by Noske, A, Rummler, B., Schlegel, M..

**Series:** 1996-28, Preprints

- MSC:
- 35J20 Variational methods for second-order elliptic equations
- 68U05 Computer graphics; computational geometry
- 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 between

two 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 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 Pois-

son equation for the unknown pressure field by taking the divergence of the

Navier-Stokes equations. The Poisson equation is supplemented with peri-

odic 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 equa-

tion 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.

**Keywords:**

Couette flow,pressure field, Poisson equation