Thermally developing Poiseuille flow with a non-uniform entrance temperature when the viscous heat generation is significant

The classical problem of thermally developing Poiseuille flow in the presence of viscous dissipation (the 'Graetz–Brinkman problem') is reviewed in this paper. Taking into account that the traditional assumptions of a uniform entrance temperature and of Poiseuille velocity profile are mutually exclusive concepts when the frictional heat generation is significant, in the present paper a non-uniform entrance temperature profile is considered. This non-uniform 'initial condition' is not prescribed, but it is deduced from the energy balance equation in a consistent way as the fully developed temperature profile of the Poiseuille flow under isothermal boundary conditions. Both for the dependence of the local Nusselt number on the Brinkman number and the developing temperature field, substantial differences have been found compared with the traditional case. The consequences of this sensitive dependence on the initial condition are discussed in detail.