Stability of convection in containers of arbitrary shape

Abstract

When a container of fluid of arbitrary shape is heated from below and the temperature gradient exceeds a critical value ([Rscr ]_c²) the conduction solution with no motion becomes unstable and is replaced by convection. The convection may have two forms: one with ‘upflow’ at the centre of the container and one with ‘downflow’ there. Here we study the stability of the two forms of convection. Both forms are here shown to be stable to infinitesimal disturbances. When the viscosity varies with the temperature or the conduction profile is not linear, etc., the steady convection can be driven with finite amplitudes |ε| at subcritical values of the temperature contrast ([Rscr ]²< [Rscr ]_c²). This subcritical convection is stable when the convection is strong (|ε| > |ε_*| > 0) but is unstable when the convection is feeble (|ε| > |ε_*|). Hence, when |ε| > |ε_*| and [Rscr ]²< [Rscr ]_c²either ‘upflow’ or ‘downflow’, but not both, is stable. When [Rscr ]²> [Rscr ]_c², however, both the ‘upflow’ and the ‘downflow’ can be stable. This contrasts with the corresponding situation which is known to hold when the container is an unbounded layer. In the layer there is only one stable form of convection. The difference between the bounded domain with two forms of convection and the layer with just one stable form is traced to the mathematical property of simplicity of [Rscr ]_c²when viewed as an eigenvalue of the linear stability problem for the conduction solution. It is argued that [Rscr ]_c²is a simple eigenvalue in most domains, but in the layer [Rscr ]_c²can have infinite multiplicity. The explanation of the transition from the bounded domain to the unbounded layer is sought (1) in the chaotic conditions which frequently prevail at the edges of a ‘bounded’ layer and (2) in the fact that in the layer of large horizontal extent, the higher eigenvalues crowd [Rscr ]_c². In the course of the explanation, a new exact solution of the linear Bénard problem in a cylinder with a rigid side wall and a stress-free top and bottom is derived.