Advertisement

Applied Mathematics and Mechanics

, Volume 33, Issue 6, pp 749–764 | Cite as

MHD axisymmetric flow of third grade fluid between porous disks with heat transfer

  • T. Hayat
  • A. Shafiq
  • M. NawazEmail author
  • A. Alsaedi
Article

Abstract

The magnetohydrodynamic (MHD) flow of the third grade fluid between two permeable disks with heat transfer is investigated. The governing partial differential equations are converted into the ordinary differential equations by suitable transformations. The transformed equations are solved by the homotopy analysis method (HAM). The expressions for square residual errors are defined, and the optimal values of convergence-control parameters are selected. The dimensionless velocity and temperature fields are examined for various dimensionless parameters. The skin friction coefficient and the Nusselt number are tabulated to analyze the effects of dimensionless parameters.

Key words

heat transfer axisymmetric flow third grade fluid porous disk skin friction coefficient Nusselt number 

Chinese Library Classification

O357.1 O361.3 

2010 Mathematics Subject Classification

80A20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Fetecau, C., Mahmood, A., and Jamil, M. Exact solutions for the flow of a viscoelastic fluid induced by a circular cylinder subject to a time dependent shear stress. Communications in Nonlinear Science and Numerical Simulation, 15(12), 3931–3938 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    Jamil, M., Fetecau, C., and Imran, M. Unsteady helical flows of Oldroyd-B fluids. Communications in Nonlinear Science and Numerical Simulation, 16(3), 1378–1386 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    Jamil, M., Rauf, A., Fetecau, C., and Khan, N. A. Helical flows of second grade fluid to constantly accelerated shear stresses. Communications in Nonlinear Science and Numerical Simulation, 16(4), 1959–1969 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    Tan, W. C. and Masuoka, T. Stability analysis of a Maxwell fluid in a porous medium heated from below. Physics Letters A, 360(3), 454–460 (2007)CrossRefGoogle Scholar
  5. [5]
    Tan, W. C. and Masuoka, T. Stokes first problem for an Oldroyd-B fluid in a porous half space. Physics of Fluids, 17(2), 023101–023107 (2005)MathSciNetCrossRefGoogle Scholar
  6. [6]
    Sajid, M. and Hayat, T. Non-simliar series solution for boundary layer flow of a third-order fluid over a stretching sheet. Applied Mathematics and Computation, 189, 1576–1585 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    Sajid, M., Hayat, T., and Asghar, S. Non-similar analytic solution for MHD flow and heat transfer in a third order fluid over a stretching sheet. International Journal of Heat and Mass Transfer, 50(9–10), 1723–1736 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    Hayat, T., Mustafa, M., and Asghar, S. Unsteady flow with heat and mass transfer of a third grade fluid over a stretching surface in the presence of chemical reaction. Nonlinear Analysis: Real World Applications, 11(4), 3186–3197 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    Abbasbandy, S. and Hayat, T. On series solution for unsteady boundary layer equations in a special third grade fluid. Communications in Nonlinear Science and Numerical Simulation, 16(8), 3140–3146 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    Sahoo, B. Heimenz flow and heat transfer of a third grade fluid. Communications in Nonlinear Science and Numerical Simulation, 14(3), 811–826 (2009)CrossRefGoogle Scholar
  11. [11]
    Sahoo, B. and Do, Y. Effects of slip on sheet driven flow and heat transfer of a third-grade fluid past a stretching sheet. International Communication in Heat and Mass Transfer, 37(8), 1064–1071 (2010)CrossRefGoogle Scholar
  12. [12]
    Von Karman, T. Uber laminare und turbulente Reibung. Zeitschrift für Angewandte Mathematik und Mechanik (ZAMM), 1(4), 233–2555 (1921)zbMATHCrossRefGoogle Scholar
  13. [13]
    Cochran, W. G. The flow due to a rotating disk. Mathematical Proceedings of the Cambridge Philosophical Society, 30(3), 365–375 (1934)zbMATHCrossRefGoogle Scholar
  14. [14]
    Benton, E. R. On the flow due to a rotating disk. Journal of Fluid Mechanics, 24(781–800) 133–137 (1966)Google Scholar
  15. [15]
    Takhar, H. S., Chamkha, A. J., and Nath, G. Unsteady mixed convection flow from a rotating verticle cone with a magnetic field. Heat and Mass Transfer, 39(4), 297–304 (2003)Google Scholar
  16. [16]
    Maleque, K. A. and Sattar, M. A. The effects of variable properties and Hall current on steady MHD laminar convective fluid flow due to a porous rotating disk. International Journal of Heat and Mass Transfer, 48(23-24), 4963–4972 (2005)zbMATHCrossRefGoogle Scholar
  17. [17]
    Stuart, J. T. On the effect of uniform suction on the steady flow due to a rotating disk. The Quarterly Journal of Mechanics and Applied Mathematics, 7(4), 446–457 (1954)MathSciNetCrossRefGoogle Scholar
  18. [18]
    Sparrow, E. M., Beavers, G. S., and Hung, L. Y. Flow about a porous-surfaced rotating disk. International Journal of Heat and Mass Transfer, 14(7), 993–996 (1971)CrossRefGoogle Scholar
  19. [19]
    Miklavcic, M. and Wang, C. Y. The flow due to a rough rotating disk. Zeitschrift für Angewandte Mathematik und Physik (ZAMP), 55(2), 235–2466 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    Hayat, T. and Nawaz, M. Unsteady stagnation point flow of viscous fluid caused by an impulsively rotating disk. Journal of the Taiwan Institute of Chemical Engineers, 42(1), 41–49 (2011)CrossRefGoogle Scholar
  21. [21]
    Liao, S. J. Beyond Perturbation: Introduction to Homotopy Analysis Method, Chapman and Hall, CRC Press, Florida (2003)CrossRefGoogle Scholar
  22. [22]
    Liao, S. J. On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet. Journal of Fluid Mechanics, 488, 189–212 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  23. [23]
    Liao, S. J. On the homotopy analysis method for nonlinear problems. Applied Mathematics and Computation, 147, 499–513 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  24. [24]
    Liao, S. J. Notes on the homotopy analysis method: some definitions and theorems. Communications in Nonlinear Science and Numerical Simulation, 14(4), 983–997 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    Liao, S. J. An optimal homotopy analysis approach for strongly nonlinear differential equations. Communications in Nonlinear Science and Numerical Simulation, 15(8), 2003–2016 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    Rashidi, M. M., Domairry, G., and Dinarvand, S. Approximate solutions for the Burger and regularized long wave equations by means of the homotopy analysis method. Communications in Nonlinear Science and Numerical Simulation, 14(3), 708–717 (2009)CrossRefGoogle Scholar
  27. [27]
    Abbasbandy, S. and Shirzadi, A. A new application of the homotopy analysis method: solving the Sturm-Liouville problems. Communications in Nonlinear Science and Numerical Simulation, 16(1), 112–126 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  28. [28]
    Bataineh, A. S., Noorani, M. S. M., and Hashim, I. Approximate analytical solutions of systems of PDEs by homotopy analysis method. Computer and Mathematics with Applications, 55(12), 2913–2923 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    Hashim, I., Abdulaziz, O., and Momani, S. Homotopy analysis method for fractional IVPs. Communications in Nonlinear Science and Numerical Simulation, 14(3), 674–684 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  30. [30]
    Hayat, T., Nawaz, M., and Obaidat, S. Axisymmetric magnetohydrodynamic flow of micropolar fluid between unsteady stretching surfaces. Applied Mathematics and Mechanics (English Edition), 32(3), 361–374 (2011) DOI 10.1007/s10483-011-1421-8MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    Fosdick, R. L. and Rajagopal, K. R. Thermodynamics and stability of fluids of third grade. Proceedings of the Royal Society A, 369(1738), 351–377 (1980)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsQuaid-i-Azam UniversityIslamabadPakistan
  2. 2.Department of Humanities and SciencesInstitute of Space TechnologyIslamabadPakistan
  3. 3.Department of Mathematics, Faculty of ScienceKing Abdulaziz UniversityJeddahSaudi Arabia

Personalised recommendations