The effects of boundary imperfections on convection in a saturated porous layer: non-resonant wavelength excitation

Abstract

The onset of Rayleigh-Bénard convection in a horizontally unbounded saturated porous layer is considered when the temperatures of both horizontal boundaries vary periodically in one direction about their respective mean values. Attention is focused on small-amplitude thermal modulations with a wavenumber not close to the critical value for the perfect layer. A stability analysis of weakly nonlinear convection is performed and the effects of different wavenumbers and symmetries of the thermal modulations are deduced systematically. It is shown that there are many special cases to be considered and that the convection patterns depend crucially on the particular configuration. Intuitively it might be expected that one-dimensional thermal modulation would always stimulate a two-dimensional motion. Surprisingly, however, for a wide range of modulation wavenumber a three-dimensional motion with a rectangular planform results from a resonant interaction between a pair of oblique rolls and the boundary forcing.