Non-similar solutions of the boundary layers equations with favorable and adverse pressure gradients, isothermal wall and slip boundary conditions: Application to Falkner–Skan gaseous flow

Abstract

Abstract This article presents the non-similar solutions of the steady incompressible two-dimensional boundary layers equations of momentum and energy for Falkner–Skan gaseous flow with favorable and adverse pressure gradient under slip boundary conditions at a relatively low Mach number. By considering the slip boundary conditions at the fluid–wallinterface for the velocity and temperature, the non-dimensional boundary layer equations are solved numerically by using the centered implicit finite difference Keller Box methods. The results of the effects of different dimensionless parameters involved in the problem, namely the modified boundary layer Knudsen number, i.e., the slip parameter K and the shape parameter β on the local hydrodynamic and heat transfer characteristics of the flow are investigated and discussed. The results of these characteristics obtained from non-similar solutions are compared to those obtained from local similarity approach often used by several authors in the last decades. We show that for small shape parameter β and moderate or large values of the slip parameter, the self-similarity of the Falkner–Skan flows is generally lost because of slip boundary conditions. Therefore, the use of the local similarity approach may produce substantial errors for hydrodynamic and thermal characteristics of the flow as compared to non-similar solution. But in contrast, when β becomes larger than 2/3 up to 1, the flow regains its self-similarity character, in this case, the local similarity approach remains valid and may be used. While for β > 1 up to 2, the solutions of the boundary layer equations again become non-similar. Furthermore, a special attention was made to evaluate the separation point under rarefaction, the results show that the effect of slip parameter leads to decrease the separation point β s and it is concluded that separation of the boundary layer should be delayed in the slip flow regime.