by Rummler, B..

**Series:** 1996-31, Preprints

- MSC:
- 76E05 Parallel shear flows
- 35Q30 ~Navier-Stokes equations
- 76F10 Shear flows

**Abstract:**

We study problem of energetic stability of the plane parallel Couette flow

of an incompressible Newtonian 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 conditions for the velocity field. The boundary conditions

are supplemented with periodical conditions for the sought velocity field in

the former unbounded directions.

We use the ideas of v. Wahl [1[] and follow the way of considerations of the

plane parallel Couette flow used by the author and A.Noske in [11[]. We

derive out of the eigenvalue problem of the energetic stability in the infinite-

dimensional case in form of the Euler-Lagrange system by separation of

the variables a correspondent eigenvalue problem of a system of ordinary

differential equations.

Additionally, we formulate the problem of energetic stability in the finite-

dimensional case for the Galerkin-approximations of [11[] as an algebraic

problem.

**Keywords:**

Couette flow, energetic stability