Non-axisymmetric forced and free flow in a vertical circular duct

The fully developed laminar mixed convection in a vertical circular duct is studied analytically, with reference to non-axisymmetric boundary conditions such that the fluid temperature does not change along the axial direction. The Boussinesq approximation is applied by taking the average temperature in a duct section as the reference fluid temperature. The dimensionless momentum and energy balance equations are solved by employing Fourier series expansions of the temperature and the velocity fields. The solution shows that the temperature field is not influenced by the velocity distribution and that the Fanning friction factor is not affected by buoyancy. On the other hand, the velocity field is strongly influenced by the buoyancy forces and may display flow reversal phenomena. Two special cases are studied in detail: a duct with a sinusoidal wall temperature distribution; a duct subjected to an external convection heat transfer with two environments having different reference temperatures.