New classes of similar solutions for laminar free convection problems

As shown by the senior author, the proper formulation of free convection boundary-layer theory depends on order of magnitude of the Eckert number defined as Ec = Hg/cpΔT, the conventional theory being valid in the limit Ec → 0. The present paper investigates the solutions of the laminar on flat plate problem, over the entire Ec-range, for the case in which similarity prevails. It is shown that for Ec ≡ O(1) the similar solutions are attainable for linearly varying wall temperature (in particular constant) whereas in the limit for Ec → ∞ any wall temperature distribution leads to similar solutions. Similar profiles for Ec ≡ O(1) depend on the Prandtl number and on the ratio (Ec/β′) where β′ is the constant wall temperature gradient. Similar profiles for Ec → ∞ are universal insofar as they do not depend on any parameter. Universal profiles are given in closed form. Numerical solutions for Pr = 0.72 and several values of (Ec/β′) are presented and analysed in terms of velocity and temperature profiles, wall shear stress and Nusselt number. In particular the paper shows that the results of conventional theory cannot be used for β′ smaller than (0.05-0.