Analytical and numerical solutions for torsional flow between coaxial discs with heat transfer

We consider nonisothermal torsional flow between two coaxial parallel plates with heat transfer at the upper rotating plate, constant temperature on the lower stationary plate, and no heat loss at the fluid-air interface. Viscous heating is modelled by a Nahme law with exponential dependence on temperature. Due to the highly nonlinear nature of the governing equations an exact solution is not feasible. Therefore we solve the problem using both numerical and perturbation methods. Specifically, analytical solutions are obtained using asymptotic expansions based on the aspect ratio and the Nahme-Griffith number, a measure of viscous heating, as perturbation parameters. The numerical solutions are obtained by a finite element method. Good agreement is found between the analytical and numerical solutions in the appropriate parameter range. In viscometric applications the torque exerted by the fluid on the lower plate is an important quantity. For isothermal flow the dimensionless torque can easily be calculated. In this paper we obtain an analytical formula that can be used to calculate nonisothermal correction to the torque.