The Determination of the pressure for the plane parallel Couette flow

We study the plane parallel Couette flow of an incompressible Newtonian fluid within an unbounded layer in R"3 of the thickness 2 between two parallel walls moved in opposite directions. We demand constant non-dimensionalized velocities #+-#(1, 0, 0) of the walls and suppose nonslip conditions for the velocity field. The boundary conditions are supplemented with periodical conditions for the sought velocity field in the former unbounded directions. We suppose that the Galerkin-approximations of the velocity fields - that means the solutions of the initial-value problem of the autonomous system of ordinary differential equations for the coefficients of the eigenfunctions of the Stokes operator as the basic elements of the Galerkin-approximation space - are known. It is our aim to reconstruct the pressure-field from these known Galerkin-approximations of the velocity fields. We derive a Poisson equation for the unknown pressure field by taking the divergence of the Navier-Stokes equations. The Poisson equation is supplemented with periodic and Neumann boundary conditions at the rigid walls which comes from the boundary values of the Laplacian applied on the eigenfunctions of the Stokes operator. We solve this boundary value problem of the Poisson equation in two steps. We decompose the pressure field in a part fulfilling the inhomogeneous Neumann boundary conditions and the Laplace equation and in the solution of the Poisson equation with homogeneous Neumann boundary conditions. We solve both problems by spectral methods and get the pressure as a function of the coefficients of the eigenfunctions of the Stokes operator. Finally we give the implementation and illustrations of our solution. (orig.)SIGLEAvailable from TIB Hannover: RR 4487(1996,28) / FIZ - Fachinformationszzentrum Karlsruhe / TIB - Technische InformationsbibliothekDEGerman