Hostname: page-component-848d4c4894-xm8r8 Total loading time: 0 Render date: 2024-06-25T08:51:45.211Z Has data issue: false hasContentIssue false
  • English
  • Français

Inertial settling of a sphere through an interface. Part 1. From sphere flotation to wake fragmentation

Published online by Cambridge University Press:  28 November 2017

Jean-Lou Pierson  and
Jacques Magnaudet Open the ORCID record for Jacques Magnaudet [Opens in a new window]
Jean-Lou Pierson
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, INPT, UPS, Toulouse, France
Jacques Magnaudet*
Affiliation:
Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS, INPT, UPS, Toulouse, France
*
†Email address for correspondence: jmagnaud@imft.fr
Get access Rights & Permissions [Opens in a new window]

Abstract

Experiments are performed to better understand the characteristics of the flow induced by the gravity-driven settling of a rigid sphere through a two-layer arrangement of immiscible Newtonian fluids, mostly in inertia-controlled regimes. High-speed video imaging is employed to follow the sphere motion and the deformation of the interface separating the two fluids. The viscosity ratio between the lower and upper fluids is varied by four orders of magnitude, making it possible to observe highly contrasting interface patterns. Depending on the properties of the sphere and the fluids, the sphere may either float steadily at the interface or cross it by pulling a column of the upper fluid into the lower one. This column, which may be axisymmetric or three-dimensional depending on the relative magnitude of inertia effects in the upper fluid, generally pinches off at some position located either close to the initial interface or, more frequently, close to the sphere. Its lower part then recedes towards the sphere, forming a drop which remains attached to its top half. However, when inertia effects in the lower fluid are large enough and the upper fluid is not ‘too’ viscous, the tail quickly undergoes a complete fragmentation, giving birth to a large quantity of filaments and droplets. These various interface configurations are qualitatively analysed using the five independent dimensionless parameters characterizing the system, and regime maps based on the most relevant of them are provided. The influence of several of these parameters on four specific features observed in the course of the experiments, namely the pinch-off position, the floating/sinking transition, the volume of the attached drops and the average size of the droplets formed during the fragmentation process, is examined in detail. A simple model providing qualitative or quantitative predictions is established in each case, and its validity and limitations are assessed against experimental observations.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface) Multiphase and Particle-laden Flows: Multiphase and Particle-laden Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 835 , 25 January 2018 , pp. 762 - 807
Check for updates
Copyright
© 2017 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.)

Footnotes

‡

Present address: IFP Energies nouvelles, BP 3, 69360 Solaize, France.

References

Abaid, N., Adalsteinsson, D., Agyapong, A. & McLaughlin, R. M. 2004 An internal splash: levitation of falling spheres in stratified fluids. Phys. Fluids 16, 1567–1580.Google Scholar
Aristoff, J. M. & Bush, J. W. M. 2009 Water entry of small hydrophobic spheres. J. Fluid Mech. 619, 45–78.Google Scholar
Aristoff, J. M., Truscott, T. T., Techet, A. H. & Bush, J. W. M. 2010 The water entry of decelerating spheres. Phys. Fluids 22, 032102.Google Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Birkhoff, G. & Zarantonello, E. H. 1957 Jets, Wakes and Cavities. Academic Press.Google Scholar
Bonhomme, R., Magnaudet, J., Duval, F. & Piar, B. 2012 Inertial dynamics of air bubbles crossing a horizontal fluid–fluid interface. J. Fluid Mech. 707, 405–443.Google Scholar
Burns, P. & Chemel, C. 2015 Interactions between downslope flows and a developing cold-air pool. Boundary-Layer Meteorol. 154, 57–80.Google Scholar
Calabrese, R. V., Chang, T. P. K. & Dang, P. T. 1986 Drop breakup in turbulent stirred-tank contactors. Part I. Effect of dispersed-phase viscosity. AIChE J. 32, 657–666.Google Scholar
Camassa, R., Falcon, C., Lin, J., McLaughlin, R. M. & Mykins, N. 2010 A first-principle predictive theory for a sphere falling through sharply stratified fluid at low Reynolds number. J. Fluid Mech. 664, 436–465.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.Google Scholar
Chandrasekar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Clift, R., Grace, J. R. & Weber, M. E. 1978 Bubbles, Drops and Particles. Academic Press.Google 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
Coulaloglou, C. A. & Tavlarides, L. L. 1977 Description of interaction processes in agitated liquid–liquid dispersions. Chem. Engng Sci. 32, 1289–1297.CrossRefGoogle Scholar
Darwin, S. C. 1953 Note on hydrodynamics. Proc. Camb. Phil. Soc. 49, 342–354.Google Scholar
De Folter, J. W. J., De Villeneuve, V. W. A., Aarts, D. G. A. L. & Lekkerkerker, N. H. W. 2010 Rigid sphere transport through a colloidal gas–liquid interface. New J. Phys. 12, 023013.Google Scholar
De Gennes, P. G., Brochard-Wyart, F. & Quéré, D. 2003 Capillarity and Wetting Phenomena: Drops, Bubbles, Pearls, Waves. Springer.Google 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, 225–256.CrossRefGoogle Scholar
Dietrich, N., Poncin, S. & Li, H. Z. 2011 Dynamical deformation of a flat liquid–liquid interface. Exp. Fluids 50, 1293–1303.Google Scholar
Do-Quang, M. & Amberg, G. 2009 The splash of a solid sphere impacting on a liquid surface: numerical simulation of the influence of wetting. Phys. Fluids 21, 022102.Google Scholar
Duclaux, V., Caillé, F., Duez, C., Ybert, C., Bocquet, L. & Clanet, C. 2007 Dynamics of transient cavities. J. Fluid Mech. 591, 1–19.Google Scholar
Duez, C., Ybert, C., Clanet, C. & Bocquet, L. 2007 Making a splash with water repellency. Nat. Phys. 3, 180–183.CrossRefGoogle Scholar
Eastwood, C. D., Armi, L. & Lasheras, J. C. 2004 The breakup of immiscible fluids in turbulent flows. J. Fluid Mech. 502, 309–333.Google Scholar
Fabre, D., Tchoufag, J. & Magnaudet, J. 2012 The steady oblique path of buoyancy-driven disks and spheres. J. Fluid Mech. 707, 24–36.CrossRefGoogle Scholar
Frisch, U. 1995 Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press.Google Scholar
Geller, A. S., Lee, S. H. & Leal, L. G. 1986 The creeping motion of a spherical particle normal to a deformable interface. J. Fluid Mech. 169, 27–69.Google Scholar
Grumstrup, T., Keller, J. B. & Belmonte, A. 2007 Cavity ripples observed during the impact of solid objects into liquids. Phys. Rev. Lett. 99, 114502.Google Scholar
Hartland, S. 1968 The approach of a rigid sphere to a deformable liquid/liquid interface. J. Colloid Interface Sci. 26, 383–394.CrossRefGoogle Scholar
Hartland, S. 1969 The profile of the draining film between a rigid sphere and a deformable fluid–liquid interface. Chem. Engng Sci. 24, 987–995.Google Scholar
Hinze, J. O. 1955 Fundamentals of the hydrodynamic mechanism of splitting in dispersion processes. AIChE J. 1, 289–295.Google Scholar
James, D. F. 1974 The meniscus on the outside of a small circular cylinder. J. Fluid Mech. 63, 657–664.CrossRefGoogle Scholar
Jeffreys, G. V. & Davies, G. A. 1971 Coalescence of liquid droplets and liquid dispersion. In Recent Advances in Liquid–Liquid Extraction (ed. Hanson, C.), pp. 495–584. Pergamon Press, chap. 14.Google Scholar
Johnson, R. E. 1981 Stokes flow past a sphere coated with a thin fluid film. J. Fluid Mech. 110, 217–238.Google Scholar
Jones, A. F. & Wilson, S. D. R. 1978 The film drainage problem in droplet coalescence. J. Fluid Mech. 87, 263–288.Google Scholar
Keller, J. B. 1998 Surface tension force on a partly submerged body. Phys. Fluids 10, 3009–3010.Google Scholar
Kellogg, W. W. 1980 Aerosols and climate. In Interaction of Energy and Climate (ed. Bach, W., Pankrath, J. & Williams, J.), pp. 281–303. Reidel.CrossRefGoogle Scholar
Kolmogorov, A. N. 1949 On the disintegration of drops in a turbulent flow. Dokl. Akad. Nauk SSSR 66, 825–828.Google Scholar
Lawrence, C. J. & Mei, R. 1995 Long-time behaviour of the drag on a body in impulsive motion. J. Fluid Mech. 283, 307–327.CrossRefGoogle Scholar
Lee, D. G. & Kim, H. Y. 2008 Impact of a superhydrophobic sphere onto water. Langmuir 24, 142–145.Google Scholar
Lee, D. G. & Kim, H. Y. 2011 Sinking of small sphere at low Reynolds number through interface. Phys. Fluids 23, 072104.Google Scholar
Lovalenti, P. M. & Brady, J. F. 1993 The hydrodynamic force on a rigid particle undergoing arbitrary time-dependent motion at small Reynolds number. J. Fluid Mech. 256, 561–605.CrossRefGoogle Scholar
Luo, H. & Svendsen, H. F. 1996 Theoretical model for drop and bubble breakup in turbulent dispersions. AIChE J. 42, 1225–1233.Google Scholar
MacIntyre, S., Alldredge, A. L. & Gotschalk, C. C. 1995 Accumulation of marine snow at density discontinuities in the water column. Limnol. Oceanogr. 40, 449–468.Google Scholar
Manga, M., Stone, H. A. & O’Connell, R. L. 1993 The interaction of plume heads with compositional discontinuities in the Earth’s mantle. J. Geophys. Res. 98, 19979–19990.Google Scholar
Mansfield, E. H., Sepangi, H. R. & Eastwood, E. A. 1997 Equilibrium and mutual attraction or repulsion of objects supported by surface tension. Phil. Trans. R. Soc. Lond. A 355, 869–919.Google Scholar
Marmottant, P. & Villermaux, E. 2004 On spray formation. J. Fluid Mech. 498, 73–111.Google Scholar
Martinez-Bazán, C., Montañes, J. L. & Lasheras, J. C. 1999 On the breakup of an air bubble injected into a fully developed turbulent flow. Part 2. Size p.d.f. of the resulting daughter bubbles. J. Fluid Mech. 401, 183–207.Google Scholar
Maru, H. C., Wasan, D. T. & Kintner, R. C. 1971 Behavior of a rigid sphere at a liquid–liquid interface. Chem. Engng Sci. 26, 1615–1628.Google Scholar
Milne-Thomson, L. M. 1962 Theoretical Hydrodynamics. MacMillan.Google Scholar
O’Brien, S. B. G. 1996 The meniscus near a small sphere and its relationship to line pinning of contact lines. J. Colloid Interface Sci. 183, 51–56.Google Scholar
Peters, I. R., Madonia, M., Lohse, D. & Van Der Meer, D. 2016 Volume entrained in the wake of a disc intruding into an oil–water interface. Phys. Rev. Fluids 1, 033901.Google Scholar
Pierson, J. L. & Magnaudet, J. 2017b Inertial settling of a sphere through an interface. Part 2. Sphere and tail dynamics. J. Fluid Mech. 835, 808–851.Google Scholar
Pitois, O., Moucheront, P. & Weill, C. 1999 Franchissement d’interface et enrobage d’une sphère. Comptes Rendus Acad. Sci. Ser. II-B 327, 605–611.Google Scholar
Poggi, D., Minto, M. & Davenport, W. G. 1969 Mechanisms of metal entrapment in slags. J. Met. 21, 40–45.Google Scholar
Riebesell, U. 1992 The formation of large marine snow and its sustained residence in surface waters. Limnol. Oceeanogr. 37, 63–67.Google Scholar
Sakamoto, H. & Haniu, H. 1991 A study of vortex shedding from spheres in a uniform flow. Trans. ASME J. Fluids Engng 113, 183–189.Google Scholar
Shannon, G., White, L. & Sridhar, S. 2008 Modeling inclusion approach to the steel/slag interface. Mater. Sci. Eng. A – Struct. Mater. Prop. Microstruct. Process. 495, 310–315.Google Scholar
Shoukry, E., Hafez, M. & Hartland, S. 1975 Separation of drops from wetted surfaces. J. Colloid Interface Sci. 53, 261–270.Google Scholar
Smith, P. G. & Van de Den, T. G. M. 1985 The separation of a liquid drop from a stationary solid sphere in a gravitational field. J. Colloid Interface Sci. 105, 7–20.Google Scholar
Smith, P. G. & Van de Ven, T. G. M. 1984 The effect of gravity on the drainage of a thin liquid film between a solid sphere and a liquid/fluid interface. J. Colloid Interface Sci. 100, 456–464.CrossRefGoogle Scholar
Srdic-Mitrovic, 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
Stechemesser, H. & Nguyen, A. V. 1999 Time of gas–solid–liquid three-phase contact expansion in flotation. Intl J. Miner. Process. 56, 117–132.Google Scholar
Tan, B. C. W., Vlaskamp, J. H. A., Denissenko, P. & Thomas, P. J. 2016 Cavity formation in the wake of falling spheres submerging into a stratified two-layer system of immiscible liquids. J. Fluid Mech. 790, 33–56.CrossRefGoogle 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
Truscott, T. T., Epps, B. P. & Belden, J. 2014 Water entry of projectiles. Annu. Rev. Fluid Mech. 46, 355–378.Google Scholar
Tsai, S. S., Wexler, J. S., Wan, J. & Stone, H. A. 2011 Conformal coating of particles in microchannels by magnetic forcing. Appl. Phys. Lett. 99, 153509.Google Scholar
Tsouris, C. & Tavlarides, L. L. 1994 Breakage and coalescence models for drops in turbulent dispersions. AIChE J. 40, 395–406.CrossRefGoogle Scholar
Vella, D. 2015 Floating versus sinking. Annu. Rev. Fluid Mech. 47, 115–135.Google Scholar
Vella, D., Lee, D. G. & Kim, H. Y. 2006 The load supported by small floating objects. Langmuir 22, 5979–5981.Google Scholar
Villermaux, E. 2007 Fragmentation. Annu. Rev. Fluid Mech. 39, 419–446.Google Scholar
Wang, C. Y. & Calabrese, R. V. 1986 Drop breakup in turbulent stirred-tank contactors. Part II. Relative influence of viscosity and interfacial tension. AIChE J. 32, 667–676.Google Scholar
Weinstein, S. J. & Palmer, H. J. 1997 Capillary hydrodynamics and interfacial phenomena. In Liquid Film Coating (ed. Kistler, S. F. & Schweizer, P. M.), Chapman & Hall.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

Pierson and Magnaudet supplementary movie 1a

A 14mm steel sphere is settling in a fluid setup with silicon oil on top of a water bath (lambda=0.21, corresponding to configuration 18(b) in figure 4). Massive fragmentation takes place in the tail.

Download Pierson and Magnaudet supplementary movie 1a(Video)
Video 2.8 MB

Pierson and Magnaudet supplementary movie 1b

Zooom of movie 1a allowing a better tracking of the tail dynamics.

Download Pierson and Magnaudet supplementary movie 1b(Video)
Video 1.6 MB

Pierson and Magnaudet supplementary movie 2a

Same as in Movie 1a-b, except that the silicon oil is ten times more viscous (lambda=0.02, corresponding to configuration 27(a) in figure 5). Again, massive fragmentation takes place in the tail.

Download Pierson and Magnaudet supplementary movie 2a(Video)
Video 2.4 MB

Pierson and Magnaudet supplementary movie 2b

Zooom of movie 2a.

Download Pierson and Magnaudet supplementary movie 2b(Video)
Video 935.2 KB

Pierson and Magnaudet supplementary movie 3a

Same as in Movie 1a-b, except that the silicon oil is a hundred times more viscous (lambda=0.002, corresponding to configuration 27(b) in figure 5).

Download Pierson and Magnaudet supplementary movie 3a(Video)
Video 2 MB

Pierson and Magnaudet supplementary movie 3b

Zooom of movie 3a.

Download Pierson and Magnaudet supplementary movie 3b(Video)
Video 1.1 MB