The simultaneous development of the velocity and temperature profiles in circular tubes is analysed, accounting for axial diffusion of both momentum and heat. The axisymmetric problem is formulated in variational form by the weighted residuals Galerkin method and then numerically solved by the finite element method. Results obtained in the case of vanishing axial diffusion effects satisfactorily compare with previous boundary layer solutions ; in the presence of axial diffusion effects the solutions presented in the literature are scanty and do not always agree. The isothermal flow results, presented for Reynolds numbers ranging from 1 to 1000, show the appearance of the overshoots in the axial velocity profile, as well as the dependence on the axial coordinate and Reynolds number of the various axial momentum transfer processes—radial diffusion at the wall, convection and axial diffusion. The non-isothermal flow results, presented for the Prandtl number ranging from 0.1 to 100 and the Péclet number from 5 to 500, point out the effect of axial diffusion of momentum and, above all, of heat, on the convective heat transfer rate.