Some Results About a Boundary Value Problem on Mixed Convection

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M. Boulekbache
M. Aiboudi
K. Boudjema Djeffal

Abstract

The purpose of this paper is to study the autonomous third order non linear differential equation f''' + ff'' + g(f') = 0 on [0, +∞[ with g(x) = βx(x - 1) and β > 1, subject to the boundary conditions f(0) = a ∈ R, f'(0) = b < 0 and f'(t) → λ ∈ {0, 1} as t → +∞. This problem arises when looking for similarity solutions to problems of boundary-layer theory in some contexts of fluids mechanics, as mixed convection in porous medium or flow adjacent to a stretching wall. Our goal, here is to investigate by a direct approach this boundary value problem as completely as possible, say study existence or non-existence and uniqueness solutions and the sign of this solutions according to the value of the real parameter β.

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References

  1. M. Aiboudi, I. Bensari-Khellil, B. Brighi, Similarity solutions of mixed convection boundary-layer flows in a porous medium. Differ. Equ. Appl. 9(1)(2017), 69-85.
  2. M. Aiboudi, K. Boudjema Djeffal, B. Brighi, On the convex and convex-concave solutions of opposing mixed convection boundary layer flow in a porous medium, Abstr. Appl. Anal. 2018 (2018), ID 4340204.
  3. M. Aiboudi, B. Brighi, On the solutions of a boundary value problem arising in free convection with prescribed heat flux, Arch. Math. 93 (2009), 165-174.
  4. E.H. Aly, L. Elliott, D.B. Ingham, Mixed convection boundary-layer flows over a vertical surface embedded in a porous medium. Eur. J. Mech. B, Fluids, 22 (2003), 529-543.
  5. Z. Belhachemi, B. Brighi, K. Taous, On a family of differential equations for boundary layer approximations in porous media, Eur. J. Appl. Math. 12(4) (2001), 513-528.
  6. B. Brighi, A. Fruchard, T. Sari, On the Blasius problem, Adv. Differ. Equ. 13 (5-6)(2008), 509-600.
  7. B.Brighi, J.-D. Hoernel, On general similarity boundary layer equation, Acta Math. Univ. Comenian. 77 (2008), 9-22.
  8. B. Brighi, J.-D. Hoernel, On the concave and convex solutions of mixed convection boundary layer approximation in a porous medium. Appl. Math. Lett. 19 (1) (2006), 69-74.
  9. B.Brighi, Sur un probleme aux limites associé a` l'équation différentielle f''' + f f'' + 2f'' = 0, Ann. Sci. Math. Québec, 33 (1) (2012), 355-391.
  10. B. Brighi, The equation f''' + ff'' + g(f' ) = 0 and the associated boundary value problems, Results Math. 61 (3-4) (2012), 355-391.
  11. M. Guedda, Multiple solutions of mixed convection boundary-layer approximations in a porous medium. Appl. Math. Lett. 19 (1) (2006), 63-68.
  12. M. Guedda, Nonuniqueness of solutions to differential equations for boundary layer approximations in porous media, C. R. Mecanique, 330 (2002), 279-283.
  13. S. P. Hastings and W. C. Troy, Oscillating solutions of the Falkner-Skan equations for positive β, J. Differ. Equ. 71(1) (1988), 123-144.
  14. G.C. Yang, An extension result of the opposing mixed convection problem arising in boundary layer theory. Appl. Math. Lett. 38 (1) (2014), 180-185.
  15. G.C.Yang, An upper bound on the critical value β * involved in the Blasius problem, J. Inequal. Appl. 2010 (2010), ID 960365.
  16. G.C. Yang, L. Zhang, L.F.Dang, Existence and nonexistence of solutions on opposing mixed convection problems in boundary layer theory. Eur. J. Mech. B, Fluids, 43 (2014), 148-153.