Forced convection with viscous dissipation in the thermal entrance region of a circular duct with prescribed wall heat flux

The thermal entrance forced convection in a circular duct with a prescribed wall heat flux distribution is studied under the assumptions of a fully developed laminar flow and of a negligible axial heat conduction in the fluid, by taking into account the effect of viscous dissipation. The solution of the local energy balance equation is obtained analytically by employing the Laplace transform method. The effect of viscous dissipation is taken into account also in the region upstream of the entrance cross-section, by assuming an adiabatic preparation of the fluid. The latter hypothesis implies that the initial condition in the entrance cross-section is a non-uniform radial temperature distribution. Two special cases are investigated in detail: an axially uniform wall heat flux, a wall heat flux varying linearly in the axial direction.