Poiseuille flow with viscous dissipation in a plane channel: a consistent analytical solution for the thermal entrance region

Poiseuille flow forced convection in a plane-parallel channel is considered in this paper. Taking into account that the traditional assumption 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 entrance condition is not “prescribed”, but it is obtained in a consequent way as the fully developed temperature profile of the plane Poiseuille flow under isothermal boundary conditions. Compared with the case of uniform entrance condition, both for the dependence of the local Nusselt number on the Brinkman number as well as for the developing temperature field, substantial differences have been found. The paper discusses the consequences of this sensitive dependence on the entrance condition in detail. The local energy balance equation is solved analytically by employing the method of separation of variables and the solution of the corresponding eigenvalue problem is given in terms of the confluent hypergeometric functions.