Given a flow field $\vec{u}$, and it transmits a materials with $c_p$ heat capacity, and $k$ heat conduction coefficient. It results the following equation, which should be solved for the $T$ temperature distribution.

$c_p \frac{\partial T}{\partial t} = \vec{\nabla} k \vec{\nabla} T - c_p \vec{u} \vec{\nabla} T + \dot{q}$