Creeping three-dimensional convective motion in a layer with velocity field of a special type

Problem of three-dimensional motion of a heat-conducting fluid in a channel with solid parallel walls is considered. Given temperature distribution is maintained on solid walls. The liquid temperature depends quadratically on the horizontal coordinates, and the velocity field has a special form. The resulting initial-boundary value problem for the Oberbeck–Boussinesq model is inverse and reduced to a system of five integro-differential equations. For small Reynolds numbers (creeping motion), the resulting system becomes linear. A stationary solution has been found for this system, and a priori estimates have been obtained. On the basis of these estimates, sufficient conditions for exponential convergence of a smooth non-stationary solution to a stationary solution have been established. The solution of the inverse problem has been found in the form of quadratures for the Laplace images under weaker conditions for the temperature regime on the walls of the layer. Behaviour of the velocity field for a specific liquid medium have been presented. The results were obtained with the use of numerical inversion of the Laplace transform.