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Levitation of non-magnetizable droplet inside ferrofluid

  • Chamkor Singh (a1) (a2), Arup K. Das (a3) and Prasanta K. Das (a1)

Abstract

The central theme of this work is that a stable levitation of a denser non-magnetizable liquid droplet, against gravity, inside a relatively lighter ferrofluid – a system barely considered in ferrohydrodynamics – is possible, and exhibits unique interfacial features; the stability of the levitation trajectory, however, is subject to an appropriate magnetic field modulation. We explore the shapes and the temporal dynamics of a plane non-magnetizable droplet levitating inside a ferrofluid against gravity due to a spatially complex, but systematically generated, magnetic field in two dimensions. The coupled set of Maxwell’s magnetostatic equations and the flow dynamic equations is integrated computationally, utilizing a conservative finite-volume-based second-order pressure projection algorithm combined with the front-tracking algorithm for the advection of the interface of the droplet. The dynamics of the droplet is studied under both the constant ferrofluid magnetic permeability assumption as well as for more realistic field-dependent permeability described by Langevin’s nonlinear magnetization model. Due to the non-homogeneous nature of the magnetic field, unique shapes of the droplet during its levitation, and at its steady state, are realized. The complete spatio-temporal response of the droplet is a function of the Laplace number $La$ , the magnetic Laplace number $La_{m}$ and the Galilei number $Ga$ ; through detailed simulations we separate out the individual roles played by these non-dimensional parameters. The effect of the viscosity ratio, the stability of the levitation path and the possibility of existence of multiple stable equilibrium states is investigated. We find, for certain conditions on the viscosity ratio, that there can be developments of cusps and singularities at the droplet surface; we also observe this phenomenon experimentally and compare with the simulations. Our simulations closely replicate the singular projection on the surface of the levitating droplet. Finally, we present a dynamical model for the vertical trajectory of the droplet. This model reveals a condition for the onset of levitation and the relation for the equilibrium levitation height. The linearization of the model around the steady state captures that the nature of the equilibrium point goes under a transition from being a spiral to a node depending upon the control parameters, which essentially means that the temporal route to the equilibrium can be either monotonic or undulating. The analytical model for the droplet trajectory is in close agreement with the detailed simulations.

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Corresponding author

Email address for correspondence: pkd@mech.iitkgp.ernet.in

References

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Afkhami, S., Renardy, Y., Renardy, M., Riffle, J. S. & St Pierre, T. 2008 Field-induced motion of ferrofluid droplets through immiscible viscous media. J. Fluid Mech. 610, 363380.
Afkhami, S., Tyler, A. J., Renardy, Y., Renardy, M., St Pierre, T. G., Woodward, R. C. & Riffle, J. S. 2010 Deformation of a hydrophobic ferrofluid droplet suspended in a viscous medium under uniform magnetic fields. J. Fluid Mech. 663, 358384.
Bacri, J.-C., Cebers, A. O. & Perzynski, R. 1994 Behavior of a magnetic fluid microdrop in a rotating magnetic field. Phys. Rev. Lett. 72 (17), 27052708.
Bacri, J.-C. & Salin, D. 1983 Bistability of ferrofluid magnetic drops under magnetic field. J. Magn. Magn. Mater. 39 (1), 4850.
Bashtovoi, V., Pogirnitskaya, S. & Reks, A. 1999 Dynamics of deformation of magnetic fluid flat drops in a homogeneous longitudinal magnetic field. J. Magn. Magn. Mater. 201 (1), 300302.
Beysens, D. A. & van Loon, J. J. W. A. 2015 Generation and Applications of Extra-terrestrial Environments on Earth. River Publishers.
Chen, C.-Y. & Cheng, Z.-Y. 2008 An experimental study on Rosensweig instability of a ferrofluid droplet. Phys. Fluids 20 (5), 054105.
Chen, C.-Y. & Li, C.-S. 2010 Ordered microdroplet formations of thin ferrofluid layer breakups. Phys. Fluids 22 (1), 014105.
Dunne, P. A., Hilton, J. & Coey, J. M. D. 2007 Levitation in paramagnetic liquids. J. Magn. Magn. Mater. 316 (2), 273276.
Duplat, J. & Mailfert, A. 2013 On the bubble shape in a magnetically compensated gravity environment. J. Fluid Mech. 716, R11.
Fattah, A. R. A., Ghosh, S. & Puri, I. K. 2016 Printing microstructures in a polymer matrix using a ferrofluid droplet. J. Magn. Magn. Mater. 401, 10541059.
Geim, A. K., Simon, M. D., Boamfa, M. I. & Heflinger, L. O. 1999 Magnet levitation at your fingertips. Nature 400 (6742), 323324.
Gondret, P. & Rabaud, M. 1997 Shear instability of two-fluid parallel flow in a Hele-Shaw cell. Phys. Fluids 9 (11), 32673274.
Gu, Y., Bragheri, F., Valentino, G., Morris, K., Bellini, N. & Osellame, R. 2015 Ferrofluid-based optofluidic switch using femtosecond laser-micromachined waveguides. Appl. Opt. 54 (6), 14201425.
Gu, Y., Chow, H. & Morris, K. 2016 Motion of ferrofluid droplets under oscillating magnetic field. In Bulletin of the American Physical Society, APS March Meeting 2016, vol. 61. American Physical Society.
Halbach, K. 1985 Application of permanent magnets in accelerators and electron storage rings. J. Appl. Phys. 57 (8), 36053608.
Huber, F. & Littke, W. 1996 Technology experiments for magnetic levitation in transparent ferrofluids. In Space Station Utilisation, Symposium Proceedings, Darmstadt, vol. 385, pp. 479481.
Ikezoe, Y., Hirota, N., Nakagawa, J. & Kitazawa, K. 1998 Making water levitate. Nature 393 (6687), 749750.
Jackson, D. P. 2005 Theory, experiment, and simulations of a symmetric arrangement of quasi-two-dimensional magnetic fluid drops. J. Magn. Magn. Mater. 289, 188191.
Jackson, D. P. & Miranda, J. A. 2007 Confined ferrofluid droplet in crossed magnetic fields. Eur. Phys. J. E 23 (4), 389396.
Kim, D., Yu, S., Kang, B.-G. & Yun, K.-S. 2015 Liquid-based electrostatic energy harvester using rotational motion of ferrofluid droplets. In 18th International Conference on Solid-State Sensors, Actuators and Microsystems (TRANSDUCERS), 2015 Transducers, pp. 5961. IEEE.
Kim, D. & Yun, K. S. 2015 Energy harvester using contact-electrification of magnetic fluid droplets under oscillating magnetic field. J. Phys.: Conf. Ser. 660, 012108.
Kim, H. & Lim, H. 2015 Mode pattern of internal flow in a water droplet on a vibrating hydrophobic surface. J. Phys. Chem. B 119 (22), 67406746.
Koh, W. H., Lok, K. S. & Nguyen, N.-T. 2013 A digital micro magnetofluidic platform for lab-on-a-chip applications. Trans. ASME J. Fluids Engng 135 (2), 021302.
Korlie, M. S., Mukherjee, A., Nita, B. G., Stevens, J. G., Trubatch, A. D. & Yecko, P. 2008 Modeling bubbles and droplets in magnetic fluids. J. Phys.: Condens. Matter 20 (20), 204143.
Kovalchuk, N. M. & Vollhardt, D. 2001 A numerical study of surface tension auto-oscillations. Effect of surfactant properties. J. Phys. Chem. B 105 (20), 47094714.
Limbach, C. M., Robinson, R., Adams, D., Wilbanks, M. & Yalin, A. P. 2016 Toward a microscopic study of laser interactions with levitated liquid fuel droplets. In 47th AIAA Plasmadynamics and Lasers Conference. AIAA.
Lira, S. A. & Miranda, J. A. 2016 Ferrofluid patterns in Hele-Shaw cells: exact, stable, stationary shape solutions. Phys. Rev. E 93 (1), 013129.
Liu, J., Tan, S.-H., Yap, Y. F., Ng, M. Y. & Nguyen, N.-T. 2011a Numerical and experimental investigations of the formation process of ferrofluid droplets. Microfluid. Nanofluid. 11 (2), 177187.
Liu, J., Yap, Y. F. & Nguyen, N.-T. 2011b Numerical study of the formation process of ferrofluid droplets. Phys. Fluids 23 (7), 072008.
Liu, S., Yi, X., Leaper, M. & Miles, N. J. 2014 Horizontal deflection of single particle in a paramagnetic fluid. Eur. Phys. J. E 37 (6), 19.
Mirica, K. A., Shevkoplyas, S. S., Phillips, S. T., Gupta, M. & Whitesides, G. M. 2009 Measuring densities of solids and liquids using magnetic levitation: fundamentals. J. Am. Chem. Soc. 131 (29), 1004910058.
Mugele, F., Baret, J. C. & Steinhauser, D. 2006 Microfluidic mixing through electrowetting-induced droplet oscillations. Appl. Phys. Lett. 88 (20), 204106.
Nguyen, N.-T. 2013 Deformation of ferrofluid marbles in the presence of a permanent magnet. Langmuir 29 (45), 1398213989.
Olaru, R., Petrescu, C. & Arcire, A. 2013 Maximizing the magnetic force generated by an actuator with non-magnetic body in a ferrofluid pre-magnetized by permanent magnets. Intl Rev. Elec. Eng. (IREE) 8 (2), 904911.
Pamme, N. 2006 Magnetism and microfluidics. Lab on a Chip 6 (1), 2438.
Price, C. J., Giltrap, S., Stuart, N. H., Parker, S., Patankar, S., Lowe, H. F., Smith, R. A., Donnelly, T. D., Drew, D., Gumbrell, E. T. et al. 2015 An in-vacuo optical levitation trap for high-intensity laser interaction experiments with isolated microtargets. Rev. Sci. Instrum. 86 (3), 033502.
Rosenkilde, C. E. 1969 A dielectric fluid drop an electric field. Proc. R. Soc. Lond. A 312, 473494.
Rosensweig, R. E. 1966 Buoyancy and stable levitation of a magnetic body immersed in a magnetizable fluid. Nature 210, 613614.
Rosensweig, R. E. 1985 Ferrohydrodynamics. Cambridge University Press.
Rowghanian, P., Meinhart, C. D. & Campàs, O. 2016 Dynamics of ferrofluid drop deformations under spatially uniform magnetic fields. J. Fluid Mech. 802, 245262.
Ruyer-Quil, C. 2001 Inertial corrections to the darcy law in a Hele-Shaw cell. C. R. Acad. Sci. Ser. II B 329 (5), 337342.
Sandre, O., Browaeys, J., Perzynski, R., Bacri, J.-C., Cabuil, V. & Rosensweig, R. E. 1999 Assembly of microscopic highly magnetic droplets: magnetic alignment versus viscous drag. Phys. Rev. E 59 (2), 1736.
Sero-Guillaume, O. E., Zouaoui, D., Bernardin, D. & Brancher, J. P. 1992 The shape of a magnetic liquid drop. J. Fluid Mech. 241, 215232.
Sherwood, J. D. 1988 Breakup of fluid droplets in electric and magnetic fields. J. Fluid Mech. 188, 133146.
Singh, C., Das, A. K. & Das, P. K. 2016a Flow restrictive and shear reducing effect of magnetization relaxation in ferrofluid cavity flow. Phys. Fluids 28 (8), 087103.
Singh, C., Das, A. K. & Das, P. K. 2016b Single-mode instability of a ferrofluid-mercury interface under a nonuniform magnetic field. Phys. Rev. E 94 (1), 012803.
Stone, H. A., Lister, J. R. & Brenner, M. P. 1999 Drops with conical ends in electric and magnetic fields. Proc. R. Soc. Lond. A 455, 329347.
Tan, S.-H., Nguyen, N.-T., Yobas, L. & Kang, T. G. 2010 Formation and manipulation of ferrofluid droplets at a microfluidic t-junction. J. Micromech. Microengng 20 (4), 045004.
Taylor, G. 1964 Disintegration of water drops in an electric field. Proc. R. Soc. Lond. A 280, 383397.
Timonen, J. V. I., Latikka, M., Leibler, L., Ras, R. H. A. & Ikkala, O. 2013 Switchable static and dynamic self-assembly of magnetic droplets on superhydrophobic surfaces. Science 341 (6143), 253257.
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), 20592065.
Tryggvason, G., Scardovelli, R. & Zaleski, S. 2011 Direct Numerical Simulations of Gas–Liquid Multiphase Flows. Cambridge University Press.
Ueno, K., Higashitani, M. & Kamiyama, S. 1995 Study on single bubbles rising in magnetic fluid for small Weber number. J. Magn. Magn. Mater. 149 (1–2), 104107.
Ueno, K., Nishita, T. & Kamiyama, S. 1999 Numerical simulation of deformed single bubbles rising in magnetic fluid. J. Magn. Magn. Mater. 201 (1), 281284.
Unverdi, S. O. & Tryggvason, G. 1992 A front-tracking method for viscous, incompressible, multi-fluid flows. J. Comput. Phys. 100 (1), 2537.
Verkouteren, R. M. & Verkouteren, J. R. 2011 Inkjet metrology II: resolved effects of ejection frequency, fluidic pressure, and droplet number on reproducible drop-on-demand dispensing. Langmuir 27 (15), 96449653.
Vojtíšek, M., Tarn, M. D., Hirota, N. & Pamme, N. 2012 Microfluidic devices in superconducting magnets: on-chip free-flow diamagnetophoresis of polymer particles and bubbles. Microfluid. Nanofluid. 13 (4), 625635.
Whitehill, J., Neild, A., Ng, T. W., Martyn, S. & Chong, J. 2011 Droplet spreading using low frequency vibration. Appl. Phys. Lett. 98 (13), 133503.
Wohlhuter, F. K. & Basaran, O. A. 1993 Effects of physical properties and geometry on shapes and stability of polarizable drops in external fields. J. Magn. Magn. Mater. 122 (1–3), 259263.
Wojciechowski, K. & Kucharek, M. 2009 Interfacial tension oscillations without surfactant transfer. J. Phys. Chem. B 113 (41), 1345713461.
Wu, Y., Fu, T., Ma, Y. & Li, H. Z. 2013 Ferrofluid droplet formation and breakup dynamics in a microfluidic flow-focusing device. Soft Matt. 9 (41), 97929798.
Zhu, G.-P., Nguyen, N.-T., Ramanujan, R. V. & Huang, X.-Y. 2011a Nonlinear deformation of a ferrofluid droplet in a uniform magnetic field. Langmuir 27 (24), 1483414841.
Zhu, T. 2013 Microfluidic Continuous-flow Manipulation of Particles and Cells Inside Ferrofluids. Uga.
Zhu, T., Lichlyter, D. J., Haidekker, M. A. & Mao, L. 2011b Analytical model of microfluidic transport of non-magnetic particles in ferrofluids under the influence of a permanent magnet. Microfluid. Nanofluid. 10 (6), 12331245.
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