Viscous-dissipation effects on the heat transfer in a Poiseuille flow

In this study, laminar heat-convection in a Poiseuille flow of a Newtonian fluid with constant properties is analyzed by taking the viscous dissipation into account. At first, both hydrodynamically and thermally fully-developed flow case is investigated. Then, consideration is given to thermally-developed laminar forced-convection. The axial heat-conduction in the fluid is neglected. Two different thermal boundary-conditions are considered: the constant heat-flux and the constant wall-temperature. Both the hot-wall and the cold-wall cases are considered. In the literature, the viscous-dissipation effect is commonly represented by the Brinkman number. Several different definitions of the Brinkman number arise depending on the thermal boundary conditions. Either for the thermally fully-developed case or the thermally-developing case (the Graetz problem), temperature distributions and the Nusselt numbers are analytically determined as functions of the Brinkman number.