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Shape and fission instabilities of ferrofluids in non-uniform magnetic fields

Published online by Cambridge University Press:  13 February 2018

Thibault Vieu Open the ORCID record for Thibault Vieu [Opens in a new window]  and
Clément Walter
Thibault Vieu*
Affiliation:
Magistère de Physique Fondamentale, Université Paris-Saclay, Bât. 470, F-91405 Orsay, France
Clément Walter*
Affiliation:
Magistère de Physique Fondamentale, Université Paris-Saclay, Bât. 470, F-91405 Orsay, France
*
†Email addresses for correspondence: thibault.vieu@u-psud.fr, clement.walter@u-psud.fr
†Email addresses for correspondence: thibault.vieu@u-psud.fr, clement.walter@u-psud.fr
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Abstract

We study static distributions of ferrofluid submitted to non-uniform magnetic fields. We show how the normal-field instability is modified in the presence of a weak magnetic field gradient. Then we consider a ferrofluid droplet and show how the gradient affects its shape. A rich phase transition phenomenology is found. We also investigate the creation of droplets by successive splits when a magnet is vertically approached from below and derive theoretical expressions which are solved numerically to obtain the number of droplets and their aspect ratio as a function of the field configuration. A quantitative comparison is performed with previous experimental results, as well as with our own experiments, and yields good agreement with the theoretical modelling.

JFM classification

Drops and Bubbles: Drops MHD and Electrohydrodynamics: Magnetic fluids Phase change: Morphological instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 840 , 10 April 2018 , pp. 455 - 497
Copyright
© 2018 Cambridge University Press 

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Footnotes

‡

T.V. and C.W. have equally contributed to this work.

References

Abou, B.1998 Instabilité interfaciale d’une couche de ferrofluide sous champ magnétique normal; étude de la transition hexagones – carrés. PhD thesis, Université Pierre et Marie Curie – Paris VI.Google Scholar
Abou, B., de Surgy, G. N. & Wesfreid, J. E. 1997 Dispersion relation in a ferrofluid layer of any thickness and viscosity in a normal magnetic field; asymptotic regimes. J. Phys. II France 7 (8), 1159–1171.Google Scholar
Abou, B., Wesfreid, J.-E. & Roux, S. 2000 The normal field instability in ferrofluids: hexagon-square transition mechanism and wavenumber selection. J. Fluid Mech. 416, 217–237.CrossRefGoogle Scholar
Afkhami, S., Tyler, A. J., Renardy, Y., Renardy, M., Pierre, T. G. St., 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, 358–384.CrossRefGoogle Scholar
Andelman, D. & Rosensweig, R. E. 2009 The phenomenology of modulated phases: from magnetic solids and fluids to organic films and polymers. In Polymers, Liquids and Colloids in Electric Fields: Interfacial Instabilites, Orientation and Phase Transitions, pp. 1–56. World Scientific Publishing Co.Google Scholar
Bacri, J.-C. & Elias, F. 2011 Ferrofluids: a model system of self-organised equilibrium. In Morphogenesis: Origins of Patterns and Shapes (ed. Bourgine, P. & Lesne, A.). Springer.Google Scholar
Bacri, J.-C. & Salin, D. 1982 Instability of ferrofluid magnetic drops under magnetic field. J. Phys. Lett. 43 (17), 649–654.CrossRefGoogle Scholar
Bacri, J.-C. & Salin, D. 1984 First-order transition in the instability of a magnetic fluid interface. J. Phys. Lett. 45 (11), 559–564.CrossRefGoogle Scholar
Barkov, Yu. D. & Berkovsky, B. M. 1980 Breakup of a drop of magnetic fluid. Magnetohydrodynamics 16 (3), 228–230.Google Scholar
Bashtovoi, V. G., Krakov, M. S. & Reks, A. G. 1985 Instability of a flat layer of magnetic liquid for supercritical magnetic fields. Magnetohydrodynamics 21 (1), 14–18.Google Scholar
Behrens, S., Bönnemann, H., Modrow, H., Kempter, V., Riehemann, W., Wiedenmann, A., Odenbach, S., Will, S., Thrams, L., Hergt, R. et al. 2009 Synthesis and characterization. In Colloidal Magnetic Fluids: Basics, Development and Application of Ferrofluids (ed. Odenbach, S.), Lecture Notes in Physics, vol. 763. Springer.CrossRefGoogle Scholar
Beleggia, M., De Graef, M. & Millev, Y. 2006 Demagnetization factors of the general ellipsoid: an alternative to the Maxwell approach. Phil. Mag. 86 (16), 2451–2466.CrossRefGoogle Scholar
Berkovsky, B. & Bashtovoi, V. 1980 Instabilities of magnetic fluids leading to a rupture of continuity. IEEE Trans. Magn. 16, 288–297.CrossRefGoogle Scholar
Berkovsky, B., Bashtovoi, V., Mikhalev, V. & Rex, A. 1987 Experimental study of the stability of bounded volumes of magnetic fluid with a free surface. J. Magn. Magn. Mater. 65 (2), 239–241.CrossRefGoogle Scholar
Berkovsky, B. M. & Kalikmanov, V. I. 1985 Topological instability of magnetic fluids. J. Phys. Lett. 46 (11), 483–491.CrossRefGoogle Scholar
Berkovsky, B. M., Medvedev, V. F. & Krakov, M. S. 1993 Magnetic Fluids: Engineering Applications. Oxford University Press.Google Scholar
Blums, E., Cebers, A. & Maiorov, M. M. 1997 Magnetic Fluids. Walter de Gruyter.Google Scholar
Brancher, J. P. & Zouaoui, D. 1987 Equilibrium of a magnetic liquid drop. J. Magn. Magn. Mater. 65 (2), 311–314.CrossRefGoogle Scholar
Bushueva, C. A., Kostarev, K. G. & Lebedev, A. V. 2011 Evolution of a ferrofluid floating layer under the influence of an inhomogeneous magnetic field. Magnetohydrodynamics 47 (2), 207–212.CrossRefGoogle Scholar
Camacho, J. M. & Sosa, V. 2013 Alternative method to calculate the magnetic field of permanent magnets with azimuthal symmetry. Revista Mexicana de Física E 59, 8–17.Google Scholar
Catherall, A. T., Benedict, K. A., King, P. J. & Eaves, L. 2003 Surface instabilities on liquid oxygen in an inhomogeneous magnetic field. Phys. Rev. E 68, 037302.Google Scholar
Cebers, A. & Maiorov, M. M. 1980 Magnetostatic instabilities in plane layers of magnetizable liquids. Magnetohydrodynamics 16 (1), 21–27.Google Scholar
Chen, C.-Y. & Cheng, Z.-Y. 2008 An experimental study on Rosensweig instability of a ferrofluid droplet. Phys. Fluids 20 (5), 054105.CrossRefGoogle Scholar
Cowley, M. D. & Rosensweig, R. E. 1967 The interfacial stability of a ferromagnetic fluid. J. Fluid Mech. 30 (4), 671–688.CrossRefGoogle Scholar
Friedrichs, R. & Engel, A. 2000 Statics and dynamics of a single ferrofluid-peak. Eur. Phys. J. B 18 (2), 329–335.CrossRefGoogle Scholar
Friedrichs, R. & Engel, A. 2001 Pattern and wave number selection in magnetic fluids. Phys. Rev. E 64 (2), 021406.CrossRefGoogle ScholarPubMed
Gailitis, A. 1977 Formation of the hexagonal pattern on the surface of a ferromagnetic fluid in an applied magnetic field. J. Fluid Mech. 82 (3), 401–413.CrossRefGoogle Scholar
Gollwitzer, C., Rehberg, I. & Richter, R. 2006 Via hexagons to squares in ferrofluids: experiments on hysteretic surface transformations under variation of the normal magnetic field. J. Phys.: Condens. Matter 18 (38), S2643.Google Scholar
Huebner, A., Sharma, S., Srisa-Art, M., Hollfelder, F., Edel, J. B. & deMello, A. J. 2008 Microdroplets: a sea of applications? Lab on a Chip 8, 1244–1254.CrossRefGoogle ScholarPubMed
Ivanov, A., Kantorovich, S., N Reznikov, E., Holm, C., Pshenichnikov, A., V Lebedev, A., Chremos, A. & J Camp, P. 2007 Magnetic properties of polydisperse ferrofluids: a critical comparison between experiment, theory, and computer simulation. Phys. Rev. E 75, 061405.Google ScholarPubMed
Knieling, H., Richter, R., Rehberg, I., Matthies, G. & Lange, A. 2007 Growth of surface undulations at the Rosensweig instability. Phys. Rev. E 76, 066301.Google ScholarPubMed
Lange, A., Gollwitzer, C., Maretzki, R., Rehberg, I. & Richter, R. 2016 Retarding the growth of the Rosensweig instability unveils a new scaling regime. Phys. Rev. E 93, 043106.Google ScholarPubMed
Lange, A., Langer, H. & Engel, A. 2000 Dynamics of a single peak of the Rosensweig instability in a magnetic fluid. Physica D 140 (3), 294–305.CrossRefGoogle Scholar
Lange, A., Reimann, B. & Richter, R. 2001 Wave number of maximal growth in viscous ferrofluids. Magnetohydrodynamics 37, 261–267.Google Scholar
Lange, A., Richter, R. & Tobiska, L. 2007 Linear and nonlinear approach to the Rosensweig instability. GAMM-Mitteilungen 30 (1), 171–184.CrossRefGoogle Scholar
Mahr, T. & Rehberg, I. 1998 Nonlinear dynamics of a single ferrofluid-peak in an oscillating magnetic field. Physica D 111 (1), 335–346.CrossRefGoogle Scholar
Neveu-Prin, S., Tourinho, F. A., Bacri, J.-C. & Perzynski, R. 1993 Magnetic birefringence of cobalt ferrite ferrofluids. Colloid Surf. A 80 (1), 1–10.CrossRefGoogle Scholar
Newell, A. J., Williams, W. & Dunlop, D. J. 1993 A generalization of the demagnetizing tensor for nonuniform magnetization. J. Geophys. Res. 98 (B6), 9551–9555.CrossRefGoogle Scholar
Petit, P. A., de Albuquerque, M. P., Cabuil, V. & Molho, P. 1993 Patterns in a ferrofluid film under normal fields: shape of thickness modulations and hysteresis. J. Magn. Magn. Mater. 122 (1), 271–276.CrossRefGoogle Scholar
Rannacher, D. & Engel, A. 2006 Cylindrical Korteweg–de Vries solitons on a ferrofluid surface. New J. Phys. 8 (6), 108.CrossRefGoogle Scholar
Richter, R. & Lange, A. 2009 Surface instabilities of ferrofluids. In Colloidal Magnetic Fluids: Basics, Development and Application of Ferrofluids (ed. Odenbach, S.), Lecture Notes in Physics, vol. 763. Springer.CrossRefGoogle Scholar
Rosensweig, R. E. 1985 Ferrohydrodynamics. Cambridge University Press.Google Scholar
Rosensweig, R. E. 1987 Magnetic fluids. Annu. Rev. Fluid Mech. 19, 437–463.CrossRefGoogle Scholar
Rowghanian, P., Meinhart, C. D. & Campàs, O. 2016 Dynamics of ferrofluid drop deformations under spatially uniform magnetic fields. J. Fluid Mech. 802, 245–262.CrossRefGoogle Scholar
Rupp, P., Richter, R. & Rehberg, I. 2003 Critical exponents of directed percolation measured in spatiotemporal intermittency. Phys. Rev. E 67, 036209.Google ScholarPubMed
Salin, D. 1993 Wave vector selection in the instability of an interface in a magnetic or electric field. Europhys. Lett. 21 (6), 667.CrossRefGoogle Scholar
Séro-Guillaume, O. E., Zouaoui, D., Bernardin, D. & Brancher, J. P. 1992 The shape of a magnetic liquid drop. J. Fluid Mech. 241, 215–232.CrossRefGoogle Scholar
Shliomis, M. I. 1974 Magnetic fluids. Sov. Phys. Uspekhi 17 (2), 153.CrossRefGoogle Scholar
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.CrossRefGoogle Scholar
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, 253–257.CrossRefGoogle ScholarPubMed
Tyler, A.2010 Ferrofluid droplets in uniform magnetic fields: evidence for field-dependent interfacial tension. PhD thesis, University of Western Australia.Google Scholar
Vislovich, A. N. 1990 Phenomenological equation of static magnetization of magnetic fluids. Magnetohydrodynamics 26, 178–183.Google Scholar
Yan, Q., Xuan, S., Ruan, X., Wu, J. & Gong, X. 2015 Magnetically controllable generation of ferrofluid droplets. Microfluid. Nanofluid. 19 (6), 1377–1384.CrossRefGoogle Scholar
Zelazo, R. E. & Melcher, J. R. 1969 Dynamics and stability of ferrofluids: surface interactions. J. Fluid Mech. 39 (1), 1–24.CrossRefGoogle Scholar
Zhu, G.-P., Nguyen, N.-T., Ramanujan, R. V. & Huang, X.-Y. 2011 Nonlinear deformation of a ferrofluid droplet in a uniform magnetic field. Langmuir 27 (24), 14834–14841.CrossRefGoogle Scholar