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

**Series:** 1996-30, Preprints

- MSC:
- 35J20 Variational methods for second-order elliptic equations
- 42B05 Fourier series and coefficients
- 76F10 Shear flows

**Abstract:**

We investigate the flow of an incompressible Newtonian fluid within an

unbounded layer in R 3 of the thickness 2 between two parallel walls. Firstly,

we demand nonslip conditions for the velocity field at the rigid walls of the

unbounded layer. We supplement the boundary conditions for the velocity

field with periodical conditions in the former unbounded directions.

Now, we suppose that the Galerkin-approximations of the velocity fields -

that means the solutions of the initial-value problem of the autonomous sys-

tem of ordinary differential equations for the coefficients of the eigenfunctions

of the Stokes operator as the basic elements of the Galerkin-approximation

space - are calculated in the way, written down by two of the authors in

a foregoing paper. It is the aim of our considerations , to reconstruct the

remaining pressure-field from these known Galerkin-approximations of the

velocity fields. We follow the way used by authors to determine the pressure

in the case of plane parallel Couette flow. So, we receive 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 Neu-

mann boundary conditions at the rigid and impermeable walls which comes

from the boundary values of the Laplacian applied on the eigenfunctions of

the Stokes operator. The solution of this boundary value problem of the

Poisson equation is calculated in two steps. We decompose the remaining

pressure field in a part fulfilling the inhomogeneous Neumann boundary con-

ditions 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 receive the remaining pressure as a function of the

coefficients of the eigenfunctions of the Stokes operator. Finally we describe

some specific features of the implementation.

**Keywords:**

channel flow, pressure, Poisson equation