Thermal entrance heat transfer of an adiabatically prepared fluid with viscous dissipation in a tube with isothermal wall

Forced convection in the thermal entrance region of a circular duct is analyzed. Viscous dissipation effects are taken into account under conditions of laminar hydrodynamically developed flow. The duct wall is assumed to be isothermal in the region downstream of the entrance cross section. The prescription of the initial condition at the entrance cross section is coherent with the assumption of a non-negligible viscous heating in the whole duct. The special case of an adiabatic-wall preparation of the fluid in the upstream region is considered. This adiabatic preparation results in a non-uniform entrance temperature distribution. The governing equations are solved analytically by separation of variables. Important differences are pointed out in the comparison of the solution with those available in the literature, which are based on the assumption of a uniform temperature distribution in the entrance cross section.