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1996-28

The Determination of the Presure for Plane Parallel Couette Flow

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