Nonuniqueness of solutions to differential equations for boundary-layer approximations in porous media

Mohammed Guedda

doi:10.1016/S1631-0721(02)01458-4

Nonuniqueness of solutions to differential equations for boundary-layer approximations in porous media
[Non unicité de solutions des équations différentielles pour un problème de couches limites en milieux poreux]

Mohammed Guedda ¹

¹ LAMFA, CNRS UMR 6140, Université de Picardie Jules Verne, Faculté de mathématiques et d'informatique, 33, rue Saint-Leu 80039 Amiens, France

Comptes Rendus. Mécanique, Volume 330 (2002) no. 4, pp. 279-283.

Résumé
Abstract

La modélisation d'un phénomène de convection naturelle dans un milieu poreux, occupant un domaine non borné, nous conduit à l'équation différentielle $f ‴ + \frac{α + 1}{2} ff ″ - α f'^{2} = 0$ , dans (0,+∞). Nous montrons que, pour $α \in (- \frac{1}{3}, 0),$ cette équation avec les conditions initiales $f (0) = a \in ℝ, f' (0) = 0$ ou 1, admet une infinité de solutions globales, dont les dérivées d'ordre un et deux convergent vers 0 à l'infini.

The free convection, along a vertical flat plate embedded in a porous medium, can be described in terms of solutions to $f ‴ + \frac{α + 1}{2} ff ″ - α f'^{2} = 0,$ for all t∈(0,+∞). The purpose of this Note is to study the nonuniqueness of solutions to this problem, with the initial conditions, $f (0) = a \in ℝ$ and f′(0)∈{0,1}, where $α \in (- \frac{1}{3}, 0) .$ No assumption at infinity is imposed. We show that this problem has an infinite number of unbounded global solutions. Moreover, we prove that the first and the second derivative of solutions tend to 0 as t approaches infinity.

Reçu le : 2001-10-22
Accepté le : 2002-03-04
Publié le : 2002-01-01

DOI : 10.1016/S1631-0721(02)01458-4

Keywords: porous media, boundary layer, existence and nonuniqueness
Mot clés : milieu poreux, couche limite, existence et non unicité

Affiliations des auteurs :

Mohammed Guedda ¹

¹ LAMFA, CNRS UMR 6140, Université de Picardie Jules Verne, Faculté de mathématiques et d'informatique, 33, rue Saint-Leu 80039 Amiens, France

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Mohammed Guedda. Nonuniqueness of solutions to differential equations for boundary-layer approximations in porous media. Comptes Rendus. Mécanique, Volume 330 (2002) no. 4, pp. 279-283. doi : 10.1016/S1631-0721(02)01458-4. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/S1631-0721(02)01458-4/

Bibliographie
Cité par

[1] P. Cheng; W.J. Minkowycz Free convection about a vertical flat plate embedded in a porous medium with application to heat transfer from a dike, J. Geophys. Res., Volume 82 (1977) no. 14, pp. 2040-2044

[2] D.B. Ingham; S.N. Brown Flow past a suddenly heated vertical plate in a porous medium, J. Proc. Roy. Soc. London Ser. A, Volume 403 (1986), pp. 51-80

[3] Z. Belhachmi; B. Brighi; K. Taous On a family of differential equation for boundary laer approximations in porous media, European J. Appl. Math., Volume 12 (2001), pp. 513-528

[4] M.A. Chaudhary; J.H. Merkin; I. Pop Similarity solutions in free convection boundary-layer flows adjacent to vertical permeable surfaces in porous media. I. Prescribed surface temperature, European J. Mech. B Fluids, Volume 14 (1995) no. 2, pp. 217-237

[5] E. Magyari; B. Keller Exact solutions for self-similar boundary-layer flows induced by permeable stretching walls, European J. Mech. B Fluids, Volume 19 (2000) no. 1, pp. 109-122

[6] W.H.H. Banks; M.B. Zaturska Eigensolutions in boundary-lary flow adjacent to a stretching wall, IMA J. Appl. Math., Volume 36 (1986), pp. 375-392

[7] W.H.H. Banks Similarity solutions of the boundary layer equations for a stretching wall, J. Méc. Théorie Appl., Volume 2 (1983), pp. 375-392

[8] Z. Belhachmi; B. Brighi; K. Taous Solutions similaires pour un problème de couches limites en milieux poreux, C. R. Acad. Paris, Série IIb, Volume 328 (2000), pp. 407-410

[9] B. Brighi, Private communication

[10] W.A. Coppel On a differential equation of boundary layer theory, Philos. Trans. Roy. Soc. London Ser. A, Volume 253 (1960), pp. 101-136

[11] M. Guedda, Similarity solutions to a boundary-layer problem in porous media (in preparation)

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