Heat transfer: boundary conditions with fluid velocity

Question

The following equation is considered: $$ \frac{\partial u}{\partial t} - a\Delta u + \mathbf v \cdot \nabla u = f. $$ I have difficulties in formulating boundary conditions for this equation. If $\mathbf v$ is absent, the boundary conditions are $$ a\frac{\partial u}{\partial n} + \beta(u - u_b) = 0 $$ where $u_b$ is prescribed temperature field on the boundary. But if $\mathbf v$ is present? If we suggest the following boundary condition: $$ a\frac{\partial u}{\partial n} - (\mathbf v \cdot \mathbf n) + \beta(u - u_b) = 0 $$ then we will have problems with analysing the equation when $(\mathbf v \cdot \mathbf n) > 0$ (where the fluid outflows). How to set correct boundary conditions?

Answer 1

The boundary condition $$ a\frac{\partial u}{\partial n} - (\mathbf v \cdot \mathbf n)u + \beta(u - u_b) = 0 $$ is correct and cannot cause a problem. Indeed, $\mathbf v\cdot\mathbf n=0$ at the boundary whenever it happens to be impermeable to fluid. Otherwise, condition $(\mathbf v\cdot\mathbf n) > 0$ cannot hold on the whole boundary according to the principle of conservation of mass implying that the total flux of fluid through the boundary equals zero.