Solution of the Graetz-Brinkman problem with the Laplace transform Galerkin method

The present study concentrates on the effects of viscous dissipation in laminar forced convection. A power law fluid rheology model is applied and the effect of heat conduction in the axial direction is considered negligible. The physical properties are considered constant. Assuming fully developed velocity profile, the development of the temperature profile and its asymptotic behavior are investigated. For the solution of the problem the Laplace transform Galerkin technique is used. The method allows for the most general boundary conditions. A detailed comparison with previously published results provides a verification of the numerical technique. An important feature of the approach is that derivatives and integrals with respect to the axial location can be obtained through the operational rules of the Laplace transformation and hence no numerical derivation or integration is needed. As an application of the numerical model, we focus on the natural cooling regime, when the viscous dissipation of energy is counter-balanced by keeping the wall temperature at the ambient value. We derive a correlation for the asymptotic behavior of the Nusselt number in the natural cooling regime. This correlation reproduces the known value for the Newtonian case and provides a convenient means to normalize the Nusselt number for a wide range of flow behavior indices.