Immersed Interface Methods for Stokes Flow with Elastic Boundaries or Surface Tension

A second-order accurate interface tracking method for the solution of incompressible Stokes flow problems with moving interfaces on a uniform Cartesian grid is presented. The interface may consist of an elastic boundary immersed in the fluid or an interface between two different fluids. The interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed. The Stokes equations are then discretized using the secondorder accurate finite difference methods for elliptic equations with singular sources developed in our previous paper [SIAM J. Numer. Anal., 31(1994), pp. 1019–1044]. The resulting velocities are interpolated to the interface to determine the motion of the interface. An implicit quasi-Newton method is developed that allows reasonable time steps to be used.