We present a similarity solution for mass/heat transfer in laminar forced convection at high Peclet numbers. The classical boundary layer solution of the Graetz-Nusselt problem, valid for straight channels or pipes, is generalized to an axisymmetric microchannel with circular cross-section, whose radius R(z) varies continuously along the axial coordinate z. The case of fixed wall concentration/temperature is analyzed. The advection/diffusion transport problem is solved by taking into account both the tangential and normal velocity components (and their scaling behaviours as a function of the wall normal distance), in order to obtain an accurate description of the concentration/temperature profile in the boundary layer. The analytical solution of the local Sherwood/Nusselt number is compared with finite elements numerical results for a truncated cone and a wavy sinusoidal channel. (C) 2013 Elsevier Ltd. All rights reserved.