Effect of quasi-periodic gravitational modulation on the convective instability in Hele-Shaw cell

The aim of the present paper is to examine the effect of a quasi-periodic gravitational modulation on the onset of convective instability in Hele-Shaw cell. The quasi-periodic modulation considered here consists in a modulation having two incommensurate frequencies. This study is an extension of a previous work by Aniss et al. [Asymptotic study of the convective parametric instability in Hele-Shaw cell, Phys. Fluids 12 (2) (2000) 262–268] in which only a periodic gravitational modulation was considered. We have shown that for Pr=O(1)Pr=O(1) or Pr⪢1Pr⪢1, the gravitational modulation has no effect on the convective threshold as expected. However, for Pr=O(ε2)Pr=O(ε2), it turns out that a modulation with two incommensurate frequencies has a stabilizing or a destabilizing effect strongly depending on the frequencies ratio.