On the Convex and Convex-Concave Solutions of Opposing Mixed Convection Boundary Layer Flow in a Porous Medium

Abstract

In this paper, we are concerned with the solution of the third-order nonlinear differential equation $f^{″\mathrm{\prime }}+f{f}^{″}+\beta f^{\mathrm{\prime }}(f^{\mathrm{\prime }}-\mathrm{1})=\mathrm{0}$, satisfying the boundary conditions $f(\mathrm{0})=a\in \mathbb{R}$, , and $f^{\mathrm{\prime }}(t)\to \lambda$, as $t\to +\mathrm{\infty }$, where $\lambda \in \{\mathrm{0,1}\}$ and The problem arises in the study of the opposing mixed convection approximation in a porous medium. We prove the existence, nonexistence, and the sign of convex and convex-concave solutions of the problem above according to the mixed convection parameter and the temperature parameter .