Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-09T22:21:31.021Z Has data issue: false hasContentIssue false
  • English
  • Français

Inertial drag on a sphere settling in a stratified fluid

Published online by Cambridge University Press:  24 September 2018

R. Mehaddi ,
F. Candelier  and
B. Mehlig Open the ORCID record for B. Mehlig [Opens in a new window]
R. Mehaddi
Affiliation:
Université de Lorraine, UMR CNRS 7563, LEMTA (Laboratoire d’Énergétique et Mécanique Théorique et Appliqué), F-54500 Vandoeuvre-Les-Nancy, France
F. Candelier
Affiliation:
Université d’Aix-Marseille, UMR CNRS 7343, IUSTI (Institut Universitaire des Systèmes Thermiques et Industriels), F-13013 Marseille, France
B. Mehlig*
Affiliation:
Department of Physics, Gothenburg University, SE-41296 Gothenburg, Sweden
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We compute the drag force on a sphere settling slowly in a quiescent, linearly stratified fluid. Stratification can significantly enhance the drag experienced by the settling particle. The magnitude of this effect depends on whether fluid-density transport around the settling particle is due to diffusion, to advection by the disturbance flow caused by the particle or due to both. It therefore matters how efficiently the fluid disturbance is convected away from the particle by fluid-inertial terms. When these terms dominate, the Oseen drag force must be recovered. We compute by perturbation theory how the Oseen drag is modified by diffusion and stratification. Our results are in good agreement with recent direct numerical simulation studies of the problem.

JFM classification

Multiphase and Particle-laden Flows: Multiphase and Particle-laden Flows Multiphase and Particle-laden Flows: Multiphase flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 855 , 25 November 2018 , pp. 1074 - 1087
Check for updates
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

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.Google Scholar
Alias, A. A. & Page, M. A. 2017 Low-Reynolds-number diffusion-driven flow around a horizontal cylinder. J. Fluid Mech. 825, 1035–1055.Google Scholar
Ardekani, A. M., Doostmohammadi, A. & Desai, N. 2017 Transport of particles, drops, and small organisms in density stratified fluids. Phys. Rev. Fluids 2 (10), 100503.Google Scholar
Ardekani, A. M. & Stocker, R. 2010 Stratlets: low Reynolds number point-force solutions in a stratified fluid. Phys. Rev. Lett. 105, 084502.Google Scholar
Bergström, B. & Strömberg, J.-O. 1997 Behavioural differences in relation to pycnoclines during vertical migration of the euphausiids meganyctiphanes norvegica (m. sars) and thysanoessa raschii (m. sars). J. Plankton Res. 19 (2), 255–261.Google Scholar
Blanchette, F. & Shapiro, A. M. 2012 Drops settling in sharp stratification with and without marangoni effects. Phys. Fluids 24 (4), 042104.Google Scholar
Bloomfield, L. J. & Kerr, R. C. 1998 Turbulent fountains in a stratified fluid. J. Fluid Mech. 358, 335–356.Google 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 (3), 031702.Google Scholar
Candelier, F., Mehaddi, R. & Vauquelin, O.2013 Note on the method of matched-asymptotic expansions for determining the force acting on a particle. arXiv:1307.6314.Google Scholar
Candelier, F., Mehaddi, R. & Vauquelin, O. 2014 The history force on a small particle in a linearly stratified fluid. J. Fluid Mech. 749, 183–200.Google Scholar
Candelier, F., Mehlig, B. & Magnaudet, J.2018 Time-dependent lift and drag on a rigid body in a viscous steady linear flow. arXiv:1806.05734 (submitted to J. Fluid Mech.).Google Scholar
Chadwick, R. S. & Zvirin, Y. 1974a The effect of ambient density stratification on the oseen drag of small spherical particles. Israel J. Technol. 12, 262–267.Google Scholar
Chadwick, R. S. & Zvirin, Y. 1974b Slow viscous flow of an incompressible stratified fluid past a sphere. J. Fluid Mech. 66, 377–383.Google Scholar
De Pietro, M., van Hinsberg, M. A. T., Biferale, L., Clercx, H. J. H., Perlekar, P. & Toschi, F. 2015 Clustering of vertically constrained passive particles in homogeneous isotropic turbulence. Phys. Rev. E 91, 053002.Google Scholar
Doostmohammadi, A., Dabiri, S. & Ardekani, A. M. 2014 A numerical study of the dynamics of a particle settling at moderate Reynolds numbers in a linearly stratified fluid. J. Fluid Mech. 750, 5–32.Google Scholar
Doostmohammadi, A., Stocker, R. & Ardekani, A. M. 2012 Low-Reynolds-number swimming at pycnoclines. Proc. Natl Acad. Sci. USA 109 (10), 3856–3861.Google Scholar
Durham, W. M., Climent, E., Barry, M., de Lillo, F., Boffetta, G., Cencini, M. & Stocker, R. 2013 Turbulence drives microscale patches of motile phytoplankton. Nat. Commun. 4 (2148), 1–7.Google Scholar
Franks, P. J. S. & Jaffe, J. S. 2008 Microscale variability in the distributions of large fluorescent particles observed in situ with a planar laser imaging fluorometer. J. Mar. Sys. 69, 254–270.Google Scholar
Gray, D. D. & Giorgini, A. 1976 The validity of the Boussinesq approximation for liquids and gases. Intl J. Heat Mass Transfer 19 (5), 545–551.Google Scholar
Guasto, J. S., Rusconi, R. & Stocker, R. 2012 Fluid mechanics of planktonic microorganisms. Annu. Rev. Fluid Mech. 44, 373–400.Google Scholar
Gustavsson, K., Berglund, F., Jonsson, P. R. & Mehlig, B. 2016 Preferential sampling and small-scale clustering of gyrotactic microswimmers in turbulence. Phys. Rev. Lett. 116, 108104.Google Scholar
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics. Kluwer Academic Publishers.Google Scholar
Jacobson, M. Z. 2005 Fundamentals of Atmospheric Modeling. Cambridge University Press.Google Scholar
Jephson, T. & Carlsson, P. 2009 Species-and stratification-dependent diel vertical migration behaviour of three dinoflagellate species in a laboratory study. J. Plankton Res. 31 (11), 1353–1362.Google Scholar
Jonsson, P. 1989 Vertical distributions of planktonic ciliates – an experimental analysis of swimming behaviour. Mar. Ecol. Prog. Ser. 52, 39–53.Google Scholar
Kessler, J. O. 1985 Hydrodynamic focusing of motile algal cells. Nature 313, 218–220.Google Scholar
Leelőssy, Á., Molnár, F., Izsák, F., Havasi, Á., Lagzi, I. & Mészáros, R. 2014 Dispersion modeling of air pollutants in the atmosphere: a review. Centr. Eur. J. Geosci. 6, 257–278.Google Scholar
Linden, P. F. 1999 The fluid mechanics of natural ventilation. Annu. Rev. Fluid Mech. 31 (1), 201–238.Google Scholar
Lovalenti, P. M. & Brady, J. F. 1993 The force on a bubble, drop or particle in arbitrary time-dependent motion at small Reynolds number. Phys. Fluids 5 (9), 2104–2116.Google Scholar
Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26, 883–889.Google Scholar
Maxworthy, T., Leilich, J., Simpson, J. E. & Meiburg, E. H. 2002 The propagation of a gravity current into a linearly stratified fluid. J. Fluid Mech. 453, 371–394.Google Scholar
McDougall, T. J. 1978 Bubble plumes in stratified environments. J. Fluid Mech. 85 (4), 655–672.Google Scholar
Mehaddi, R., Candelier, F. & Vauquelin, O. 2013 Naturally bounded plumes. J. Fluid Mech. 717, 472–483.Google Scholar
Mehaddi, R., Vauquelin, O. & Candelier, F. 2012 Analytical solutions for turbulent boussinesq fountains in a linearly stratified environment. J. Fluid Mech. 691, 487–497.Google Scholar
Meibohm, J., Candelier, F., Rosen, T., Einarsson, J., Lundell, F. & Mehlig, B. 2016 Angular velocity of a spheroid log rolling in a simple shear at small Reynolds number. Phys. Rev. Fluids 1 (8), 084203.Google Scholar
Mowbray, D. E. & Rarity, B. S. H. 1967 A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid. J. Fluid Mech. 28, 1–16.Google Scholar
Oseen, C. W. 1910 Über die Stokes’sche Formel und über eine verwandte Aufgabe in der Hydrodynamik. Ark. Mat. Astron. Fys. 6, 143–152.Google Scholar
Pierson, J.-L. & Magnaudet, J. 2018 Inertial settling of a sphere through an interface. Part 2. Sphere and tail dynamics. J. Fluid Mech. 835, 808–851.Google Scholar
Proudman, I. & Pearson, J. R. A. 1957 Expansions at small Reynolds numbers for the flow past a sphere and circular cylinder. J. Fluid Mech. 22 (2), 385–400.Google Scholar
Roberts, A. M. 1970 Geotaxis in motile micro-organisms. J. Exp. Biol. 53, 687–699.Google Scholar
Saffman, P. G. 1965 The lift on a small sphere in a slow shear flow. J. Fluid Mech. 22 (2), 385–400.Google Scholar
Salazar, A. 2003 On thermal diffusivity. Eur. J. Phys. 24 (4), 351–358.Google Scholar
Sozza, A., De Lillo, F., Musacchio, S. & Boffetta, G. 2016 Large-scale confinement and small-scale clustering of floating particles in stratified turbulence. Phys. Rev. Fluids 1 (5), 052401.Google 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.Google Scholar
Torres, C. R., Hanazaki, H., Ochoa, J., Castillo, J. & Van Woert, M. 2000 Flow past a sphere moving vertically in a stratified diffusive fluid. J. Fluid Mech. 417, 211–236.Google Scholar
Turner, J. S. 1979 Buoyancy Effects in Fluids. Cambridge University Press.Google Scholar
Van Aartrijk, M. & Clercx, H. J. H. 2010 Vertical dispersion of light inertial particles in stably stratified turbulence: the influence of the basset force. Phys. Fluids 22 (1), 013301.Google Scholar
Wang, S. & Ardekani, A. M. 2012a Inertial squirmer. Phys. Fluids 24 (10), 101902.Google Scholar
Wang, S. & Ardekani, A. M. 2012b Unsteady swimming of small organisms. J. Fluid Mech. 702, 286–297.Google Scholar
Woods, A. W. 2010 Turbulent plumes in nature. Annu. Rev. Fluid Mech. 42, 391–412.Google 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 numbers. J. Fluid Mech. 632, 49–68.Google Scholar
Zhang, J., Mercier, M. & Magnaudet, J. 2017 Wake of a vertically moving sphere in a linearly stratified fluid. In 16th Eur. Turbulence Conf., Stockholm, Sweden.Google Scholar
Zvirin, Y. & Chadwick, R. S. 1975 Settling of an axially symmetric body in a viscous stratified fluid. Intl J. Multiphase Flow 1, 743–752.Google Scholar