K. V. Gubareva, E. Yu. Prosviryakov

EXACT ANALYTICAL SOLUTION TO THE PROBLEM OF STATIONARY CONVECTION IN THE BOUSSINESQ APPROXIMATION WITH ACCOUNT FOR VISCOUS DISSIPATION

An exact analytical solution is obtained for the system of equations governing stationary convection of a viscous incompressible fluid, accounting for the buoyancy force (within the Boussinesq approximation) and viscous dissipation. The flow in a plane layer between two parallel plates is considered. It is found that the system admits two mutually exclusive classes of solutions. One describes a thermogravitational flow with a linear dependence of temperature on the longitudinal coordinate and a velocity dependent only on the transverse coordinate. The other class represents a generalized shear flow, combining the Couette and Poiseuille profiles with temperature depending solely on the transverse coordinate and explicitly accounting for dissipative heating. It is shown that the energy equation prohibits the simultaneous existence of a transverse velocity shear and a longitudinal temperature gradient. For the latter class, an explicit closed-form solution is constructed for a particular case, namely a generalized Couette–Poiseuille flow with a constant pressure gradient and isothermal boundaries, which includes closed-form expressions for velocity, temperature, and pressure. All the solutions strictly satisfy the equations of motion and energy, as well as the specified boundary conditions.

Acknowledgment: The research was performed under the state assignment of the Russian Ministry of Science and Higher Education for the IES UB RAS, theme No. 124020600042-9.

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