The Graetz–Brinkman problem in a plane-parallel channel with adiabatic-to-isothermal entrance

An analysis of laminar hydrodynamically developed forced convection in the thermal entrance region of a plane-parallel channel with isothermal walls is presented. The effect of viscous dissipation is taken into account in a self-consistent way, i.e. by assuming that this effect is important not only in the region downstream of the entrance section, but also in the region upstream. In the latter region, the channel is assumed to be thermally insulated. As a consequence, the thermal entrance temperature distribution is non-uniform. The temperature field and the local Nusselt number are evaluated.