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Taylor’s swimming sheet in a yield-stress fluid

Published online by Cambridge University Press:  30 August 2017

D. R. Hewitt Open the ORCID record for D. R. Hewitt [Opens in a new window]  and
N. J. Balmforth Open the ORCID record for N. J. Balmforth [Opens in a new window]
D. R. Hewitt*
Affiliation:
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, Vancouver, BC, V6T 1Z2, Canada
*
†Email address for correspondence: drh39@cam.ac.uk
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Abstract

A yield stress is added to Taylor’s (Proc. R. Soc. Lond. A, vol. 209, 1951, pp. 447–461) model of a two-dimensional flexible sheet swimming through a viscous fluid. Both transverse waves along the sheet, as in Taylor’s original model, and longitudinal waves are considered as means of locomotion. In each case, numerical solutions are provided over a range of the two key parameters of the problem: the wave amplitude relative to the wavelength and a Bingham number which describes the strength of the yield stress. The numerical solutions are supplemented with discussions of various limits of the problem in which analytical progress is possible. When the yield stress is large, the swimming speed for low wave amplitude is exactly double that for a Newtonian fluid, for either type of wave.

JFM classification

Non-Newtonian Flows: Non-Newtonian Flows Non-Newtonian Flows: Plastic materials Biological Fluid Dynamics: Propulsion
Type
Papers
Information
Journal of Fluid Mechanics , Volume 828 , 10 October 2017 , pp. 33 - 56
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Copyright
© 2017 Cambridge University Press 

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References

Balmforth, N. J., Craster, R. V., Hewitt, D. R., Hormozi, S. & Maleki, A. 2017 Viscoplastic boundary layers. J. Fluid Mech. 813, 929–954.Google Scholar
Bansil, R., Celli, J., Hardcastle, J. & Turner, B. 2013 The influence of mucus microstructure and rheology in Helicobacter pylori infection. Front. Immunol. 4, 310.Google Scholar
Blake, J. R. 1971 Infinite models for ciliary propulsion. J. Fluid Mech. 49 (02), 209–222.Google Scholar
Chaudhury, T. K. 1979 On swimming in a visco-elastic liquid. J. Fluid Mech. 95, 189–197.Google Scholar
Denny, M. W. 1980 The role of gastropod mucus in locomotion. Nature 285, 160–161.CrossRefGoogle Scholar
Elfring, G. J. & Goyal, G. 2016 The effect of gait on swimming in viscoelastic fluids. J. Non-Newtonian Fluid Mech. 234, 8–14.Google Scholar
Elfring, G. J., Pak, O. S. & Lauga, E. 2010 Two-dimensional flagellar synchronization in viscoelastic fluids. J. Fluid Mech. 646, 505–515.Google Scholar
Espinosa-Garcia, J., Lauga, E. & Zenit, R. 2013 Fluid elasticity increases the locomotion of flexible swimmers. Phys. Fluids 25 (3), 031701.Google Scholar
Fortin, M. & Glowinski, R. 2006 Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Value Problems. North-Holland.Google Scholar
Fu, H. C., Shenoy, V. B. & Powers, T. R. 2010 Low-Reynolds-number swimming in gels. Europhys. Lett. 91 (2), 24002.Google Scholar
Fulford, G. R., Katz, D. F. & Powell, R. L. 1998 Swimming of spermatozoa in a linear viscoelastic fluid. Biorheology 35 (4, 5), 295–309.Google Scholar
Hosoi, A. E. & Goldman, D. I. 2015 Beneath our feet: strategies for locomotion in granular media. Annu. Rev. Fluid Mech. 47, 431–453.Google Scholar
Katz, D. F. 1974 On the propulsion of micro-organisms near solid boundaries. J. Fluid Mech. 64, 33–49.Google Scholar
Krieger, M. S., Spagnolie, S. E. & Powers, T. R. 2014 Locomotion and transport in a hexatic liquid crystal. Phys. Rev. E 90, 052503.Google Scholar
Lauga, E. 2007 Propulsion in a viscoelastic fluid. Phys. Fluids 19 (8), 083104.CrossRefGoogle Scholar
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72 (9), 096601.Google Scholar
Li, G. & Ardekani, A. M. 2015 Undulatory swimming in non-Newtonian fluids. J. Fluid Mech. 784, R4.Google Scholar
Liu, Y., Balmforth, N. J., Hormozi, S. & Hewitt, D. R. 2016 Two–dimensional viscoplastic dambreaks. J. Non-Newtonian Fluid Mech. 238, 65–79.Google Scholar
McInroe, B., Astley, H. C., Gong, C., Kawano, S. M., Schiebel, P. E., Rieser, J. M., Choset, H., Blob, R. W. & Goldman, D. I. 2016 Tail use improves performance on soft substrates in models of early vertebrate land locomotors. Science 353 (6295), 154–158.Google Scholar
Neumeyer, M. J. & Jones, B. D. 1965 The marsh screw amphibian. J. Terramech. 2, 83–88.Google Scholar
Pegler, S. S. & Balmforth, N. J. 2013 Locomotion over a viscoplastic film. J. Fluid Mech. 727, 1–29.Google Scholar
Prager, W. & Hodge, P. G. 1951 Theory of Perfectly Plastic Solids. Wiley.Google Scholar
Riley, E. E. & Lauga, E. 2014 Enhanced active swimming in viscoelastic fluids. Europhys. Lett. 108, 34003.Google Scholar
Sauzade, M., Elfring, G. J. & Lauga, E. 2011 Taylor’s swimming sheet: analysis and improvement of the perturbation series. Physica D 240 (20), 1567–1573.Google Scholar
Taylor, G. I. 1951 Analysis of the swimming of microscopic organisms. Proc. R. Soc. Lond. A 209, 447–461.Google Scholar
Tuck, E. O. 1968 A note on a swimming problem. J. Fluid Mech. 31, 305–308.Google Scholar
Valdastri, P., Simi, M. & Webster, R. J. 2012 Advanced technologies for gastrointestinal endoscopy. Annu. Rev. Biomed. Engng 14, 397–429.Google Scholar
Vélez-Cordero, J. R. & Lauga, E. 2013 Waving transport and propulsion in a generalized Newtonian fluid. J. Non-Newtonian Fluid Mech. 199, 37–50.Google Scholar
Vinay, G., Wachs, A. & Agassant, J.-F. 2005 Numerical simulation of non-isothermal viscoplastic waxy crude oil flows. J. Non-Newtonian Fluid Mech. 128, 144–162.Google Scholar