A method is described for solving steady-state fluid flow and heat transfer problems which are governed by elliptic-type differential equations. A contrived transient version of the steady-state problem is constructed by appending time derivatives to all the participating equations, regardless of whether or not such terms have physical reality. Each time derivative is multiplied by a fictive diffusivity coefficient which is varied during the course of an explicit marching procedure in order to achieve rapid, stable convergence to the steady state. The solution method is applied to a three-component laminar flow in a cylindrical enclosure having one rotating wall and coolant throughflow. Recirculation patterns are set up in the enclosure due to the shearing action of the throughflow and to the rotation of the disk. The surface heat transfer is found to decrease as the Reynolds number of the throughflow increases.