Analytical and numerical study of mixed convection flow in a vertical circular duct with circumferentially variable wall heat flux and viscous dissipation

Mixed convection in a vertical circular duct is studied under the assumption of fully developed, laminar and parallel flow. A non-axisymmetric thermal boundary condition such that the duct wall is half subjected to a uniform heat flux and half adiabatic is considered. First, by neglecting the effect of viscous dissipation, the dimensionless governing equations of the problem are solved analytically by means of Fourier series expansions of the temperature field and of the velocity field. The dimensionless temperature and velocity distributions as well as the average Nusselt number are obtained. Then, the effect of viscous dissipation is taken into account and the non-linear dimensionless governing equations are solved numerically by means of a finite element method.