Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-22T09:09:54.786Z Has data issue: false hasContentIssue false
  • English
  • Français

Viscoplastic slender-body theory

Published online by Cambridge University Press:  11 October 2018

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
Get access Rights & Permissions [Opens in a new window]

Abstract

The theory of slow viscous flow around a slender body is generalized to the situation where the ambient fluid has a yield stress. The local flow around a cylinder that is moving along or perpendicular to its axis, and rotating, provides a first step in this theory. Unlike for a Newtonian fluid, the nonlinearity associated with the viscoplastic constitutive law precludes one from linearly superposing solutions corresponding to each independent component of motion, and instead demands a full numerical approach to the problem. This is accomplished for the case of a Bingham fluid, along with a consideration of some asymptotic limits in which analytical progress is possible. Since the yield stress of the fluid strongly localizes the flow around the body, the leading-order slender-body approximation is rendered significantly more accurate than the equivalent Newtonian problem. The theory is applied to the sedimentation of inclined cylinders, bent rods and helices, and compared with some experimental data. Finally, the theory is applied to the locomotion of a cylindrical filament driven by helical waves through a viscoplastic fluid.

JFM classification

Non-Newtonian Flows: Non-Newtonian Flows Non-Newtonian Flows: Plastic materials Low-Reynolds-number flows: Slender-body theory
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 856 , 10 December 2018 , pp. 870 - 897
Check for updates
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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
Gray, J. & Hancock, G. J. 1979 The propulsion of sea-urchin spermatoza. Biophys. J. 25, 113–127.Google Scholar
Hancock, G. J. 1953 The self-propulsion of microscopic organisms through liquids. Proc. R. Soc. Lond. A 217, 96–121.Google Scholar
Hewitt, D. R. & Balmforth, N. J. 2017 Taylor’s swimming sheet in a yield-stress fluid. J. Fluid Mech. 828, 33–56.Google Scholar
Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.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
Jalaal, M., Balmforth, N. J. & Stoeber, B. 2015 Slip of spreading viscoplastic droplets. Langmuir 31, 12071–12075.Google Scholar
Jossic, L. & Magnin, A. 2001 Drag and stability of objects in a yield stress fluid. AIChE J. 47, 2666–2672.Google Scholar
Keller, J. B. & Rubinow, S. I. 1976 Slender-body theory for slow viscous flow. J. Fluid Mech. 75 (4), 705–714.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72 (9), 096601.Google Scholar
Lighthill, S. J. 1975 Mathematical Biofluiddynamics. SIAM.Google Scholar
Madani, A., Storey, S., Olson, J. A., Frigaard, I. A., Salmela, J. & Martinez, D. M. 2010 Fractionation of non-Brownian rod-like particle suspensions in a viscoplastic fluid. Chem. Engng Sci. 65 (5), 1762–1772.Google Scholar
Oldroyd, J. G. 1947 Two-dimensional plastic flow of a Bingham solid: a plastic boundary-layer theory for slow motion. Proc. Camb. Phil. Soc. 43, 383–395.Google Scholar
Piau, J.-M. 2002 Viscoplastic boundary layer. J. Non-Newtonian Fluid Mech. 102, 193–218.Google Scholar
Poumaere, A., Moyers-González, M., Castelain, C. & Burghelea, T. 2014 Unsteady laminar flows of a Carbopol gel in the presence of wall slip. J. Non-Newtonian Fluid Mech. 205, 28–40.Google Scholar
Putz, A. M. V., Burghelea, T. I., Frigaard, I. A. & Martinez, D. M. 2008 Settling of an isolated spherical particle in a yield stress shear thinning fluid. Phys. Fluids 20 (3), 033102.Google Scholar
Randolph, M. F. & Houlsby, G. T. 1984 The limiting pressure on a circular pile loaded laterally in cohesive soil. Géotechnique 34, 613–623.Google Scholar
Roquet, N. & Saramito, P. 2003 An adaptive finite element method for Bingham fluid flows around a cylinder. Comput. Meth. Appl. Mech. Engng 192, 3317–3341.Google Scholar
Saramito, P. & Wachs, A. 2017 Progress in numerical simulation of yield stress fluid flows. Rheol. Acta 56, 211–230.Google Scholar
Tabuteau, H., Coussot, P. & de Bruyn, J. R. 2007 Drag force on a sphere in steady motion through a yield-stress fluid. J. Rheol. 51, 125–137.Google Scholar
Taylor, G. I. 1952 The action of waving cylindrical tails in propelling microscopic organisms. Proc. R. Soc. Lond. A 211, 225–239.Google Scholar
Tokpavi, D. L., Magnin, A. & Jay, P. 2008 Very slow flow of Bingham viscoplastic fluid around a circular cylinder. J. Non-Newtonian Fluid Mech. 154, 65–76.Google Scholar
Tokpavi, D. L., Magnin, A., Jay, P. & Jossic, L. 2009 Experimental study of the very slow flow of a yield stress fluid around a circular cylinder. J. Non-Newtonian Fluid Mech. 164, 35–44.Google Scholar
Tornberg, A.-K. & Shelley, M. J. 2004 Simulating the dynamics and interactions of flexible fibers in Stokes flows. J. Comput. Phys. 196 (1), 8–40.Google Scholar
Wolgemuth, C. W., Charon, N. W., Goldstein, S. F. & Goldstein, R. E. 2006 The flagellar cytoskeleton of the spirochetes. J. Mol. Microbiol. Biotechnol. 11, 221–227.Google Scholar