PHYSICAL REVIEW E 86, 046601 (2012)
Boundary effects on effective conductivity of random heterogeneous media with spherical inclusions
A. Rabinovich, G. Dagan, and T. Miloh
School of Mechanical Engineering, Tel Aviv University, Tel Aviv, Israel
(Received 1 May 2012; revised manuscript received 11 September 2012; published 3 October 2012)
It is common to determine the effective conductivity of heterogeneous media by assuming stationarity of
the random local properties. This assumption is not obeyed in a boundary layer of a body of finite size. The
effect of different types of boundaries is examined for a two-phase medium with spherical inclusions of given
conductivity distributed randomly in a matrix of a different conductivity. Exact solutions are derived for the
apparent conductivity and the boundary layer thickness. The interaction between the spheres and the boundaries
is fully incorporated in the solutions using a spherical harmonics expansion and the method of images. As
applications, the corrections for the effective conductivity are given for two cases of finite bodies: the Maxwell
sphere and a cylinder of flow parallel to the axis.
DOI: 10.1103/PhysRevE.86.046601 PACS number(s): 41.20.Cv, 44.35.+c, 47.56.+r
I. INTRODUCTION
The classical problem of determining the effective con-
ductivity of a medium composed of spherical inclusions of
one substance embedded in a matrix of a different material
has been investigated extensively for more than a century.
Many approximate relationships have been proposed since and
comprehensive and up-to-date compendia have been published
recently [1,2]. We consider a process that is described
mathematically by q =−K ∇T , ∇ · q = 0, where q is the
flux vector, x = (x,y,z) is a Cartesian coordinate, K (x) is the
spatially variable conductivity, and T drives the flow. There
are at least eight different physical processes that obey such a
law [3], e.g., electrical conduction, magnetism, flow in porous
media, dielectrics, etc., but our primary interest is in steady
heat flow through solids for which q is the heat flux, K is the
thermal conductivity, and T is the temperature. Without loss
of generality we shall maintain this nomenclature.
We regard K as a random isotropic space function and pur-
sue determining 〈q〉 and 〈T 〉, the mean flux and temperature,
respectively. Most of the literature dealt with stationary K
fields and mean uniform flow, i.e., 〈q〉 and 〈∇T 〉 are constant.
With the effective conductivity defined by 〈q〉=−K
eff
〈∇T 〉,
one of the central problems of physics is to derive its
dependence on the statistical properties of K [1,2]. Strictly
speaking, stationarity of K and mean flow uniformity imply
an unbounded spatial domain. However, in applications, as
well as in numerical or experimental determination of K
eff
, the
domain is bounded. It is generally assumed that the impact of
the boundary is felt only in a thin layer, which has a negligible
effect upon K
eff
of the bulk. Our aim is to quantify this effect
and to provide estimates of the error incurred by the presence
of boundaries onto K
eff
. Such estimates may prove to be of
importance in applications in which the heterogeneity scale is
not much smaller than other scales, e.g., for natural formations
or thin composite layers.
The few past attempts to investigate this problem in
the specialized literature dealt with particular configurations
and under limiting assumptions. Thus, the works of [4–6]
derive analytical expressions for the mean fields and effective
conductivity near a boundary. However, the conductivity
is regarded as a continuous random function and weak
heterogeneity is assumed. Other relevant studies such as [7–9]
deal with heterogeneous media of periodic nature using the
various periodic homogenization techniques. To the best of
our knowledge, the only work addressing the classical problem
of randomly distributed spherical inclusions near a boundary
is [10]. However their approach is quite different from ours
for a number of reasons: (i) they use a Green’s function
formulation to arrive at the averaged properties (see [11]),
which is not straightforward and involves a tedious mathe-
matical derivation; (ii) while accounting for the presence of
the boundary by using images of the inclusions, the nonlinear
interaction between a sphere and its image is determined in
an approximate manner, making the solution imprecise for
spheres close to the boundary; (iii) only a flow normal to the
boundary of a given temperature is considered. In contrast,
in this work we use a simple formulation, based on the exact
solution of an isolated sphere near a boundary, to solve with
great precision, a few types of flow and boundary conditions.
II. FORMULATION
The structure we consider here is a matrix of conductivity
K
0
with nonoverlapping spherical inclusions of radius R and
of conductivity K . The domain of the heterogeneous medium
is the half space z 0; the influence of the boundary at
z = 0 is similar to the one of a finite body, provided that the
radius of curvature and other length scales are much larger than
R. Furthermore, we consider the dilute limit, for which the
volume density n (average volume of inclusions per volume of
media) is much smaller than unity. This substantially simplifies
the problem and yet has been shown to be applicable for a
significant range of values of n ([12] and [2], Sec. 19.1.3). The
randomness of K (x) =K
0
+
∑
j
(K
0
− K )I (x− x
j
), where the
unit function I = 1 inside the sphere of center coordinate x
j
and I = 0 outside, stems from the randomness of x
j
. An ideal
random distribution is defined as the one for which centers
are scattered uniformly and independently in space. Such a
setup is not possible for relatively dense systems due to the
nonoverlap requirement. The random generation of centers
for nonoverlapping inclusions is treated in a considerable
body of literature (see, for instance, recent discussions in [13]
and [14]). In any case the marginal probability density function
(PDF) f ( x) is uniform and the nonoverlapping condition
046601-1 1539-3755/2012/86(4)/046601(6) ©2012 American Physical Society