General slow viscous flows in a two-fluid system

Abstract

A general solution of the creeping flow equations suitable for a flow that is bounded by a non-deforming planar interface is presented. New compact representations for the velocity and pressure fields are given in terms of two scalar functions which describe arbitrary Stokes flow. A general reflection theorem is derived for a fluid-fluid interface problem containing Lorentz reflection formula as a particular case. The theorem allows a better interpretation of the image system for various singularities in the presence of a planar interface. The general solution is further used to describe the first-order approximation of the deformed interface by performing normal stress balance. It is found that the normal stress imbalance and the interface displacement are independent of the viscosity ratio of two fluids (!) and only depend on the location of initial singularity.