Theory of Images in Spherical-Layered Axisymmetric Viscous Hydrodynamics

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.