Heat-conducting fluids in domains with open boundaries

We study a Boussinesq system with mixed boundary conditions in a bounded domain, for steady and non-steady cases. The special feature of the problem is that the domain boundary is assumed to have an "open part" where the fluid can leave or re-enter. At this outlet, the fluid flow is assumed to satisfy a classical do-nothing condition and a new artificial nonlinear condition that couples the fluid velocity and temperature is considered for the heat transfer. In particular, the latter depends on a bounded function $\beta$ that may be chosen according to the problem of interest. For the stationary case, we prove the existence and uniqueness of a weak solution in two or three dimensions provided $\beta$ is Lipschitz continuous, and show that existence is retained if $\beta$ is only continuous. For the time evolutionary problem, we provide existence and uniqueness of a weak solution for $\beta$ Lipschitz in the two-dimensional case via a specific choice of the state space. All results are obtained under a "small" data assumption and restrictions on Reynolds, Prandtl, and Grashof numbers. We further present numerical tests that show increased accuracy of the new boundary condition when compared to other standard ones.