Forced convection in circular ducts with slug flow and viscous dissipation is analyzed. The temperature field and the local Nusselt number in the thermal entrance region are determined analytically when the tube wall exchanges heat with an external fluid by a uniform convection coefficient. The evaluations are performed for an arbitrary axial distribution of the external-fluid reference temperature. The temperature field and the local Nusselt number for a prescribed axial distribution of wall temperature are obtained by taking the limit of an infinite Blot number. Two cases are analyzed in detail : the reference temperature of the external fluid is uniform; the reference temperature of the external fluid changes linearly in the axial direction.