On the Horton–Rogers–Lapwood convective instability with vertical vibration: Onset of convection

Abstract

We present a numerical and analytical study of diffusive convection in a rectangular saturated porous cell heated from below and subjected to high frequency vibration. The configuration of the Horton–Rogers–Lapwood problem is adopted. The classical Darcy model is shown to be insufficient to describe the vibrational flow correctly. The relevant system is described by time-averaged Darcy–Boussinesq equations. These equations possess a pure diffusive steady equilibrium solution provided the vibrations are vertical. This solution is linearly stable up to a critical value of the stability parameter depending on the strength of the vibration. The solutions in the neighborhood of the bifurcation point are described analytically as a function of the strength of vibration, and the larger amplitude states are computed numerically using a spectral collocation method. Increasing the vibration amplitude delays the onset of convection and may even create subcritical solutions. The majority of primary bifurcations are of a special type of symmetry-breaking bifurcation even if the system is subjected to vertical vibration