Mean values and non-periodic pressure in convection problems between plates or with stress-free boundaries

Burkhard J. Schmitt and Wolf von Wahl

Vol. 185 (1998), No. 2, 347–362

DOI: 10.2140/pjm.1998.185.347

Abstract

When considering the Oberbeck-Boussinesq equations in an infinite layer it is mostly assumed that the pressure π is periodic in the plane, whereas the equations only require ∇π to be periodic. We study here the influence the general admissible form of the pressure may have on the velocity field u below the onset of convection, a question which is closely connected with the mean flow. This is a vector field which depends on z only and which is given by the mean values of u_x, u_y, u_z over the plane periodicity cell. — The mean value of u over the layer is constant under stress-free boundary conditions and periodic pressure. If this constant c is not 0 there is in most cases no longer an exchange of stability on the onset for the linearization around c. We study its spectrum on the onset.

Milestones

Received: 27 January 1997

Revised: 5 August 1997

Published: 1 October 1998