On a Similarity Boundary Layer Equation

Abstract

The purpose of this paper is to study the autonomous third order nonlinear differential equation $f^{\prime \prime \prime }+\frac{m+1}{2}f{f}^{\prime \prime }–m{f}^{\prime 2}=0$ on $(0,\infty )$, subject to the boundary conditions $f(0)=a\in \mathbb{R},f^{\prime }(0)=1$ and $f^{\prime }(t)\to 0$ as $t\to \infty$. This problem arises when looking for similarity solutions to problems of boundary-layer theory in some contexts of fluids mechanics, as free convection in porous medium or flow adjacent to a stretching wall. Our goal here is to investigate by a direct approach this boundary value problem as completely as possible, say studying existence or non-existence and uniqueness or non-uniqueness of solutions according to the values of the real parameter m. In particular, we will emphasize similarities and differences between the cases $a=0$ and $a\ne 0$ in the boundary condition $f(0)=a$.