An exact solution is found for a non-linear problem with thermomechanical coupling, the steady flow of a fluid with viscosity exponentially dependent on temperature, which is sheared between an adiabatic, fixed, inner cylinder and a thermostatted, rotating, outer cylinder. There is a maximum torque above which no steady flow is possible and below which two flows are possible, a high shear and a low shear steady flow for each value of torque.