The Boussinesq system with mixed non-smooth boundary conditions and do-nothing boundary flow

Abstract

A stationary Boussinesq system for an incompressible viscous fluid in a bounded domain with a nontrivial condition at an open boundary is studied. The problem is motivated by modeling energy systems in rooms that possess an outlet where the fluid can freely flow, known as an “open boundary”. Numerical and experimental investigations available from the literature on heated cavities with open boundaries suggest that the heat transfer at the outlet depends on the temperature at the boundary, the velocity of the fluid, and the outside temperature. Aiming to include this feature in the model, we propose a novel non-smooth boundary condition which is deduced from physical assumptions. We show that this condition is compatible with two approaches at dealing with the “do-nothing” boundary condition for the fluid: (1) the “directional do-nothing” condition and (2) the “do-nothing” condition together with an integral bound for the backflow. Well-posedness of variational formulations is proved for each problem.