Analysis and Mathematical Physics

, Volume 7, Issue 4, pp 417–435 | Cite as

Global regularity for MHD Sisko fluid in annular pipe

Article

Abstract

The flow of Sisko fluid in an annular pipe is considered. The governing nonlinear equation of an incompressible Sisko fluid is modelled. The purpose of present paper is to obtain the global classical solutions for unsteady flow of magnetohydrodynamic Sisko fluid in terms of the bounded mean oscillations norm. Uniqueness of solution is also verified.

Keywords

Nonlinear problem Sisko fluid Global classical solutions MHD Sisko fluid 

References

  1. 1.
    Farooq, U., Hayat, T., Alsaedi, A., Liao, S.: Heat and mass transfer of two-layer flows of third-grade nano-fluids in a vertical channel. Appl. Math. Comput. 242, 528–540 (2014)MathSciNetMATHGoogle Scholar
  2. 2.
    Baoku, I.G., Olajuwon, B.I., Mustapha, A.O.: Heat and mass transfer on a MHD third grade fluid with partial slip flow past an infinite vertical insulated porous plate in a porous medium. Int. J. Heat Fluid Flow 40, 81–88 (2013)CrossRefGoogle Scholar
  3. 3.
    Hayat, T., Shafiq, A., Alsaedi, A., Awais, M.: MHD axisymmetric flow of third grade fluid between stretching sheets with heat transfer. Comput. Fluids 86, 103–108 (2013)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Aziz, T., Mahmood, F.M., Ayub, M., Mason, D.P.: Non-linear time-dependent flow models of third grade fluids: a conditional symmetry approach. Int. J. Nonlinear Mech. 54, 55–65 (2013)CrossRefGoogle Scholar
  5. 5.
    Okoya, S.S.: Disappearance of criticality for reactive third-grade fluid with Reynold’s model viscosity in a flat channel. Int. J. Nonlinear Mech. 46, 1110–1115 (2011)CrossRefGoogle Scholar
  6. 6.
    Makinde, O.D., Chinyoka, T.: Numerical study of unsteady hydromagnetic Generalized Couette flow of a reactive third-grade fluid with asymmetric convective cooling. Comput. Math. Appl. 61, 1167–1169 (2011)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Hatami, M., Hatami, J., Ganji, D.D.: Computer simulation of MHD blood conveying gold nanoparticles as a third grade non-Newtonian nanofluid in a hollow porous vessel. Comput. Methods Programs Biomed. 113, 632–641 (2014)CrossRefGoogle Scholar
  8. 8.
    Zhaoa, C., Lianga, Y., Zhaob, M.: Upper and lower bounds of time decay rate of solutions to a class of incompressible third grade fluid eqs. Nonlinear Anal. Real World Appl. 15, 229–238 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Danish, M., Kumar, S., Kumar, S.: Exact analytical solutions for the Poiseuille and Couette-Poiseuille flow of third grade fluid between parallel plates. Commun. Nonlinear Sci. Numer. Simul. 17, 1089–1097 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Hayat, T., Naza, R., Asghar, S., Mesloub, S.: Soret-Dufour effects on three-dimensional flow of third grade fluid. Nuclear Eng. Design 243, 1–14 (2012)CrossRefGoogle Scholar
  11. 11.
    Hayat, T., Moitsheki, R.J., Abelman, S.: Stokes first problem for Sisko fluid over a porous wall. Appl. Math. Comp. 217, 622–628 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Molati, M., Hayat, T., Mahomed, F.: Rayleigh problem for a MHD Sisko fluid. Nonlinear Anal. Real World Appl. 10, 3428–3434 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Mekheimer, KhS, Kot, M.A.El: Mathematical modelling of unsteady flow of a Sisko fluid through an anisotropically tapered elastic arteries with time-variant over lapping stenosis. Appl. Math. Model. 36, 5393–5407 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Wang, Y., Hayat, T., Ali, N., Oberlack, M.: Magnetohydrodynamic peristaltic motion of a Sisko fluid in a symmetric or asymmetric channel. Phys. A. 387, 347–362 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Khan, M., Shahzad, A.: On axisymmetric flow of a Sisko fluid over a radially stretching sheet. Int. J. Nonlinear Mech. 47, 999–1007 (2012)CrossRefGoogle Scholar
  16. 16.
    Zhou, Y., Fan, J.: Logarithmically improved regularity criteria for the 3D viscous MHD equations. Forum Math. 24, 691–708 (2012)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Jia, X., Zhou, Y.: Regularity criteria for the 3D MHD equations involving partial components. Nonlinear Anal. Real World Appl. 13, 410–418 (2012)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Rahman, S.: Regularity criterion for 3D MHD fluid passing through the porous medium in terms of gradient pressure. J. Comput. Appl. Math. 270, 88–99 (2014)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Kusaka, K.: Classical solvability of a stationary free boundary problem for an incompressible viscous fluid describing the process of interface formation. Anal. Math. Phys. 5, 67–86 (2015)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Zhou, Y.: Remarks on regularities for the 3D MHD equations. Discrete Contin. Dyn. Syst. 12, 881–886 (2005)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Zhou, Y.: Regularity criteria for the generalized viscous MHD equations. Ann. Inst. H. Poincare Anal. Non Lineaire. 24, 491–505 (2007)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Zhou, Y., Gala, S.: Regularity criteria for the solutions to the 3D MHD equations in the multiplier space. Z. Angew. Math. Phys. 61, 193–199 (2010)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Jia, X., Zhou, Y.: Regularity criteria for the 3D MHD equations via partial derivatives. Kinet. Relat. Models 5, 505–516 (2012)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Jia, X., Zhou, Y.: Regularity criteria for the 3D MHD equations via partial derivatives. II. Kinet. Relat. Models 7, 291–304 (2014)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Fan, J., Malaikah, H., Monaquel, S., Nakamura, G., Zhou, Y.: Global Cauchy problem of 2D generalized MHD equations. Monatshefte Math. 175, 127–131 (2014)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Fan, J., Jia, X., Nakamura, G., Zhou, Y.: On well-posedness and blowup criteria for the magnetohydrodynamics with the hall and ion-slip effects. Z. Angew. Math. Phys. 66, 1695–1706 (2015)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Jia, X., Zhou, Y.: Ladyzhenskaya-Prodi-Serrin type regularity criteria for the 3D incompressible MHD equations in terms of 3\(\times \)3 mixture matrices. Nonlinearity 28, 3289–3307 (2015)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Jia, X., Zhou, Y.: On regularity criteria for the 3D incompressible MHD equations involving one velocity component. J. Math. Fluid Mech. 18, 187–206 (2016)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Jiang, Zaihong, Wang, Yanan, Zhou, Yong: On regularity criteria for the 2D generalized MHD system. J. Math. Fluid Mech. 18, 331–341 (2016)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Solonnikov, V.A.: Estimates of the solutions of the nonstationary Navier-Stokes system, in ”Boundary Value Problems of Mathematical Physics and Related Questions in the Theory of Functions, 7,”. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 38, 153–231 (1973)Google Scholar
  31. 31.
    Fefferman, C., Stein, E.M.: \(H^{p}\) spaces of several variables. Acta Math. 129, 137–193 (1972)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of MathematicsCOMSATS Institute of Information TechnologyAbbottabadPakistan
  2. 2.Department of MathematicsQuaid-I-Azam UniversityIslamabadPakistan
  3. 3.Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia

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