Non-axisymmetric mixed convection with viscous dissipation in a vertical circular duct

The laminar mixed convection in a vertical circular duct is studied, with reference to non-axisymmetric boundary conditions such that the wall temperature is uniform along the axial direction and is an arbitrary function of the angular coordinate. With reference to a fully-developed parallel flow, the dimensionless momentum and energy balance equations are solved by applying the Boussinesq approximation. The effect of viscous dissipation inside the fluid is taken into account. The solution is found numerically, by means of software FlexPDE (© PDESolutions, Inc.). It is shown that either the dimensionless temperature field, or the dimensionless velocity field, or the Fanning friction factor, or the Nusselt number are strongly influenced by the Brinkman number and by the ratio between the Grashof number and the Reynolds number. Two special cases are studied in detail: a duct which undergoes a wall temperature distribution which is sinusoidal with respect to the angular coordinate; a duct whose wall is half adiabatic and half isothermal.