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The interaction between rotationally oscillating spheres and solid boundaries in a Stokes flow

Published online by Cambridge University Press:  26 June 2018

F. Box Open the ORCID record for F. Box [Opens in a new window] ,
K. Singh  and
T. Mullin Open the ORCID record for T. Mullin [Opens in a new window]
F. Box*
Affiliation:
Manchester Centre for Nonlinear Dynamics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
K. Singh
Affiliation:
Oxford Centre for Collaborative Applied Mathematics, Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
T. Mullin
Affiliation:
Manchester Centre for Nonlinear Dynamics, School of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK
*
†Email address for correspondence: finn.box@maths.ox.ac.uk
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Abstract

We present the results of an experimental and theoretical investigation into the influence of proximate boundaries on the motion of an rotationally oscillating sphere in a viscous fluid. The angular oscillations of the sphere are controlled using the magnetic torque generated by a spatially uniform, oscillatory magnetic field which interacts with a small magnet embedded within the sphere. We study the motion of the sphere in the vicinity of stationary walls that are parallel and perpendicular to the rotational axis of the sphere, and near a second passive sphere that is non-magnetic and free to move. We find that rigid boundaries introduce viscous resistance to motion that acts to suppress the oscillations of the driven sphere. The amount of viscous resistance depends on the orientation of the wall with respect to the axis of rotation of the oscillating sphere. A passive sphere also introduces viscous resistance to motion, but for this case the rotational oscillations of the active sphere establish a standing wave that imparts vorticity to the fluid and induces oscillations of the passive sphere. The standing wave is analogous to the case of an oscillating plate in a viscous fluid; the amplitude of the wave decays exponentially with radial distance from the surface of the oscillating sphere. The standing wave introduces a phase lag between the motion of the active sphere and the response of the passive sphere which increases linearly with separation distance.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 849 , 25 August 2018 , pp. 834 - 859
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Copyright
© 2018 Cambridge University Press 

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References

Ambari, A., Gauthier-Manuel, B. & Guyon, E. 1984 Wall effects on a sphere translating at constant velocity. J. Fluid Mech. 149, 235–253.Google Scholar
Ardekani, A. & Rangel, R. 2006 Unsteady motion of two solid spheres in Stokes flow. Phys. Fluids 18, 103306.Google Scholar
Besseris, G., Miller, I. F. & Yeates, D. B. 1999 Rotational magnetic particle microrheometry: the Newtonian case. J. Rheol. 43, 591–608.Google Scholar
Blake, J. R. & Chwang, A. T. 1974 Fundamental singularities of viscous flow. J. Engng Maths 8, 23–29.Google Scholar
Box, F., Han, E., Tipton, C. R. & Mullin, T. 2017 On the motion of linked spheres in a Stokes flow. Exp. Fluids 58, 29.Google Scholar
Box, F., Thompson, A. B. & Mullin, T. 2015 Torsional oscillations of a sphere in a Stokes flow. Exp. Fluids 56, 209.Google Scholar
Buchanan, J. 1891 The oscillations of a spheroid in a viscous liquid. Proc. Lond. Math. Soc. 22, 181–214.Google Scholar
Cichocki, B., Felderhof, B. U., Hinsen, K., Wajnryb, E. & Blawzdziewicz, J. 1994 Friction and mobility of many spheres in a Stokes flow. J. Chem. Phys. 100, 3780–3790.Google Scholar
Cichocki, B., Felderhof, B. U. & Schmitz, R. 1988 Hydrodynamic interactions between two spherical particles. Physico-Chem. Hydrodyn. 10, 383–403.Google Scholar
Cox, R. G. & Brenner, H. 1967 The slow motion of a sphere through a viscous fluid towards a plane surface. II. Small gap widths, including inertial effects. Chem. Engng Sci. 22, 1753–1777.Google Scholar
Crocker, J. C. 1997 Measurement of the hydrodynamic correction to the Brownian motion of two colloidal spheres. J. Chem. Phys. 106, 2837–2840.Google Scholar
Dean, W. R. & O’Neill, M. E. 1963 A slow motion of viscous liquid caused by the rotation of a solid sphere. Mathematika 10, 13–24.Google Scholar
Faxén, H. 1922 The resistance against the movement of a rigour sphere in viscous fluids, which is embedded between two parallel layered barriers. Ann. Phys. 68, 89–119.Google Scholar
Goldman, A. J., Cox, R. G. & Brenner, H. 1967 Slow viscous motion of a sphere parallel to a plane wall. I. Motion through a quiescent fluid. Chem. Engng Sci. 22, 637–651.Google Scholar
Guazzelli, E. & Hinch, J. 2010 Fluctuations and instability in sedimentation. Annu. Rev. Fluid Mech. 43, 97–116.Google Scholar
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media, 1st edn. Springer.Google Scholar
Henderson, S. I., Mitchell, S. J. & Bartlett, P. 2001 Direct measurements of colloidal friction coefficients. Phys. Rev. E 64, 061403.Google Scholar
Henderson, S. I., Mitchell, S. J. & Bartlett, P. 2002 Propagation of hydrodynamic interactions in colloidal suspensions. Phys. Rev. Lett. 88, 088302.Google Scholar
Jeffery, G. B. 1915 On the steady rotation of a solid of revolution in a viscous fluid. Proc. Lond. Math. Soc. 14, 327338.Google Scholar
Jeffrey, D. J. 1992 The calculation of the low Reynolds-number resistance functions for two unequal spheres. Phys. Fluids 4, 16–29.Google Scholar
Jeffrey, D. J. & Onishi, Y. 1984a Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds number flow. J. Fluid Mech. 139, 261–290.Google Scholar
Jeffrey, D. J. & Onishi, Y. 1984b The forces and couples acting on two nearly touching spheres in low-Reynolds number flow. Z. Angew. Math. Phys. 35, 634–641.Google Scholar
Kestin, J. & Persen, L. N.1954 Small oscillations of bodies of revolution in a viscous fluid. Brown University Report AF-891/2, Contract A718(600)-891.Google Scholar
Kim, I. & Miffin, R. T. 1985 The resistance and mobility functions of two equal spheres in low-Reynolds-number flow. Phys. Fluids 28, 2033–2045.Google Scholar
Kim, S. & Karrila, S. J. 1991 Microhydrodynamics: Principles and Selected Applications, 1st edn. Butterworth-Heinemann.Google Scholar
Kim, S. & Russel, W. 1985 Modelling of porous media by renormalization of the Stokes equations. J. Fluid Mech. 154, 269–286.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Dover.Google Scholar
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601.Google Scholar
Liu, Q. & Prosperetti, A. 2010 Wall effects on a rotating sphere. J. Fluid Mech. 657, 1–21.Google Scholar
Meiners, J. C. & Quake, S. R. 1999 Direct measurement of the hydrodynamic cross-correlations between two particles in an external potential. Phys. Rev. Lett. 82, 2211–2214.Google Scholar
Parkin, S. J., Knoner, G., Nieminen, T. A., Heckenberg, N. R. & Rubinsztein-Dunlop, H. 2007 Picoliter viscometry using optically rotated particles. Phys. Rev. E 76, 041507.Google Scholar
Pozrikidis, C. 1989 A singularity method for unsteady linearized flow. Phys. Fluids 1, 1508.Google Scholar
Steinbach, G., Gemmin, S. & Erbe, A. 2016 Rotational friction of dipolar colloids measured by driven torsional oscillations. Sci. Rep. 6, 34193.Google Scholar
Stimson, M. & Jeffrey, G. B. 1926 The motion of two spheres in a viscous fluid. Proc. R. Soc. Lond. A 111, 110–116.Google Scholar
Stuart, J. T. 1963 Unsteady boundary layers. In Laminar Boundary Layers (ed. Rosenhead, L.), pp. 349–408. Oxford University Press.Google Scholar
Trankle, B., Speidel, M. & Rohrbach, A. 2012 Interaction dynamics of two colloids in a single optical potential. Phys. Rev. E 86, 021401.Google Scholar
Ye, Z., Diller, E. & Sitti, M. 2012 Micro-manipulation using rotational fluid flows induced by remote magnetic micro-manipulators. J. Appl. Phys. 112, 064912.Google Scholar