Thermovibrational low-mode model of convection in a horizontal layer with longitudinal vibrations

Thermovibrational convection in a horizontal layer of fluid between isothermal solid boundaries heated to different temperatures in the presence of longitudinal vibrations is considered in this paper. Stability and supercritical bifurcation of convection is investigated in a low-mode approximation. Bifurcation diagrams of supercritical modes are analytically obtained in the area of stability of supercritical convection. The analysis of diagrams shows that vibrations can lead to the rigid type of the occurrence of convection when upper boundary is heated. In addition, the hysteresis between stationary states is observed. The size of hysteresis interval of the Rayleigh numbers increases with the growth of the Gershuni number. A numerical study of the linear stability of the supercritical vibration-convective flows in the interval of Prandtl numbers $1 \leqslant \mathrm{Pr} \leqslant 10$ is conducted in the context of the proposed model. The region of flow stability decreases with increasing the Prandtl number. For any value of the Prandtl number from the given interval drastic excitation of stationary vibrational convection with hysteresis is possible.