96-31
Remarks to the Stability of the Kinetic Energy of the Plane Parallel Couette Flow
by Rummler, B.
Preprint series: 96-31, Preprints
- MSC:
- 76E05 Stability of parallel flows
- 35Q30 Stokes and Navier-Stokes equations, See also {76D05, 76D07, 76N10}
- 76F10 Shear flows
Abstract: We study problem of energetic stability of the plane parallel Couette flowof an incompressible Newtonian fluid within an unbounded layer in R 3 ofthe thickness 2 between two parallel walls moved in opposite directions. Wedemand constant non-dimensionalized velocities \Sigma(1; 0; 0) of the walls andsuppose nonslip conditions for the velocity field. The boundary conditionsare supplemented with periodical conditions for the sought velocity field inthe former unbounded directions.We use the ideas of v. Wahl [1[] and follow the way of considerations of theplane parallel Couette flow used by the author and A.Noske in [11[]. Wederive out of the eigenvalue problem of the energetic stability in the infinite-dimensional case in form of the Euler-Lagrange system by separation ofthe variables a correspondent eigenvalue problem of a system of ordinarydifferential equations.Additionally, we formulate the problem of energetic stability in the finite-dimensional case for the Galerkin-approximations of [11[] as an algebraicproblem.
Keywords: Couette flow, energetic stability
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