On exact nonstationary solutions of equations of vibrational convection
In this paper, we consider a class of exact non-stationary solutions of the Boussinesq equations, which describe the motion of an inhomogeneous fluid in a vessel performing periodic linear vibrations of a finite frequency. The inhomogeneity of the medium implies the existence of the density gradient, which can occur due to different factors (external or internal). An important condition for obtaining an exact solution in the closed form is the orthogonality of the density gradient and the direction of vibrations, which should be maintained at any time moment during the vibration period. If this condition is fulfilled, then there exists a class of exact unsteady solutions describing the laminar flow of fluid in the direction of vibrations. In this case, the velocity profile can have a complicated dependence on the coordinates, which are transverse to the fluid motion. This functional dependence is determined by the character of the density inhomogeneity. Finally, the inertial field, varying in time, differently affects the laminar layers of various densities and defines the main physical mechanism of the fluid flow. The final result of the calculations also depends essentially on the return flow condition. As examples, the following problems of thermo- and chemovibrational convection have been considered: the flow of a viscous fluid in a plane layer heated from the side and performing periodic harmonic vibrations along the layer; the flow of a viscous heat-generating fluid in a plane layer under the action of periodic vibrations directed along the layer; the flow of a viscous fluid in a plane layer at the boundary of which a constant gradient of the reactant is assigned, the chemical reaction of the first order occurs, and the layer itself performs longitudinal periodic vibrations; the flow of a viscous heat-generating fluid filling a cylindrical channel that performs periodic oscillations in the axis direction. In each case, we present analytical expressions for fluid velocity, pressure, temperature, and reagent concentration. A general procedure for finding exact expressions for a given class of solutions is discussed.