The stability of finite amplitude cellular convection and its relation to an extremum principle

Abstract

The stability of cellular convection flow in a layer heated from below is discussed for Rayleigh number R close to the critical value Rc. It is shown that in this region the stable stationary solution is determined by a minimum of the integral \[ \int_0^{H_0}R(H)\,dH, \] where R(H) is a functional of arbitrary convective velocity fields which satisfy the boundary conditions. For the stationary solutions R(H) is equal to the Rayleigh number. H0 is a given value of the convective heat transport. In a second part of the paper explicit results are derived for the convection problem with deviations from the Boussinesq approximation owing to the temperature dependence of the material properties.