Advertisement

Fluid Mechanics of Ciliary Propulsion

  • John Blake
Part of the The IMA Volumes in Mathematics and its Applications book series (IMA, volume 124)

Abstract

Cilia have many functions in the animal kingdom, some of these being cleansing, feeding, excretion, locomotion and reproduction. They occur in all phyla of the animal kingdom with the possible exception of the class Nematoda. This lecture will discuss the development of fluid mechanical models and theories that help with our understanding and interpretation of locomotion of protozoa, mucous transport in the lung, filter feeding in bivalve molluscs and gamete transport.

The theoretical development for the fluid mechanics requires obtaining the fundamental singularities and image systems pertinent to the system under study, the physical interpretation of them and their constructive use to model the flow fields generated by fields of cilia.

These theories allow estimates of the flow fields in the cilia sublayer and for a greater understanding of propulsive mechanisms in both micro-organisms and mucous transport in the lower respiratory tract. The sophisticated models in turn allow us to develop better approximations for simplified models that provide an improved understanding of more complicated flows involving filter feeding in bivalve molluscs and with ovum transport in the oviduct.

Finally, studies of possible filter feeding strategies in the sessile organism, Vorticella, which alters the length of its stalk periodically, has led to the development of some interesting non-linear mathematics in a simplified ‘blinking stokeslet’ model of this filter feeding phenomenon. We shall demonstrate that this can lead to chaotic dynamics, which has been shown to enhance mixing and hence improve the efficiency of feeding currents. The continuous system is reduced to an area-preserving map, which allows for greater analytical progress to be made in this inertia-free system. Poincaré sections and Lyapunov exponents are used alongside other chaotic measures to determine the nature and extent of the chaos. Effects of molecular diffusion are mimicked via the incorporation of white noise in the map and enhanced feeding levels are predicted.

The author of this paper acknowledges, with gratitude, the enormous influence that Sir James Lighthill has had on his life and academic career. This paper is dedicated to his memory and is based on research work conducted by the author of this review over the last 30 years with the material in this article being taken from papers over this period.

Keywords

Muscular Activity Volume Flow Rate Mucous Layer Recovery Stroke Poincare Section 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Aderogba, K. and Blake, J.R., 1978. Action of a force near the planar surface between semi-infinite immiscible liquids at very low Reynolds numbers: Addendum. Bull. Aust. Math. Soc. 19, 309–318.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Aiello, E. and Sleigh, M.A., 1972. The metachronal wave of Mytilus edulis, J. Cell. Biol. 54, 493–506.CrossRefGoogle Scholar
  3. [3]
    Aref, H., 1984. Stirring by chaotic advection, J. Fluid Mech. 143, 1–21.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Aref, H. and Balachandar, S., 1986. Chaotic advection in a Stokes flow. Phys. Fluids 29, 3515–3521.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Blake, J.R., 1972. A model for the microstructure in ciliated organisms, J. Fluid Mech. 555, 1–23.CrossRefGoogle Scholar
  6. [6]
    Blake, J.R., 1975. On the movement of mucus in the lung, J. Biomech. 8,179–190.CrossRefGoogle Scholar
  7. [7]
    Blake, J.R., 1975. Fluid flows in fields of resistance. Bull. Aust. Math. Soc. 13, 129–145.CrossRefzbMATHGoogle Scholar
  8. [8]
    Blake, J.R., 1984. Mechanics of muco-ciliary transport, IMA J. Appl. Math. 32, 69–87.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Blake, J.R. and Fulford, G.R., 1984. Mechanics of muco-ciliary transport, Physico. Chem. Hydrodynamics 5, 401–411.Google Scholar
  10. [10]
    Blake, J.R. and Fulford, G.R., 1995. Hydrodynamics of filter feeding. In: Biological Fluid dynamics, Editors: C.P. Ellington and T.J. Pedley. Soc. Exp. Biol., pp. 183–197.Google Scholar
  11. [11]
    Blake, J.R., Liron, N., and Aldis, G.K., 1982. Flow patterns around micro-organisms and in ciliated ducts. J. Theor. Biol. 98, 127–141.MathSciNetCrossRefGoogle Scholar
  12. [12]
    Blake, J.R. and Otto, S.R., 1996 Ciliary propulsion, chaotic filtration and a blinking stokeslet. J. Eng. Math. 30, 151–168.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    Blake, J.R., Otto, S.R., and Blake, D.A., 1998. Filter feeding, chaotic filtration and a blinking stokeslet. Theor. Comp. Fluid Dyn. 10 23–36.CrossRefzbMATHGoogle Scholar
  14. [14]
    Blake, J.R. and Sleigh, M.A., 1974. Mechanics of ciliary locomotion, Biol. Rev. 49, 85–125.CrossRefGoogle Scholar
  15. [15]
    Blake, J.R., Vann, P., and Winet, H., 1983. A model for ovum transport, J. Theor. Biol. 102, 145–166.CrossRefGoogle Scholar
  16. [16]
    Blake, J.R. and Winet, H., 1980. On the mechanics of muco-ciliary transport, Biorheol. 17, 125.Google Scholar
  17. [17]
    Brennen, C. and Winet, H., 1977. Fluid mechanics of propulsion by cilia and flagella, Ann. Rev. Fluid Mech. 9, 339–398.CrossRefGoogle Scholar
  18. [18]
    Brosens, I.A. and Vasquez, G., 1976. J. Reprod. Med. 16, 17.Google Scholar
  19. [19]
    Cheer, A.Y.L. and Koehl, M.A.R., 1987. Paddles and Rakes: Fluid flow through bristled appendages of small organisms. J. Theor. Biol. 129, 17–39.CrossRefGoogle Scholar
  20. [20]
    Croxatto, H.B., Oritz, M.E., Diaz, S., and Hess, R., 1979. J. Reprod. Pert. 55, p. 231.CrossRefGoogle Scholar
  21. [21]
    Dresdner, R.D., Katz, D.F., and Berger, S.A., 1980. The propulsion by large amplitude waves of uniflagellar micro-organisms of finite length, J. Fluid Mech. 97, 591–621.CrossRefzbMATHGoogle Scholar
  22. [22]
    De Frutos, J. and Sanz-Serna, J.M., 1992. An easily implementable fourth-order method for the time integration of wave problems. J. Comp. Phys. 103, 160–168.CrossRefzbMATHGoogle Scholar
  23. [23]
    Fulford, G.R. and Blake, J.R., 1986. Muco-ciliary transport in the lung, J. Theor. Biol. 121, 381–402.CrossRefGoogle Scholar
  24. [24]
    Gray, J. and Hancock, G.J., 1955. The propulsion of sea urchin spermatozoa, J. Exp. Biol. 32, 802–814.Google Scholar
  25. [25]
    Halbert, S.A., Tam, P.Y., and LandAU, R.J., 1976. Science 191, 1052.CrossRefGoogle Scholar
  26. [26]
    Hancock, G.J., 1953. The self propulsion of microscopic organisms through liquids, Proc. Roy. Soc. A219, 96–121.MathSciNetGoogle Scholar
  27. [27]
    Happel, J. and Brenner, H., 1965. Low Reynolds Number Hydrodynamics, Prentice-Hall, Englewood Cliffs, N.J.Google Scholar
  28. [28]
    Harper, J.F. and Wake, G.C., 1983. Stokes flow between parallel plates due to a transversely moving end wall. IMA J. Appl. Math. 30, 141–149.MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    HÉNON, M., 1969. Numerical Study of Quadratic Area-Preserving Mappings. Quart. Appl. Maths 27, 291–312.zbMATHGoogle Scholar
  30. [30]
    Higdon, J.J.L., 1979. A hydrodynamic analysis of flagellar propulsion, J. Fluid Mech. 90, 685–711.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    Higdon, J.J.L., 1979. The generation of feeding currents by flagellar motions J. Fluid Mech. 94, 305–330.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    Higdon, J.J.L., 1979. The hydrodynamics of flagellar propulsion: helical waves, J. Fluid Mech. 94, 331–351.MathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    Hodgson, B.J., Talo, A., and Pauerstein, C.J., 1977. Biol. Reprod. 16, p. 394.CrossRefGoogle Scholar
  34. [34]
    Jansen, R.P.S., 1980. Am. J. Obstet. Gynaecol. 136, p. 292.Google Scholar
  35. [35]
    Johnson, R.E. and Brokaw, C.J., 1979. Flagellar hydrodynamics. A comparison between resistive force theory and slender body theory, Biophys. J. 25, 113–127.CrossRefGoogle Scholar
  36. [36]
    Jones, H.D., Richards, O.G., and Southern, T.A., 1992. Gill dimensions, water pumping rate and body size in the mussel Mytilus edulis. L. J. Exp. Mar. Biol. Ecol. 155, 213–237.CrossRefGoogle Scholar
  37. [37]
    Jørgenson, C.B., 1981. A hydromechanical principle for particle retention in Mytilus edulis and other ciliary suspension feeders. Marine Biol. 61, 277–282.CrossRefGoogle Scholar
  38. [38]
    Jørgenson, C.B., 1981. Feeding and cleaning mechanisms in the suspension feeding bivalve Mytilus edulis. Marine Biol. 65, 159–163.CrossRefGoogle Scholar
  39. [39]
    Jørgenson, C.B., 1982. Fluid mechanics of the mussel gill: The lateral cilia. Marine Biol. 70, 275–281.CrossRefGoogle Scholar
  40. [40]
    Jørgenson, C.B., 1991. Bivalve Filter Feeding: Hydrodynamics, Bioenergetics, Physiology and Ecology, Denmark: Olsen and Olsen.Google Scholar
  41. [41]
    Jørgenson, C.B., Larsen, P.S., Mohlenberg, F., and Riisgard, H.V., 1988. The mussel pump: properties and modelling, Mar. Ecol. Prog. S er. 45, 205–216.CrossRefGoogle Scholar
  42. [42]
    Jørgenson, C.B., Mohlenberg, F., and Sten-Knudson, O., 1986. Nature of relation between ventilation and oxygen consumption in filter feeders. Mar. Ecol. Prog. Ser. 29, 73–88.CrossRefGoogle Scholar
  43. [43]
    Jørgenson, C.B., Famme, P., Kristensen, H.S., Larsen, P.S., Mohlenberg, F., and Riisgard, H.U., 1986. The bivalve pump. Mar. Ecol. Prog. Ser. 34, 69–77.CrossRefGoogle Scholar
  44. [44]
    Joseph, D.D. and Sturges, L., 1978. The convergence of biorthogonal series for biharmonic and Stokes’ flow edge problems. Part II. SIAM J. Appl. Math. 34, 7–26.MathSciNetCrossRefzbMATHGoogle Scholar
  45. [45]
    Katz, D.F. and Blake, J.R., 1975. Flagellar motions near walls, In: Swimming and Flying in Nature, Eds. Wu, T.Y., Brokaw, C.J. and Brennen, C. Plenum, New York, pp. 173–184.Google Scholar
  46. [46]
    Knight-Jones, E.W., 1954. Relations between metachronism and the direction of ciliary beat ion Metazoa, Quart. J. Microsc. Sci. 95, 503–521.Google Scholar
  47. [47]
    Lighthill, M.J., 1970. Aquatic animal propulsion of high hydromechanical efficiency. J. Fluid Mech. 44, 265–301.CrossRefzbMATHGoogle Scholar
  48. [48]
    Lighthill, M.J., 1976. Flagellar hydrodynamics, SIAM Rev. 18, 161–230.MathSciNetCrossRefzbMATHGoogle Scholar
  49. [49]
    Liron, N. and Blake, J.R., 1981. Existence of viscous eddies near boundaries. J. Fluid Mech. 107, 109–129.MathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    Liron, N. and Mochon, S., 1976. The discrete cilia approach to propulsion of ciliated micro-organisms, J. Fluid Mech. 75, 593–607.MathSciNetCrossRefzbMATHGoogle Scholar
  51. [51]
    Litt, M., 1970. Mucus rheology, Archs. Intern. Med. 126, 417–423.CrossRefGoogle Scholar
  52. [52]
    Matsui, H., RandELL, S.H., Peretti, S.W., Davis, C.W., and Boucher, R.C., 1998. Coordinated clearance of periciliary liquid and mucus from airway surfaces. J. Clin. Invest. 102, 1125–1131.CrossRefGoogle Scholar
  53. [53]
    Meleshko, V.V. and Aref, H., 1996. A blinking rotlet model for chaotic advection. Phys. Fluids A 8, 3215–3217.CrossRefzbMATHGoogle Scholar
  54. [54]
    Otto, S.R., Yannacopolous, A.N. and Blake, J.R., 2000. Transport and mixing in Stokes flow: the effect of chaotic dynamics on the blinking stokeslet. J. Fluid Mech. (in press).Google Scholar
  55. [55]
    Pauerstein, C.J. and Eddy, C.A., 1979. J. Reprod. Fert. 55, 223.CrossRefGoogle Scholar
  56. [56]
    Pedley, T.J., Schröter, R.C., and Sudlow, M.F., 1971. Flow and pressure drop in systems of repeatedly branching tubes, J. Fluid Mech. 46, 365–383.CrossRefGoogle Scholar
  57. [57]
    Sade, J., Eliezer, N., Silberberg, A., and Nero, A.C., 1970. The rôle of mucus in transport by cilia, Am. Rev. Resp. Dis. 102, 48–52.Google Scholar
  58. [58]
    Sanderson, M.J. and Sleigh, M.A., 1981. Ciliary activity of cultured rabbit tracheal epithelium: Beat pattern and metachrony, J. Cell. Sci. 47, 331–347.Google Scholar
  59. [59]
    Satir, P., Hamasaki, T., and Holwill, M., 1998. Modeling outer dynein arm activity and its relation to the ciliary beat cycle. In: Cilia, Mucus and Mucociliary Interactions, Eds. G.L. Baum, Z. Priel, Y. Roth, N. Liron and E.J. Ostfeld. Marcel Dekker.Google Scholar
  60. [60]
    Shack, W.J., Fray, C.S., and Lardner, T.J., 1971. Observations of the hydrodynamics and swimming motions of mammalian spermatozoa, MIT Preprint.Google Scholar
  61. [61]
    Silberberg, A., 1982. Rheology of mucus, muco-ciliary interaction, and ciliary activity, Cell. Not. (Suppl.) 1, 25–28.Google Scholar
  62. [62]
    Silvester, N.R., 1988. Hydrodynamics of flow in Mytilus gills. J. Exp. Mar. Biol. Ecpl. 120 171–182CrossRefGoogle Scholar
  63. [63]
    Silvester, N.R. and Sleigh, M.A., 1984. Hydrodynamic aspects of particle capture by Mytilus, J. Mar. Biol. Assoc. U.K. 64, 859–879.CrossRefGoogle Scholar
  64. [64]
    Sleigh, M.A., 1962. The Biology of Cilia and Flagella, Pergamon.Google Scholar
  65. [65]
    Sleigh, M.A., 1973. The Biology of Protozoa, London, Edward Arnold, p. 315.Google Scholar
  66. [66]
    Sleigh, M.A., 1977. The nature and action of respiratory tract cilia, In: Respiratory Defence Mechanisms, (Brain, J.D., Proctor, D.F. and Reid, L. Eds.) pp. 247–288, New York: Marcel Dekker.Google Scholar
  67. [67]
    Sleigh, M.A., 1981. Ciliary function in mucus transport, Chest 805, 791–795.Google Scholar
  68. [68]
    Sleigh, M.A., 1982. Movement and coordination of tracheal cilia and the relation of these to mucus transport, Cell. Not. (Suppl) 1, 19–24.Google Scholar
  69. [69]
    Sleigh, M.A. and Blake, J.R., 1975. Hydromechanical aspects of ciliary propulsion. In: Swimming and Flying in Nature, Eds. Wu, T.Y., Brokaw, C.J. and Brennen, C. Plenum, New York, pp. 185–210.Google Scholar
  70. [70]
    Sleigh, M.A., Blake, J.R., and Liron, N., 1988. The propulsion of mucus by cilia. Am. Rev. Resp. Dis. 137, 726–741.Google Scholar
  71. [71]
    Spence, D.A., 1982. A note on the eigenfunction expansion for the elastic strip. SIAM J. Appl. Math. 42, 155–174.MathSciNetCrossRefzbMATHGoogle Scholar
  72. [72]
    Spence, D.A., 1983. A class of biharmonic end-strip problems arising in elasticity and Stokes flow. IMA J. Appl. Math. 30, 107–139.MathSciNetCrossRefzbMATHGoogle Scholar
  73. [73]
    Talo, A., 1974. Biol. Reprod. 11, p. 335.CrossRefGoogle Scholar
  74. [74]
    Taylor, G.I., 1951. Analysis of the swimming of microscopic organisms, Proc. Roy. Soc. A209, 447–461.Google Scholar
  75. [75]
    Teodorescu, P.P., 1960. Sur le problème de la demibande plane élastique. Arch. Mech. Stos. 12, 313–331.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • John Blake
    • 1
  1. 1.School of Mathematics and StatisticsThe University of BirminghamEdgbaston, BirminghamUK

Personalised recommendations