Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2025-01-06T00:09:40.708Z Has data issue: false hasContentIssue false
  • English
  • Français

Sedimentation of a surfactant-laden drop under the influence of an electric field

Published online by Cambridge University Press:  18 June 2018

Antarip Poddar ,
Shubhadeep Mandal ,
Aditya Bandopadhyay  and
Suman Chakraborty Open the ORCID record for Suman Chakraborty [Opens in a new window]
Antarip Poddar
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal - 721302, India
Shubhadeep Mandal
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal - 721302, India Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 Göttingen, Germany
Aditya Bandopadhyay*
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal - 721302, India
Suman Chakraborty*
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal - 721302, India
*
†Email addresses for correspondence: [email protected], [email protected]
†Email addresses for correspondence: [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The sedimentation of a surfactant-laden deformable viscous drop acted upon by an electric field is considered theoretically. The convection of surfactants in conjunction with the combined effect of electrohydrodynamic flow and sedimentation leads to a locally varying surface tension, which subsequently alters the drop dynamics via the interplay of Marangoni, Maxwell and hydrodynamic stresses. Assuming small capillary number and small electric Reynolds number, we employ a regular perturbation technique to solve the coupled system of governing equations. It is shown that when a leaky dielectric drop is sedimenting in another leaky dielectric fluid, the Marangoni stress can oppose the electrohydrodynamic motion severely, thereby causing corresponding changes in the internal flow pattern. Such effects further result in retardation of the drop settling velocity, which would have otherwise increased due to the influence of charge convection. For non-spherical drop shapes, the effect of Marangoni stress is overcome by the ‘tip-stretching’ effect on the flow field. As a result, the drop deformation gets intensified with an increase in sensitivity of the surface tension to the local surfactant concentration. Consequently, for an oblate type of deformation the elevated drag force causes a further reduction in velocity. For similar reasons, prolate drops experience less drag and settle faster than the surfactant-free case. In addition to this, with increased sensitivity of the interfacial tension to the surfactant concentration, the asymmetric deformation about the equator gets suppressed. These findings may turn out to be of fundamental significance towards designing electrohydrodynamically actuated droplet-based microfluidic systems that are intrinsically tunable by varying the surfactant concentration.

JFM classification

Drops and Bubbles: Drops and Bubbles
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 849 , 25 August 2018 , pp. 277 - 311
Check for updates
Copyright
© 2018 Cambridge University Press 

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

Ajayi, O. O. 1978 A note on Taylor’s electrohydrodynamic theory. Proc R. Soc. Lond. A 364, 499–507.Google Scholar
Anna, S. L. 2016 Droplets and bubbles in microfluidic devices. Annu. Rev. Fluid Mech. 48 (1), 285–309.Google Scholar
Arp, P. A., Foister, R. T. & Mason, S. G. 1980 Some electrohydrodynamic effects in fluid dispersions. Adv. Colloid Interface Sci. 12 (4), 295–356.Google Scholar
Bandopadhyay, A., Mandal, S., Kishore, N. K. & Chakraborty, S. 2016 Uniform electric-field-induced lateral migration of a sedimenting drop. J. Fluid Mech. 792 (2016), 553–589.Google Scholar
Baret, J. C. 2011 Surfactants in droplet-based microfluidics. Lab on a Chip 12 (422), 422–433.Google Scholar
Basaran, O. A. 2002 Small-scale free surface flows with breakup: drop formation and emerging applications. AIChE J. 48 (9), 1842–1848.Google Scholar
Borhan, A., Haj-Hariri, H. & Nadim, A. 1992 Effect of surfactants on the thermocapillary migration of a concentric compound drop. J. Colloid Interface Sci. 149 (2), 553–560.Google Scholar
Chan, P. C.-H. & Leal, L. G. 1979 The motion of a deformable drop in a second-order fluid. J. Fluid Mech. 92 (01), 131–170.Google Scholar
Das, S., Mandal, S. & Chakraborty, S. 2017a Cross-stream migration of a surfactant-laden deformable droplet in a Poiseuille flow. Phys. Fluids 29 (8), 082004.Google Scholar
Das, S., Mandal, S., Som, S. K. & Chakraborty, S. 2017b Migration of a surfactant-laden droplet in non-isothermal Poiseuille flow. Phys. Fluids 29 (1), 012002.Google Scholar
De Bruijn, R. A. 1993 Tipstreaming of drops in simple shear flows. Chem. Engng Sci. 48 (2), 277–284.Google Scholar
Eggleton, C. D., Pawar, Y. P. & Stebe, K. J. 1999 Insoluble surfactants on a drop in an extensional flow: a generalization of the stagnated surface limit to deforming interfaces. J. Fluid Mech. 385, 79–99.Google Scholar
Ervik, Å., Penne, T. E., Hellesø, S. M., Munkejord, S. T. & Müller, B. 2018 Influence of surfactants on the electrohydrodynamic stretching of water drops in oil. Intl J. Multiphase Flow 98, 96–109.Google Scholar
Esmaeeli, A. & Sharifi, P. 2011 Transient electrohydrodynamics of a liquid drop. Phys. Rev. E 84 (3), 036308.Google 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.Google Scholar
Flumerfelt, R. W. 1980 Effects of dynamic interfacial properties on drop deformation and orientation in shear and extensional flow fields. J. Colloid Interface Sci. 76 (2), 330–349.Google Scholar
Ha, J. W. & Yang, S. M. 1995 Effects of surfactant on the deformation and stability of a drop in a viscous fluid in an electric field. J. Colloid Interface Sci. 175 (2), 385–396.Google Scholar
Ha, J.-W. & Yang, S.-M. 1998 Effect of nonionic surfactant on the deformation and breakup of a drop in an electric field. J. Colloid Interface Sci. 206 (1), 195–204.Google Scholar
Hanna, J. A. & Vlahovska, P. M. 2010 Surfactant-induced migration of a spherical drop in Stokes flow. Phys. Fluids 22 (1), 1–7.Google Scholar
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics. Martinus Nijhoff.Google Scholar
Hoburg, J. F. & Melcher, J. R. 1977 Electrohydrodynamic mixing and instability induced by co-linear fields and conductivity gradients. Phys. Fluids 20 (6), 903–911.Google Scholar
Jensen, O. E. & Grotberg, J. B. 1992 Insoluble surfactant spreading on a thin viscous film: shock evolution and film rupture. J. Fluid Mech. 240, 259–288.Google Scholar
Jensen, O. E. & Grotberg, J. B. 1993 The spreading of heat or soluble surfactant along a thin liquid film. Phys. Fluids A 5 (1), 58–68.Google Scholar
Johnson, R. E. & Sadhal, S. S. 1983 Stokes flow past bubbles and drops partially coated with thin films. Part 2. Thin films with internal circulation – a perturbation solution. J. Fluid Mech. 132, 295–318.Google Scholar
Kim, H. S.1988 Surfactant effects on the thermocapillary migration of a droplet. PhD thesis, Clarkson University.Google Scholar
Kim, H. S. & Subramanian, R. S. 1989 Thermocapillary migration of a droplet with insoluble surfactant. I. Surfactant cap. J. Colloid Interface Sci. 127 (2), 417–428.Google Scholar
Lac, E. & Homsy, G. M. 2007 Axisymmetric deformation and stability of a viscous drop in a steady electric field. J. Fluid Mech. 590 (May), 239–264.Google Scholar
Lanauze, J. A., Walker, L. M. & Khair, A. S. 2015 Nonlinear electrohydrodynamics of slightly deformed oblate drops. J. Fluid Mech. 774, 245–266.Google Scholar
Leal, L. G. 2007 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press.Google Scholar
Levich, V. G. 1962 Physicochemical Hydrodynamics. Prentice Hall.Google Scholar
Li, X. & Pozrikidis, C. 1997 The effect of surfactants on drop deformation and on the rheology of dilute emulsions in Stokes flow. J. Fluid Mech. 341 (1997), 165–194.Google Scholar
Mandal, S., Bandopadhyay, A. & Chakraborty, S. 2016a Dielectrophoresis of a surfactant-laden viscous drop. Phys. Fluids 28 (6), 062006.Google Scholar
Mandal, S., Bandopadhyay, A. & Chakraborty, S. 2016b Effect of surface charge convection and shape deformation on the dielectrophoretic motion of a liquid drop. Phys. Rev. E 93 (4), 012101.Google Scholar
Mandal, S., Bandopadhyay, A. & Chakraborty, S. 2016c The effect of uniform electric field on the cross-stream migration of a drop in plane Poiseuille flow. J. Fluid Mech. 809 (2016), 726–774.Google Scholar
Mandal, S., Bandopadhyay, A. & Chakraborty, S. 2017a The effect of surface charge convection and shape deformation on the settling velocity of drops in nonuniform electric field. Phys. Fluids 29 (1), 012101.Google Scholar
Mandal, S., Chakrabarti, S. & Chakraborty, S. 2017b Effect of nonuniform electric field on the electrohydrodynamic motion of a drop in Poiseuille flow. Phys. Fluids 29 (5), 052006.Google Scholar
Mazutis, L., Gilbert, J., Ung, W. L., Weitz, D. A., Griffiths, A. D. & Heyman, J. A. 2013 Single-cell analysis and sorting using droplet-based microfluidics. Nat. Protoc. 8 (5), 870–891.Google Scholar
Melcher, J. R. & Taylor, G. I. 1969 Electrohydrodynamics: a review of the role of interfacial shear stresses. Annu. Rev. Fluid Mech. 1 (1), 111–146.Google Scholar
Mhatre, S. & Thaokar, R. M. 2013 Drop motion, deformation, and cyclic motion in a non-uniform electric field in the viscous limit. Phys. Fluids 25 (7), 072105.Google Scholar
Milliken, W. J. & Leal, L. G. 1994 The influence of surfactant on the deformation and breakup of a viscous drop: the effect of surfactant solubility. J. Colloid Interface Sci. 166 (2), 275–285.Google Scholar
Nganguia, H., Young, Y. N., Vlahovska, P. M., Bławzdziewcz, J., Zhang, J. & Lin, H. 2013 Equilibrium electro-deformation of a surfactant-laden viscous drop. Phys. Fluids 25 (9), 092106.Google Scholar
Pak, O. S., Feng, J. & Stone, H. A. 2014 Viscous marangoni migration of a drop in a Poiseuille flow at low surface Péclet numbers. J. Fluid Mech. 753, 535–552.Google Scholar
Pawar, S. & Stebe, K. J. 1996 Marangoni effects on drop deformation in an extensional flow: the role of surfactant physical chemistry. I. Insoluble surfactants. Phys. Fluids 8, 1738–1751.Google Scholar
Pethig, R. 2013 Dielectrophoresis: an assessment of its potential to aid the research and practice of drug discovery and delivery. Adv. Drug Deliv. Rev. 65 (11–12), 1589–1599.Google Scholar
Ramachandran, A. & Leal, L. G. 2012 The effect of interfacial slip on the rheology of a dilute emulsion of drops for small capillary numbers. J. Rheol. 56 (6), 1555–1587.Google Scholar
Sadhal, S. S. & Johnson, R. E. 1983 Stokes flow past bubbles and drops partially coated with thin films. Part 1. Stagnant cap of surfactant film exact solution. J. Fluid Mech. 126 (1), 237–250.Google Scholar
Saville, D. A. 1997 Electrohydrodynamics: the Taylor–Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 29 (1), 27–64.Google Scholar
Sengupta, R., Walker, L. M. & Khair, A. S. 2017 The role of surface charge convection in the electrohydrodynamics and breakup of prolate drops. J. Fluid Mech. 833, 29–53.Google Scholar
Sharanya, V. & Raja Sekhar, G. P. 2015 Thermocapillary migration of a spherical drop in an arbitrary transient Stokes flow. Phys. Fluids 27 (6), 063104.Google Scholar
Shields, C. W., Reyes, C. D. & López, G. P. 2015 Microfluidic cell sorting: a review of the advances in the separation of cells from debulking to rare cell isolation. Lab on a Chip 15 (5), 1230–1249.Google Scholar
Spertell, R. B. & Saville, D. A. 1976 The roles of electrohydrodynamic phenomena in the motion of drops and bubbles. In Proceedings of the International Colloquium on Drops and Bubbles (ed. Plesset, M.), vol. 1, pp. 106–121. California Institute of Technology.Google Scholar
Stebe, K. J., Lin, S.-Y. & Maldarelli, C. 1991 Remobilizing surfactant retarded fluid particle interfaces. I. Stressfree conditions at the interfaces of micellar solutions of surfactants with fast sorption kinetics. Phys. Fluids A 3 (1), 3–20.Google Scholar
Stone, H. A. & Leal, L. G. 1990 The effects of surfactants on drop deformation and breakup. J. Fluid Mech. 220, 161–186.Google Scholar
Subramanian, R. S. & Balasubramaniam, R. 2001 The Motion of Bubbles and Drops in Reduced Gravity. Cambridge University Press.Google Scholar
Taylor, G. 1966 Studies in electrohydrodynamics. I. The circulation produced in a drop by an electric field. Proc. R. Soc. Lond. A 291 (1425), 159–166.Google Scholar
Teigen, K. E. & Munkejord, S. T. 2010 Influence of surfactant on drop deformation in an electric field. Phys. Fluids 22 (11), 112104.Google Scholar
Thaokar, R. M. 2012 Dielectrophoresis and deformation of a liquid drop in a non-uniform, axisymmetric ac electric field. Eur. Phys. J. E 35 (8), 76.Google Scholar
Tsouris, C., Culbertson, C. T., Depaoli, D. W., Jacobson, S. C., De Almeida, V. F. & Ramsey, J. M. 2003 Electrohydrodynamic mixing in microchannels. AIChE J. 49 (8), 2181–2186.Google Scholar
Tsukada, T., Katayama, T., Ito, Y. & Hozawa, M. 1993 Theoretical and experimental studies of circulations inside and outside a deformed drop under a uniform electric field. J. Chem. Engng Japan 26 (6), 698–703.Google Scholar
Vlahovska, P. M., Loewenberg, M. & Blawzdziewicz, J. 2005 Deformation of a surfactant-covered drop in a linear flow. Phys. Fluids 17 (10), 1–18.Google Scholar
Xu, X. & Homsy, G. M. 2006 The settling velocity and shape distortion of drops in a uniform electric field. J. Fluid Mech. 564, 395–414.Google Scholar
Yariv, E. & Almog, Y. 2016 The effect of surface-charge convection on the settling velocity of spherical drops in a uniform electric field. J. Fluid Mech. 797, 536–548.Google Scholar
Yeo, L. Y., Hou, D., Maheshswari, S. & Chang, H.-C. 2006 Electrohydrodynamic surface microvortices for mixing and particle trapping. Appl. Phys. Lett. 88 (23), 233512.Google Scholar
Young, N. O., Goldstein, J. S. & Block, M. J. 1959 The motion of bubbles in a vertical temperature gradient. J. Fluid Mech. 6 (3), 350–356.Google Scholar
Zeng, J. & Korsmeyer, T. 2004 Principles of droplet electrohydrodynamics for lab-on-a-chip. Lab on a Chip 4 (4), 265–277.Google Scholar
Zhang, L., He, L., Ghadiri, M. & Hassanpour, A. 2015 Effect of surfactants on the deformation and break-up of an aqueous drop in oils under high electric field strengths. J. Petrol. Sci. Engng 125, 38–47.Google Scholar
Zheng, B., Tice, J. D. & Ismagilov, R. F. 2004 Formation of droplets of alternating composition in microfluidic channels and applications to indexing of concentrations in droplet-based assays. Anal. Chem. 76 (17), 4977–4982.Google Scholar
Supplementary material: File

Poddar et al. supplementary material

Poddar et al. supplementary material 1

Download Poddar et al. supplementary material(File)
File 26.7 KB