Link to this article: http://journals.cambridge.org/abstract_S0956792501004582How to cite this article: ZAKARIA BELHACHMI, BERNARD BRIGHI and KHALID TAOUS (2001). On a family of differential equations for boundary layer approximations in porous media.Free convection along a vertical flat plate embedded in a porous medium is considered, within the framework of boundary layer approximations. In some cases, similarity solutions can be obtained by solving a boundary value problem involving an autonomous third-order nonlinear equation, depending on a parameter related to the temperature on the wall. The paper deals with existence and uniqueness questions to this problem, for every value of the parameter.

In this paper we are concerned with the solutions of the differential equationand 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.

We study the concave and convex solutions of the third order similarity differential equation f + ff + g(f ) = 0, and especially the ones that satisfies the boundary conditions f (0) = a, f (0) = b and f (t) → λ as t → +∞, where λ is a root of the function g. According to the sign of g between b and λ, we obtain results about existence, uniqueness and boundedness of solutions to this boundary value problem, that we denote by (P g;a,b,λ ). In this way, we pursue and complete the study done in 2008.Mathematics Subject Classification (2010). 34B15, 34C11, 76D10.

The goal of this paper is to introduce the approximated convex envelope of a function and to estimate how it differs from its convex envelope. Such a problem arises in various physical situations where the function considered is some energy that has to be minimized.This study is a first step toward understanding how to approximate the quasi-convex envelope of a function. The importance of this issue is due to the various applications that are encountered, in particular, in the field of material science. SIAM J. NUMER. ANAL. Vol. 31, No. 1, pp. 128-148, February 1994 ( Abstract. The goal of this paper is to introduce the approximated convex envelope of a function and to estimate how it differs from its convex envelope. Such a problem arises in various physical situations where the function considered is some energy that has to be minimized.This study is a first step toward understanding how to approximate the quasi-convex envelope of a function. The importance of this issue is due to the various applications that are encountered, in particular, in the field of material science.