On a general similarity boundary layer equation

TL;DR

This paper investigates a generalized boundary layer differential equation, establishing conditions for the existence, uniqueness, and asymptotic behavior of solutions relevant to various fluid flow and convection problems.

Contribution

It extends the analysis of boundary layer equations to a broader class including the Falkner-Skan case, providing new existence, uniqueness, and asymptotic results.

Findings

Existence of concave and convex solutions under certain conditions

Uniqueness of solutions in specified cases

Results on nonexistence and asymptotic behavior

Abstract

In this paper we are concerned with the solutions of the differential equation $f'''+ff''+g(f')=0$ on $[0,\infty)$, satisfying the boundary conditions $f(0)=\alpha$, $f'(0)=\beta\geq 0$, $f'(\infty)=\l$, and where $g$ is some given continuous function. This general boundary value problem includes the Falkner-Skan case, and can be applied, for example, to free or mixed convection in porous medium, or flow adjacent to stretching walls in the context of boundary layer approximation. Under some assumptions on the function $g$, we prove existence and uniqueness of a concave or a convex solution. We also give some results about nonexistence and asymptotic behaviour of the solution.