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Shapes and stability of pendant and sessile dielectric drops in an electric field

Published online by Cambridge University Press:  26 April 2006

Fred K. Wohlhuter  and
Osman A. Basaran
Fred K. Wohlhuter
Affiliation:
Department of Chemical Engineering, University of Tennessee, Knoxville, TN 37996, USA
Osman A. Basaran
Affiliation:
Chemical Technology Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6224, USA
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Abstract

Axisymmetric equilibrium shapes and stability of linearly polarizable dielectric (ferrofluid) drops of fixed volume which are pendant/sessile on one plate of a parallel-plate capacitor and are subjected to an applied electric (magnetic) field are determined by solving simultaneously the free boundary problem comprised of the Young-Laplace equation for drop shape and the Laplace equation for electric (magnetic) field distribution. When the contact angle that the drop makes with the plate is fixed to be 90° and the distance between the plates is infinite, the results are identical to those of a free drop immersed in a uniform field and resolve discrepancies between previously reported theoretical predictions and experimental measurements. Remarkably, regardless of the value of the ratio of the permittivity (permeability) of the drop to that of the surrounding fluid, κ, drop shapes develop conical tips as drop deformation increases. However, three types of behaviour are found, depending on the value of κ. When κ < κ1, the drop deformation grows without bound as field strength rises. On the other hand, when κ > κ2 > κ1, families of equilibrium drop shapes become unstable at turning points with respect to field strength. Beyond the turning points, the unstable families terminate: the mean curvature at the virtually conical drop tip grows without bound. However, in the range κ1 < κ < κ2, the new results predict that drop deformation exhibits hysteresis, in accord with experiments of Bacri, Salin & Massart (1982) and Bacri & Salin (1982, 1983). Such hysteresis phenomena have been surmized previously on the basis of approximate theories, though they have not been calculated systematically until now. Moreover, detailed computations reveal the importance of varying the drop size and plate spacing, and whether, along the three-phase contact line, the contact line is fixed or the contact angle is prescribed.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 235 , February 1992 , pp. 481 - 510
Copyright
© 1992 Cambridge University Press

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References

Abbott, J. P. 1978 An efficient algorithm for the determination of certain bifurcation points. J. Comput. Appl. Maths 4, 19.Google Scholar
Adornato, P. M. & Brown, R. A. 1983 Shape and stability of levitated drops. Proc. R. Soc. Lond. A 389, 101.Google Scholar
Bacri, J. C. & Salin, D. 1982 Instability of ferrofluid magnetic drops under magnetic field. J. Phys. Lett. 43, L649.Google Scholar
Bacri, J. C. & Salin, D. 1983 Dynamics of the shape transition of a ferrofluid magnetic drop. J. Phys. Lett. 44, L415.Google Scholar
Bacri, J. C., Salin, D. & Massart, R. 1982 Shape of the deformation of ferrofluid droplets in a magnetic field. J. Phys. Lett. 43, L179.Google Scholar
Basaran, O. A., Scott, T. C. & Byers, C. H. 1989 Drop oscillations in liquid-liquid systems. AIChE J. 35, 1263.Google Scholar
Basaran, O. A. & Scriven, L. E. 1982 Profiles of electrified drops and bubbles. In Proc. Second lntl Colloq. on Drops and Bubbles (ed. D. H. LeCroissette). Jet Propulsion Laboratory, Pasadena, CA.
Basaran, O. A. & Scriven, L. E. 1989a Axisymmetric shapes and stability of isolated charged drops. Phys. Fluids A 1, 795.Google Scholar
Basaran, O. A. & Scriven, L. E. 1989b Axisymmetric shapes and stability of charged drops in an external electric field. Phys. Fluids A 1, 799.Google Scholar
Basaran, O. A. & Scriven, L. E. 1990 Axisymmetric shapes and stability of pendant and sessile drops in an electric field. J. Colloid Interface Sci. 140, 10.Google Scholar
Beard, K. V., Ochs, H. T. & Kubesh, R. J. 1989 Natural oscillations of small raindrops. Nature 342, 408.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, 81.Google Scholar
Berkovsky, B. M., Bashtovoi, V., Mikhalev, V. & Rex, A. 1987 Experimental study of the stability of bounded volumes of magnetic fluid with a free surface. J. Magnetism Magn. Mater. 65, 239.Google Scholar
Berkovsky, B. M. & Kalikmanov, V. I. 1985 Topological instability of magnetic fluids. J. Phys. Lett. 46, L483.Google Scholar
Bohr, N. & Wheeler, J. A. 1939 The mechanism of fission. Phys. Rev. 56, 426.Google Scholar
Boudouvis, A. G., Puchalla, J. L. & Scriven, L. E. 1988 Magnetohydrostatic equilibria of ferrofluid drops in external magnetic fields. Chem. Engng Commun. 67, 129.Google Scholar
Brancher, J. P. & Zouaoui, D. 1987 Equilibrium of a magnetic liquid drop. J. Magnetism Magn. Mater. 65, 311.Google Scholar
Brazier-Smith, P. R. 1971 Stability and shape of isolated and pairs of water drops in an electric field. Phys. Fluids 14, 1.Google Scholar
Budnik, A. M. & Polevikov, V. K. 1987 Numerical study of equilibrium forms of magnetic fluid including magnetic field disturbances. J. Magnetism Magn. Mater. 65, 335.Google Scholar
Byers, C. H. & Perona, J. J. 1988 Drop formation from an orifice in an electric field. AIChE J. 34, 1577.Google Scholar
Carruthers, J. R. & Testardi, L. R. 1983 Materials processing in the reduced-gravity of space. Ann. Rev. Mater. Sci. 13, 247.Google Scholar
Chandrasekhar, S. 1969 Ellipsoidal Figures of Equilibrium. Yale University Press.
Chang, L. S. & Berg, J. C. 1985 The effect of interfacial tension gradients on the flow structure of single drops or bubbles translating in an electric field. AIChE J. 31, 551.Google Scholar
Cohen, S., Plasil, R., & Swiatecki, W. J. 1974 Equilibrium configurations of gravitating liquid masses with surface tension. Annls Phys. 82, 557.Google Scholar
Franklin, B. 1751 New Experiments and Observations on Electricity Made at Philadelphia in America. Printed by E. Cave at St John's Gate.
Gupte, S. M. & Tsamopoulos, J. A. 1989 Densification of porous materials by chemical vapor infiltration. J. Electrochem. Soc. 136, 555.Google Scholar
Hood, P. 1976 Frontal solution program for unsymmetric matrices. Intl J. Numer. Meth. Engng 10, 379, and Correction, Intl J. Num. Meth. Engng 11 (1977), 1055.Google Scholar
Iooss, G. & Joseph, D. D. 1990 Elementary Stability and Bifurcation Theory. Springer.
Isaacson, E. & Keller, H. B. 1966 Analysis of Numerical Methods. John Wiley.
Jeffreys, D. J. 1973 Conduction through a random suspension of spheres. Proc. R. Soc. Lond. A 355, 355.Google Scholar
Kistler, S. F. & Scriven, L. E. 1983 Coating flows. In Computational Analysis of Polymer Processing (ed. J. R. A. Pearson & S. M. Richardson). Applied Science Publishers.
Landau, L. D. & Lifshitz, E. M. 1960 Electrodynamics of Continuous Media. Pergamon.
Maxwell, J. C. 1873 Electricity and Magnetism. Clarendon.
Melcher, J. R. 1981 Continuum Electromechanics. MIT Press.
Melcher, J. R. & Taylor, G. I. 1969 Electrohydrodynamics: a review of the role of interfacial shear stresses. Ann. Rev. Fluid Mech. 1, 111.Google Scholar
Miksis, M. J., 1981 Shape of a drop in an electric field. Phys. Fluids 24, 1967.Google Scholar
Ortega, J. M. & Rheinboldt, W. C. 1970 Iterative Solution of Nonlinear Equations in Several Variables. Academic.
Pelekasis, N. A., Tsamopoulos, J. A. & Manolis, G. D. 1990 Equilibrium shapes and stability of charged and conducting drops. Phys. Fluids A 2, 1328.Google Scholar
Rayleigh, Lord 1879 The influence of electricity on colliding water drops. Proc. R. Soc. Lond. A 28, 406.Google Scholar
Rayleigh, Lord 1882 On the equilibrium of liquid conducting masses charged with electricity. Phil. Mag. 14, 184.Google Scholar
Rhim, W. K., Chung, S. K. & Elleman, D. D. 1989 Experiments on rotating charged liquid drops. In Proc. Third. Intl Colloquium on Drops and Bubbles (ed. T. G. Wang). American Institute of Physics.
Rosenkilde, C. E. 1969 A dielectric fluid drop in an electric field. Proc. R. Soc. Lond. A 312, 473.Google Scholar
Rosensweig, R. E. 1979 Fluid dynamics and science of magnetic liquids. In Advances in Electronics and Electron Physics, Vol. 48 (ed. L. Marton). Academic.
Rosensweig, R. E. 1985 Ferrohydrodynamics. Cambridge University Press.
Saito, H. & Scriven, L. E. 1981 Study of coating by the finite element method. J. Comput. Phys. 42, 53.Google Scholar
Sartor, J. D. 1969 Electricity and rain. Phys. Today 22, 45.Google Scholar
Sherwood, J. D. 1988 Breakup of fluid droplets in electric and magnetic fields. J. Fluid Mech. 188, 133.Google Scholar
Strang, G. & Fix, G. J. 1973 An Analysis of the Finite Element Method. Prentice-Hall.
Taylor, G. I. 1964 Disintegration of water drops in an electric field. Proc. R. Soc. Lond. A 280, 383.Google Scholar
Ungar, L. H. & Brown, R. A. 1982 The dependence of the shape and stability of captive rotating drops on multiple parameters. Phil. Trans. R. Soc. Lond. A 306, 347.Google Scholar
Ungar, L. H. & Brown, R. A. 1985 Cellular interface morphologies in directional solidification IV. The formation of deep cells. Phys. Rev. B 31, 5931.Google Scholar
Walters, R. A. 1980 The frontal method in hydrodynamics simulations. Comput. Fluids 8, 265.Google Scholar
Weatherburn, C. E. 1927 Differential Geometry of Three Dimensions. Cambridge University Press.
Wilson, C. T. R. & Taylor, G. I. 1925 The bursting of soap-bubbles in a uniform electric field. Proc. Camb. Phil. Soc. 22, 98.Google Scholar