of SpheresAuthor(s): D. J. JeffreySource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 335, No. 1602 (Nov. 27, 1973), pp. 355-367Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/78573 .

Accessed: 03/05/2014 23:33

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

The Royal Society is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of theRoyal Society of London. Series A, Mathematical and Physical Sciences.

http://www.jstor.org

This content downloaded from 130.132.123.28 on Sat, 3 May 2014 23:33:36 PMAll use subject to JSTOR Terms and Conditions

http://www.jstor.org/action/showPublisher?publisherCode=rsl

http://www.jstor.org/stable/78573?origin=JSTOR-pdf

http://www.jstor.org/page/info/about/policies/terms.jsp


Proc. R. Soc. Lond. A. 335, 355-367 (1973) Printedl in Great Britain

Conduction through a random suspension of spheres

BY D. J. JEFFREY

Department of Applied Mathematics and Theoretical Physics, University of Cambridge

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

The conduction of heat (or electricity) through a stationary random suspension of spheres is studied for a volume fraction of the spheres (c) which is small. The work of Maxwell (I873) is extended to calculate the flux of heat exactly to order c2 by using the method of Batchelor (1972), which reduces the problem to a consideration of interactions between pairs of spheres while avoiding the usual convergence difficulties. The result depends on the way in which pairs of spheres are distributed with respect to each other; for the case of all possible pair configurations being equally probable the coefficient of c2 is found explicitly for all values of the ratio of conductivities of the two phases. The results also apply to permittivitics and permeabilities of suspensions.

1. INTRODUCTION

This paper studies the conduction of heat through a stationary, random and

statistically homogeneous suspension of spherical particles in a matrix of uniform

conductivity. The volume fraction of the particles (c) is small. The problems of electrical conduction, electric permittivity and magnetic permeability of the sus-

pension are mathematically the same as the heat problem, and hence their solutions can be derived from the heat solution by appropriately renaming the symbols appearing in it. The solution of this problem, to order c, was given by Maxwell (1 873) exactly 100 years ago; and now a century later we consider the next terni in the series. Here the average heat flux is calculated exactly to order c2 by making use of the general method devised by Batchelor (I972) for the averaging of interactions between spheres to obtain bulk properties of suspensions. The application of the method to conduction is straightforward and so the general method is merely outlined here and for a complete exposition (in particular for details of the ensemble

average manipulations) reference should be made to the previous applications described by Batchelor (1972) and by Batchelor & Green (I972) (see the latter paper in particular in view of the general similarity of the problems of heat conduction in a two-phase disperse system and of the stress in a fluid suspension in motion). The solution shows the first effects of interactions between the spheres and so can

provide a check for calculations which use models to approximate the effects of the interactions but which are unable to provide estimates of the errors thereby intro- duced. Those not interested in the full details of the calculation should note that the solution depends on the statistics of the microstructure of the material and the numerical results given in table 1 should not be used without reading ? 4.

[ 355 ]



D. J. Jeffrey

2. THE GENERAL EXPRESSION FOR THE HEAT FLUX

A matrix of conductivity A1 contains spheres each of radius a and conductivity A2. The medium is infinite and contains a temperature field T which is a stationary random function of position and which has an average gradient G given by

G= lim- VTdV. (2.1) V->co V V

There is a resulting average flux F (for convenience the negative of the actual heat

flux) defined by

F= lirm - AVTdV. (2.2) V->co VJ v

This is converted into an integral over the volume occupied by the spheres by using (2.1)

F =AIG+ lim - (A2- Ai) VTdV, (2.3)

where KV is that part of V occupied by spheres. Explicit reference to the spheres is obtained by introducing S, defined for each sphere by

S (A- A) VT d V, (2.4)

the integration being over the volume of one sphere. S is the strength of the dipole component of the thermal disturbance produced by the sphere and will be loosely referred to as the dipole strength of the sphere. If S is the average of S over all

spheres in the suspension, (2.3) becomes

F = AG+nS, (2.5)

where n is the number of spheres per unit volume, and is related to their volume fraction by c = -7ra3n. Now the value of S for any one sphere depends on the way in which the other spheres are situated around it, and so the next step is to convert S into an ensemble average, in which attention is fixed on one sphere -the 'reference

sphere'- and the configuration of the surrounding spheres allowed to vary. The notation follows that of Batchelor (I972) except that the equations are

simplified by suppressing reference to the (large) number of spheres in the configuration. The centre of the reference sphere is the origin of the coordinate system and around it are spheres with centres at r,, r2, etc., the whole configuration, less the reference sphere, being represented by the set V = {r1, r2, r3, ...}. The term

'sphere with centre at r' will frequently be abbreviated to 'sphere at r'. Two probability density functions are associated with each configuration W: P(jlo) and P(W). The former is the probability density for the spheres in a configuration V given that the reference sphere occupies its place at the origin and the latter is the probability density for a configuration V when the reference sphere has been re-

356



Conduction through a random suspension of spheres 357

moved from the system and does not influence the configuration. Both functions are normalized in the following way:

fP(VIo)d = fP( )d= 1, (2.6)

where dC is the volume element dr1dr3dr .... When all members of c are far away from the reference sphere, the randomness and the lack of any long-range order in the suspension imply

P(WIo) P(r). (2.7)

Finally S(W) is the dipole strength of the reference sphere (additional to that of the displaced matrix material) when it is surrounded by a configuration ' and

VT(W) is the temperature gradient at the origin in the presence of a configuration V when the reference sphere has been removed from the system. In this notation the emsemble average versions of (2.1) and (2.5) are

G = VT(W) P(C)d, (2.8)

and F = AG G+n fJS(c) P(|o) d. (2.9)

3. DETERMINATION OF THE HEAT FLUX TO ORDER C2

The first to consider the problem of conduction in a suspension of spheres was Maxwell (1873) who assumed that c was small enough to make all interactions between the spheres negligible. In this approximation, S(r) is equal to S0, the value of the dipole strength of a sphere alone in an infinite matrix. The temperature field around an isolated sphere is

((1-2~ IG.x for Ixl >a 1 A2+2A \x 3/ 1 X

T(x)- 3A1 (3.1)

2+2G .x for Ixl <a, A2 +?2A1

and so with the introduction of the dimensionless numbers

A2 1A2 -A1_ cA - 1 4 A-anAd +-2 (3.2) =, h, r and + 2Aj-c+2'

the definition (2.4) gives o = 433,1 G, (3.3)

and (2.9) reduces to F = A1 G + 3,cA, G. (3.4)

If the suspension is assigned an effective conductivity A* by

F = A*G, (3.5)



as is always possible when the suspension has a statistically isotropic structure, then Maxwell's result may be stated as

A*/A* 1 + 3tc, (3.6)

Maxwell himself gave the result in the form

A*t +2+2c(a-1) (3.7) AI 'a+2-c(cx-l)

and in this form many people have treated it as though it applies to all concentra- tions, whereas in fact it is a first-order correction only.

It has long been believed in the study of dilute suspensions that a correction O(c2) to (3.4) can be obtained from (2.9) by allowing for interactions between pairs of spheres, i.e. the reference sphere and one other sphere, the tentative justification being that only spheres within a distance of O(a) of the reference sphere produce an 0(1) effect on its dipole strength and the probabilities for one sphere and for two spheres being within this distance are respectively O(c) and 0(c2). The integral in (2.9) is multiplied by n and so it would appear that only interactions with one other sphere need be considered to 0(c2). To modify (2.9) in line with this belief, one is tempted to rewrite it as

F = AG + 3?fcA G +n [S()- So] P(Wio) d, (3.8)

and take S(W) to be given approximately by SO + Sl(r), where S1 is the additional dipole strength of the reference sphere due to a second sphere at r. Then after integration with respect to the positions of all spheres in ' except one, (3.8) reduces to

F Ai G 3/?8cAi G+nf S(r)P(rlo)dr, (3.9)

where P(rlo) is the probability density for a second sphere being at r. For large r the integrand varies as r-3 (see (4.2) and (5.8) below) and the integral is only conditionally convergent. An integral that is not absolutely convergent may still yield a finite result on integration and such an integral is termed conditionally convergent. The result found, however, is not unique and depends on the mode of integration; for example, if the integral is evaluated for a finite region which is then allowed to become infinite without change of shape, the result will depend on the shape of the volume chosen. Thus (3.9) is indeterminate as it stands and no result derived from it is valid without additional justification for the mode of integration. employed.

The first to encounter this difficulty was Rayleigh (i 892) in the related problem of regular arrays of spheres. After noting that integrating over a cube gave zero for his integral, he chose instead a volume indefinitely extended in the direction of the mean temperature gradient and continued unabashed. Peterson & Hermans (I969) obtained (3.9) but without the probability density function, apparently

358 D. J. Jeffrey




by assuming implicitly equation (4.3) below. They realized the danger of integrating over all space and fell back upon the existence of plates bounding the sample and

supplying the temperature gradient. Finite samples are still, however, shape de-

pendent as Brown (i955, I965) pointed out. Brown, and later Davies (i97i), took a finite sample and explicitly eliminated the shape-dependent factors, but their work is not of further relevance here as they used a different method for describing the statistics of the medium. Finally there are two papers by Finkel'berg (1964a, b) which seem to have remained largely unnoticed even though they are available in translation; in the first paper Finkel'berg obtains the form for the c2-term derived below (i.e. (3.13)) as part of more general considerations but does not evaluate it, while in the second he rederives the c2-term using a method which, by his own assessment, is inferior to that of the first paper and then partially evaluates it.

The present method (following Batchelor I972) is to use an overall constraint of the problem to modify (3.8) so that a step equivalent to (3.9) can be taken without

ending up with a conditionally convergent integral. The way this is to be done can be inferred from the behaviour of the integrand of (3.9) for large r. From (5.9) below,

S,(r)- t4a3 3,A,V (r)+O(r-6), (3.10)

where G + Vq(r) is the temperature gradient at the origin due to an isolated sphere at r. Since P(rlo) ~ P(r) for large r, an integrand of the form

S,(r) P(rlo) - 47aa3 fAl VO(r)P(r)

will give an absolutely convergent integral. To obtain such an integrand, (3.8) is modified by rewriting (2.8) as

S[VT(W)-G]P(W6 )d'] = 0, (3.11)

and subtracting 47ca3f,Al times this from (3.8) giving

F = A G + 3/?cAi G + n {[S(W) -So] P(| o)- 47a3flA, [VT(W) - G] P(W)} dM.

(3.12)

This equation is exact and is in a suitable state for reduction to the two-sphere approximation

F = A G + 3fcA G+ n j{Sl(r) P(rl o)-4xca3flAl Vq(r) P(r)}dr+o(c2). (3.13)

The integral being absolutely convergent, (3.13) is a perfectly respectable equation and is the required alternative to (3.9). Also, because the integrand goes as r-6 for r -> oo, the order of magnitude estimate suggested above is validated, for now it is indeed only spheres that are within distances of O(a) of the reference sphere which have an 0(1) effect on the integrand and so (3.13) is correct to O(c2).

359



360 D. J. Jeffrey

If the spheres are not identical but have radii distributed according to a density function g(a), with fg(a) da = 1, then (3.13) can be put in the more general form

F = Al G+3f/cA G

+ n jf{IS(a; r, b) P(r, blo, a)- 47a3flA1Vq(r, b) P(r ) ) (b() b)da db dr,

(3.14)

where the additions to the notation indicate that the reference sphere has radius a and the other sphere radius b.

4. THE PROBABILITY DENSITY FUNCTIONS

For a statistically homogeneous material in which the spheres have number

density n, the probability density for a sphere being at r is simply

P(r) = n. (4.1)

Things are not so simple for P(rjo), however, because its form depends on the statistical structure of the medium and hence will vary from material to material.

Although for deforming fluid suspensions the form of P(rlo) is, in principle, de- ducible (Batchelor & Green I972), for stationary suspensions it is determined by the method of manufacture and hence must be regarded as a given condition of the

problem. Thus a suitable form of P(rlo) must be chosen for the evaluation of (3.13). Any choice for P(rlo) must satisfy

P(rlo) 0 for r < 2a,4

P(rlo) n for r a, (.

the first condition being that the spheres do not overlap and the second the equivalent of (2.7). The invariable choice in previous work has been

(O for r 2a,c P(rjo) = for

r< 2a(4.3)

in for r> 2a.j

This simple distribution corresponds to the notion of a 'well-stirred' suspension in that the second sphere occupies all allowable positions with equal probability; it also serves as a well-defined standard or reference state which can be used when there is no external agent in the problem to compel the suspension to adopt a particular distribution. This form will be used here. Since (4.3) is an isotropic distribution, one immediate consequence is that the suspension will still have a scalar effective conductivity.

5. SOLUTION OF TWO-SPHERE PROBLEM

There are two standard methods for solving problems involving two spheres, that using bispherical coordinates invented by Jeffery (I 9 z 2) and that using twin spherical expansions first formalized by Ross (1968). The later method proves to be the better




for this problem. Two systems of spherical polar coordinates are taken as shown in figure 1, the origins are at the centres of the two spheres, the 0 coordinate is common to both systems, and the 0 = 0 plane is the plane of G.

If G has components Go and G1 parallel to and perpendicular to the line of centres then the temperature inside sphere number i can be expressed as

TW = G- x+ J E md (i) pm (cos Oi) cosmo, m=O n=m a

1 o (1)

n+l

m=O n=m

(5.1)

pm (cos 01) + - Pn (cos2)}cos mOS.

(5.2)

FIGURE 1. Coordinates for twin expansions. 0 is common to both systems.

The coefficients are found from the boundary conditions which are continuity of temperature and normal heat flux across each spherical interface. The boundary conditions are applied with the aid of the identity

() P,n(cos ,)) = ? +m) ( )i pm(os 3)-

The resulting equations for the coefficients are

d( = gm()n (n + g (m3- (i) $m =m =

(5.3)

(5.4)

( 1cc~i+ \ 0

n/M) ( s \ n+s+1 mnd(i) +- - g(m)- E (n + = (- l)i(m-1) (a

- 1) )aln, (5.5)

$s=\I =m \n+-m/

where a is defined in (3.2). Since the spheres are equal, symmetry allows the g(i to be replaced by

and outside as

361

agmn = g(1) - ( _ 1)m- g(2) ym ~1 vmn ~ m \ ?/ Vn? (5.6)



362 D. J. Jeffrey

The elimination of d() from (5.4) and (5.5) and the introduction of

r = n(oc- )/(nc + n+ 1)

(note from (3.2) that f/? = f) gives

- gmn + E (n +m) gs ln (57)

This can be solved by expressing gnn as a power series in a/I. The dipole strength can now be calculated from (2.4) and the final result for S( = So + S) can be written as

S(R) = -7a3(3, Al, G-3flA, A )s (R) (A G-B_ 3 2 G

R, R (5.8) 3 1C3 W \ n

where the AP and BP are known coefficients, the first few being

A3=l; A4=A5=0; A6=-f?; A 0; A8g=-3/IfA; A-9g -

B3=33A;B=B = O; B -0 7 =; B8 = 63?8; B9=; ; 9

From (3.1) it is easily verified that (5.8) is equivalent to

S(R) = S0+-tca33flA i V(R)- 4ra33flil1 E A G-B 2 R) , (5.9) 3 P=6 \)1/ \2

where G + V?(R) is the temperature gradient at the origin due to a sphere at R.

6. EVALUATION OF THE INTEGRAL IN (3.13)

The integration in (3.13) can now be performed. Because P(rlo) is zero for r < 2a, the region of integration falls naturally into two parts: r < 2a and r > 2a. The contribution from the first part is

r2a -c 3f?AiVlV(r)ndr = 3f2C2A, G. (6.1)

J =0

This term has great significance for the physics of the problem. In the first place it is the O(c2) term in the solution Rayleigh (X892) gav e for regular arrays of spheres, and as will now be shown it is also the c2-term for a random, 'well-separated' array, i.e. an array in which all spheres are far from each other. In a well-separated array the distance between neighbours is O(ac-?) and so P(rlo) is zero for r < ac-W and 0(n) as before, for r > ac-?. Thus (3.13) now falls into three parts: (i) r < 2a which is (6.1), (ii) 2a < r < ac-~ which integrates to zero and (iii) r > ac-i which is o(c2), so (6.1) is the c2-term for a well-separated array. Another way of obtaining (6.1) involves only Maxwell's solution (3.4) and applies to both random and ordered suspensions. To obtain (3.4) it was assumed that within every sphere the temperature gradient was uniform and equal to 3G/(O + 2) + O(c), which implies that within the matrix the average temperature gradient is G +/ cG +O(c2). Now to obtain (3.4)




it was assumed that each sphere interacted only with the average gradient G and nothing else, but as just seen this assumption leads to a change in the average gradient within the matrix, so a second-order correction to (3.4) can be obtained by supposing that the gradient seen by each sphere is not G but instead G +,8cG. The flux is then

F = A1 G + 3fcA.(G +fcG) - Al G+ 3fcA1 G+3fl2c2Al G. (6.2)

This result also applies to suspensions containing mixtures of sphere sizes, provided each sphere is a distance ac- (where a is its own radius) from its neighbours. All well-separated suspensions are isotropic to O(c2). The physical importance of (6.1) now becomes clear; it is a result of the reference sphere interacting with every other sphere in the suspension in a particular way, and it is only after this has been calculated that the details of interactions with one other sphere need be considered. To put it another way, the assumption that led people to the improper expression (3.9), namely that the c2-term depends only on two-sphere interactions, is wrong because it leaves out the role played by multiple interactions in the entire suspension. This also explains why regular arrays and well-separated random arrays have the same c2-term: for both, two-sphere interactions are o(c2). The analogous behaviour of fluid suspensions has been described fully by Batchelor & Green (1972). Walpole (x97z) obtained the result corresponding to (6.2) for an elastic suspension of well- separated spheres, but did not consider the contribution from two-sphere interactions.

The contribution from the second part of the region of integTation can now be evaluated using (5.9) and the final expression for the effective conductivity is

L l ?f 2(, 00 Br - 3A \

A- -= 1 + 3,c + 31 3+ (-3 2-.3)

= 1+3flc+c2(32+ C + 16 2 +3+6+ . .) (6.4)

Where fi = ( - 1)/(a + 2) as before. The series in (6.3) is only slowly convergent and the numerical values given in table I required the summation of over 100 terms before being correct to three significant figures. In figure 2 the coefficient is plotted against log a. If the mean conductivity A = cA2 + (1-c)A1 is used in place of A1 in (6.3), the expression becomes

A* / J? B - 3A\ -A-= 1l-(a-1l)/c+ 3/32+3f Z

(p_3)2P+ (a-1)2)C2 (6.5)

The coefficient of c is always negative, as one would expect since A is an upper bound of A*. For large a, both the c and c2 coefficients become very large and in this limit

(6.5) is not very useful; for small a, however, the coefficients are smaller than those of (6.3), and (6.5) might then prove to be the more convenient (see figure 3).

Vol. 335. A.

363

24



D. J. Jeffrey

7. COMPARISON WITH OTHER WORK

The first comparison must be with other exact calculations of the coefficient of c2. Finkel'berg (i964b) adopted the distribution (4.3) and obtained the form (6.4), except for a numerical factor in the third term, but did not sum it. Peterson & Hermans (I969) gave a closed form for the coefficient by using the method of reflexions to solve the two-sphere problem and their result is tabulated in table 1

coefficient of c2

4.51 -l

4-

2-

gC a

FIGURE 2. Graph of coefficient of c2 in (6.3) plotted against Ig a, where a = A2/A1.

TABLE 1. VALUES OF THE COEFFICIENT OF C2 FOR VARIOUS Oa AS CALCULATED FROM

(6.3) AND AS GIVEN BY PETERSON & H-ERMANS (1969)

The last column gives value of 8 for which (7.4) agrees with (6.3). c2 coefficient

a

0 0.02 0.1 0.5 1.0 2.0 5.0

50.0 oo

fl - 0.500 -0.485 -0.429 -0.200

0 0.25 0.571 0.942 1

from1

(6.3) 0.588 0.558 0.450 0.110 0 0.208 1.23 3.90 4.51

Peterson & Hlermans

0.656 0.620 0.492 0.114 0 0.199 1.12 3.30 3.76

0.22 0.22 0.21 0.21 0.21 0.22 0.22 0.25 0.25

for comparison with the values computed from (6.3). The next point of comparison should be with experimental work, but there is only one paper, Sundstrom & Chen (1970), which examines a suspension for which c takes small values. They present

364




their measurements graphically together with a plot of Peterson & Hermans's calculations but give no numerical data. All that can be said is that Peterson & Hermans's curve is slightly below the observed values.

Any calculated value of the coefficient must be compatible with the bound obtained from the variational approach of Hashin and Shtrikman (1962). This states

A*/All< 1 + 31/c+ 3/2c2+ . .., (7.1)

where the upper sign applies for a > 1, the lower for c < 1. This is satified by (6.3). The appearance of (6.1) again, this time as an extremum of the c2 coefficient requires

2-

(1)

*I I /~~~~~~~~~~ .(3)

. ,J

0 0.1 0.2

FIGURE 3. Graph of effective conductivity A* as given in (6.3) and (6.5) plotted against c, the volume fraction of the spheres, for a = 0 and a = oo. (1) A*/A1 for a = oo, (2) A*/A for a = 0, (3) A*/A1 for a = 0.

comment. To order c2, a well-separated array realizes the bound (7.1), this can be seen in Hashin & Shtrikman's proof that their bounds are the best possible within their restrictions. To prove that (7.1) is the best possible bound for heterogeneous isotopic media, Hashin & Shtrikman constructed a medium made up wholly of

'composite spheres' of varying size. As can be seen from figure 4, each composite sphere contains a sphere of conductivity A2 surrounded by a shell of conductivity A1. Thus the composite sphere medium is a suspension of spheres of conductivity A2 and further each sphere is separated from every other sphere by a distance ac-~, where a is the radius of the particular sphere. The medium is thus a well-separated sus-

pension of spheres of varying size. But to order c2 the distribution of sizes is not

important, so the composite sphere medium has the same conductivity (to order c2) as a well-separated array. For the case a s 1, (6.3) must agree with the expression derived by Beran (I965) and Landau & Lifshitz (I960) for any material in which the fluctuations of conductivity are weak:

A X-(A -)2/3X, (7.2) 24-2

365



where A is the mean conductivity defined above. The relation (7.2) is valid to O(e2), where = a- 1 < 1, and can be rewritten as

At/A, - 1 + (e - 62) c + 1e2c2 + O(e3). (7.3)

To 0(62), (6.3) agrees with this, as

/ = ?e-6 62 + 0O(3).

Finally, some interesting results come from a comparison of the work here with the self-consistent scheme reviewed by Hashin (I968). The self-consistent scheme

FIGURE 4. Composite sphere for Hashin-Shtrikman bound and for self-consistent scheme. For Hashin-Shtrikman bound, p = ac-?. For self-consistent scheme p is not specified and

= a3/p3 (see (7.4)).

attempts to model the effect of the suspension on the reference sphere by considering the reference sphere to be embedded in a continuous matrix of conductivity A*, but separated from it by a shell of conductivity A1 (see figure 4 again). Hashin

pointed out that the radius of this shell defined an unknown parameter (called c' in his paper but 8 here) that prevented the scheme from giving a unique solution.

Expanded as a series in c, Hashin's expression for A* is

A*/A 1 + 3pc + 3P 22( + 28 2) + .... (7.4)

In table 1, the value of 8 which makes (7.4) agree with (6.3) is given for various values of a. The approximate uniformity of the values of 6 invites an explanation, but none can be offered here.

It is a pleasant duty to thank Gonville and Caius College for the Gonville research studentship which has supported me during this work; the many references in the text bear witness to the help received from my supervisor Professor G. K. Batchelor, F.R.S.

366 D. J. Jeffrey



Conduction through a random suspension of spheres 367

REFERENCES

Batchelor, G. K. 1972 J. Fluid Mech. 52, 245. Batchelor, G. K. & Green, J. T. I972 J. Fluid Mech. 56, 461. Beran, M. J. 1965 Suppl. Nuovo Cim. 3, 448. Brown, W. F. 1955 J. chem. Phys. 23, 1514. Brown, W. F. 1965 Trans. Soo. Rheol. 9, 1, 357. Davies, W. E. A. I971 J. Phys. D 4, 318. Finkel'berg, V.M. i964a Doklady 8, 907 (Russian original 152, 320). Finkel'berg, V. M. I964b Tech. Phys. 9, 396 (Russian original 34, 509). Hashin, Z. I968 J. composite Materials 2, 284. Hashin, Z. & Shtrikman, S. I962 J. appl. Phys. 33, 3125. Jeffery, G. B. 1912 Proc. R. Soc. Lond. A 87, 109. Landau, L. D. & Lifshitz, E. M. I960 Electrodynamics of continuous media. ?9. Pergamon

Press. Maxwell, J. C. 1873 Electricity and magnetism (1st ed.). Clarendon Press. Peterson, J. M. & Hermans, J. J. 1969 J. composite Materials 3, 338. Lord Rayleigh 1892 Phil. Mag. 34, 481. Ross, D. K. 1968 Aust. J. Phys. 21, 817. Sundstrom, D. W. & Chen, S. Y. 1970 J. composite Materials 4, 113. Walpole, L. J. 1972 Q. Jl mech. appl. Math. 25, 153.



conduction through a random suspension of spheres

Documents