Vibration effect on the onset of thermal convection in an inhomogeneous porous layer underlying a fluid layer

The linear stability problem for a two-layer system consisting of a horizontal pure fluid layer and an inhomogeneous porous layer saturated with the same fluid and heated from below in the presence of vertical high-frequency and small-amplitude vibrations in a gravitational field is studied in the framework of the averaging method. The problem is solved numerically with a shooting method. Bimodal neutral curves of the stability of mechanical equilibrium characterized by the absence of an averaged fluid flow in layers are obtained for the linear profile of porosity within a porous layer and permeability versus porosity through the Carman-Kozeny relation. We consider a competition between perturbations of shorter wavelength (short-wave perturbations) localized mainly in the fluid layer and those of longer wavelength (long-wave perturbations) spanning both layers near the stability threshold in the case when the intensity of vibrations and porosity gradient change. It is shown that when porosity increases with depth, instability is associated with the development of large-scale and long-wave perturbations. The stability threshold of these perturbations weakly depends on the vibration intensity in the range of the parameters considered. Perturbations penetrate into the porous medium poorly when porosity decreases with depth. Vibrations in such a situation stabilize the short-wave instability mode noticeably and lead to an increase in the wavelength of the most dangerous perturbations of equilibrium due to different contributions of inertial effects to fluid and porous layers. It has been found that when the vibration intensity and porosity gradient increase, a hard abrupt transition from short-wave to long-wave perturbations is replaced by a smooth change in the wavelength of critical perturbations related to the loss of stability of equilibrium in the examined two-layer system.