The conservation equations describing steady, incompressible flow in a variable-area duct with mass transfer occurring at its walls are simplified by linearizing the inertial and convective terms. Solutions to a large class of problems can be obtained by means of the general method which is presented. Particular examples considered are entrance flow with heat transfer to an isothermal wall in the presence of mass addition, constant rate of injection of a foreign gas through the wall, vaporization and sublimation of a volatile wall material, and gas-phase combustion of a fuel which enters the duct from its wall. Comparison of the present results with previous work and with new experimental results is discussed for the first of these applications. It is concluded that the present results for velocity and pressure fields are likely to be accurate for small values of the Reynolds number based on wall injection velocity, that the present results for temperature and composition fields are likely to be accurate in the absence of wall mass transfer, and that in the absence of wall mass transfer the linearization technique is likely to exhibit its highest accuracy for flows with uniform entrance conditions.