Free boundary problems for stokes' flows and finite element methods

Abstract

In two dimensions a Stokes' flow is considered symmetric to the abscissa η=0 and periodic with respect to ξ. On the free boundary |η|=S(ξ) the conditions are: (i) the free boundary is a streamline, (ii) the tangential force vanishes, (iii) the normal force is proportional to the mean curvature of the boundary. By straightening the boundary, i. e. by introducing the variables x=ξ, y=η/S(ξ), the problem is reduced to one in a fixed domain. The underlying differential equations are now highly nonlinear: They consist in an elliptic system coupled with an ordinary differential equation for S. The analytic properties of the solution as well as the convergence of the proposed finite element approximation are discussed.