Numerical techniques have been used to solve the thermally developed regime for a laminar pipe flow that exchanges heat with a fluid environment in the presence of a circumferentiatly varying external heat transfer coefficient. By making use of the fact that the temperature distributions have similar shapes at successive streamwise locations, the three-dimensional temperature field was scaled to two dimensions. The resulting Two-dimensional eigenvalue problem was solved by a rapidly converging automated scheme that successively refines an initial guess. Solutions were obtained for two circumferential distributions of the external heat transfer coefficient respectively intended to model forced and natural convection cross flows. The circumferential average heat transfer coefficient was found to be quite insensitive to the imposed circumferential variations. The local wall heat flux is nearly circumferentially uniform when the mean value of the external coefficient is high. On the other hand, at low mean values of the external coefficient, the local wall heat flux tends to follow the imposed circumferential variations.