The Three-Dimensional Jump Conditions for the Stokes Equations with Discontinuous Viscosity, Singular Forces, and an Incompressible Interface

TL;DR

This paper derives detailed three-dimensional jump conditions for pressure and velocity fields across an incompressible interface in Stokes flow, accounting for discontinuous viscosity and singular forces, to improve numerical methods like the Immersed Interface Method.

Contribution

It provides the first comprehensive derivation of second-normal-derivative jump conditions for Stokes equations with discontinuous viscosity and singular forces.

Findings

Derived jump conditions for pressure and velocity fields.

Extended jump conditions up to the second normal derivative.

Facilitates development of more accurate numerical methods.

Abstract

The three-dimensional jump conditions for the pressure and velocity fields, up to the second normal derivative,across an incompressible/inextensible interface in the Stokes regime are derived herein. The fluid viscosity is only piecewise continuous in the domain while the embedded interface exerts singular forces on the surround fluids. This gives rise to discontinuous solutions in the pressure and velocity field. These jump conditions are required to develop accurate numerical methods, such as the Immersed Interface Method, for the solutions of the Stokes equations in such situations.