A spherical drop of viscosity $μ(1)$ and radius $a1$ is surrounded by a spherical shell of viscosity $μ(2)$, internal radius $a1$, and external radius $a2$, beyond which there is an unbounded matrix fluid of viscosity $μ(3)$. An arbitrary axisymmetric singularity acts inside or outside the viscous spherical shell. It is established that the Stokes’ stream function induced in this heterogeneous medium is explicitly expressible solely in terms of the corresponding stream function for the unperturbed homogeneous unbounded medium. As an application of the general solution, it is shown that there exists a homogeneous spherical drop of viscosity $μ$ and radius $a2$, which is equivalent to the spherical drop of viscosity $μ(1)$ and radius $a1$, surrounded by the spherical shell of viscosity $μ(2)$ and outer radius $a2$. A new formula is also established for the effective viscosity of a multiphase medium comprising an incompressible fluid of viscosity $μ(3)$ in which are embedded $N$ small identical spherical drops of viscosity $μ(1)$, surrounded by spherical shells of viscosity $μ(2)$, assuming that the interference effects of these coated spherical drops are negligible, and that their arrangement is axisymmetrical. The corresponding two-dimensional results are also presented, by slightly modifying the three-dimensional results.

1.
Milne-Thomson
,
L. M.
, 1940, “
On Hydrodynamical Images
,”
Proc. Cambridge Philos. Soc.
0068-6735,
36
, pp.
246
248
.
2.
Weiss
,
P.
, 1944, “
Arbitrary Irrotational Flow Disturbed by a Sphere
,”
Proc. Cambridge Philos. Soc.
0068-6735,
40
, pp.
259
263
.
3.
Butler
,
S. E. J.
, 1953, “
A Note for Stokes’ Stream Function for Motion With a Spherical Boundary
,”
Proc. Cambridge Philos. Soc.
0068-6735,
49
, pp.
169
171
.
4.
Collins
,
W. D.
, 1958, “
Viscous Stokes Flow Past a Rigid Sphere
,”
Mathematika
0025-5793,
5
, pp.
118
124
.
5.
Stokes
,
G. G.
, 1851, “
On the Effect of the Internal Friction of Fluids on the Motion of Pendulums
,”
Trans. Cambridge Philos. Soc.
0371-5779,
9
, pp.
8
96
.
6.
Maxwell
,
J. C.
, 1873,
A Treatise on Electricity and Magnetism
, Vol.
1
,
Dover
,
New York
, pp.
244
450
.
7.
,
K.
, 2003, “
An Image Treatment of Elastostatic Transmission From an Interface Layer
,”
J. Mech. Phys. Solids
0022-5096,
51
, pp.
267
279
.
8.
,
K.
, 2007, “
Theory of Images in Plane-Layered Axisymmetric Elastostatics
,”
Int. J. Solids Struct.
0020-7683,
44
, pp.
3920
3927
.
9.
,
K.
, 1972, “
Some Results for Two-Phase Elastic Plates
,”
Proc. R. Soc. Edinburgh, Sect. A: Math. Phys. Sci.
0080-4541,
71
, pp.
21
35
.
10.
Eshelby
,
J. D.
, 1957, “
The Determination of the Elastic Field of an Ellipsoidal Inhomogeneity and Related Problems
,”
Proc. R. Soc. London, Ser. A
0950-1207,
241
, pp.
376
396
.
11.
Eshelby
,
J. D.
, 1961, “
Inclusions and Inhomogeneities
,”
Progress in Solid Mechanics
,
2
, pp.
89
140
.