Computing corresponding values of the Neumann and Dirichlet boundary values for incompressible Stokes flow

Abstract

We consider the Stokes equation for a flow through a partially obstructed channel and determine the relationship between Dirichlet boundary values (velocities) and Neumann boundary values (forces) for the FEM discrete form. For the steady state case we find a linear relationship. For the transient case the relationship depends on the time stepping procedure and includes the relationship at prior states. We resolve the issue for trapezoid and Adams–Bashford-2 time stepping. Since Stokes flow may be considered as the startup phase of Navier–Stokes flow, we give particular attention to a flow with a startup function.