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A first-principle predictive theory for a sphere falling through sharply stratified fluid at low Reynolds number

Published online by Cambridge University Press:  12 October 2010

ROBERTO CAMASSA ,
CLAUDIA FALCON ,
JOYCE LIN ,
RICHARD M. McLAUGHLIN  and
NICHOLAS MYKINS
ROBERTO CAMASSA
Affiliation:
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA
CLAUDIA FALCON
Affiliation:
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA
JOYCE LIN*
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA
RICHARD M. McLAUGHLIN
Affiliation:
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA
NICHOLAS MYKINS
Affiliation:
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA
*
†Email address for correspondence: [email protected]
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Abstract

A sphere exhibits a prolonged residence time when settling through a stable stratification of miscible fluids due to the deformation of the fluid-density field. Using a Green's function formulation, a first-principles numerically assisted theoretical model for the sphere–fluid coupled dynamics at low Reynolds number is derived. Predictions of the model, which uses no adjustable parameters, are compared with data from an experimental investigation with spheres of varying sizes and densities settling in stratified corn syrup. The velocity of the sphere as well as the deformation of the density field are tracked using time-lapse images, then compared with the theoretical predictions. A settling rate comparison with spheres in dense homogeneous fluid additionally quantifies the effect of the enhanced residence time. Analysis of our theory identifies parametric trends, which are also partially explored in the experiments, further confirming the predictive capability of the theoretical model. The limit of infinite fluid domain is considered, showing evidence that the Stokes paradox of infinite fluid volume dragged by a moving sphere can be regularized by density stratifications. Comparisons with other possible models under a hierarchy of additional simplifying assumptions are also presented.

JFM classification

Low-Reynolds-number flows: Stokesian dynamics Geophysical and Geological Flows: Stratified flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 664 , 10 December 2010 , pp. 436 - 465
Copyright
Copyright © Cambridge University Press 2010

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