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Peristaltic pumping of viscous fluid 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 model is derived for long peristaltic waves propagating steadily down a fluid-filled, axisymmetric tube. The waves are driven by imposing a radial force of prescribed form on the tube. The resulting deformation of the tube wall is modelled using linear elasticity and the internal flow using the lubrication approximation. Numerical solutions for periodic wave trains and solitary waves are presented, along with asymptotic solutions at both small and large forcing amplitudes. Large-amplitude periodic waves are characterized by narrow blisters adjoining long occluded sections of the tube, whereas a solitary wave of strong contraction produces a long inflated bow wave that propels a large quantity of fluid. A measure of pumping efficacy is given by the ratio of the net fluid displacement to the power input, and is highest for a large-amplitude solitary wave.

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. 196 - 218
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Argentina, M., Skotheim, J. & Mahadevan, L. 2007 Settling and swimming of flexible fluid-lubricated foils. Phys. Rev. Lett. 99 (22), 224503.CrossRefGoogle ScholarPubMed
Ashmore, J., Hosoi, A. E. & Stone, H. A. 2003 The effect of surface tension on rimming flows in a partially filled rotating cylinder. J. Fluid Mech. 479, 65–98.CrossRefGoogle Scholar
Balmforth, N. J., Coombs, D. & Pachman, S. 2010 Microelastohydrodynamics of swimming organisms near solid boundaries in complex fluids. Q. J. Mech. Appl. Maths 63, 267–294.Google Scholar
Bohme, G. & Friedrich, R. 1983 Peristaltic flow of viscoelastic liquids. J. Fluid Mech. 128, 109–122.CrossRefGoogle Scholar
Brasseur, J. 1987 A fluid mechanical perspective on esophageal bolus transport. Dysphagia 2 (1), 32–39.CrossRefGoogle ScholarPubMed
Carew, E. O. & Pedley, T. J. 1997 An active membrane model for peristaltic pumping. Part I. Periodic activation waves in an infinite tube. J. Biomech. Engng 119, 66–76.CrossRefGoogle Scholar
Chan, D., Balmforth, N. J. & Hosoi, A. 2005 Building a better snail: lubrication theory and adhesive locomotion. Phys. Fluids, 17, 113101.CrossRefGoogle Scholar
Cowley, S. J. 1982 Elastic jumps on fluid-filled elastic tubes. J. Fluid Mech. 116, 459–473.Google Scholar
Fung, Y. C. 1971 Peristaltic pumping: a bioengineering model. In Urodynamics: Hydrodynamics of the Ureter and Renal Pelvis (ed. Boyarsky, S. et al. .), pp. 178–198. Academic.Google Scholar
Fung, Y. C. & Yih, C. S. 1968 Peristaltic transport. J. Appl. Mech. 35, 669–675.CrossRefGoogle Scholar
Griffiths, D. J. 1987 Dynamics of the upper urinary tract. Part I. Peristaltic flow through a distensible tube of limited length. Phys. Med. Biol. 32, 813–822.CrossRefGoogle Scholar
Griffiths, D. J. 1989 Flow of urine through the ureter: a collapsible, muscular tube undergoing peristalsis. J. Biomech. Engng 111, 206–211.CrossRefGoogle ScholarPubMed
Katz, D. F. 1974 On the propulsion of micro-organisms near solid boundaries. J. Fluid Mech. 64, 33–49.CrossRefGoogle Scholar
Keller, J. & Falkovitz, M. 1983 Crawling of worms. J. Theor. Biol. 104 (3), 417–442.Google Scholar
Kerr, A. D. 1984 On the formal development of elastic foundation models. Ing.-Archiv. 64, 455–464.CrossRefGoogle Scholar
Kriegsmann, J. J., Miksis, M. J. & Vanden-Broeck, J. M. 1998 Pressure driven disturbances on a thin viscous film. Phys. Fluids 10, 1249.CrossRefGoogle Scholar
Li, M. & Brasseur, J. G. 1993 Non-steady peristaltic transport in finite-length tubes. J. Fluid Mech. 248, 129–151.CrossRefGoogle Scholar
Lister, J. R. 1992 Viscous flows down an inclined plane from point and line sources. J. Fluid Mech. 242, 631–653.CrossRefGoogle Scholar
Lykoudis, P. & Roos, R. 1970 The fluid mechanics of the ureter from a lubrication theory point of view. J. Fluid Mech. 43 (4), 661–674.CrossRefGoogle Scholar
Miftakhov, R. & Wingate, D. 1994 Numerical simulation of the peristaltic reflex of the small bowel. Biorheology 31 (4), 309–325.CrossRefGoogle ScholarPubMed
Pozrikidis, C. 1987 A study of peristaltic flow. J. Fluid Mech. 180, 515–527.CrossRefGoogle Scholar
Shapiro, A. H., Jaffrin, M. Y. & Weinberg, S. L. 1969 Peristaltic pumping with long wavelengths at low Reynolds number. J. Fluid Mech. 37, 799–825.CrossRefGoogle Scholar
Skotheim, J. & Mahadevan, L. 2005 Soft lubrication: the elastohydrodynamics of conforming and non-conforming contacts. Phys. Fluids 17, 092101.Google Scholar
Szeri, A. J., Park, S. C., Verguet, S., Weiss, A. & Katz, D. F. 2008 A model of transluminal flow of an anti-HIV microbicide vehicle: combined elastic squeezing and gravitational sliding. Phys. Fluids 20, 083101.CrossRefGoogle Scholar
Takabatake, S., Ayukawa, K. & Mori, A. 1969 Peristaltic pumping in circular cylindrical tubes: a numerical study of fluid transport and its efficiency. J. Fluid Mech. 193, 267–283.Google Scholar
Takagi, D. 2009 Nonlinear peristaltic waves: a bitter pill to swallow. In Proceedings of the 2009 Summer Study Program in Geophysical Fluid Dynamics, Woods Hole Oceanographic Institution, USA.Google Scholar
Takagi, D. & Balmforth, N. J. 2011 Peristaltic pumping of rigid objects in an elastic tube. J. Fluid Mech. 672, 219–244.Google Scholar
Tang, D. & Rankin, S. 1993 Numerical and asymptotic solutions for peristaltic motion of nonlinear viscous flows with elastic free boundaries. SIAM J. Sci. Comput. 14, 1300–1319.CrossRefGoogle Scholar
Taylor, G. 1951 Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond. A 209, 447–461.Google Scholar
Timoshenko, S. P. & Woinowsky-Krieger, S. 1959 Theory of Plates and Shells. McGraw-Hill.Google Scholar
Vajravelu, K., Sreenadh, S. & RameshBabu, V. Babu, V. 2005 Peristaltic transport of a Herschel–Bulkley fluid in an inclined tube. Intl J. Nonlinear Mech. 40 (1), 83–90.CrossRefGoogle Scholar
Walker, S. W. & Shelley, M. J. 2010 Shape optimization of peristaltic pumping. J. Comput. Phys. 229, 1260–1291.CrossRefGoogle Scholar
Yin, F. C. & Fung, Y. C. 1971 Mechanical properties of isolated mammalian ureteral segments. Am. J. Physiol. 221, 1484–1493.CrossRefGoogle ScholarPubMed