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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: joyce.lin@utah.edu
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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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References

REFERENCES

Abaid, N., Adalsteinsson, D., Agyapong, A. & McLaughlin, R. M. 2004 An internal splash: levitation of falling spheres in stratified fluids. Phys. Fluids 16 (5), 1567–1580.CrossRefGoogle Scholar
Akers, B. & Belmonte, A. 2006 Impact dynamics of a solid sphere falling into a viscoelastic micellar fluid. J. Non-Newtonian Fluid Mech. 135, 97–108.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bercovici, D. & Mahoney, J. 1994 Double flood basalts and plume head separation at the 660-kilometer discontinuity. Science 266, 1367–1369.CrossRefGoogle ScholarPubMed
Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Engng Sci. 16, 242–251.CrossRefGoogle Scholar
Camassa, R., Falcon, C., Lin, J., McLaughlin, R. M. & Parker, R. 2009 Prolonged residence times for particles settling through stratified miscible fluids in the Stokes regime. Phys. Fluids 21, 031702-1–4.CrossRefGoogle Scholar
Camassa, R., McLaughlin, R. M., Moore, M. & Vaidya, A. 2008 Brachistochrone paths in potential and stokes flow past a sphere. Phys. Lett. A 372, 6742–6749.CrossRefGoogle Scholar
Condie, S. A. & Bormans, M. 1997 The influence of density stratification on particle settling, dispersion and population growth. J. Theor. Biol. 187, 65–75.CrossRefGoogle Scholar
Darwin, C. 1953 Note on hydrodynamics. Proc. Camb. Phil. Soc. 49, 342–254.CrossRefGoogle Scholar
Denman, K. L. & Gargett, A. E. 1995 Biological–physical interactions in the upper ocean: the role of vertical and small scale transport processes. Annu. Rev. Fluid Mech. 27, 225–255.CrossRefGoogle Scholar
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics with Special Applications to Particulate Media. Noordhoff International Publishing.Google Scholar
Happel, J. & Byrne, B. J. 1954 Motion of a sphere and fluid in a cylindrical tube. Ind. Engng Chem. 46 (6), 1181–1186.CrossRefGoogle Scholar
Higdon, J. J. L. 1979 A hydrodynamic analysis of flagellar propulsion. J. Fluid Mech. 90 (4), 685–711.CrossRefGoogle Scholar
Jurine, D., Jaupart, C., Brandeis, G. & Tackley, P. J. 2005 Penetration of mantle plumes through depleted lithosphere. J. Geophys. Res. 110, B10104.1–B10104.18.Google Scholar
Kellogg, W. W. 1980 Aerosols and climate. In Interaction of Energy and Climate (ed. Bach, W., Prankrath, J. & Williams, J.), pp. 281–296. D. Reidel.CrossRefGoogle Scholar
Lin, J. 2009 An experimental and mathematical study on the prolonged residence time of a sphere falling through stratified fluids at low Reynolds number. PhD thesis, University North Carolina, North Carolina, NC.Google Scholar
Linton, C. M. 1995 Multipole methods for boundary-value problems involving a sphere in a tube. IMA J. Appl. Maths 55, 187–204.CrossRefGoogle Scholar
Ma, T. & Wang, S. 2001 A generalized Poincaré–Hopf index formula and its applications to 2-D incompressible flows. Nonlinear Anal. Real World Appl. 2, 467–482.Google Scholar
MacIntyre, S., Alldredge, A. L. & Gottschalk, C. C. 1995 Accumulation of marine snow at density discontinuities in the water column. Limnol. Oceanogr. 40 (3), 449–468.CrossRefGoogle Scholar
Manga, M. & Stone, H. A. 1995 Low Reynolds number motion of bubbles, drops and rigid spheres through fluid–fluid interfaces. J. Fluid Mech. 287, 279–298.CrossRefGoogle Scholar
Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a non-uniform flow. Phys. Fluids 26 (4), 883–889.CrossRefGoogle Scholar
Oseen, C. W. 1927 Hydrodynamik. Academische Verlagsgesellschaft M. B. H.Google Scholar
Parsons, J. D., Bush, J. W. M. & Syvitski, J. P. M. 2001 Hyperpycnal plume formation from riverine outflows with small sediment concentrations. Sedimentology 48, 465–478.CrossRefGoogle Scholar
Rard, J. A. & Miller, D. G. 1979 The mutual diffusion coefficients of NaCl–H2O and CaCl2–H2O at 25°C from Rayleigh interferometry. J. Sol. Chem. 8, 701–716.CrossRefGoogle Scholar
Srdić-Mitrović, A. N., Mohamed, N. A. & Fernando, H. J. S. 1999 Gravitational settling of particles through density interfaces. J. Fluid Mech. 381, 175–198.CrossRefGoogle Scholar
Sutor, M. M. & Dagg, M. J. 2008 The effects of vertical sampling resolution on estimates of Plankton biomass and rate calculations in stratified water columns. Estuar. Coast. Shelf Sci. 78, 107–121.CrossRefGoogle Scholar
Tanner, R. I. 1963 End effects in falling-ball viscometry. J. Fluid Mech. 17, 161–170.CrossRefGoogle Scholar
Taylor, G. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. A 219, 186–203.Google Scholar
Turco, R. P., Toon, O. B., Ackerman, T. P., Pollack, J. B. & Sagan, C. 1990 Climate and smoke: an appraisal of nuclear winter. Science 247 (4939), 662–665.CrossRefGoogle ScholarPubMed
Widder, E. A., Johnsen, S., Bernstein, S. A., Case, J. F. & Neilson, D. J. 1999 Thin layers of bioluminescent copepods found at density discontinuities in the water column. Mar. Biol. 134, 429–437.CrossRefGoogle Scholar
Yick, K. Y., Torres, C. R., Peacock, T. & Stocker, R. 2009 Enhanced drag of a sphere settling in a stratified fluid at small Reynolds Number. J. Fluid. Mech. 632, 49–68.CrossRefGoogle Scholar