**Series:** 1995-28, Preprints

- MSC:
- 76F10 Shear flows
- 35Q30 ~Navier-Stokes equations
- 42C15 General harmonic expansions, frames
- 47A75 Eigenvalue problems
- 76H05 Transonic flows

**Abstract:**

We investigate the Couette flow of an Newtonian fluid within a domain

between two parallel walls moved in opposite directions.

We demand constant velocities of the walls and suppose nonslip

conditions of the fluid at the walls of the plane channel.

We formulate the initial-boundary value problem for the velocity of the

fluid by the Navier-Stokes equations for the unbounded domain between the

walls in $ {\bf R}^{3} $.

We obtain by transition to non-dimensionalized quantities and equations a system

of the Navier-Stokes equations , where the physical properties of the

movement are included in a parameter $ R $ (the half of the Reynolds number

$ Re $ ).

The restriction of the domain on an open

bounded rectangular parallelepiped of $ {\bf R}^{3} $ supplemented

with periodical

conditions for the sought velocity field in

the former unbounded directions and the decomposition of the velocity field

in two parts - the laminar flow fulfilling the nonzero boundary conditions on

the walls and the remaining velocity with homogeneous Dirichlet conditions

on the walls provide equations for the determination of the

remaining velocity.

We choose a fix period $ 2l{ }= 2\times 2,69 $ for the first investigations .

By the use of the Galerkin method we obtain the Galerkin equations of

the weak solution of these

equations as an 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.\\

We regard 356 Stokes eigenfunctions

to involve at least all the Stokes eigenfunctions to eigenvalues $\lambda

\le 4\pi^{2} $ in our considerations .

We solve the corresponding system of ordinary differential equations for several

values of the parameter $ R $ and a set of initial values.

We use the kinetic energy of the approximated remaining velocity as a measure

of turbulence .\\

Our numerical investigations provide good agreements with

experimental results in the vicinity of the critical Reynolds number

at the study of the transition from laminar to turbulent flows and we

obtain satisfactory results for the mean velocities of the turbulent flow.

**Keywords:**

Navier-Stokes equations, eigenfunctions of the Stokes operator, Galerkin method