Thermal entrance region for laminar forced convection in a circular tube with a power-law wall heat flux

Laminar forced convection in a circular tube is investigated with a boundary condition of prescribed axially varying wall heat flux, under the assumptions of hydrodynamically developed flow and of negligible axial conduction and viscous dissipation in the fluid. A condition on the asymptotic behaviour of the axial distribution of wall heat flux is established which is fulfilled by power-law varying axial distributions and which guarantees the existence of a thermally developed regime. It is proved that for all the axial distributions which fulfil this condition the asymptotic value reached by the local Nusselt number is 48 11, i.e. the same which holds for a uniform wall heat flux. For some power-law varying axial distributions of wall heat flux, a finite difference determination of the thermal entrance region is performed. In every numerical solution, the local Nusselt number tends asymptotically to 48/11.