Coupled Problems for Viscous Incompressible Flow in Exterior Domains

Abstract

The formulation of the {\it fluid flow in an unbounded exterior domain} $\Omega$ is not always convenient for computations and, therefore, the problem is often truncated to a bounded domain $\Omega^-\subset\Omega$ with an artificial exterior boundary $\Gamma$. Then the problem of the choice of suitable "transparent" boundary conditions on $\Gamma$ appears. Another possibility is to simulate the presence of the fluid in the domain $\Omega^+$ exterior to $\Gamma$ with the use of a suitable (preferably linear) approximation of the equations describing the flow. The interior and exterior problems are coupled with the aid of {\it transmission conditions} on the interface $\Gamma$. Here we briefly describe the formulation and analysis of the coupling of the interior Navier--Stokes problem and the exterior Stokes problem and Oseen problem. At the end we give the reformulation of the coupled problems with the aid of integral equations on the artificial interface. Here we briefly describe the formulation and analysis of the coupling of the interior Navier-Stokes problem and the exterior Stokes problem and Oseen problem. At the end we give the reformulation of the coupled problems with the aid of integral equations on the artificial interface.