Fluid flow and thermal transport have been numerically investigated for a heated plate moving with and within a uniform forced flow. Two forced-flow circumstances involving a plate moving in a channel are considered in this study, one with uniform flow at the inlet and one in a uniform free stream in an extensive medium. These circumstances are of interest in a wide variety of manufacturing processes, such as continuous casting, extrusion, wire drawing, and fiber drawing. A detailed numerical study is carried out, assuming two-dimensional, laminar, transient flow. The governing full elliptic equations are solved by employing finite-difference and finite-volume methods. The transport in the solid material is coupled with that in the fluid through the boundary conditions, and simultaneous solutions for the two are obtained. The numerical simulation of such conjugate transport processes is discussed, and the considerations important to obtain accurate results for a wide variety of practical problems that involve a moving material subjected to heat transfer are outlined. Numerical results are obtained for the flow field and for the temperature distributions in both the solid and the flow. The numerical imposition of the boundary conditions is shown to be a very important aspect of the simulation. The two forced-flow circumstances considered have to be treated quite differently because of the nature of the resulting flow. The effect on the results is also determined for the relevant parameters that arise in the numerical scheme. The numerical results obtained indicate that the penetration of the conductive effects upstream of the point of emergence is significant. When a plate moves in a channel flow, the effect of channel width on the resulting heat transfer rate is significant, at least over ranges of practical interest. However, when the channel width increases, the resulting heat transfer in the case of a plate moving in a channel flow approaches the value obtained in the case of a plate moving in a free stream, as expected.