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Numerical-asymptotic models for the manipulation of viscous films via dielectrophoresis

Published online by Cambridge University Press:  02 September 2020

David J. Chappell Open the ORCID record for David J. Chappell [Opens in a new window]  and
Reuben D. O'Dea Open the ORCID record for Reuben D. O'Dea [Opens in a new window]
David J. Chappell
Affiliation:
School of Science and Technology, Nottingham Trent University, Clifton Campus, NottinghamNG11 8NS, UK
Reuben D. O'Dea*
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, NottinghamNG7 2RD, UK
*
†Email address for correspondence: reuben.odea@nottingham.ac.uk
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Abstract

The effect of an externally applied electric field on the motion of an interface between two viscous dielectric fluids is investigated. We first develop a powerful, efficient and widely applicable boundary integral method to compute the interface dynamics in a general multiphysics model comprising coupled Laplace and Stokes flow problems in a periodic half-space. In particular, we exploit the relevant Stokes and Laplace Green's functions to reduce the problem to one defined on the interfacial part of the domain alone. Secondly, motivated by recent experimental work that seeks to underpin the development of switchable liquid optical devices, we concentrate on a fluid–air interface and derive asymptotic approximations suitable to describe the behaviour of a thin film of fluid above an array of electrodes. In this case, the problem is reduced to a single nonlinear partial differential equation describing the film height, coupled to the electrostatic problem via suitable numerical solution or via an asymptotic formula for electrostatic forcing. Comparison against numerical simulations of the full problem shows that the reduced models successfully capture key features of the film dynamics in appropriate regimes; all three approaches are shown to reproduce experimental results.

JFM classification

Interfacial Flows (free surface): Thin films Mathematical Foundations: Computational methods Complex Fluids: Dielectrics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 901 , 25 October 2020 , A35
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Atkinson, K. E. 1997 The Numerical Solution of Integral Equations of the Second Kind. Cambridge University Press.CrossRefGoogle Scholar
Barrett, J. W., Blowey, J. F. & Garcke, H. 1998 Finite element approximation of a fourth order nonlinear degenerate parabolic equation. Numer. Maths 80, 525–556.CrossRefGoogle Scholar
Brown, C. V., McHale, G. & Mottram, N. J. 2011 Analysis of a static undulation on the surface of a thin dielectric liquid layer formed by dielectrophoresis forces. J. Appl. Phys. 110 (2), 024107.CrossRefGoogle Scholar
Brown, C. V., Wells, G. G., Newton, M. I. & McHale, G. 2009 Voltage-programmable liquid optical interface. Nat. Photonics 3 (7), 403.CrossRefGoogle Scholar
Chappell, D. J. 2015 Boundary integral solution of potential problems arising in the modelling of electrified oil films. J. Integral Equ. Appl. 27 (3), 407–430.CrossRefGoogle Scholar
Corson, L. T., Mottram, N. J., Duffy, B. R., Wilson, S. K., Tsakonas, C. & Brown, C. V. 2016 Dynamic response of a thin sessile drop of conductive liquid to an abruptly applied or removed electric field. Phys. Rev. E 94 (13), 043112.CrossRefGoogle ScholarPubMed
Corson, L. T., Tsakonas, C., Duffy, B. R., Mottram, N. J., Sage, I. C., Brown, C. V. & Wilson, S. K. 2014 Deformation of a nearly hemispherical conducting drop due to an electric field: theory and experiment. Phys. Fluids 26 (12), 122106.CrossRefGoogle Scholar
Craster, R. V. & Matar, O. K. 2005 Electrically induced pattern formation in thin leaky dielectric films. Phys. Fluids 17, 032104.CrossRefGoogle Scholar
Ding, Z. & Willis, A. P. 2019 Relative periodic solutions in conducting liquid films flowing down vertical fibres. J. Fluid Mech. 873, 835–855.CrossRefGoogle Scholar
Falk, R. S. & Walker, S. W. 2013 A mixed finite element method for EWOD that directly computes the position of the moving interface. SIAM J. Numer. Anal. 51 (2), 1016–1040.CrossRefGoogle Scholar
Ha, Y., Kim, Y.-J. & Myers, T. G. 2008 On the numerical solution of a driven thin film equation. J. Comput. Phys. 227, 7246–7263.CrossRefGoogle Scholar
Kalogirou, A., Cîmpeanu, R., Keaveny, E. E. & Papageorgiou, D. T. 2016 Capturing nonlinear dynamics of two-fluid Couette flows with asymptotic models. J. Fluid Mech. 806, R1.CrossRefGoogle Scholar
Kress, R. 1989 Linear Integral Equations. Springer.CrossRefGoogle Scholar
Kress, R. & Sloan, I. H. 1993 On the numerical solution of a logarithmic integral equation of the first kind for the Helmholtz equation. Numer. Math. 66 (1), 199–214.CrossRefGoogle Scholar
Li, Y. & Kim, J. 2013 Numerical investigations on self-similar solutions of the nonlinear diffusion equation. Eur. J. Mech. B/Fluids 42, 30–36.CrossRefGoogle Scholar
Linton, C. M. 1999 Rapidly convergent representations for Green's functions for Laplace's equation. Proc. R. Soc. Lond. A 455 (1985), 1767–1797.CrossRefGoogle Scholar
Momoniat, E., Harley, C. & Adlem, E. 2010 Numerical investigation of the generalized lubrication equation. Appl. Maths Comput. 217 (6), 2631–2638.CrossRefGoogle Scholar
Newhouse, L. A. & Pozrikidis, C. 1990 The Rayleigh–Taylor instability of a viscous liquid layer resting on a plane wall. J. Fluid Mech. 217, 615–638.CrossRefGoogle Scholar
O'Dea, R. D. & Waters, S. L. 2006 Flow and solute uptake in a twisting tube. J. Fluid Mech. 562, 173–182.CrossRefGoogle Scholar
Orchard, S. E. 1963 On surface levelling in viscous liquids and gels. Appl. Sci. Res. A 11 (4–6), 451–464.CrossRefGoogle Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (3), 931.CrossRefGoogle Scholar
den Otter, M. W. 2002 Approximate expressions for the capacitance and electrostatic potential of interdigitated electrodes. Sensors Actuators A 96 (2–3), 140–144.CrossRefGoogle Scholar
Papageorgiou, D. T. 2019 Film flows in the presence of electric fields. Annu. Rev. Fluid Mech. 51, 155–187.CrossRefGoogle Scholar
Papageorgiou, D. T., Maldarelli, C. & Rumschitzki, D. S. 1990 Nonlinear interfacial stability of core-annular film flows. Phys. Fluids A 2 (3), 340–352.CrossRefGoogle Scholar
Papageorgiou, D. T. & Petropoulos, P. G. 2004 Generation of interfacial instabilities in charged electrified viscous liquid films. J. Engng Maths 50, 223–240.CrossRefGoogle Scholar
Pease, L. F. & Russel, W. B. 2002 Linear stability analysis of thin leaky dielectric films subjected to electric fields. J. Non-Newtonian Fluid Mech. 102 (2), 233–250.CrossRefGoogle Scholar
Pohl, H. A. 1958 Some effects of nonuniform fields on dielectrics. J. Appl. Phys. 29 (8), 1182–1188.CrossRefGoogle Scholar
Pohl, H. A. 1978 Dielectrophoresis: The Behavior of Neutral Matter in Nonuniform Electric Fields. Cambridge University Press.Google Scholar
Pozrikidis, C. 1992 Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press.CrossRefGoogle Scholar
Preston, M. D., Chamberlain, P. G. & Chandler-Wilde, S. N. 2011 A Nyström method for a boundary value problem arising in unsteady water wave problems. IMA J. Numer. Anal. 31 (3), 1123–1153.CrossRefGoogle Scholar
Saxena, A. 2017 Anisotropic dielectrophoresis: Nematic liquid crystals. Ph.D. thesis, Nottingham Trent University, School of Science and Technology.Google Scholar
Secomb, T. W. 1978 Flow in a channel with pulsating walls. J. Fluid Mech. 88 (2), 273–288.CrossRefGoogle Scholar
Shankar, V. & Sharma, A. 2004 Instability of the interface between thin fluid films subjected to electric fields. J. Colloid Interface Sci. 274 (1), 294–308.CrossRefGoogle ScholarPubMed
Sorgentone, C., Tornberg, A.-K. & Vlahovska, P. M. 2019 A 3D boundary integral method for the electrohydrodynamics of surfactant-covered drops. J. Comput. Phys. 389, 111–127.CrossRefGoogle Scholar
Tomlin, R. J., Cîmpeanu, R. & Papageorgiou, D. T. 2020 Instability and dripping of electrified liquid films flowing down inverted substrates. Phys. Rev. Fluids 5, 013703.CrossRefGoogle Scholar
Tsakonas, C., Corson, L. T., Sage, I. C. & Brown, C. V. 2014 Electric field induced deformation of hemispherical sessile droplets of ionic liquid. J. Electrostat. 72 (6), 437–440.CrossRefGoogle Scholar
Tseluiko, D., Blyth, M. G., Papageorgiou, D. T. & Vanden-Broeck, J.-M. 2008 Electrified viscous thin film flow over topography. J. Fluid Mech. 597, 449–475.CrossRefGoogle Scholar
Tseluiko, D., Blyth, M. G., Papageorgiou, D. T. & Vanden-Broeck, J.-M. 2010 Electrified falling-film flow over topography in the presence of a finite electrode. J. Engng Maths 68 (3–4), 339–353.CrossRefGoogle Scholar
Tseluiko, D. & Papageorgiou, D. T. 2006 Wave evolution on electrified falling films. J. Fluid Mech. 556, 361–386.CrossRefGoogle Scholar
Tseluiko, D. & Papageorgiou, D. T. 2007 Nonlinear dynamics of electrified thin liquid films. SIAM J. Appl. Maths 67 (5), 1310–1329.CrossRefGoogle Scholar
Volkov, D., Papageorgiou, D. T. & Petropoulos, P. G. 2005 Accurate and efficient boundary integral methods for electrified liquid bridge problems. SIAM J. Sci. Comput. 26 (6), 2102–2132.CrossRefGoogle Scholar
Walker, S. W. & Shapiro, B. 2006 Modeling the fluid dynamics of electrowetting on dielectric (EWOD). J. Microelectromech. Syst. 15 (4), 986–1000.CrossRefGoogle Scholar
Wang, Q. & Papageorgiou, D. T. 2016 Using electric fields to induce patterning in leaky dielectric fluids in a rod-annular geometry. IMA J. Appl. Maths 83 (1), 24–52.Google Scholar
Yeo, L. Y. & Chang, H.-C. 2006 Electrowetting films on parallel line electrodes. Phys. Rev. E 73 (1), 011605.CrossRefGoogle ScholarPubMed
Zhornitskaya, L. & Bertozzi, A. L. 1999 Positivity-preserving numerical schemes for lubrication-type equations. SIAM J. Numer. Anal. 37, 523–555.CrossRefGoogle Scholar