Analysis of the effect of viscous dissipation for laminar flow in stadium-shaped ducts

The fully-developed laminar forced convection of a Newtonian fluid in a duct with stadium-shaped cross section has been analyzed. The effect of viscous dissipation has been taken into account. Three different thermal boundary conditions have been considered: (T) uniform wall temperature distribution; (H1) axially uniform wall heat flux distribution with peripherally uniform wall temperature distribution; (H2) axially and peripherally uniform wall heat flux distribution. The adiabatic-wall boundary condition has also been analyzed as a special case of the H2 boundary condition. The velocity and temperature distributions in the fluid, as well as the Fanning friction factor and the Nusselt number, have been evaluated numerically, by employing a Galerkin finite element method. As expected, the numerical evaluation of the dimensionless temperature distribution and of the Nusselt number reveals that increasing discrepancies between the H1 and H2 boundary conditions exist if the stadium-shaped duct is gradually flattened.