MODELING OF FLOWS THROUGH A CHANNEL BY THE NAVIER–STOKES VARIATIONAL INEQUALITIES

Abstract

We deal with a mathematical model of a flow of an incompressible Newtonian fluid through a channel with an artificial boundary condition on the outflow. We explain how several artificial boundary conditions formally follow from appropriate variational formulations and the way one expresses the dynamic stress tensor. As the boundary condition of the “do nothing”–type, that is predominantly considered to be the most appropriate from the physical point of view, does not enable one to derive an energy inequality, we explain how this problem can be overcome by using variational inequalities. We derive a priori estimates, which are the core of the proofs, and present theorems on the existence of solutions in the unsteady and steady cases.

Keywords

• variational inequality
• navier--stokes equation
• “do nothing” outflow boundary condition