On the solutions of a boundary value problem arising in free convection with prescribed heat flux

TL;DR

This paper investigates a third-order nonlinear differential equation related to free convection boundary problems, establishing conditions for solution existence, uniqueness, and asymptotic behavior, with implications for fluid mechanics and boundary layer theory.

Contribution

It proves the existence of a critical parameter value ensuring solutions decay at infinity, addressing key questions in boundary value problems in fluid mechanics.

Findings

Existence of a threshold parameter for solution existence.

Solutions tend to zero as time approaches infinity.

Results apply to boundary layer theory in fluid mechanics.

Abstract

For given $a\in\R$, c<0, we are concerned with the solution $f^{}_b$ of the differential equation $f^{\prime\prime\prime}+ff^{\prime\prime}+\g(f^{\prime})=0$, satisfying the initial conditions $f(0)=a$, $f'(0)=b$, $f''(0)=c< 0$, where g is some nonnegative subquadratic locally Lipschitz function. It is proven that there exists $b_*>0$ such that $f^{}_b$ exists on $[0,+\infty)$ and is such that $f'_b(t)\to 0$ as $t\to+\infty$, if and only if $b\geq b_*$. This allows to answer questions about existence, uniqueness and boundedness of solutions to a boundary value problem arising in fluid mechanics, and especially in boundary layer theory.