by Rummler, B..

**Series:** 1999-35, Preprints

- MSC:
- 34A34 Nonlinear equations and systems, general
- 35Q30 ~Navier-Stokes equations

**Abstract:**

We regard a general class of boundary-pressure-driven flows of

incompressible Newtonian fluids in unbounded layers and in unbounded pipes

in $ {\bf R}^{3} $ with thickness $2R$ or radius $R$, which marginal cases are

e.g. plane Couette flows and Poiseuille flows in channels,

respectively.

We avail the incompressible nonstationary Navier-Stokes equations as

description of the physical process. \\

We def\/ine energetic Reynolds numbers. Using Galerkin approximations

based on Stokes eigenfunctions on open bounded subdomains

in $ {\bf R}^{3} $ furnished with periodical conditions in the at first

unbounded spatial directions, we get an autonomous system of

ordinary differential equations for the time-dependent coeff\/icients

of the Stokes eigenfunctions from the dimensionless

Navier-Stokes equations for the difference $\bu $

between the velocity and the laminar velocity.

For the Galerkin method we utilize f\/ixed periods $ 2l $ and the f\/irst

$N(l)$ Stokes eigenfunctions. We apply the kinetic energy of $\bu$

as a measure of turbulence.

The numerical calculations yield satisfactory results in comparison with

measurements keeping in mind the small dimension of our approximation spaces

and hope raising results for investigations of bifurcations.

**Keywords:**

Navier-Stokes equations, Stokes eigenfunctions, Galerkin methods, transition to turbulence