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Sedimentation of spheroidal bodies near walls in viscous fluids: glancing, reversing, tumbling and sliding

Published online by Cambridge University Press:  11 May 2015

William H. Mitchell  and
Saverio E. Spagnolie
William H. Mitchell
Affiliation:
Department of Mathematics, University of Wisconsin – Madison, 480 Lincoln Drive, Madison, WI 53706, USA
Saverio E. Spagnolie*
Affiliation:
Department of Mathematics, University of Wisconsin – Madison, 480 Lincoln Drive, Madison, WI 53706, USA
*
†Email address for correspondence: spagnolie@math.wisc.edu
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Abstract

The sedimentation of a rigid particle near a wall in a viscous fluid has been studied numerically by many authors, but analytical solutions have been derived only for special cases such as the motion of spherical particles. In this paper the method of images is used to derive simple ordinary differential equations describing the sedimentation of arbitrarily oriented prolate and oblate spheroids at zero Reynolds number near a vertical or inclined plane wall. The differential equations may be solved analytically in many situations, and full trajectories are predicted which compare favourably with complete numerical simulations. The simulations are performed using a novel double-layer boundary integral formulation, a method of stresslet images. The conditions under which the glancing and reversing trajectories, first observed by Russel et al. (J. Fluid Mech., vol. 83, 1977, pp. 273–287), occur are studied for bodies of arbitrary aspect ratio. Several additional trajectories are also described: a periodic tumbling trajectory for nearly spherical bodies, a linearly stable sliding trajectory which appears when the wall is slightly inclined, and three-dimensional glancing, reversing and wobbling.

JFM classification

Low-Reynolds-number flows: Boundary integral methods Low-Reynolds-number flows: Low-Reynolds-number flows Low-Reynolds-number flows: Stokesian dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 772 , 10 June 2015 , pp. 600 - 629
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Copyright
© 2015 Cambridge University Press 

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Mitchell supplementary movie

Sedimentation near walls in viscous fluids. Six trajectories are shown: three-dimensional glancing and three-dimensional reversing of both prolate and oblate bodies, and periodic tumbling with a lateral wobble of nearly spherical prolate and oblate bodies.

Download Mitchell supplementary movie(Video)
Video 5.7 MB