Integral equation methods for unsteady Stokes flow in two dimensions

We present an integral equation formulation fo r the unsteady Stokes equations in two dimensions. This problem is of interest in its own right as a model for slow viscous flow, but perhaps more importantly as an ingredient in the solution of the full, incompressible Navier-Stokes equations. Using the unsteady Green's function, the velocity evolves analytically as a divergence-free vector field. This avoids the need for either the solution of coupled field equations (as in fully implicit PDE-based marching schemes) or the projection of the velocity field onto a divergence-free field at each time step (as in operator splitting methods). In addition to discussing the analytic properties of the operators that arise in the integral formulation, we describe a family of high order accurate numerical schemes and illustrate their performance with several examples.