A numerical and analytical study of the transport process arising due to the movement of a continuous heated body has been carried out. The relevant heat transfer mechanisms are of interest in a wide variety of practical applications, such as continuous casting, extrusion, hot rolling, and crystal growing. The conjugate problem, which involves a coupling between the heat transfer in the moving material and the transport in the fluid, is considered. The thermal fields in the material and in the fluid are computed. The temperature level is found to decay gradually with distance along the moving material, as expected. Results are obtained for a wide range of governing parameters, particularly the Peclet number Pe and the parameter R , which depends on the properties of the fluid and the material. The results obtained are compared with those for the idealized cases of an assumed surface heat transfer coefficient and of a moving isothermal surface. Of particular interest were the nature of the flow generated by the moving surface and the resulting thermal transport. The results obtained are also considered in terms of the underlying physical processes in the problem.