Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-26T14:54:11.946Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-layer electrified pressure-driven flow in topographically structured channels

Published online by Cambridge University Press:  02 February 2017

Elizaveta Dubrovina ,
Richard V. Craster  and
Demetrios T. Papageorgiou
Elizaveta Dubrovina
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Richard V. Craster
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Demetrios T. Papageorgiou*
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: d.papageorgiou@imperial.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

The flow of two stratified viscous immiscible perfect dielectric fluids in a channel with topographically structured walls is investigated. The flow is driven by a streamwise pressure gradient and an electric field across the channel gap. This problem is explored in detail by deriving and studying a nonlinear evolution equation for the interface valid for large-amplitude long waves in the Stokes flow regime. For flat walls, the electrified flow is long-wave unstable with a critical cutoff wavenumber that increases linearly with the magnitude of the applied voltage. In the nonlinear regime, it is found that the presence of pressure-driven flow prevents electrostatically induced interface touchdown that has been observed previously – time-modulated nonlinear travelling waves emerge instead. When topography is present, linearly stable uniform flows become non-uniform spatially periodic steady states; a small-amplitude asymptotic theory is carried out and compared with computations. In the linearly unstable regime, intricate nonlinear structures emerge that depend, among other things, on the magnitude of the wall corrugations. For a low-amplitude sinusoidal boundary, time-modulated travelling waves are observed that are similar to those found for flat walls but are influenced by the geometry of the wall and slide over it without touching. The flow over a high-amplitude sinusoidal pattern is also examined in detail and it is found that for sufficiently large voltages the interface evolves to large-amplitude waves that span the channel and are subharmonic relative to the wall. A type of ‘walking’ motion emerges that causes the lower fluid to wash through the troughs and create strong vortices over the peaks of the lower boundary. Non-uniform steady states induced by the topography are calculated numerically for moderate and large values of the flow rate, and their stability is analysed using Floquet theory. The effect of large flow rates is also considered asymptotically to find solutions that compare very well with numerical computations.

JFM classification

Interfacial Flows (free surface): Interfacial Flows (free surface) Low-Reynolds-number flows: Low-Reynolds-number flows Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 814 , 10 March 2017 , pp. 222 - 248
Check for updates
Copyright
© 2017 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

Akrivis, G., Papageorgiou, D. T. & Smyrlis, Y.-S. 2011 Linearly implicit methods for a semi linear parabolic system arising in two-phase flows. IMA J. Numer. Anal. 31, 299–321.Google Scholar
Barannyk, L. L., Papageorgiou, D. T., Petropoulos, P. G. & Vanden-Broeck, J.-M. 2015 Nonlinear dynamics and wall touch-up in unstably stratified multilayer flows in horizontal channels under the action of electric fields. SIAM J. Appl. Maths 75, 92–113.Google Scholar
Bielarz, C. & Kalliadasis, S. 2003 Time-dependent free-surface thin film flows over topography. Phys. Fluids 15 (9), 2512–2524.Google Scholar
Chou, S. Y. & Zhuang, L. 1999 Lithographically induced self-assembly of periodic polymer micropillar arrays. J. Vac. Sci. Technol. B 17 (6), 3197–3202.CrossRefGoogle Scholar
Cimpeanu, R. & Papageorgiou, D. T. 2015 Electrostatically induced mixing in confined stratified multi-fluid systems. Intl J. Multiphase Flow 75, 194–204.Google Scholar
Craster, R. V. & Matar, O. K. 2005 Electrically induced pattern formation in thin leaky dielectric films. Phys. Fluids 17 (3), 032104.CrossRefGoogle Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81 (3), 1131–1198.CrossRefGoogle Scholar
Davis, M. J., Gratton, M. B. & Davis, S. H. 2010 Suppressing van der Waals driven rupture through shear. J. Fluid Mech. 661, 522–539.CrossRefGoogle Scholar
Erneaux, T. & Davis, S. H. 1993 Non linear rupture of free films. Phys. Fluids A 5, 1117–1122.CrossRefGoogle Scholar
Heier, J., Groenewold, J. & Steiner, U. 2009 Pattern formation in thin polymer films by spatially modulated electric fields. Soft Matt. 5, 3997–4005.Google Scholar
Kalpathy, S. K., Francis, L. F. & Kumar, S. 2010 Shear-induced suppression of rupture in two-layer thin liquid films. J. Colloid Interface Sci. 348, 271–279.Google Scholar
Karapetsas, G. & Bontozoglou, V. 2015 Non-linear dynamics of a viscoelastic film subjected to a spatially periodic electric field. J. Non-Newtonian Fluid Mech. 217, 1–13.Google Scholar
Lenz, R. D. & Kumar, S. 2007 Steady two-layer flow in a topographically patterned channel. Phys. Fluids 19 (10), 102103.CrossRefGoogle Scholar
Li, F., Ozen, O., Aubry, N., Papageorgiou, D. T. & Petropoulos, P. G. 2007 Linear stability of a two-fluid interface for electrohydrodynamic mixing in a channel. J. Fluid Mech. 583, 347–377.CrossRefGoogle Scholar
Luo, H. & Pozrikidis, C. 2006 Shear-driven and channel flow of a liquid film over a corrugated or indented wall. J. Fluid Mech. 556, 167–188.Google Scholar
Maehlmann, S. & Papageorgiou, D. T. 2011 Interfacial instability in electrified plane Couette flow. J. Fluid Mech. 666, 155–188.Google 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
Nie, Z. & Kumacheva, E. 2008 Patterning surfaces with functional polymers. Nat. Mater. 7 (4), 277–290.Google Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980.Google Scholar
Ozen, O., Aubry, N., Papageorgiou, D. T. & Petropoulos, P. G. 2006 Electrohydrodynamic linear stability of two immiscible fluids in channel flow. Electrochim. Acta 51, 5316–5323.CrossRefGoogle Scholar
Papaefthymiou, E. S., Papageorgiou, D. T. & Pavliotis, G. A. 2013 Nonlinear interfacial dynamics in stratified multilayer channel flows. J. Fluid Mech. 734, 114–143.Google Scholar
Pease, L. F. & Russell, 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
Petzold, L. R. 1983 A description of DASSL: a differential/algebraic system solver. In Scientific Computing (ed. Stepleman, R. S.), pp. 65–68. North Holland.Google Scholar
Pollak, T. & Aksel, N. 2013 Crucial flow stabilisation and multiple instability branches of gravity-driven films over topography. Phys. Fluids 25, 024103.Google Scholar
Ramkrishnan, A. & Kumar, S. 2014 Electrohydrodynamic deformation of thin liquid films near surfaces with topography. Phys. Fluids 26 (12), 122110.Google Scholar
Roberts, S. A. & Kumar, S. 2009 AC electrohydrodynamic instabilities in thin liquid films. J. Fluid Mech. 631, 255–279.Google Scholar
Roberts, S. A. & Kumar, S. 2010 Electrohydrodynamic instabilities in thin liquid trilayer films. Phys. Fluids 22 (12), 122102.Google Scholar
Savettaseranee, K., Papageorgiou, D. T., Petropoulos, P. G. & Tilley, B. S. 2003 The effect of electric fields on the rupture of thin viscous films by van der Waals forces. Phys. Fluids 15, 641–652.Google Scholar
Saville, D. A. 1997 Electrohydrodynamics: the Taylor–Melcher leaky dielectric model. Annu. Rev. Fluid Mech. 29 (1), 27–64.Google Scholar
Schaffer, E., Thurn-Albrecht, T., Russell, T. P. & Steiner, U. 2000 Electrically induced structure formation and pattern transfer. Nature 403 (6772), 874–877.Google Scholar
Schaffer, E., Thurn-Albrecht, T., Russell, T. P. & Steiner, U. 2001 Electrohydrodynamic instabilities in polymer films. Europhys. Lett. 53 (4), 518.CrossRefGoogle Scholar
SooHoo, J. R. & Walker, G. M. 2009 Microfluidic aqueous two phase system for leukocyte concentration from whole blood. Biomed. Microdevices 11 (2), 323–329.Google Scholar
Thaokar, R. M. & Kumaran, V. 2005 Electrohydrodynamic instability of the interface between two fluids confined in a channel. Phys. Fluids 17, 084104.Google Scholar
Toner, M. & Irimia, D. 2005 Blood-on-a-chip. Annu. Rev. Biomed. Engng 7, 77.CrossRefGoogle ScholarPubMed
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.Google Scholar
Tseluiko, D. & Papageorgiou, D. T. 2006 Wave evolution on electrified falling films. J. Fluid Mech. 556, 361–386.Google Scholar
Tseluiko, D. & Papageorgiou, D. T. 2007a Dynamics of a viscous thread surrounded by another viscous fluid in a cylindrical tube under the action of a radial electric field: breakup and touchdown singularities. SIAM J. Appl. Maths 67, 1310–1329.Google Scholar
Tseluiko, D. & Papageorgiou, D. T. 2007b Nonlinear dynamics of electrified thin liquid films. SIAM J. Appl. Math. 67, 1310–1329.Google Scholar
Uguz, A. K. & Aubry, N. 2008 Quantifying the linear stability of a flowing electrified two-fluid layer in a channel for fast electric times for normal and parallel electric fields. Phys. Fluids 20, 092103.Google Scholar
Wu, N. & Russell, W. B. 2009 Micro- and nano-patterns created via electrohydrodynamic instabilities. Nano Today 4 (2), 180–192.Google Scholar
Yih, C.-S. 1967 Instability due to viscosity stratification. J. Fluid Mech. 27, 337–352.Google Scholar
Zhou, C. & Kumar, S. 2012 Two-dimensional two-layer channel flow near a step. Chem. Engng Sci. 81, 38–45.Google Scholar

Dubrovina et al. supplementary movie

Flow past a low amplitude sinusoidal wall showing with travelling waves spanning the channel and following the topography.

Download Dubrovina et al. supplementary movie(Video)
Video 385.7 KB

Dubrovina et al. supplementary movie

Flow past a large amplitude sinusoidal wall at a voltage that is just below the bifurcation of the second unstable mode - a "walking" solution is found at large time.

Download Dubrovina et al. supplementary movie(Video)
Video 785.9 KB

Dubrovina et al. supplementary movie

Flow past a sinusoidal wall of amplitude 0.4 at a voltage $V_b=5.6$ that is just above the bifurcation of the second unstable mode - a travelling wave forms following the topography with time periodic oscillations that reach the upper wall.

Download Dubrovina et al. supplementary movie(Video)
Video 1 MB

Dubrovina et al. supplementary movie

Flow past a sinusoidal wall of amplitude 0.4 at a voltage $V_b=8.5$, before the bifurcation of the third unstable mode - the wave gets caught by the topography and oscillates in time before loosing stability to a "walking" solution spanning the whole channel.

Download Dubrovina et al. supplementary movie(Video)
Video 668.2 KB