On low-Prandtl-number convection in an inclined layer of liquid mercury

Abstract

This paper reports on a theoretical analysis of convection in an inclined layer of mercury, a common low-Prandtl-number fluid ($Pr=0.025$). The investigation is based on the standard Oberbeck–Boussinesq equations, which are explored as a function of the inclination angle $\unicode[STIX]{x1D6FE}$ and for Rayleigh numbers $R$ in the vicinity of the convection onset. Along with the conventional Galerkin methods to study convection rolls and their secondary instabilities, we employ direct numerical simulations for fluid layers with quite large aspect ratios. It turns out that, even for small inclination angles $\unicode[STIX]{x1D6FE}\lesssim 6^{\circ }$, the secondary instabilities of the basic rolls lead either to oscillatory three-dimensional patterns or to stationary ones, which appear alternately with increasing $\unicode[STIX]{x1D6FE}$. Due to the competition of these instabilities the patterns may show a complex dynamics.