Similarity solutions of differential equations for boundary layer approximations in porous media

Abstract.

This paper is concerned with the ordinary differential equation

$${{{f}\ifmmode{'}\else$'$\fi}\ifmmode{'}\else$'$\fi}\ifmmode{'}\else$'$\fi + mf\,{{f}\ifmmode{'}\else$'$\fi}\ifmmode{'}\else$'$\fi - \alpha {f}\ifmmode{'}\else$'$\fi^{2} = 0,$$

on (0, + ∞), subject to the boundary conditions

$$f(0) = a,\quad {f}\ifmmode{'}\else$'$\fi(0) = b,\quad {f}\ifmmode{'}\else$'$\fi(\infty ) = {\mathop {\lim }\limits_{t \to \infty } }{f}\ifmmode{'}\else$'$\fi(t) = 0,$$

in which a and b are reals, m > 0 and α < 0. Such problem, with $m = \frac{{\alpha + 1}}{2},\;a = 0\;{\text{ and }}\;b = 1,$ arises in the study of the free convection, along a vertical flat plate embedded in a porous medium.

The analysis deals with existence, non–uniqueness and large–t behaviour of solutions of the above problem under favourable conditions on m, α, a and b.