96-30
The Determination of the Pressure for the Plane Channel Flow
by Noske, A; Rummler, B.; Schlegel, M.
Preprint series: 96-30, 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 anunbounded 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 theunbounded layer. We supplement the boundary conditions for the velocityfield 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 eigenfunctionsof the Stokes operator as the basic elements of the Galerkin-approximationspace - are calculated in the way, written down by two of the authors ina foregoing paper. It is the aim of our considerations , to reconstruct theremaining pressure-field from these known Galerkin-approximations of thevelocity fields. We follow the way used by authors to determine the pressurein the case of plane parallel Couette flow. So, we receive a Poisson equationfor the unknown pressure field by taking the divergence of the Navier-Stokesequations. The Poisson equation is supplemented with periodic and Neu-mann boundary conditions at the rigid and impermeable walls which comesfrom the boundary values of the Laplacian applied on the eigenfunctions ofthe Stokes operator. The solution of this boundary value problem of thePoisson equation is calculated in two steps. We decompose the remainingpressure field in a part fulfilling the inhomogeneous Neumann boundary con-ditions and the Laplace equation and in the solution of the Poisson equationwith homogeneous Neumann boundary conditions. We solve both problemsby spectral methods and receive the remaining pressure as a function of thecoefficients of the eigenfunctions of the Stokes operator. Finally we describesome specific features of the implementation.
Keywords: channel flow, pressure, Poisson equation
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