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Translating and squirming cylinders in a viscoplastic fluid

  • R. Supekar (a1), D. R. Hewitt (a2) and N. J. Balmforth (a3)


Three related problems of viscoplastic flow around cylinders are considered. First, translating cylinders with no-slip surfaces appear to generate adjacent rotating plugs in the limit where the translation speed becomes vanishingly small. In this plastic limit, analytical results are available from plasticity theory (slipline theory) which indicate that no such plugs should exist. Using a combination of numerical computations and asymptotic analysis, we show that the plugs of the viscoplastic theory actually disappear in the plastic limit, albeit very slowly. Second, when the boundary condition on the cylinder is replaced by one that permits sliding, the plastic limit corresponds to a partially rough cylinder. In this case, no plasticity solution has been previously established; we provide evidence from numerical computations and slipline theory that a previously proposed upper bound (Martin & Randolph, Geotechnique, vol. 56, 2006, pp. 141–145) is actually the true plastic solution. Third, we consider how a prescribed surface velocity field can propel cylindrical squirmers through a viscoplastic fluid. We determine swimming speeds and contrast the results with those from the corresponding Newtonian problem.


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Adachi, K. & Yoshioka, N. 1973 On creeping flow of a visco-plastic fluid past a circular cylinder. Chem. Engng Sci. 28 (1), 215226.
Balmforth, N. J., Craster, R. V., Hewitt, D. R., Hormozi, S. & Maleki, A. 2017 Viscoplastic boundary layers. J. Fluid Mech. 813, 929954.
Balmforth, N. J., Frigaard, I. A. & Ovarlez, G. 2014 Yielding to stress: recent developments in viscoplastic fluid mechanics. Annu. Rev. Fluid Mech. 46, 121146.
Barnes, H. A. 1995 A review of the slip (wall depletion) of polymer solutions, emulsions and particle suspensions in viscometers: its cause, character, and cure. J. Non-Newtonian Fluid Mech. 56 (3), 221251.
Blake, J. R. 1971a A spherical envelope approach to ciliary propulsion. J. Fluid Mech. 46 (1), 199208.
Blake, J. R. 1971b Self propulsion due to oscillations on the surface of a cylinder at low Reynolds number. Bull. Aust. Math. Soc. 5 (2), 255264.
Brookes, G. F. & Whitmore, R. L. 1969 Drag forces in bingham plastics. Rheol. Acta 8 (4), 472480.
Chaparian, E. & Frigaard, I. A. 2017 Yield limit analysis of particle motion in a yield-stress fluid. J. Fluid Mech. 819, 311351.
Clarke, R. J., Finn, M. D. & MacDonald, M. 2014 Hydrodynamic persistence within very dilute two-dimensional suspensions of squirmers. Proc. R. Soc. Lond. A 470 (2167), 20130508.
Crowdy, D. G. & Or, Y. 2010 Two-dimensional point singularity model of a low-Reynolds-number swimmer near a wall. Phys. Rev. E 81 (3), 036313.
Ding, Y., Gravish, N. & Goldman, D. I. 2011 Drag induced lift in granular media. Phys. Rev. Lett. 106 (2), 028001.
Harlen, O. G. 2002 The negative wake behind a sphere sedimenting through a viscoelastic fluid. J. Non-Newtonian Fluid Mech. 108 (1–3), 411430.
Hewitt, D. R. & Balmforth, N. J. 2017 Taylor’s swimming sheet in a yield-stress fluid. J. Fluid Mech. 828, 3356.
Hewitt, D. R. & Balmforth, N. J. 2018 Viscoplastic slender body theory. J. Fluid Mech. 856, 870897.
Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.
Hosoi, A. E. & Goldman, D. I. 2015 Beneath our feet: strategies for locomotion in granular media. Annu. Rev. Fluid Mech. 47, 431453.
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72 (9), 096601.
Lighthill, M. J. 1952 On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers. Commun. Pure Appl. Maths 5 (2), 109118.
Martin, C. M. & Randolph, M. F. 2006 Upper-bound analysis of lateral pile capacity in cohesive soil. Geotechnique 56 (2), 141145.
Murff, J. D., Wagner, D. A. & Randolph, M. F. 1989 Pipe penetration in cohesive soil. Géotechnique 39 (2), 213229.
Ouyang, Z., Lin, J. & Ku, X. 2018 The hydrodynamic behavior of a squirmer swimming in power-law fluid. Phys. Fluids 30 (8), 083301.
Ozogul, H., Jay, P. & Magnin, A. 2015 Slipping of a viscoplastic fluid flowing on a circular cylinder. J. Fluids Engng 137 (7), 071201.
Pedley, T. J. 2016 Spherical squirmers: models for swimming micro-organisms. IMA J. Appl. Maths 81 (3), 488521.
Prager, W. & Hodge, P. G. 1951 Theory of Perfectly Plastic Solids. Wiley.
Randolph, M. F. & Houlsby, G. T. 1984 The limiting pressure on a circular pile loaded laterally in cohesive soil. Géotechnique 34, 613623.
Roquet, N. & Saramito, P. 2003 An adaptive finite element method for Bingham fluid flows around a cylinder. Comput. Meth. Appl. Mech. Engng 192, 33173341.
Spagnolie, S. E. & Lauga, E. 2010 Jet propulsion without inertia. Phys. Fluids 22 (8), 081902.
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, 6576.
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, 3544.
Ultman, J. S. & Denn, M. M. 1971 Slow viscoelastic flow past submerged objects. Chem. Engng J. 2 (2), 8189.
Yazdi, S., Ardekani, A. M. & Borhan, A. 2014 Locomotion of microorganisms near a no-slip boundary in a viscoelastic fluid. Phys. Rev. E 90 (4), 111.
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Translating and squirming cylinders in a viscoplastic fluid

  • R. Supekar (a1), D. R. Hewitt (a2) and N. J. Balmforth (a3)


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