On the Boussinesq approximation in the problems of convection induced by high–frequency vibration

The applicability of Boussinesq approximation to the problems of thermovibrational convection in closed volumes is analyzed. The limit of high frequency and small amplitude is considered on the basis of averaging approach. The magnitudes of oscillatory and averaged flow fields are estimated. It is found that the dependence of the Reynolds number for averaged motion on the ratio of Gershuni and Prandtl numbers obeys linear and square root laws for small and large Reynolds numbers, respectively. It provides new essential information about the intensity of averaged flows in a wide range of vibration stimuli. Taking into account the obtained estimations, the basic assumptions of Boussinesq approach are applied to the momentum, continuity, and energy equations for a compressible, viscous heat–conducting fluid. The contribution of viscous energy dissipation and pressure work to the energy balance is also taken into account. The order of magnitude analysis provides a number of dimensionless parameters, the smallness of which guarantees the validity of Boussinesq approximation.