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Peristaltic pumping of rigid objects in an elastic tube

Published online by Cambridge University Press:  24 February 2011

D. TAKAGI  and
N. J. BALMFORTH
D. TAKAGI
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
N. J. BALMFORTH*
Affiliation:
Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, V6T 1Z2, Canada Department of Earth and Ocean Science, University of British Columbia, 6339 Stores Road, Vancouver, V6T 1Z4, Canada
*
†Email address for correspondence: [email protected]
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Abstract

A mathematical model is developed for long peristaltic waves propelling a suspended rigid object down a fluid-filled axisymmetric tube. The fluid flow is described using lubrication theory and the deformation of the tube using linear elasticity. The object is taken to be either an infinitely long rod of constant radius or a parabolic-shaped lozenge of finite length. The system is driven by a radial force imposed on the tube wall that translates at constant speed down the tube axis and with a form chosen to generate a periodic wave train or a solitary wave. These waves exert a traction on the enclosed object, forcing it into motion. Periodic waves drive the infinite rod at a speed that attains a maximum at a moderate forcing amplitude and approaches approximately one quarter of the wave speed in the large-amplitude limit. The finite lozenge can be entrained and driven at the same speed as a solitary wave or periodic wave train if the forcing is sufficiently strong. For weaker forcing, the lozenge is either left behind the solitary wave or interacts repeatedly with the waves in the periodic train to generate stuttering forward progress. The threshold forcing amplitude for entrainment increases weakly with the radial span of the enclosed object, but strongly with the axial length, with entrainment becoming impossible if the object is too long.

JFM classification

Low-Reynolds-number flows: Lubrication theory Biological Fluid Dynamics: Peristaltic pumping Waves/Free-surface Flows: Solitary waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 672 , 10 April 2011 , pp. 219 - 244
Copyright
Copyright © Cambridge University Press 2011

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References

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