Vibration effect on double-diffusive instability in an inhomogeneous porous layer underlying a binary fluid layer

Abstract

A linear stability problem for mechanical equilibrium in an inhomogeneous fluid-saturated porous layer underlying a horizontal binary fluid layer is numerically simulated. The layers are subjected to high-frequency vertical vibration in a gravity field. Different values of temperature and concentration are fixed at the outer impermeable boundaries of the two-layer system. Porosity of a porous layer linearly depends on a transverse z-coordinate. Permeability is given by the Carman-Kozeny formula. To describe a fluid flow in layers, we use the vibrational convection equations written in the Boussinesq approximation and obtained by the averaging method. It is shown that when the fluid is heated from below, vibration effectively increases the equilibrium stability threshold and wavelength of its most dangerous perturbations. If a temperature gradient coincides with the direction of a concentration gradient for the heavier component of binary fluid, average convection is excited in an oscillatory manner. In the opposite case, there is a monotonic instability of equilibrium. Effects of the dimensionless porosity gradient mz and vibrational parameter pv on the instability threshold are studied. For pv<0.0134 and a fixed solutal Rayleigh number (-5⩽Rmc⩽15) an abrupt jump-like transition from the long-wave to short-wave most dangerous perturbations occurs as mz grows in the range of 0.030–0.162. Long-wave perturbations penetrate both layers. Short-wave perturbations mainly locate in the fluid layer. The transition is smoothed for pv>0.0134. When the fluid is heated from above, wavelength of critical perturbations smoothly varies with a change in pv and mz. A convective fluid flow arises monotonously and mostly in the form of long wavelength rolls (a wave number is of k<3.3). Vibration weakly lowers the equilibrium stability threshold in layers. The effect is most pronounced at high enough Rayleigh numbers (Rm=-40) for a porous medium with the porosity increasing with depth at mz=-0.2.