Similarity solutions of differential equations for boundary layer approximations in porous media. (English) Zbl 1084.34034
Summary: This paper is concerned with the ordinary differential equation
\[
f''' + mff'' - \alpha {f'}^2 = 0
\]
on \((0,+\infty)\), subject to the boundary conditions
\[
f(0) = a, \quad f'(0) = b, \quad f'(\infty) = \lim_{t\to\infty} f'(t) = 0,
\]
in wich \(a\) and \(b\) are reals, \(m > 0\) and \(\alpha < 0\). Such problem, with \(m = \frac{\alpha+1}2\), \(a = 0\), 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, nonuniqueness and large-\(t\) behaviour of solutions of the above problem under favourable conditions on \(m,\alpha,a\) and \(b\).
The analysis deals with existence, nonuniqueness and large-\(t\) behaviour of solutions of the above problem under favourable conditions on \(m,\alpha,a\) and \(b\).
MSC:
34B40 | Boundary value problems on infinite intervals for ordinary differential equations |
34B15 | Nonlinear boundary value problems for ordinary differential equations |
34B60 | Applications of boundary value problems involving ordinary differential equations |
76D10 | Boundary-layer theory, separation and reattachment, higher-order effects |