On a Similarity Boundary Layer Equation

Abstract

The purpose of this paper is to study the autonomous third order nonlinear differential equation $f'''+ \frac{m+1}{2} f f'' – mf'^2 = 0$ on $(0, \infty)$, subject to the boundary conditions $f(0) = a \in \mathbb R, f'(0) = 1$ and $f'(t) \rightarrow 0$ as $t \rightarrow \infty$. 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 $a \neq 0$ in the boundary condition $f(0) = a$.