Dual solutions for viscous mixed convection flows in a vertical circular duct: a numerical benchmark

Forced and free convection flow in a vertical isothermal circular duct is studied by taking into accountthe internal heating due to viscous dissipation. The momentum and energy balance equations lead to a nonlinear boundary value problem that is solved numerically by means of the finite-element software Comsol Multiphysics 3.3a (Comsol, Inc.). For any prescribed value of the average velocity smaller than a maximum, two solutions exist which are completely different in thermal and mechanical characteristics (dual solutions). It is shown how the software can be used to determine the velocity and temperature fields and the main interesting physical quantities for both the solutions. The numerical results are compared with the analytical results available in the literature, thus performing a benchmark of the numerical solution