Effect of viscous dissipation on the Darcy free convection boundary-layer flow over a vertical plate with exponential temperature distribution in a porous medium

The title problem is investigated for an upward projecting hot plate (“upflow”) and for its downward projecting cold counterpart (“downflow”). When viscous dissipation is negligible, these two cases are physically equivalent, but the heat released by viscous friction breaks the equivalence between the upflow and downflow cases, and substantial differences occur. In particular, we find that, for self-similar flows, downflow is possible for all nonnegative values of the temperature exponent, but upflow only exists above a critical value of this parameter, which equals the half of the Gebhart number of the fluid. Each two upflow and downflow solution branches were found, respectively. All the corresponding solutions decay exponentially with increasing distance from the plate. It could be shown that these up and downflow solution branches do not represent in fact two isolated solutions but, they are the limiting cases of respective families of intermediate solutions which are bounded between the branches, and which decay algebraically in the transversal far field. This paper investigates in detail the heat transfer characteristics of all these self-similar free convection flows analytically and numerically.