Steady stokes flow in and around a droplet calculated using viscosity smoothened across interface

Velocity continuity and force balance are Usually required at the interface of two fluid phases in conventional hydrodynamics, where the viscosity is assumed to be constant in a single fluid phase. These boundary conditions connect the pressure and velocity fields across the interface. An alternative way to achieve this connection, where the viscosity is assumed to smoothly change across a thin interfacial region, was proposed to facilitate the numerical study of colloidal dynamics. We study the steady Stokes flow in and around a single droplet by use of the smoothened viscosity, imposing a purely extensional flow far from the droplet. In the limit of the thin interfacial region, we analytically obtain a set of connection formulas, which yields the fields that are different from those obtained in conventional hydrodynamics unless the droplet is a rigid body.