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Modelling levitated 2-lobed droplets in rotation using Cassinian oval curves

Published online by Cambridge University Press:  15 May 2018

Haruki Ishikawa Open the ORCID record for Haruki Ishikawa [Opens in a new window]  and
Katsuhiro Nishinari
Haruki Ishikawa*
Affiliation:
Department of Aeronautics and Astronautics, School of Engineering, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan
Katsuhiro Nishinari
Affiliation:
Research Center for Advanced Science and Technology, The University of Tokyo, 4-6-1, Komaba, Meguro-ku, Tokyo 153-8904, Japan
*
†Email address for correspondence: ishikawa@jamology.rcast.u-tokyo.ac.jp
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Abstract

A simple model of rotating 2-lobed droplets is proposed by setting the outline shape of the droplet to the Cassinian oval, a mathematical curve that closely resembles in shape. By deriving the governing equation of the proposed model and obtaining its stationary solutions, the relationship between the angular velocity of rotation and the maximum deformation length is explicitly and precisely calculated. The linear stability analysis is performed for the stationary solutions, and it is demonstrated that the stability of the solutions depends only on the ratio of the deformation length to the radius of the central cross-section of the droplet, which is independent of the physical properties of the droplet. Via comparison with an experimental study, it is observed that the calculated result is consistent with the deformation behaviour of actual 2-lobed droplets in the range where the stationary solution of the proposed model is linearly stable. Therefore, the proposed model is a suitable model for reproducing the steady deformation behaviour of 2-lobed droplets in a wide range of viscosities, surface tensions, densities and initial radii of the droplet, and especially if the viscosity of the droplet is low, the entire process of deformation of the 2-lobed droplet, including the unsteady breakup process, can be very well reproduced by the proposed model.

JFM classification

Drops and Bubbles: Drops Drops and Bubbles: Drops and Bubbles
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 846 , 10 July 2018 , pp. 1088 - 1113
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Copyright
© 2018 Cambridge University Press 

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References

Aussillous, P. & Quéré, D. 2004 Shapes of rolling liquid drops. J. Fluid Mech. 512, 133–151.CrossRefGoogle Scholar
Baldwin, K. A., Butler, S. L. & Hill, R. J. A. 2015 Artificial tektites: an experimental technique for capturing the shapes of spinning drops. Sci. Rep. 5, 7660.CrossRefGoogle ScholarPubMed
Basset, A. B. 1901 An Elementary Treatise on Cubic and Quartic Curves. Deighton, Bell.Google Scholar
Beaugnon, E., Fabregue, D., Billy, D., Nappa, J. & Tournier, R. 2001 Dynamics of magnetically levitated droplets. Physica B 294, 715–720.CrossRefGoogle Scholar
Beaugnon, E. & Tournier, R. 1991 Levitation of water and organic substances in high static magnetic fields. J. Phys. III 1 (8), 1423–1428.Google Scholar
Benner, R. E., Basaran, O. A. & Scriven, L. E. 1991 Equilibria, stability and bifurcations of rotating columns of fluid subjected to planar disturbances. Proc. R. Soc. Lond. A 433 (1887), 81–99.Google Scholar
Bhat, P. P., Appathurai, S., Harris, M. T., Pasquali, M., McKinley, G. H. & Basaran, O. A. 2010 Formation of beads-on-a-string structures during break-up of viscoelastic filaments. Nat. Phys. 6 (8), 625–631.CrossRefGoogle Scholar
van der Bos, A., van der Meulen, M.-J., Driessen, T., van den Berg, M., Reinten, H., Wijshoff, H., Versluis, M. & Lohse, D. 2014 Velocity profile inside piezoacoustic inkjet droplets in flight: comparison between experiment and numerical simulation. Phys. Rev. Appl. 1 (1), 014004.CrossRefGoogle Scholar
Brandt, E. H. 1989 Levitation in physics. Science 243 (4889), 349–355.CrossRefGoogle ScholarPubMed
Brown, R. A. & Scriven, L. E. 1980 The shape and stability of rotating liquid drops. Proc. R. Soc. Lond. A 371 (1746), 331–357.Google Scholar
Butler, S. L., Stauffer, M. R., Sinha, G., Lilly, A. & Spiteri, R. J. 2011 The shape distribution of splash-form tektites predicted by numerical simulations of rotating fluid drops. J. Fluid Mech. 667, 358–368.CrossRefGoogle Scholar
Castrejón-Pita, J. R., Castrejón-Pita, A. A., Thete, S. S., Sambath, K., Hutchings, I. M., Hinch, J., Lister, J. R. & Basaran, O. A. 2015 Plethora of transitions during breakup of liquid filaments. Proc. Natl Acad. Sci. 112 (15), 4582–4587.CrossRefGoogle ScholarPubMed
Chandrasekhar, S. 1965 The stability of a rotating liquid drop. Proc. R. Soc. Lond. A 286 (1404), 1–26.Google Scholar
Chandrasekhar, S. 1967 Ellipsoidal figures of equilibrium–an historical account. Commun. Pure Appl. Maths 20 (2), 251–265.CrossRefGoogle Scholar
Clasen, C., Phillips, P. M., Palangetic, L. & Vermant, J. 2012 Dispensing of rheologically complex fluids: the map of misery. AIChE J. 58 (10), 3242–3255.CrossRefGoogle Scholar
Czerski, H. & Deane, G. B. 2010 Contributions to the acoustic excitation of bubbles released from a nozzle. J. Acoust. Soc. Am. 128 (5), 2625–2634.CrossRefGoogle Scholar
Day, R. F., Hinch, E. J. & Lister, J. R. 1998 Self-similar capillary pinchoff of an inviscid fluid. Phys. Rev. Lett. 80 (4), 704–707.CrossRefGoogle Scholar
Eggers, J. 1993 Universal pinching of 3d axisymmetric free-surface flow. Phys. Rev. Lett. 71 (21), 3458–3460.CrossRefGoogle ScholarPubMed
Eggers, J. 1997 Nonlinear dynamics and breakup of free-surface flows. Rev. Mod. Phys. 69, 865–930.CrossRefGoogle Scholar
Eggers, J. & Dupont, T. F. 1994 Drop formation in a one-dimensional approximation of the navier–stokes equation. J. Fluid Mech. 262, 205–221.CrossRefGoogle Scholar
Eggers, J. & Fontelos, M. A. 2005 Isolated inertialess drops cannot break up. J. Fluid Mech. 530, 177–180.CrossRefGoogle Scholar
Elkins-Tanton, L. T., Aussillous, P., Bico, J., Quere, D. & Bush, J. W. M. 2003 A laboratory model of splash-form tektites. Meteorit. Planet. Sci. 38 (9), 1331–1340.CrossRefGoogle Scholar
Foresti, D., Nabavi, M., Klingauf, M., Ferrari, A. & Poulikakos, D. 2013 Acoustophoretic contactless transport and handling of matter in air. Proc. Natl Acad. Sci. USA 110 (31), 12549–12554.CrossRefGoogle ScholarPubMed
Frohn, A. & Roth, N. 2000 Dynamics of Droplets. Springer.CrossRefGoogle Scholar
Heine, C.-J. 2006 Computations of form and stability of rotating drops with finite elements. IMA J. Numer. Anal. 26 (4), 723–751.CrossRefGoogle Scholar
Hill, R. J. A. & Eaves, L. 2008 Nonaxisymmetric shapes of a magnetically levitated and spinning water droplet. Phys. Rev. Lett. 101, 234501.CrossRefGoogle ScholarPubMed
Landau, L. D. & Lifshits, E. M. 1959 Fluid Mechanics. Pergamon Press.Google Scholar
Ohsaka, K., Rednikov, A., Sadhal, S. S. & Trinh, E. H. 2002 Noncontact technique for determining viscosity from the shape relaxation of ultrasonically levitated and initially elongated drops. Rev. Sci. Instrum. 73 (5), 2091–2096.CrossRefGoogle Scholar
Papageorgiou, D. T. 1995 On the breakup of viscous liquid threads. Phys. Fluids 7 (7), 1529–1544.CrossRefGoogle Scholar
Patzek, T. W., Basaran, O. A., Benner, R. E. & Scriven, L. E. 1995 Nonlinear oscillations of two-dimensional, rotating inviscid drops. J. Comput. Phys. 116 (1), 3–25.CrossRefGoogle Scholar
Poincaré, H. 1885 Sur l’équilibre d’une masse fluide animée d’un mouvement de rotation. Acta Mathematica 7 (1), 259–380.CrossRefGoogle Scholar
Qian, J. & Law, C. K. 1997 Regimes of coalescence and separation in droplet collision. J. Fluid Mech. 331, 59–80.CrossRefGoogle Scholar
Rayleigh, Lord 1879 On the capillary phenomena of jets. Proc. R. Soc. Lond. 29 (196–199), 71–97.Google Scholar
Rhim, W.-K., Chung, S. K. & Elleman, D. D. 1990 Experiments on rotating charged liquid drops. AIP Conf. Proc. 197 (1), 91–105.CrossRefGoogle Scholar
Rhim, W.-K., Collender, M., Hyson, M. T., Simms, W. T. & Elleman, D. D. 1985 Development of an electrostatic positioner for space material processing. Rev. Sci. Instrum. 56 (2), 307–317.CrossRefGoogle Scholar
Sun, K., Zhang, P., Law, C. K. & Wang, T. 2015 Collision dynamics and internal mixing of droplets of non-newtonian liquids. Phys. Rev. Appl. 4 (5), 054013.CrossRefGoogle Scholar
Tanaka, R., Matsumoto, S., Kaneko, A. & Abe, Y. 2013 Viscosity measurement using breakup of a levitated droplet by rotation. Interfacial Phenom. Heat Transfer 1 (2), 181–194.CrossRefGoogle Scholar
Tirtaatmadja, V., McKinley, G. H. & Cooper-White, J. J. 2006 Drop formation and breakup of low viscosity elastic fluids: effects of molecular weight and concentration. Phys. Fluids 18 (4), 043101.CrossRefGoogle Scholar
Trinh, E. H. 1985 Compact acoustic levitation device for studies in fluid dynamics and material science in the laboratory and microgravity. Rev. Sci. Instrum. 56 (11), 2059–2065.CrossRefGoogle Scholar
Wang, T. G., Anilkumar, A. V., Lee, C. P. & Lin, K. C. 1994 Bifurcation of rotating liquid drops: results from usml-1 experiments in space. J. Fluid Mech. 276, 389–403.CrossRefGoogle Scholar
Wang, T. G., Trinh, E. H., Croonquist, A. P. & Elleman, D. D. 1986 Shapes of rotating free drops: spacelab experimental results. Phys. Rev. Lett. 56, 452–455.CrossRefGoogle ScholarPubMed