The analytical solution of a two-equation model presented in an earlier study is examined. Heat transfer characterization is classified into two regimes which are dominated by fluid conduction or solid conduction and interstitial heat exchange, respectively by using the entransy dissipation concept. The computed pattern of variation of thermal resistance with shape factor S at a fixed Brinkman number for a low ratio of the fluid to solid effective thermal conductivities implies the occurrence of temperature gradient bifurcation as S decreases. Therefore, the thermal diffusion term in the fluid phase in the two-equation model is not negligible for both regimes. (C) 2012 Elsevier Ltd. All rights reserved.