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Numerical study of electric field effects on the deformation of two-dimensional liquid drops in simple shear flow at arbitrary Reynolds number

Published online by Cambridge University Press:  10 May 2009

STEFAN MÄHLMANN  and
DEMETRIOS T. PAPAGEORGIOU
STEFAN MÄHLMANN
Affiliation:
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA
DEMETRIOS T. PAPAGEORGIOU*
Affiliation:
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*
†E-mail address for correspondence: [email protected]
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Abstract

The effect of an electric field on a periodic array of two-dimensional liquid drops suspended in simple shear flow is studied numerically. The shear is produced by moving the parallel walls of the channel containing the fluids at equal speeds but in opposite directions and an electric field is generated by imposing a constant voltage difference across the channel walls. The level set method is adapted to electrohydrodynamics problems that include a background flow in order to compute the effects of permittivity and conductivity differences between the two phases on the dynamics and drop configurations. The electric field introduces additional interfacial stresses at the drop interface and we perform extensive computations to assess the combined effects of electric fields, surface tension and inertia. Our computations for perfect dielectric systems indicate that the electric field increases the drop deformation to generate elongated drops at steady state, and at the same time alters the drop orientation by increasing alignment with the vertical, which is the direction of the underlying electric field. These phenomena are observed for a range of values of Reynolds and capillary numbers. Computations using the leaky dielectric model also indicate that for certain combinations of electric properties the drop can undergo enhanced alignment with the vertical or the horizontal, as compared to perfect dielectric systems. For cases of enhanced elongation and alignment with the vertical, the flow positions the droplets closer to the channel walls where they cause larger wall shear stresses. We also establish that a sufficiently strong electric field can be used to destabilize the flow in the sense that steady-state droplets that can exist in its absence for a set of physical parameters, become increasingly and indefinitely elongated until additional mechanisms can lead to rupture. It is suggested that electric fields can be used to enhance such phenomena.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 626 , 10 May 2009 , pp. 367 - 393
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Adalsteinsson, D. & Sethian, J. 1995 The fast level set method for propagating interfaces. J. Comp. Phys. 118, 269–277.CrossRefGoogle Scholar
Adalsteinsson, D. & Sethian, J. 1999 The fast construction of extension velocities in level set methods. J. Comp. Phys. 148, 2–22.CrossRefGoogle Scholar
Allan, R. S. & Mason, S. G. 1962 Particle behaviour in shear and electric fields. 1. Deformation and burst of fluid drops. Proc. R. Soc. Lond. Ser. A 267, 45–61.Google Scholar
Brackbill, J. U., Kothe, D. B. & Zemach, C. 1992 A continuum method for modeling surface tension. J. Comp. Phys. 100, 335–354.CrossRefGoogle Scholar
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166–188.CrossRefGoogle Scholar
Burcham, C. L. & Saville, D. A. 2000 The electrohydrodynamic stability of a liquid bridge: microgravity experiments on a bridge suspended in a dielectric gas. J. Fluid Mech. 405, 37–56.CrossRefGoogle Scholar
Charles, R. & Pozrikidis, C. 1998 Effect of the dispersed phase viscosity on the simple shear flow of suspensions of liquid drops. J. Fluid Mech. 365, 205–233.CrossRefGoogle Scholar
Chopp, D. L. 2001 Some improvements of the fast marching method. SIAM J. Sci. Comp. (USA) 23, 230–244.CrossRefGoogle Scholar
Chorin, A. 1968 Numerical solution of the Navier–Stokes equations. Math. Comput. 22, 745–762.CrossRefGoogle Scholar
Collins, R. T., Jones, J. J., Harris, M. T. & Basaran, O. A. 2008 Electrohydrodynamic tip streaming and emission of charged drops from liquid cones. Nature Phys. 4, 149–154.CrossRefGoogle Scholar
Cristini, V. & Tan, Y.-C. 2004 Theory and numerical simulation of droplet dynamics in complex flows – a review. Lab Chip 4, 257–264.CrossRefGoogle ScholarPubMed
De Menech, M. 2006 Modeling of droplet breakup in a microfluidic T-shaped junction with a phase-field model. Phys. Rev. E 73 (3), 1–9.CrossRefGoogle Scholar
Feng, J. Q. 1999 Electrohydrodynamic behaviour of a drop subjected to a steady uniform electric field at finite electric reynolds number. Proc. R. Soc. Lond. A 455, 2245–2269.CrossRefGoogle Scholar
Feng, J. Q. 2002 A 2d electrohydrodynamic model for electrorotation of fluid drops. J. Colloid Interface Sci. 246, 112–121.CrossRefGoogle ScholarPubMed
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
Fernández, A., Tryggvason, G., Che, J. & Ceccio, S. L. 2005 The effects of electrostatic forces on the distribution of drops in a channel flow: two-dimensional oblate drops. Phys. Fluids 17 (9), 1–15.CrossRefGoogle Scholar
Kennedy, M. R., Pozrikidis, C. & Skalak, R. 1994 Motion and deformation of liquid drops, and the rheology of dilute emulsions in simple shear flow. Comp. Fluids 23, 251–278.CrossRefGoogle Scholar
Khismatulin, D., Renardy, Y. & Cristini, V. 2003 Inertia-induced breakup of highly viscous drops subjected to simple shear. Phys. Fluids 15, 1351–1354.CrossRefGoogle Scholar
Lac, E. & Homsy, G. M. 2007 Axisymmetric deformation and stability of a viscous drop in a steady electric field. J. Fluid Mech. 590, 239–264.CrossRefGoogle Scholar
Lee, J. & Pozrikidis, C. 2006 Effect of surfactants on the deformation of drops and bubbles in Navier–Stokes flow. Comp. Fluids 35 (1), 43–60.CrossRefGoogle Scholar
Li, X., Charles, R. & Pozrikidis, C. 1996 Simple shear flow of suspensions of liquid drops. J. Fluid Mech. 320, 395–416.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
Li, X., Zhou, H. & Pozrikidis, C. 1995 Numerical study of the shearing motion of emulsions and foams. J. Fluid Mech. 286, 379–404.CrossRefGoogle Scholar
Melcher, J. R. & Taylor, G. I. 1969 Electrohydrodynamics: a review of the role of interfacial shear stresses. Annu. Rev. Fluid Mech. 1, 111–146.CrossRefGoogle Scholar
O'Konski, C. T. & Thacher, H. C. Jr. 1953 The distortion of aerosol droplets by an electric field. J. Phys. Chem. 57, 955–958.CrossRefGoogle Scholar
Osher, S. & Sethian, J. 1988 Front propagating with curvature dependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comp. Phys. 79, 12–49.CrossRefGoogle Scholar
Osher, S. & Shu, C.-W. 1991 High order essentially non-oscillatory schemes for Hamilton Jacobi equations. SIAM J. Num. Anal. 28, 907–922.CrossRefGoogle Scholar
Ozen, O., Aubry, N., Papageorgiou, D. T. & Petropoulos, P. G. 2006 Monodisperse drop formation in square microchannels. Phys. Rev. Lett. 96, Art. 144501.CrossRefGoogle ScholarPubMed
Peskin, C. S. 1977 Numerical analysis of blood flow in the heart. J. Comp. Phys. 25, 220–252.CrossRefGoogle Scholar
Pozrikidis, C. 1997 Numerical studies of singularity formation at free surfaces and fluid interfaces in two-dimensional Stokes flow. J. Fluid Mech. 331, 145–167.CrossRefGoogle Scholar
Pozrikidis, C. 1998 Numerical studies of cusp formation at fluid interfaces in Stokes flow. J. Fluid Mech. 357, 29–57.CrossRefGoogle Scholar
Rallison, J. M. 1981 A numerical study of the deformation and burst of a viscous drop in general shear flows. J. Fluid Mech. 109, 465–482.CrossRefGoogle Scholar
Rallison, J. M. & Acrivos, A. 1978 A numerical study of the deformation and burst of a viscous drop in an extensional flow. J. Fluid Mech. 89, 191–200.CrossRefGoogle Scholar
Renardy, Y. 2007 The effects of confinement and inertia on the production of droplets. Rheol. Acta 46, 521–529.CrossRefGoogle Scholar
Renardy, Y. & Cristini, V. 2001 Scalings for fragments produced from drop breakup in shear flow with inertia. Phys. Fluids 13, 2161–2164.CrossRefGoogle Scholar
Rhodes, P. H., Snyder, R. S. & Roberts, G. O. 1989 Electrohydrodynamic distortion of sample streams in continuous flow electrophoresis. J. Colloid Interface Sci. 129, 78–90.CrossRefGoogle Scholar
Ryskin, G. & Leal, L. G. 1984 Numerical solution of free-boundary problems in fluid mechanics. Part 3. Bubble deformation in an axisymmetric straining flow. J. Fluid Mech. 148, 37–43.CrossRefGoogle Scholar
Saville, D. A. 1997 Electrohydrodynamics: the Taylor–Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 29, 27–64.CrossRefGoogle Scholar
Sethian, J. 1996 A fast marching level set method for monotonically advancing fronts. Proc. Acad. Nat. Sci. 93, 1591–1595.CrossRefGoogle ScholarPubMed
Sethian, J. 1999 Fast marching methods. SIAM Rev. 41, 199–235.CrossRefGoogle Scholar
Shankar, V. & Sharma, A 2004 Instability on the interface between thin fluid films subjected to electric fields. J. Colloid Interface Sci. 274, 294–308.CrossRefGoogle ScholarPubMed
Sheth, K. & Pozrikidis, C. 1995 Effects of inertia on the deformation of liquid drops in simple shear flow. Comp. Fluids 24 (2), 101–119.CrossRefGoogle Scholar
Sibillo, V., Pasquariello, G., Simeone, M., Cristini, V. & Guido, S. 2006 Drop deformation in microconfined shear flow. Phys. Rev. Lett. 97, 0545021–0545024.CrossRefGoogle ScholarPubMed
Smith, C. V. & Melcher, J. R. 1967 Electrohydrodynamically induced spatially periodic cellular Stokes-flow. Phys. Fluids 10, 2315–2322.CrossRefGoogle Scholar
Song, H., Chen, D. L. & Ismagilov, R. F. 2006 Reactions in droplets in microfluidic channels. Angewandte Chemie – International Edition 45 (44), 7336–7356.CrossRefGoogle ScholarPubMed
Stone, H. A. 1994 Dynamics of drop deformation and breakup in viscous fluids. Annu. Rev. Fluid Mech. 26 (1), 65–102.CrossRefGoogle Scholar
Stone, H. A., Stroock, A. D. & Ajdari, A. 2004 Engineering flows in small devices: microfluidics toward a lab-on-a-chip. Annu. Rev. Fluid Mech. 36, 381–411.CrossRefGoogle Scholar
Sussman, M. & Fatemi, E. 1999 An efficient interface-preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow. SIAM J. Sci. Comp. 20, 1165–1191.CrossRefGoogle Scholar
Sussman, M., Fatemi, E., Smereka, P. & Osher, S. 1994 A level set approach for computing solutions to incompressible two-phase flows. J. Comp. Phys. 100, 146–159.CrossRefGoogle Scholar
Taylor, G. I. 1934 The formation of emulsions in definable fields of flow. Proc. R. Soc. Lond. 146, 501–523.Google Scholar
Taylor, G. I. 1966 Studies in electrohydrodynamics: I. The circulation produced in a drop by an electric field. Proc. R. Soc. Lond. Ser. A 291, 159–166.Google Scholar
Tomar, G., Gerlach, D., Biswas, G., Alleborn, N., Sharma, A., Durst, F., Welch, S. W. J. & Delgado, A. 2007 Two-phase electrohydrodynamic simulations using a volume of fluid approach. J. Comp. Phys. 227, 1267–1285.CrossRefGoogle Scholar
Unverdi, O. & Tryggvason, G. 1992 A front-tracking method for viscous incompressible, multi-fluid flows. J. Comp. Phys. 100, 25–37.CrossRefGoogle Scholar
Wagner, A. J., Wilson, L. M. & Cates, M. E. 2003 Role of inertia in two-dimensional deformation and breakdown of a droplet. Phys. Rev. E 68, 453011–453014.CrossRefGoogle ScholarPubMed
Ward, T. & Homsy, G. M. 2001 Electrohydrodynamically driven chaotic mixing in a translating drop. Phys. Fluids 13, 3521–3525.CrossRefGoogle Scholar
Ward, T. & Homsy, G. M. 2003 Electrohydrodynamically driven chaotic mixing in a translating drop. II. Experiments. Phys. Fluids 15, 2987–2994.CrossRefGoogle Scholar
Zhou, H. & Pozrikidis, C. 1993 a The flow of suspensions in channels: single files of drops. Phys. Fluids A 5, 311–323.CrossRefGoogle Scholar
Zhou, H. & Pozrikidis, C. 1993 b The flow of ordered and random suspensions of two-dimensional drops in a channel. J. Fluid Mech. 255, 103–127.CrossRefGoogle Scholar
Zhou, H. & Pozrikidis, C. 1994 Pressure-driven flow of suspensions of liquid drops. Phys. Fluids 6, 80–94.CrossRefGoogle Scholar