This paper provides an analytical solution for the combined diffusive and convective mass transport from a surface film of arbitrary shape at a given uniform concentration to a pure solvent flowing in the creeping regime through microchannels. This problem arises e.g. in the study of swelling and dissolution of polymeric thin films under the tangential flow of solvent, modeling the oral thin film dissolution for drug release towards the buccal mucosa or oral cavity. We present a similarity solution for mass transfer in laminar forced convection. The classical boundary layer solution of the Graetz-Nusselt problem, valid for straight channels or pipes, is generalized to a microchannel with rectangular cross-section varying continuously along the axial coordinate. Close to the curved releasing boundary h(s)(s), parametrized by a curvilinear abscissa s, both the tangential v(t)(r,s) and the normal velocity v(n)(r,s) components play a role in the dissolution process, and their scaling behavior as a function of wall normal distance r should be taken into account for an accurate description of the concentration profile in the boundary layer. An analytic expression for the local Sherwood number as a function of the curvilinear abscissa and the Peclet number is presented. (C) 2013 Elsevier Ltd. All rights reserved.