As part of a general study of the mechanical behavior of materials with temperature‐dependent properties, idealized Couette flow is considered here. The model fluid is Newtonian and incompressible. Inertial terms are neglected in the analysis. Time‐dependent temperature and velocity distributions in the fluid are computed for: (1) the constant stress case, (2) the constant rate of stress increase case, and (3) the constant boundary velocity case. In each case, a range of thermal boundary conditions is also considered. The results are compared with closed form analytical solutions that can be obtained in special cases. The temperature distributions can influence the stability of laminar flow and cavitation in liquids. The derived velocity distributions influence the drag and protective action of lubricants. Furthermore, it appears that some apparently non‐Newtonian flows can be described more simply when the deviation is attributed to heating than when it is attributed to intrinsic, non‐Newtonian behavior. A thermistor circuit analog is demonstrated as an aid to the solution of the Couette flow equations with thermal feedback.