Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-19T11:31:50.383Z Has data issue: false hasContentIssue false
  • English
  • Français

Deformation of a hydrophobic ferrofluid droplet suspended in a viscous medium under uniform magnetic fields

Published online by Cambridge University Press:  08 September 2010

S. AFKHAMI ,
A. J. TYLER ,
Y. RENARDY ,
M. RENARDY ,
T. G. St. PIERRE ,
R. C. WOODWARD  and
J. S. RIFFLE
S. AFKHAMI
Affiliation:
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA
A. J. TYLER
Affiliation:
School of Physics, M013, The University of Western Australia, Crawley, WA 6009, Australia
Y. RENARDY*
Affiliation:
Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA
M. RENARDY
Affiliation:
Department of Mathematics, Virginia Tech, Blacksburg, VA 24061, USA
T. G. St. PIERRE
Affiliation:
School of Physics, M013, The University of Western Australia, Crawley, WA 6009, Australia
R. C. WOODWARD
Affiliation:
School of Physics, M013, The University of Western Australia, Crawley, WA 6009, Australia
J. S. RIFFLE
Affiliation:
Department of Chemistry and the Macromolecules and Interfaces Institute, Virginia Tech, Blacksburg, VA 24061, USA
*
†Email address for correspondence: renardy@vt.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The effect of applied magnetic fields on the deformation of a biocompatible hydrophobic ferrofluid drop suspended in a viscous medium is investigated numerically and compared with experimental data. A numerical formulation for the time-dependent simulation of magnetohydrodynamics of two immiscible non-conducting fluids is used with a volume-of-fluid scheme for fully deformable interfaces. Analytical formulae for ellipsoidal drops and near-spheroidal drops are reviewed and developed for code validation. At low magnetic fields, both the experimental and numerical results follow the asymptotic small deformation theory. The value of interfacial tension is deduced from an optimal fit of a numerically simulated shape with the experimentally obtained drop shape, and appears to be a constant for low applied magnetic fields. At high magnetic fields, on the other hand, experimental measurements deviate from numerical results if a constant interfacial tension is implemented. The difference can be represented as a dependence of apparent interfacial tension on the magnetic field. This idea is investigated computationally by varying the interfacial tension as a function of the applied magnetic field and by comparing the drop shapes with experimental data until a perfect match is found. This estimation method provides a consistent correlation for the variation in interfacial tension at high magnetic fields. A conclusion section provides a discussion of physical effects which may influence the microstructure and contribute to the reported observations.

JFM classification

Drops and Bubbles: Drops
Type
Papers
Information
Journal of Fluid Mechanics , Volume 663 , 25 November 2010 , pp. 358 - 384
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

Afkhami, S., Renardy, Y., Renardy, M., Riffle, J. S. & St. Pierre, T. G. 2008 a Field-induced motion of ferrofluid droplets through immiscible viscous media. J. Fluid Mech. 610, 363–380.CrossRefGoogle Scholar
Afkhami, S., Renardy, Y., Renardy, M., Riffle, J. S. & St. Pierre, T. G. 2008 b Numerical modeling of ferrofluid droplets in magnetic fields. In Proceedings of XVth International Congress on Rheology. American Institute of Physics.Google Scholar
Arkhipenko, V. I., Barkov, Y. D. & Bashtovoi, V. G. 1978 Investigation of the form of a drop of a magnetizing liquid in a homogeneous magnetic field. Magnetohydrodynamics 14, 131–134.Google Scholar
Bacri, J. C. & Salin, D. 1982 Instability of ferrofluid magnetic drops under magnetic field. J. Phys. Lett. 43, 649–654.CrossRefGoogle Scholar
Basaran, O. A. & Wohlhuter, F. K. 1992 Effect of nonlinear polarization on shapes and stability of pendant and sessile drops in an electric (magnetic) field. J. Fluid Mech. 244, 1–16.CrossRefGoogle Scholar
Blums, E., Cebers, A. & Maiorov, M. M. 1997 Magnetic Fluids. De Gruyter.Google Scholar
Brackbill, J. U., Kothe, D. B. & Zemach, C. 1992 A continuum method for modeling surface tension. J. Comput. Phys. 100, 335–354.CrossRefGoogle Scholar
Cebers, A. 1982 Role of surface interactions in stratification of magnetic fluids. Magnetohydrodynamics 18, 345–350.Google Scholar
Cebers, A. 1985 Virial method of investigation of statics and dynamics of drops of magnetizable liquids. Magnetohydrodynamics 21, 19–26.Google Scholar
Cebers, A. 2002 The anisotropy of the surface tension at the magnetic-field-induced phase transformations. J. Magn. Magn. Mat. 252, 259–261.CrossRefGoogle Scholar
Dames, P., Gleich, B., Flemmer, A., Hajek, K., Seidl, N., Wiekhorst, F., Eberbeck, D., Bittmann, I., Bergemann, C., Weyh, T., Trahms, L., Rosenecker, J. & Rudolph, C. 2007 Targeted delivery of magnetic aerosol droplets to the lung. Nature Nanotech. 2, 495–499.CrossRefGoogle ScholarPubMed
Dodgson, N. & Sozou, C. 1987 The deformation of a liquid drop by an electric field. J. Appl. Math. Phys. (ZAMP) 38, 424–432.CrossRefGoogle Scholar
Drozdova, V. I., Skrobotova, T. V. & Chekanov, V. V. 1979 Experimental study of the hydrostatics characterizing the interphase boundary in a ferrofluid. Magnetohydrodynamics 15, 12–14.Google Scholar
Feng, J. Q. & Scott, T. C. 1996 A computational analysis of electrohydrodynamics of a leaky dielectric drop in an electric field. J. Fluid Mech. 311, 289–326.CrossRefGoogle Scholar
Garton, C. G. & Krasucki, Z. 1964 Bubbles in insulating liquids: stability in an electric field. Proc. R. Soc. Lond. A 280, 211–226.Google Scholar
Holligan, D. L., Gillies, B. T. & Dailey, J. P. 2003 Magnetic guidance of ferrofluidic nanoparticles in an in vitro model of intraocular retinal repair. Nanotechnology 14 (6), 661–666.CrossRefGoogle Scholar
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, 204143.Google ScholarPubMed
Lafaurie, B., Nardone, C., Scardovelli, R., Zaleski, S. & Zanetti, G. 1994 Modelling merging and fragmentation in multiphase flows with SURFER. J. Comput. Phys. 113, 134–147.CrossRefGoogle Scholar
Landau, L. D. & Lifshitz, E. M. 1984 Electrodynamics of Continuous Media. Pergamon.Google Scholar
Lavrova, O., Matthies, G., Mitkova, T., Polevikov, V. & Tobiska, L. 2006 Numerical treatment of free surface problems in ferrohydrodynamics. J. Phys.: Condens. Matter 18 (38), S2657–S2669.Google Scholar
Lavrova, O., Matthies, G., Polevikov, V. & Tobiska, L. 2004 Numerical modeling of the equilibrium shapes of a ferrofluid drop in an external magnetic field. Proc. Appl. Math. Mech. 4, 704–705.CrossRefGoogle Scholar
Li, J. & Renardy, Y. 1999 Direct simulation of unsteady axisymmetric core-annular flow with high viscosity ratio. J. Fluid Mech. 391, 123–149.CrossRefGoogle Scholar
Li, J., Renardy, Y. & Renardy, M. 2000 Numerical simulation of breakup of a viscous drop in simple shear flow through a volume-of-fluid method. Phys. Fluids 12, 269–282.CrossRefGoogle Scholar
Liggieri, L., Sanfeld, A. & Steinchen, A. 1994 Effects of magnetic and electric fields on surface tension of liquids. Physica A 206, 299–331.CrossRefGoogle Scholar
Liu, X., Kaminski, M. D., Riffle, J.S., Chen, H., Torno, M., Finck, M. R., Taylor, L. & Rosengart, A. J. 2007 Preparation and characterization of biodegradable magnetic carriers by single emulsion-solvent evaporation. J. Magn. Magn. Mat. 311, 84–87.CrossRefGoogle Scholar
Markov, K. Z. 1999 Stability and instability in viscous fluids. In Heterogeneous Media: Modelling and Simulation (ed. Markov, K. Z. & Preziosi, L.), chap. 1, pp. 1–162. Birkhäuser.Google Scholar
Martin, J. E., Venturini, E. L. & Huber, D. L. 2008 Giant magnetic susceptibility enhancement in field-structured nanocomposites. J. Magn. Magn. Mat. 320, 2221–2227.CrossRefGoogle Scholar
Mefford, O. T. 2007 Physical properties of macromolecule-metal oxide nanoparticle complexes: magnetophoretic mobility, size, and interparticle potentials. PhD thesis, Department of Chemistry, Virginia Tech.Google Scholar
Mefford, O. T., Carroll, M. R. J., Vadala, M. L., Goff, J.D., Mejia-Ariza, R., Saunders, M., Woodward, R. C., St. Pierre, T. G., Davis, R. M. & Riffle, J. S. 2008 a Size analysis of PDMS-magnetite nanoparticle complexes: experiment and theory. Chem. Mater. 20 (6), 2184–2191.CrossRefGoogle Scholar
Mefford, O. T., Vadala, M. L., Carroll, M.R. J., Mejia-Ariza, R., Caba, B. L., St. Pierre, T. G., Woodward, R. C., Davis, R. M. & Riffle, J. S. 2008 b Stability of polydimethylsiloxane-magnetite nanoparticles against flocculation: interparticle interactions of polydisperse materials. Langmuir 24 (9), 5060–5069.CrossRefGoogle ScholarPubMed
Mefford, O. T., Woodward, R. C., Goff, J.D., Vadala, T. P., St. Pierre, T. G., Dailey, J. P. & Riffle, J. S. 2007 Field-induced motion of ferrofluids through immiscible viscous media: testbed for restorative treatment of retinal detachment. J. Magn. Magn. Mater. 311, 347–353.CrossRefGoogle Scholar
Mendelev, V. S. & Ivanov, A. O. 2004 Ferrofluid aggregation in chains under the influence of a magnetic field. Phys. Rev. E 70, 051502–1–051502–10.CrossRefGoogle ScholarPubMed
Mikkelsen, C. I. 2005 Magnetic separation and hydrodynamic interactions in microfluidic systems. PhD thesis, Technical University of Denmark.Google Scholar
Miksis, M. J. 1981 Shape of a drop in an electric field. Phys. Fluids 24 (11), 1967–1972.CrossRefGoogle Scholar
Patankar, S. V. 1980 Numerical Heat Transfer and Fluid Flow. McGraw-Hill.Google Scholar
Rallison, J. M. 1980 Note on the time-dependent deformation of a viscous drop which is almost spherical. J. Fluid Mech. 98 (3), 625–633.CrossRefGoogle Scholar
Rosensweig, R. E. 1985 Ferrohydrodynamics. Cambridge University Press.Google Scholar
Scardovelli, R. & Zaleski, S. 1999 Direct numerical simulation of free surface and interfacial flow. Annu. Rev. Fluid Mech. 31, 567–604.CrossRefGoogle Scholar
Shahnazian, H. & Odenbach, S. 2007 Yield stress in ferrofluids? Intl J. Mod. Phys. B 21, 4806–4812.CrossRefGoogle Scholar
Sherwood, J. D. 1991 The deformation of a fluid drop in an electric field: a slender-body analysis. J. Phys. A 24, 4047–4053.CrossRefGoogle Scholar
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, 329–347.CrossRefGoogle Scholar
Taylor, G. I. 1964 Disintegration of water drops in an electric field. Proc. R. Soc. Lond. A 280, 383–397.Google Scholar
Tyatyushkin, A. N. & Verlarde, M. G. 2001 On the interfacial deformation of a magnetic liquid drop under the simultaneous action of electric and magnetic fields. J. Colloid Intl Sci. 235, 46–58.CrossRefGoogle ScholarPubMed
Tyler, A. J. 2008 Presentation: dynamic behaviour of magnetic fluid droplets in non-uniform magnetic fields.Google Scholar
Tyler, A. J., Woodward, R. C. & St. Pierre, T. G. 2010 Variation in the surface energy of a ferrofluid droplet in an applied magnetic field (in preparation).Google Scholar
Voltairas, P. A., Fotiadis, D. I. & Massalas, L. K. 2001 Elastic stability of silicone ferrofluid internal tamponade (SFIT) in retinal detachment surgery. J. Magn. Mag. Mat. 225, 248–255.CrossRefGoogle Scholar
Voltairas, P. A., Fotiadis, D. I. & Michalis, L. K. 2002 Hydrodynamics of magnetic drug targeting. J. Biomech. 35, 813–821.CrossRefGoogle ScholarPubMed
Wohlhuter, F. K. & Basaran, O. A. 1992 Shapes and stability of pendant and sessile dielectric drops in an electric field. J. Fluid Mech. 235, 481–510.CrossRefGoogle Scholar
Woodward, R. C., Heeris, J., St. Pierre, T. G., Saunders, M., Gilbert, E. P., Rutnakornpituk, M., Zhang, Q. & Riffle, J. S. 2007 A comparison of methods for the measurement of the particle-size distribution of magnetic nanoparticles. J. Appl. Cryst. 40, S495–S500.CrossRefGoogle Scholar