A study on the unsteady flow of two immiscible micropolar and Newtonian fluids through a horizontal channel: A numerical approach

  • M. Devakar
  • Ankush Raje
Regular Article


The unsteady flow of two immiscible micropolar and Newtonian fluids through a horizontal channel is considered. In addition to the classical no-slip and hyper-stick conditions at the boundary, it is assumed that the fluid velocities and shear stresses are continuous across the fluid-fluid interface. Three cases for the applied pressure gradient are considered to study the problem: one with constant pressure gradient and the other two cases with time-dependent pressure gradients, viz. periodic and decaying pressure gradient. The Crank-Nicolson approach has been used to obtain numerical solutions for fluid velocity and microrotation for diverse sets of fluid parameters. The nature of fluid velocities and microrotation with various values of pressure gradient, Reynolds number, ratio of viscosities, micropolarity parameter and time is illustrated through graphs. It has been observed that micropolarity parameter and ratio of viscosities reduce the fluid velocities.


  1. 1.
    A.C. Eringen, J. Math. Mech. 16, 1 (1966)MathSciNetGoogle Scholar
  2. 2.
    A.C. Eringen, Microcontinnum Field Theories: II. Fluent Media (Springer, New York, 2001)Google Scholar
  3. 3.
    J. Peddieson Jr., Int. J. Eng. Sci. 10, 23 (1972)CrossRefGoogle Scholar
  4. 4.
    G. Bayada, N. Benhaboucha, M. Chambat, Math. Mod. Methods Appl. Sci. 15, 343 (2005)CrossRefGoogle Scholar
  5. 5.
    A. Siddangouda, Lubr. Sci. 24, 339 (2012)CrossRefGoogle Scholar
  6. 6.
    J.R. Lin, T.C. Hung, T.L. Chou, L.J. Liang, Tribol. Int. 66, 150 (2013)CrossRefGoogle Scholar
  7. 7.
    N.B. Naduvinamani, S.S. Huggi, J. Eng. Tribol. 223, 1179 (2009)Google Scholar
  8. 8.
    C.K. Kang, A.C. Eringen, B. Math. Biol. 38, 135 (1976)CrossRefGoogle Scholar
  9. 9.
    M. Devakar, T.K.V. Iyenger, Eur. Phys. J. Plus 128, 41 (2013)CrossRefGoogle Scholar
  10. 10.
    Kh.S. Mekheimer, M.A.El. Kot, Acta Mech. Sin. 24, 637 (2008)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    D. Srinivasacharya, M. Shiferaw, Arab. J. Sci. Eng. 39, 5085 (2014)MathSciNetCrossRefGoogle Scholar
  12. 12.
    L. Wang, Y. Jian, F. Li, Eur. Phys. J. Plus 131, 338 (2016)CrossRefGoogle Scholar
  13. 13.
    X. Si, L. Zheng, P. Lin, X. Zhang, Y. Zhang, Int. J. Heat Mass Transfer 67, 885 (2013)CrossRefGoogle Scholar
  14. 14.
    Z. Ziabakhsh, G. Domairry, H. Bararnia, J. Taiwan Inst. Chem. Eng. 40, 443 (2009)CrossRefGoogle Scholar
  15. 15.
    M.S. Abdel-wahed, Eur. Phys. J. Plus 132, 195 (2017)ADSCrossRefGoogle Scholar
  16. 16.
    T. Ariman, M.A. Turk, N.D. Sylvester, Int. J. Eng. Sci. 11, 905 (1973)CrossRefGoogle Scholar
  17. 17.
    T. Ariman, M.A. Turk, N.D. Sylvester, Int. J. Eng. Sci. 12, 273 (1974)CrossRefGoogle Scholar
  18. 18.
    G. Lukaszewicz, Micropolar Fluids: Theory and Application (Birkhauser, Basel, 1999)Google Scholar
  19. 19.
    C. Boodoo, B. Bhatt, D. Comissiong, Rheol. Acta 52, 579 (2013)CrossRefGoogle Scholar
  20. 20.
    R. Bird, W. Stewart, E.N. Lightfoot, Transport Phenomena, second edition (John Wiley & Sons, 2002)Google Scholar
  21. 21.
    J.N. Kapur, J.B. Shukla, Appl. Sci. Res. 13, 55 (1964)CrossRefGoogle Scholar
  22. 22.
    V. Srinivasan, K. Vafai, J. Fluids Eng. 116, 135 (1994)CrossRefGoogle Scholar
  23. 23.
    J. Prathap Kumar, J.C. Umavathi, A.J. Chamkha, I. Pop, Appl. Math. Model. 34, 1175 (2010)MathSciNetCrossRefGoogle Scholar
  24. 24.
    S.M. Zivojin, D.D. Nikodijevic, B.D. Blagojevic, S.R. Savic, Trans. Can. Soc. Mech. Eng. 34, 351 (2010)CrossRefGoogle Scholar
  25. 25.
    J. Srinivas, J.V. Ramana Murthy, J. Appl. Fluid Mech. 9, 501 (2016)CrossRefGoogle Scholar
  26. 26.
    J. Srinivas, J.V. Ramana Murthy, J. Eng. Thermophys. 25, 126 (2016)CrossRefGoogle Scholar
  27. 27.
    A. Borrelli, G. Giantesio, M.C. Patria, J. Fluids Eng. Trans. ASME 139, 101203 (2017)CrossRefGoogle Scholar
  28. 28.
    N. Kumar, S. Gupta, Meccanica 47, 277 (2012)MathSciNetCrossRefGoogle Scholar
  29. 29.
    J.C. Umavathi, J. Prathap Kumar, A.J. Chamkha, Can. J. Phys. 86, 961 (2008)ADSCrossRefGoogle Scholar
  30. 30.
    J.C. Umavathi, A.J. Chamkha, Can. J. Phys. 83, 705 (2005)ADSCrossRefGoogle Scholar
  31. 31.
    A.K. Singh, Indian J. Pure Appl. Phys. 43, 415 (2005)Google Scholar
  32. 32.
    K.S. Sai, Defence Sci. J. 40, 183 (1990)CrossRefGoogle Scholar
  33. 33.
    K. Vajravelu, P.V. Arunachalam, S. Sreenadh, J. Math. Anal. Appl. 196, 1105 (1995)MathSciNetCrossRefGoogle Scholar
  34. 34.
    J.C. Umavathi, A.J. Chamkha, A. Mateen, A. Al-Mudhaf, Heat Mass Transf. 42, 81 (2005)ADSCrossRefGoogle Scholar
  35. 35.
    J.C. Umavathi, I.C. Liu, M. Shekar, Appl. Math. Mech. Engl. 33, 931 (2012)CrossRefGoogle Scholar
  36. 36.
    G. Ahmadi, Int. J. Eng. Sci. 14, 639 (1976)CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsVisvesvaraya National Institute of TechnologyNagpurIndia

Personalised recommendations