The classical Graetz problem of fluid entering a long channel possessing different temperature from that of fluid is extended here to the case where the upper wall of the channel is not stationary. Formulation of the current problem incorporates the simple Couette flow driven by the channel movement and taking into account the axial heat conduction, besides the usual convection. Analytical solutions in the thermally developing regime regarding the temperature distribution as well as the heat transfer rates from both the upper moving and the lower fixed channel walls are then obtained by means of the series expansion method making use of the proper orthogonal family of eigenfunctions. Simple approximate formulae for the rates of heat transfer at the channel entrance valid for large Peclet numbers and at the far downstream locations valid for all Peclet numbers are further derived. More simplistic outcomes are also provided when the axial pressure gradient is assumed to be constant. Results are then discussed from low to high Peclet numbers, which are in compliant with the well-known results available in the literature for the channel flow within stationary walls. On the other hand, the rate of heat transfer from the moving wall is shown to be of high importance when Peclet number increases, which requires special attention in technological applications. (c) 2021 Taiwan Institute of Chemical Engineers. Published by Elsevier B.V. All rights reserved.