96-28
The Determination of the Presure for Plane Parallel Couette Flow
by Noske, A; Rummler, B.; Schlegel, M.
Preprint series: 96-28, Preprints
- MSC:
- 35J20 Variational methods for second-order, elliptic equations
- 68U05 Computer graphics; computational geometry, See also {65Y25}
- 68U10 Image processing
- 76F10 Shear flows
Abstract: We study the plane parallel Couette flow of an incompressible New-tonian fluid within an unbounded layer in R 3 of the thickness 2 betweentwo parallel walls moved in opposite directions. We demand constant non-dimensionalized velocities \Sigma(1; 0; 0) of the walls and suppose nonslip condi-tions for the velocity field. The boundary conditions are supplemented withperiodical conditions for the sought velocity field in the former unboundeddirections.We suppose that the Galerkin-approximations of the velocity fields - thatmeans the solutions of the initial-value problem of the autonomous systemof ordinary differential equations for the coefficients of the eigenfunctionsof the Stokes operator as the basic elements of the Galerkin-approximationspace - are known. It is our aim to reconstruct the pressure-field from theseknown Galerkin-approximations of the velocity fields. We derive a Pois-son equation for the unknown pressure field by taking the divergence of theNavier-Stokes equations. The Poisson equation is supplemented with peri-odic and Neumann boundary conditions at the rigid walls which comes fromthe boundary values of the Laplacian applied on the eigenfunctions of theStokes operator. We solve this boundary value problem of the Poisson equa-tion in two steps. We decompose the pressure field in a part fulfilling theinhomogeneous Neumann boundary conditions and the Laplace equationand in the solution of the Poisson equation with homogeneous Neumannboundary conditions. We solve both problems by spectral methods and getthe pressure as a function of the coefficients of the eigenfunctions of theStokes operator. Finally we give the implementation and illustrations of oursolution.
Keywords: Couette flow,pressure field, Poisson equation
The author(s) agree, that this abstract may be stored asfull text and distributed as such by abstracting services.