Excitation of Convection in a System of Layers of a Binary Solution and an Inhomogeneous Porous Medium in a High-Frequency Vibration Field

Abstract

The onset of convection in a system of horizontal layers of a binary solution and an inhomogeneous porous medium saturated with the solution is studied. The system is subjected to high-frequency transverse vibrations in the gravity field. It is assumed that the porosity of the medium depends linearly on the vertical coordinate. The permeability is estimated by the Karman–Kozeny formula for different values of the dimensionless gradient of the porosity, mz. The convection of the fluid under the action of high-frequency vibrations in the gravity field is described using the averaging method. The linear problem of the stability of the mechanical equilibrium of the fluid is solved numerically by the shooting method. The values of the critical parameters corresponding to the convection initiation threshold are determined for the system heated from below or from above. The heating from below is distinguished by a sharp change in the character of the instability with a variation in the porosity gradient or the vibration intensity. It is shown that, when the porosity increases with depth, at mz =–0.2, the instability is caused by the development of long-wave perturbations involving the fluid and the porous layers. When the porosity decreases with depth, at mz = 0.2, the most dangerous perturbations are short-wave perturbations localized in the fluid layer. For intermediate values of the porosity gradient,–0.2< mz < 0.2, the values of the minimum Rayleigh–Darcy critical numbers, which determine the equilibrium stability threshold with respect to short-wave and long-wave disturbances, approach one another. The neutral curves are bimodal. Upon heating from below, vertical vibrations effectively suppress convection in the fluid layer; therefore, with an increase in their intensity, the transition from short-wave vibrations, which are most dangerous, to long-wave perturbations is observed. A noticeable increase in the stability threshold is observed when the porosity decreases with depth. Upon heating from above, vibrations destabilize the equilibrium in the system and lead to a reduction in the wavelength of the critical perturbations. The wavelength decreases monotonically. Its maximum change is detected in layers whose porosity increases with depth.