Skip to main content
SpringerLink
Log in

Rheological fluid motion in tube by metachronal waves of cilia

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

This paper presents a theoretical study of a non-linear rheological fluid transport in an axisymmetric tube by cilia. An attempt has been made to explain the role of cilia motion in the transport of fluid through the ductus efferent of the male reproductive tract. The Ostwald-de Waele power-law viscous fluid is considered to represent the rheological fluid. We analyze pumping by means of a sequence of cilia beats from row-to-row of cilia in a given row of cells and from one row of cells to the next (metachronal wave movement). For this purpose, we consider the conditions that the corresponding Reynolds number is small enough for inertial effects to be negligible, and the wavelength-to-diameter ratio is large enough so that the pressure can be considered uniform over the cross section. Analyses and computations of the fluid motion reveal that the time-average flow rate depends on ϵ, a non-dimensional measure involving the mean radius a of the tube and the cilia length. Thus, the flow rate significantly varies with the cilia length. Moreover, the flow rate has been reported to be close to the estimated value 6×10−3 ml/h for human efferent ducts if ϵ is near 0.4. The estimated value was suggested by Lardner and Shack (Lardner, T. J. and Shack, W. J. Cilia transport. Bulletin of Mathematical Biology, 34, 325–335 (1972)) for human based on the experimental observations of flow rates in efferent ducts of other animals, e.g., rat, ram, and bull. In addition, the nature of the rheological fluid, i.e., the value of the fluid index n strongly influences various flow-governed characteristics. An interesting feature of this paper is that the pumping improves the thickening behavior for small values of ϵ or in free pumping (ΔP = 0) and pumping (ΔP > 0) regions.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Lardner, T. J. and Shack, W. J. Cilia transport. Bulletin of Mathematical Biology, 34, 325–335 (1972)

    Google Scholar 

  2. Hess, R. A. Small tubules, surprising discoveries: from efferent ductules in the turkey to the discovery that estrogen receptor alpha is essential for fertility in the male. Animal Reproduction, 12, 7–23 (2015)

    Google Scholar 

  3. Ilio, K. Y. and Hess, R. A. Structure and function of the ductuli efferentes: a review. Microscopy Research and Technique, 29, 432–467 (1994)

    Article  Google Scholar 

  4. Rivera, J. A. Cilia, Ciliated Epithelium, and Ciliary Activity, Pergamon Press, New York (1962)

    Google Scholar 

  5. Sleigh, M. A. The Biology of Cilia and Flagella, MacMillan, New York (1962)

    Book  Google Scholar 

  6. Sleigh, M. A. Patterns of Ciliary Beating, Academic Press, New York (1968)

    Google Scholar 

  7. Lucus, A. M. Ciliated epithelium. Special Cytology, 1, 409–473 (1932)

    Google Scholar 

  8. Setchell, B. P. Testicular Blood Supply, Lymphatic Drainage, and Secretion of Fluid, Academic Press, New York (1970)

    Book  Google Scholar 

  9. Winet, H. On the mechanism for flow in the efferent ducts. Journal of Andrology, 1, 304–311 (1980)

    Article  Google Scholar 

  10. Benoit, M. J. Recherches anatomiques, cytologiques et histophysiologiques sur les voies excrétrices du testicule, chez les mammifères. Archives D’anatomie, D’histologie et D’embryologie Normales et Experimentales, 5, 173–412 (1926)

    Google Scholar 

  11. Aire, T. A. and Josling, D. Ultrastructural study of the luminal surface of the ducts of the epi- didymis of gallinaceous birds. Onderstepoort Journal of Veterinary Research, 67, 191–199 (2000)

    Google Scholar 

  12. Borell, U., Milsson, O., and Westman, A. Ciliary activity in the rabbit fallopian tube during oestrus and after copulation. Acta Obstetricia et Gynecologica Scandinavica, 36, 22–28 (1957)

    Article  Google Scholar 

  13. Jahn, T. L. and Bovee, E. C. Movement and locomotion of microorganism. Annual Review of Microbiology, 19, 21–58 (1965)

    Article  Google Scholar 

  14. Jahn, T. L. and Bovee, E. C. Motile behaviour of protozoa. Research in Protozoology, 1, 41–110 (1967)

    Article  Google Scholar 

  15. Blake, J. R. A model for the microstructure in ciliated micro-organisms. Journal of Fluid Mechanics, 55, 1–23 (1972)

    Article  MATH  Google Scholar 

  16. Blake, J. R. A spherical envelope approach to ciliary propulsion. Journal of Fluid Mechanics, 46, 199–208 (1971)

    Article  MATH  Google Scholar 

  17. Miller, C. E. The kinematics and dynamics of ciliary fluid systems. Journal of Experimental Biology, 49, 617–629 (1968)

    Google Scholar 

  18. Miller, C. E. An investigation of the movement of Newtonian liquids initiated and sustained by the oscillation of mechanical cilia. Aspen Emphysema Conference, 10, 309–321 (1967)

    Google Scholar 

  19. Miller, C. E. Streamlines, steak lines and particle pathlines associate with a mechanically-induced flow holomorphic with the mammalian mucociliary system. Biorheology, 6, 127–135 (1969)

    Google Scholar 

  20. Barton, C. and Raynor, S. Analytical investigation of cilia induced mucous flow. The Bulletin of Mathematical Biophysics, 29, 419–428 (1967)

    Article  Google Scholar 

  21. Weiss, L. and Greep, R. O. Histology, McGraw-Hill, New York (1983)

    Book  Google Scholar 

  22. Blandau, R. J. Gamete Transport-Comparative Aspects, University of Chicago Press, Chicago (1969)

    Google Scholar 

  23. Sturgis, S. H. The effect of ciliary current on sperm progress in excised human fallopian tubes. Transactions of the American Society for the Study of Sterility, 3, 31–39 (1947)

    Google Scholar 

  24. Tuck, R. R., Setchel B. P., Waites, G. M. H., and Young, J. A. The composition of fluid collected by micropuncture and catherization from the seminiferous tubules and rete testis of rats. Pflügers Archiv, 318, 225–243 (1970)

    Article  Google Scholar 

  25. Waites, G. M. H. and Setchell, P. P. Some Physiological Aspects of the Function of the Testis, North-Holland Publishing Company, New York (1969)

    Google Scholar 

  26. Malek, J., Necas, J., and Rajagopal, K. R. Global existence of solutions for fluids with pressure and shear dependent viscosities. Applied Mathematics Letters, 15, 961–967 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  27. Dunn, P. F. and Picologlou, B. F. Investigation of rheological properties of human semen. Biorheology, 14, 277–292 (1977)

    Google Scholar 

  28. Mendeluk, G., Flecha, F. L. G., Castello, P. R., and Bregni, C. Factors involved in the biochemical etiology of human seminal plasma hyperviscosity. Journal of Andrology, 21, 262–267 (2000)

    Google Scholar 

  29. Xue, H. The modified Casson’s equation and its application to pipe flows of shear thickening fluid. Acta Mechanica Sinica, 21, 243–248 (2005)

    Article  MATH  Google Scholar 

  30. Misra, J. C. and Maiti, S. Peristaltic transport of rheological fluid: model for movement of food bolus through esophagus. Applied Mathematics and Mechanics (English Edition), 33, 15–32 (2012) DOI 10.1007/s10483-012-1552-7

    Article  MathSciNet  Google Scholar 

  31. Misra, J. C. and Maiti, S. Peristaltic pumping of blood through small vessels of varying cross- section. Journal of Applied Mechanics-Transactions of the ASME, 22, 061003 (2012)

    Article  Google Scholar 

  32. Misra, J. C. and Pandey, S. K. Peristaltic flow of a multi-layered power-law fluid through a cylindrical tube. International Journal of Engineering Science, 39, 387–402 (2001)

    Article  Google Scholar 

  33. Maiti, S. and Misra, J. C. Peristaltic transport of a couple stress fluid: some applications to hemodynamics. Journal of Mechanics in Medicine and Biology, 12, 1250048 (2012)

    Article  Google Scholar 

  34. Liu, Y. and Boling, G. Coupling model for unsteady MHD flow of generalized Maxwell fluid with radiation thermal transform. Applied Mathematics and Mechanics (English Edition), 37, 137–150 (2016) DOI 10.1007/s10483-016-2021-8

    Article  MathSciNet  MATH  Google Scholar 

  35. Hayat, T., Asad, S., and Alsaedi, A. Flow of Casson fluid with nanoparticles. Applied Mathematics and Mechanics (English Edition), 37, 459–470 (2016) DOI 10.1007/s10483-016-2047-9

    Article  MathSciNet  MATH  Google Scholar 

  36. Siddiqui, A. M., Ashraf, H., Walait, A., and Haroon, T. On study of horizontal thin film flow of Sisko fluid due to surface tension gradient. Applied Mathematics and Mechanics (English Edition), 36, 847–862 (2015) DOI 10.1007/s10483-015-1952-9

    Article  MathSciNet  MATH  Google Scholar 

  37. Ding, Z., Jian, Y., and Yang, L. Time periodic electroosmotic flow of micropolar fluids through microparallel channel. Applied Mathematics and Mechanics (English Edition), 36, 769–786 (2016) DOI 10.1007/s10483-016-2081-6

    Article  MathSciNet  MATH  Google Scholar 

  38. Srivastava, L. M. and Srivastava, V. P. Peristaltic transport of a power-law fluid: application to the ductus efferentes of the reproductive tract. Rheologica Acta, 27, 428–433 (1988)

    Article  Google Scholar 

  39. Usha, S. and Rao, A. R. Peristaltic transport of two-layered power-law fluids. Journal of Biomechanical Engineering, 119, 483–488 (1997)

    Article  Google Scholar 

  40. Rao, A. R. and Mishra, M., Peristaltic transport of a power-law fluid in a porous tube. Journal of Non-Newtonian Fluid Mechanics, 121, 163–174 (2004)

    Article  MATH  Google Scholar 

  41. Childress, S. Mechanics of Swimming and Flying, Cambridge University Press, Cambridge (1981)

    Book  MATH  Google Scholar 

  42. Lauga, E. and Powers, T. R. The hydrodynamics of swimming microorganisms. Reports on Progress in Physics, 72, 096601 (2009)

    Article  MathSciNet  Google Scholar 

  43. Blake, J. R. Flow in tubules due to ciliary activity. Bulletin of Mathematical Biology, 35, 513–523 (1973)

    Article  MATH  Google Scholar 

  44. Vlez-Cordero, J. R. and Lauga, E. Waving transport and propulsion in a generalized Newtonian fluid. Journal of Non-Newtonian Fluid Mechanics, 199, 37–50 (2013)

    Article  Google Scholar 

  45. Brennen, C. An oscillating-boundary-layer theory for ciliary propulsion. Journal of Fluid Mechanics, 65, 799–824 (1974)

    Article  MATH  Google Scholar 

  46. Lauga, E. Propulsion in a viscoelastic fluid. Physics of Fluids, 19, 083104 (2007)

    Article  MATH  Google Scholar 

  47. Lauga, E. Life at high Deborah number. Europhysics Letters, 86, 64001 (2009)

    Article  Google Scholar 

  48. Siddiqui, A. M., Haroon, T., Rani, R., and Ansari, A. R. An analysis of the flow of a power law fluid due to ciliary motion in an infinite channel. Journal of Biorheology, 24, 56–69 (2010)

    Article  Google Scholar 

  49. Agarwal, H. and Anawaruddin. Cilia transport of bio-fluid with variable viscosity. Indian Journal of Pure and Applied Mathematics, 15, 1128–1139 (1984)

    MATH  Google Scholar 

  50. Blake, J. R. On the movement of mucus in the lungs. Journal of Biomechanics, 8, 179–190 (1975)

    Article  Google Scholar 

  51. Satir, P. Studies on cilia: the fixation of the metachronal wave. The Journal of Cell Biology, 18, 345–365 (1963)

    Article  Google Scholar 

  52. Maiti, S. and Misra, J. C. Non-Newtonian characteristics of peristaltic flow of blood in micro- vessels. Communications in Nonlinear Science and Numerical Simulation, 18, 1970–1988 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  53. Shapiro, A. H., Jaffrin, M. Y., and Weinberg, S. L. Peristaltic pumping with long wavelength at low Reynolds number. Journal of Fluid Mechanics, 37, 799–825 (1969)

    Article  Google Scholar 

  54. Moulik, S., Gopalkrishnan, K., Hinduja, I., and Shahani, S. K. Presence of sperm antibodies and association with viscosity of semen. Human Reproduction, 4, 290–291 (1989)

    Article  Google Scholar 

  55. Fawcett, D. Cilia and Flagella, Academic Press, New York (1961)

    Book  Google Scholar 

  56. Bird, R. B., Stewart, W. E., and Lightfoot, E. N. Transport Phenomena, John Wiley and Sons, Singapore (1960)

    Google Scholar 

  57. Takabatake, S. and Ayukawa, K. Numerical study of two-dimensional peristaltic flows. Journal of Fluid Mechanics, 122, 439–465 (1982)

    Article  MATH  Google Scholar 

Download references

Acknowledgements

One of the authors, S. MAITI, is grateful to the University Grants Commission (UGC), New Delhi for awarding Dr.D. S. Kothari Post Doctoral Fellowship during this investigation.

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi, 221005, India

    S. Maiti & S. K. Pandey

  2. Department of Mathematics, The LNM Institute of Information Technology, Jaipur, 302031, India

    S. Maiti

Authors
  1. S. Maiti
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. K. Pandey
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Maiti.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Maiti, S., Pandey, S.K. Rheological fluid motion in tube by metachronal waves of cilia. Appl. Math. Mech.-Engl. Ed. 38, 393–410 (2017). https://doi.org/10.1007/s10483-017-2179-8

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-017-2179-8

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation