Journal of Biological Physics

, Volume 44, Issue 3, pp 273–300 | Cite as

Analysis of a self-propelling sheet with heat transfer through non-isothermal fluid in an inclined human cervical canal

  • Ahsan WalaitEmail author
  • A. M. Siddiqui
  • M. A. Rana


The present theoretical analysis deals with biomechanics of the self-propulsion of a swimming sheet with heat transfer through non-isothermal fluid filling an inclined human cervical canal. Partial differential equations arising from the mathematical modeling of the proposed model are solved analytically. Flow variables like pressure gradient, propulsive velocity, fluid velocity, time mean flow rate, fluid temperature, and heat-transfer coefficients are analyzed for the pertinent parameters. Striking features of the pumping characteristics are explored. Propulsive velocity of the swimming sheet becomes faster for lower Froude number, higher Reynolds number, and for a vertical channel. Temperature and peak value of the heat-transfer coefficients below the swimming sheet showed an increase by the increment of Brinkmann number, inclination, pressure difference over wavelength, and Reynolds number whereas these quantities decrease with increasing Froude number. Aforesaid parameters have shown opposite effects on the peak value of the heat-transfer coefficients below and above the swimming sheet. Relevance of the current results to the spermatozoa transport with heat transfer through non-isothermal cervical mucus filling an inclined human cervical canal is also explored.


Swimming sheet Heat transfer Non-isothermal fluid Inclined channel Spermatozoa transport 



The authors wish to express their very sincere thanks to the reviewers for their valuable suggestions and comments.

Compliance with Ethical Standards

Disclosure of potential conflicts of interest

The authors whose names are listed below certify that they have no affiliations with or involvement in any organization or entity with any financial interest or non-financial interest in the subject matter or materials discussed in this manuscript. The authors declare that they have no conflicts of interest.


Informed Consent

Informed consent was obtained from all individual participants included in the study.

Research involving human participants and/or animals

For this type of study formal consent is not required. This article does not contain any studies with human participants or animals performed by any of the authors.


  1. 1.
    Sathananthan, A.H.: Humancentriole: origin, & how it impacts fertilization, embryogenesis, infertility & cloning. Indian J. Med. Res. 129, 348–350 (2009)Google Scholar
  2. 2.
    Jones, R.E., Lopez, K.H.: Human Reproductive Biology, 3rd edn., pp.231–236. Academic Press (Elsevier), Burlington (2006)Google Scholar
  3. 3.
    Taylor, G.I.: Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond. A 209, 447–461 (1951)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Reynolds, A.J.: The swimming of minute organisms. J. Fluid Mech. 23, 241–260 (1965)ADSCrossRefGoogle Scholar
  5. 5.
    Tuck, E.O.: A note on a swimming problem. J. Fluid Mech. 31(2), 305–308 (1968)ADSCrossRefGoogle Scholar
  6. 6.
    Shack, W.J., Lardner, T.J.: A long wavelength solution for a microorganism swimming in a channel. Bull. Math. Biol. 36, 435–444 (1974)CrossRefzbMATHGoogle Scholar
  7. 7.
    Smelser, R.E., Shack, W.J., Lardner, T.J.: The swimming spermatozoa in an active channel. J. Biomech. 7, 349–355 (1974)CrossRefGoogle Scholar
  8. 8.
    Odeblad, E.: Undulation of macromolecules in physiological fluid. Int. J. Fert. 7, 313–319 (1962)Google Scholar
  9. 9.
    Shukla, J.B., Rao, B.R.P., Parihar, R.S.: Swimming of spermatozoa in cervix: effect of dynamical interaction and peripheral layer viscosity. J. Biomech. 11, 15–19 (1978)CrossRefGoogle Scholar
  10. 10.
    Sinha, P., Singh, C., Prasad, K.R.K.: A microcontinuum analysis of the self-propulsion of the spermatozoa in the cervical canal. Int. J. Eng. Sci. 20(9), 1037–1048 (1982)CrossRefzbMATHGoogle Scholar
  11. 11.
    Shukla, J.B., Chandra, P., Sharma, R., Radhakrishnamacharya, G.: Effects of peristaltic and longitudinal wave motion of the channel wall on movement of micro-organisms: application to spermatozoa transport. J. Biomech. 21(11), 947–954 (1988)CrossRefGoogle Scholar
  12. 12.
    Philip, D., Chandra, P.: Self-propulsion of spermatozoa in microcontinua: effect of transverse wave motion of channel walls. Arch. Appl. Mech. 66, 90–99 (1995)ADSCrossRefzbMATHGoogle Scholar
  13. 13.
    Radhakrishnamacharya, G., Sharma, R.: Motion of a self-propelling micro-organism in a channel under peristalsis: effects of viscosity variation. Nonlinear Anal. Model. Control 12(3), 409–418 (2007)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Pak, O.S., Lauga, E.: The transient swimming of a waving sheet. Proc. R. Soc. A: Math. Phy. Eng. Sci. 466, 107–126 (2010)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Baltz, R.H., Bingham, P.M., Drake, J.W.: Heat mutagenesis in bacteriophage T4: the transition pathway. Proc. Natl. Acad. Sci. USA 73(4), 1269–1273 (1976)ADSCrossRefGoogle Scholar
  16. 16.
    Vasudev, C., Rao, U.R., Reddy, M.V.S., Rao, G.P.: Peristaltic pumping of Williamson fluid through a porous medium in a horizontal channel with heat transfer. Am. J. Sci. Ind. Res. 1(3), 656–666 (2010)Google Scholar
  17. 17.
    Radhakrishnamacharya, G., Srinivasulu, C.: Influence of wall properties on peristaltic transport with heat transfer. C. R. Mecanique 335(7), 369–373 (2007)ADSCrossRefzbMATHGoogle Scholar
  18. 18.
    Srinivas, S., Kothandapania, M.: The influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls. Appl. Math. Comput. 213, 197–208 (2009)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Hakeem, A.M., Naby, A.E., Misery, A.E.M.E., Shamy, I.I.E.: Effects of an endoscope and fluid with variable viscosity on peristaltic motion. Appl. Math. Comput. 158, 497–511 (2004)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Misery, A.M.E., Hakeem, A.E., Naby, A.E., Nagar, A.H.E.: Effects of a fluid with variable viscosity and an endoscope on peristaltic motion. J. Physical Soc. Japan 72, 89–93 (2001)CrossRefzbMATHGoogle Scholar
  21. 21.
    Mekheimer, K.S., Elmaboud, Y.A.: The influence of heat transfer and magnetic field on peristaltic transport of a Newtonian fluid in a vertical annulus: application of an endoscope. Phys. A 372(10), 1657–1665 (2008)zbMATHGoogle Scholar
  22. 22.
    Massoudi, M., Christie, I.: Effects of variable viscosity and viscous dissipation on the flow of a third grade fluid in a pipe. Int. J. Nonlinear Mech. 30(5), 687–699 (1995)CrossRefzbMATHGoogle Scholar
  23. 23.
    Nadeem, S., Akbar, N.S.: Effects of heat transfer on the peristaltic transport of MHD Newtonian fluid with variable viscosity: application of Adomian decomposition method. Commun. Nonlinear Sci. Numer. Simul. 14, 38443855 (2009)Google Scholar
  24. 24.
    Nadeem, S., Hayat, T., Akbar, N.S., Malik, M.Y.: On the influence of heat transfer in peristalsis with variable viscosity. Int. J. Heat Mass Transf. 52(21), 4722–4730 (2009)CrossRefzbMATHGoogle Scholar
  25. 25.
    Krishna Kumari, P.S.V.H.N., Ravi Kumar, Y.V.K., Ramana Murthy, M.V., Sreenadh, S.: Peristaltic motion of a fourth-grade fluid through a porous medium under effect of a magnetic field in an inclined channel. J. Basic. Appl. Sci. Res. 1(9), 1052–1064 (2011)Google Scholar
  26. 26.
    Kumar, S.R.: Effect of couple stress fluid flow on magnetohydrodynamic peristaltic blood flow with porous medium through inclined channel in the presence of slip effect-blood flow model. Int. J. Bio-Sci. Bio-Tech. 7(5), 65–84 (2015)CrossRefGoogle Scholar
  27. 27.
    Meletis, C., Brown, L.: Enhancing Fertility, A Couple’s Guide to Natural Approaches. Accessible Publishing System PTY, Ltd (2008)Google Scholar
  28. 28.
    Ishijima, S., Oshio, S., Mohri, H.: Flagellar movement of human spermatozoa. Mol. Reprod. Dev. 13(3), 185–197 (1986)Google Scholar
  29. 29.
    Martyn, F., McAuliffe, F.M., Wingfield, M.: The role of the cervix in fertility: is it time for a reappraisal? Hum. Reprod. 29(10), 2092–2098 (2014)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsRiphah International UniversityIslamabadPakistan
  2. 2.Department of MathematicsPennsylvania State UniversityYorkUSA

Personalised recommendations