Metachronal beating of cilia under influence of Hartmann layer and heat transfer

  • Noreen Sher Akbar
  • Z. H. Khan
  • S. Nadeem
Regular Article


The propulsion system of cilia motion is investigated considering a viscous fluid model. The problem of the two-dimensional fluid motion in a symmetric channel with ciliated walls is considered. The features of ciliary structures are resolved by the supremacy of viscosity effects over inertial control by the long-wavelength and low-Reynolds-number approximation. Exact solutions for the longitudinal pressure gradient, temperature and velocities are obtained. The pressure gradient and volume flow rate for different values of the flow parameters are also predicted. The flow possessions for the viscous fluid are solved as a function of the cilia and metachronal wave velocity.


Pressure Rise Hartmann Number Ciliary Motion Brinkman Number Symmetric Channel 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


U, V

Velocity components


Hartmann number

X, Y



Wave’s amplitudes


Viscosity of the fluid




Cilia length


Slenderness parameter


Modified Reynolds number


Kinematic viscosity of the fluid




Wave speed


Stretching parameter


Flow rate


Yield stress


Eccentricity of the elliptical motion


Brinkman number


  1. 1.
    J. Gray, G. Hancock, J. Exp. Biol. 32, 802 (1955).Google Scholar
  2. 2.
    M.A. Sleigh, The Biology of Cilia and Flagella (MacMillian, New York, 1962).Google Scholar
  3. 3.
    M.A. Sleigh, Patterns if Ciliary Beating, in Aspects of Cell Motility, Soc. Expl. Biol. Sump. XX I I (Academic Press, New York 1968).Google Scholar
  4. 4.
    S. Gueron, N. Liron, Biophys J. 63, 1045 (1992).ADSCrossRefGoogle Scholar
  5. 5.
    S. Gueron, K. Levit-Gurevich, Biophys. J. 74, 1658 (1998).CrossRefGoogle Scholar
  6. 6.
    R.H. Dillon, L.J. Fauci, C. Omoto, X. Yang, Ann. N.Y. Acad. Sci. 1101, 494 (2007).ADSCrossRefGoogle Scholar
  7. 7.
    J.R. Blake, Bull. Math. Biol. 35, 513 (1973).MATHGoogle Scholar
  8. 8.
    J.R. Blake, J. Fluid Mech. 46, 199 (1971).ADSCrossRefMATHGoogle Scholar
  9. 9.
    H. Agarwal, Anwaruddin, Indian J. Pure Appl. Math. 15, 1128 (1984).Google Scholar
  10. 10.
    J.R. Blake, J. Fluid Mech. 46, 199 (1971).ADSCrossRefMATHGoogle Scholar
  11. 11.
    Noreen Sher Akbar, Z.H. Khan, S. Nadeem, Eur. Phys. J. Plus 129, 123 (2014).CrossRefGoogle Scholar
  12. 12.
    M.J. Lighthill, Mathematical Biofluid Dynamics (SIAM, Philadelphia, 1975).Google Scholar
  13. 13.
    C. Barton, S. Raynor, Bull. Math. Biophys. 29, 419 (1967).CrossRefGoogle Scholar
  14. 14.
    Noreen Sher Akbar, Heat Transfer Res. 45, 293 (2014).CrossRefGoogle Scholar
  15. 15.
    Noreen Sher Akbar, S. Nadeem, Heat Transfer Res. 45, 219 (2014).Google Scholar
  16. 16.
    Noreen Sher Akbar, Int. J. Bio. Math. 7, 1450004 (2014).Google Scholar
  17. 17.
    Noreen Sher Akbar, S. Nadeem, Meccanica 49, 325 (2014).CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Noreen Sher Akbar, IEEE Trans. Nanotechnol. 13, 357 (2014).CrossRefGoogle Scholar
  19. 19.
    Noreen Sher Akbar, J. Comput. Theor. Nanosci. 11, 1642 (2014).CrossRefGoogle Scholar
  20. 20.
    Noreen Sher Akbar, J. Comput. Theor. Nanosci. 11, 1150 (2014).CrossRefGoogle Scholar
  21. 21.
    N.T.M. Eldabe, M.F. El-Sayed, A.Y. Ghaly, H.M. Sayed, Physica A 383, 253 (2007).ADSCrossRefGoogle Scholar
  22. 22.
    Noreen Sher Akbar, S. Nadeem, Chin. Phys. B 22, 014703 (2013).ADSCrossRefGoogle Scholar
  23. 23.
    Noreen Sher Akbar, S. Nadeem, J. Comput. Theor. Nanosci. 10, 2491 (2013).CrossRefGoogle Scholar
  24. 24.
    Noreen Sher Akbar, Int. J. Biomath. 6, 350034 (2013).Google Scholar
  25. 25.
    Noreen Sher Akbar, E.N. Maraj, S. Nadeem, Eur. Phys. J. Plus 129, 149 (2014).CrossRefGoogle Scholar
  26. 26.
    R. Ellahi, A. Riaz, S. Nadeem, Indian J. Phys. 87, 1275 (2013).CrossRefGoogle Scholar
  27. 27.
    R. Ellahi, M. Mubeshir Bhatti, A. Riaz, M. Sheikholeslami, J. Porous Media 17, 1 (2014).CrossRefGoogle Scholar
  28. 28.
    R. Ellahi, S.U. Rahman, S. Nadeem, N.S. Akber, J. Appl. Nanosci., DOI:10.1007/s13204-013-0253-6 (2013).
  29. 29.
    R. Ellahi, S.U. Rahman, S. Nadeem, N.S. Akber, J. Comput. Theor. Nanosci. 11, 1156 (2014).CrossRefGoogle Scholar
  30. 30.
    Noreen Sher Akbar, M. Raza, R. Ellahi, Eur. Phys. J. Plus 129, 155 (2014).CrossRefGoogle Scholar

Copyright information

© Società Italiana di Fisica and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.DBS&H CEMENational University of Sciences and TechnologyIslamabadPakistan
  2. 2.School of Mathematical SciencesPeking UniversityBeijingP.R. China
  3. 3.Department of MathematicsUniversity of MalakandKhyber PakhtunkhwaPakistan
  4. 4.Department of MathematicsQuaid-i-Azam University 45320IslamabadPakistan

Personalised recommendations