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Stokeslet arrays in a pipe and their application to ciliary transport

Published online by Cambridge University Press:  20 April 2006

N. Liron
Affiliation:
Department of Mathematics, Technion, Israel Institute of Technology, Haifa 32000, Israel

Abstract

The problem of fluid transport by cilia in a circular cylinder is investigated. The discrete-cilia approach is used in building the model, using the Green function due to an infinite periodic Stokeslet array in a pipe. Two different expressions are obtained for the Green function, one via a residue method and the other using the Poisson summation formula each amenable for computation in a different region. Interaction of the Stokeslets is investigated to see how, as distance decreases, interaction changes from initially separated closed vortices to a continuous flow. The singular integral equations for the forces in this model are now replaced by non-singular equations, thus overcoming the numerical difficulties in earlier works. It is found that in the pipe core the flow is time-independent and varies between a plug flow and a negative parabolic profile, in the pumping range. These results are seen to be local results due to the near field. Streamlines in the sublayer show eddies near the cilia bases blending into a uniform flow near the cilia tips.

Type
Research Article
Copyright
© 1984 Cambridge University Press

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References

Abramowitz, M. A. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Blake, J. R. 1972 A model for the micro-structure in ciliated organisms. J. Fluid Mech. 55, 123.Google Scholar
Blake, J. R. 1973 Flow in tubules due to ciliary activity. Bull. Math. Biol. 35, 513523.Google Scholar
Blake, J. R. 1979 On the generation of viscous toroidal eddies in a cylinder. J. Fluid Mech. 95, 209222.Google Scholar
Blake, J. R., Liron, N. & Aldis, G. K. 1982 Flow patterns around ciliated microorganisms and in ciliated ducts. J. Theor. Biol. 98, 127141.Google Scholar
Blandau, R. J. 1969 Gamete transport-comparative aspects. In The Mammalian Oviduct (ed. E. S. E. Hafez & R. J. Blandau). University of Chicago Press.
Blum, J. J. 1974 A note on fluid transport in ciliated tubules. J. Theor. Biol. 46, 287290.Google Scholar
Friedmann, M., Gillis, J. & Liron, N. 1968 Laminar flow in a pipe at low and moderate Reynolds numbers. Appl. Sci. Res. 19, 426438.Google Scholar
Hasimoto, H. & Sano, O. 1980 Stokeslets and eddies in creeping flows. Ann. Rev. Fluid Mech. 12, 335363.Google Scholar
Lardner, T. J. & Shack, W. J. 1972 Cilia transport. Bull. Math. Biophys. 34, 325335.Google Scholar
Lighthill, M. J. 1976 Flagellar Hydrodynamics — The John von Neumann Lecture, 1975. SIAM Rev. 18, 161230.Google Scholar
Liron, N. 1978 Fluid transport by cilia between parallel plates. J. Fluid Mech. 86, 705726.Google Scholar
Liron, N. & Blake, J. R. 1981 Existence of viscous eddies near boundaries. J. Fluid Mech. 107, 109129.Google Scholar
Liron, N. & Meyer, F. A. 1980 Fluid transport in a thick layer above an active ciliated surface. Biophys. J. 30, 463472.Google Scholar
Liron, N. & Mochon, S. 1976a The discrete cilia approach to propulsion of ciliated micro-organisms. J. Fluid Mech. 75, 593607.Google Scholar
Liron, N. & Mochon, S. 1976b Stokes flow for a Stokeslet between two parallel flat plates. J. Engng Maths 10, 287303.Google Scholar
Liron, N. & Shahar, R. 1978 Stokes flow due to a Stokeslet in a pipe. J. Fluid Mech. 86, 727744.Google Scholar
Moffatt, H. K. 1964 Viscous and resistive eddies near a sharp corner. J. Fluid Mech. 18, 118.Google Scholar
Segré, G. & Silberberg, A. 1962 Behaviour of macroscopic rigid spheres in Poiseuille flow. Part 2. Experimental results and interpretation. J. Fluid Mech. 14, 137157.Google Scholar