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?



Heat transfer: boundary conditions with fluid velocity
your second boundary condition is missing a factor $u-u\_b$:


$$a\frac{\partial u}{\partial n} - \gamma(u-u\_b)(\mathbf v \cdot \mathbf n) + \beta(u - u\_b) = 0$$


the coefficient $\beta$ gives the strength of the heat transfer at the boundary; the coefficients $a$ and $\gamma$ are the same as in the diffusion-convection equation,


$$\frac{\partial u}{\partial t} - a\Delta u + \gamma \mathbf v \cdot \nabla u = f.$$


see, for example, [The convective-diffusion equation and its use in building physics](http://www.bse.polyu.edu.hk/researchCentre/Fire_Engineering/summary_of_output/journal/IJAS/V1/p.68-79.pdf) (2000).


