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Remarks to the Stability of the Kinetic Energy of the Plane Parallel Couette Flow

by Rummler, B..

Series: 1996-31, Preprints

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

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

Couette flow, energetic stability