Skip to main content Accessibility help
×
Home

Peristaltic pumping of rigid objects in an elastic tube

  • D. TAKAGI (a1) and N. J. BALMFORTH (a2) (a3)

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.

Copyright

Corresponding author

Email address for correspondence: njb@math.ubc.ca

References

Hide All
Bertuzzi, A., Salinari, S., Mancinelli, R. & Pescatori, M. 1983 Peristaltic transport of a solid bolus. J. Biomech. 16 (7), 459464.
Carew, E. O. & Pedley, T. J. 1997 An active membrane model for peristaltic pumping. Part 1. Periodic activation waves in an infinite tube. J. Biomech. Engng 119, 6676.
Connington, K., Kang, Q., Viswanathan, H., Abdel-Fattah, A. & Chen, S. 2009 Peristaltic particle transport using the lattice Boltzmann method. Phys. Fluids 21, 053301.
Cummings, L. J., Waters, S. L., Wattis, J. A. D. & Graham, S. J. 2004 The effect of ureteric stents on urine flow: reflux. J. Math. Biol. 49 (1), 5682.
Eytan, O. & Elad, D. 1999 Analysis of intra-uterine fluid motion induced by uterine contractions. Bull. Math. Biol. 61, 221238.
Fauci, L. J. 1992 Peristaltic pumping of solid particles. Comput. Fluids 21, 583598.
Fauci, L. J. & Dillon, R. 2006 Biofluidmechanics of reproduction. Annu. Rev. Fluid Mech. 38, 371394.
Fitz-Gerald, J. M. 1969 Mechanics of red-cell motion through very narrow capillaries. Proc. R. Soc. Lond. B 174 (1035), 193227.
Fung, Y. C. 1971 Peristaltic pumping: a bioengineering model. In Urodynamics: Hydrodynamics of the Ureter and Renal Pelvis (ed. Boyarsky, S., Gottschalk, C. W., Tanagho, E. A. & Zimskind, P. D.), pp. 178198. Academic.
Hung, T. K. & Brown, T. D. 1976 Solid-particle motion in two-dimensional peristaltic flows. J. Fluid Mech. 73, 7796.
Jiménez-Lozano, J., Sen, M. & Dunn, P. F. 2009 Particle motion in unsteady two-dimensional peristaltic flow with application to the ureter. Phys. Rev. E 79 (4), 41901.
Lighthill, M. J. 1968 Pressure-forcing of tightly fitting pellets along fluid-filled elastic tubes. J. Fluid Mech. 34 (1), 113143.
Shapiro, A. H., Jaffrin, M. Y. & Weinberg, S. L. 1969 Peristaltic pumping with long wavelengths at low Reynolds number. J. Fluid Mech. 37 (4), 799825.
Siggers, J. H., Waters, S., Wattis, J. & Cummings, L. 2008 Flow dynamics in a stented ureter. Math. Med. Biol. 26, 124.
Srivastava, L. M. & Srivastava, V. P. 1989 Peristaltic transport of a particle-fluid suspension. J. Biomech. Engng 111, 157165.
Skotheim, J. & Mahadevan, L. 2005 Soft lubrication: the elastohydrodynamics of conforming and non-conforming contacts. Phys. Fluids, 17, 092101.
Takagi, D. & Balmforth, N. J. 2011 Peristaltic pumping of viscous fluid in an elastic tube. J. Fluid Mech 672, 196218.
Thiele, U. & Knobloch, E. 2006 Driven drops on heterogeneous surfaces: onset of sliding motion. Phys. Rev. Lett. 97, 204501.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Peristaltic pumping of rigid objects in an elastic tube

  • D. TAKAGI (a1) and N. J. BALMFORTH (a2) (a3)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed