On the Asymptotic Behavior of an Interface Problem in a Thin Domain

  • H. BenseridiEmail author
  • Y. Letoufa
  • M. Dilmi
Research Article


Considering a mathematical model for the bilateral, frictionless contact between two Bingham fluids, establish a variational formulation for the problem and prove estimates on the velocity and pressure which are independent of the small parameter. The passage to the limit on \(\varepsilon \) permits us to obtain the existence and uniqueness of the velocity. A specific Reynolds equation associated with variational inequalities is obtained.


Asymptotic behavior Bingham fluid Non-Newtonian fluid Reynolds equation Transmission conditions Tresca law 

Mathematics Subject Classification

35R35 76F10 78M35 



  1. 1.
    Dilmi M, Benseridi H, Saadallah A (2014) Asymptotic analysis of a Bingham fluid in a thin domain with Fourier and Tresca boundary conditions. Adv Appl Math Mech 6:797–810MathSciNetCrossRefGoogle Scholar
  2. 2.
    Assemien A, Bayada G, Chambat M (1994) Inertial effects in the asymptotic behavior of a thin film flow. Asymptot Anal 9(3):177–208MathSciNetzbMATHGoogle Scholar
  3. 3.
    Dong BQ, Chen ZM (2006) Asymptotic stability of non-Newtonian flows with large perturbation in \({\mathbb{R}}^{2}\). Appl Math Comput 173(1):243–250MathSciNetGoogle Scholar
  4. 4.
    Boukrouche M, El mir R (2004) Asymptotic analysis of non-Newtonian fluid in a thin domain with Tresca law. Nonlinear Anal Theory Methods Appl 59(1–2):85–105MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bayada G, Boukrouche M (2003) On a free boundary problem for Reynolds equation derived from the Stokes system with Tresca boundary conditions. J Math Anal Appl 282(1):212–231MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Benseridi H, Dilmi M, Saadallah A (2018) Asymptotic behaviour of a nonlinear boundary value problem with friction. Proc Natl Acad Sci India Sect A Phys Sci 88(1):55–63MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Saadallah A, Benseridi H, Dilmi M, Drabla S (2016) Estimates for the asymptotic convergence of a non-isothermal linear elasticity with friction. Georgian Math J 23(3):435–446MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hemici N, Matei A (2003) A frictionless contact problem with adhesion between two elastic bodies. Ann Univ Craiova Math Comput Sci Ser 30(2):90–99MathSciNetzbMATHGoogle Scholar
  9. 9.
    Lemrabet K (1978) An interface problem in a domain of \({\mathbb{R}}^{3}\). J Math Anal Appl 63(3):549–562MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Aksentian OK (1967) Singularitties of the stress-strain state of a plate in the neighborhood of edge. PMM 31(1):178–186Google Scholar
  11. 11.
    Nicaise S, Sändig AM (1994) General interface problems, II. Math Methods Appl Sci 17(6):431–450MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Tabacman ED, Tarzia DA (1989) Sufficient and/or necessary condition for the heat transfer coefficient on \(\Gamma _{1}\) and the heat flux on \(\Gamma _{2}\) to obtain a steady-state two-phase Stefan problem. J Differ Equ 77(1):16–37ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Temam R, Ekeland I (1974) Analyse convexe et problèmes variationnels. Dunod, Gauthier-Villars, PariszbMATHGoogle Scholar
  14. 14.
    Boukrouche M, Łukaszewicz G (2004) On a lubrication problem with Fourier and Tresca boundary conditions. Math Models Methods Appl Sci 14(6):913–941MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The National Academy of Sciences, India 2019

Authors and Affiliations

  1. 1.Applied Mathematics LaboratorySetif 1 UniversitySétifAlgeria
  2. 2.Applied Mathematics LaboratoryEl Oued UniversityEl OuedAlgeria

Personalised recommendations