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The Determination of the Pressure for the Plane Channel Flow

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

Series: 1996-30, Preprints

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

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.

channel flow, pressure, Poisson equation