Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-20T01:44:31.471Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamics and stability of an annular electrolyte film

Published online by Cambridge University Press:  26 May 2010

D. T. CONROY ,
R. V. CRASTER ,
O. K. MATAR  and
D. T. PAPAGEORGIOU
D. T. CONROY
Affiliation:
Department of Chemical Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
R. V. CRASTER*
Affiliation:
Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada
O. K. MATAR
Affiliation:
Department of Chemical Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
D. T. PAPAGEORGIOU
Affiliation:
Department of Chemical Engineering, Imperial College London, South Kensington Campus, London SW7 2AZ, UK Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK
*
†Email address for correspondence: craster@ualberta.ca
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the evolution of an electrolyte film surrounding a second electrolyte core fluid inside a uniform cylindrical tube and in a core-annular arrangement, when electrostatic and electrokinetic effects are present. The limiting case when the core fluid electrolyte is a perfect conductor is examined. We analyse asymptotically the thin annulus limit to derive a nonlinear evolution equation for the interfacial position, which accounts for electrostatic and electrokinetic effects and is valid for small Debye lengths that scale with the film thickness, that is, charge separation takes place over a distance that scales with the annular layer thickness. The equation is derived and studied in the Debye-Hückel limit (valid for small potentials) as well as the fully nonlinear Poisson–Boltzmann equation. These equations are characterized by an electric capillary number, a dimensionless scaled inverse Debye length and a ratio of interface to wall electrostatic potentials. We explore the effect of electrokinetics on the interfacial dynamics using a linear stability analysis and perform extensive numerical simulations of the initial value problem under periodic boundary conditions. An allied nonlinear analysis is carried out to investigate fully singular finite-time rupture events that can take place. Depending upon the parameter regime, the electrokinetics either stabilize or destabilize the film and, in the latter case, cause the film to rupture in finite time. In this case, the final film shape can have a ring- or line-like rupture; the rupture dynamics are found to be self-similar. In contrast, in the absence of electrostatic effects, the film does not rupture in finite time but instead evolves to very long-lived quasi-static structures that are interrupted by an abrupt re-distribution of these very slowly evolving drops and lobes. The present study shows that electrokinetic effects can be tuned to rupture the film in finite time and the time to rupture can be controlled by varying the system parameters. Some intriguing and novel behaviour is also discovered in the limit of large scaled inverse Debye lengths, namely stable and smooth non-uniform steady state film shapes emerge as a result of a balance between destabilizing capillary forces and stabilizing electrokinetic forces.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 656 , 10 August 2010 , pp. 481 - 506
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

Aul, R. W. & Olbricht, W. L. 1990 Stability of a thin annular film pressure-driven, low-Reynolds number flow through a capillary. J. Fluid Mech. 215, 585–599.CrossRefGoogle Scholar
Campana, D., Di Paolo, J. & Saita, F. 2004 A two-dimensional model of Rayleigh instability in capillary tubes: surfactant effects. Intl J. Multiphase Flow 30, 431–454.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Chang, H.-C. & Demekhin, E. A. 1999 Mechanism for drop formation on a coated vertical fibre. J. Fluid Mech. 380, 233–255.CrossRefGoogle Scholar
Chang, H.-C. & Yeo, L. Y. 2010 Electrokinetically Driven Microfluidics and Nanofluidics. Cambridge University Press.Google Scholar
Chang, H.-C. & Yossifon, G. 2009 Understanding electrokinetics at the nanoscale: a perspective. Biomicrofluidics 3, 012001.CrossRefGoogle ScholarPubMed
Craster, R. V. & Matar, O. K. 2006 On viscous beads flowing down a vertical fibre. J. Fluid Mech. 553, 85–106.CrossRefGoogle Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 1131–1198.CrossRefGoogle Scholar
Everett, D. H. & Haynes, J. M. 1972 Model studies of capillary condensation. 1. Cylindrical pore model with zero contact angle. J. Colloid Interface Sci. 38, 125–137.CrossRefGoogle Scholar
Frenke, A. L., Babchin, A. J., Levich, B. G., Shlang, T. & Sivashinsky, G. I. 1987 Annular flow can keep unstable flow from breakup: nonlinear saturation of capillary instability. J. Colloid Interface Sci. 115, 225–233.CrossRefGoogle Scholar
Gauglitz, P. A. & Radke, C. J. 1988 An extended evolution equation for liquid-film break up in cylindrical capillaries. Chem. Engng Sci. 43, 1457–1465.CrossRefGoogle Scholar
Georgiou, E., Papageorgiou, D. T., Maldarelli, C. & Rumschitzki, D. S. 1991 The double layer-capillary stability of an annular electrolyte film surrounding a dielectric-fluid core in a tube. J. Fluid Mech. 226, 149–174.CrossRefGoogle Scholar
Gitlin, I., Stroock, A. D., Whitesides, G. M. & Ajdari, A. 2003 Pumping based on transverse electrokinetic effects. Appl. Phys. Lett. 83, 1486–1488.CrossRefGoogle Scholar
Glasner, K. B. & Witelski, T. P. 2003 Coarsening dynamics of dewetting films. Phys. Rev. E 67, 016302.CrossRefGoogle ScholarPubMed
Goren, S. L. 1962 The instability of an annular thread of fluid. J. Fluid Mech. 12, 309–319.CrossRefGoogle Scholar
Grotberg, J. B. & Jensen, O. E. 2004 Biofluid mechanics in flexible tubes. Annu. Rev. Fluid Mech. 36, 121–147.CrossRefGoogle Scholar
Hagerdon, J. G., Martyn, N. S. & Douglas, J. F. 2004 Breakup of a fluid thread in a confined geometry: droplet–plug transition, perturbation sensitivity, and kinetic stabilization with confinement. Phys. Rev. E 69, 056312.Google Scholar
Halpern, D. & Grotberg, J. B. 2003 Nonlinear saturation of the Rayleigh instability due to oscillatory flow in a liquid-lined tube. J. Fluid Mech. 492, 251–270.CrossRefGoogle Scholar
Hammond, P. S. 1983 Nonlinear adjustment of a thin annular film of viscous fluid surrounding a thread of another within a circular cylindrical pipe. J. Fluid Mech. 137, 363–384.CrossRefGoogle Scholar
Hosoi, A. E. & Mahadevan, L. 1999 Axial instability of a free-surface front in a partially filled horizontal rotating cylinder. Phys. Fluids 11, 97–106.CrossRefGoogle Scholar
Jensen, O. E. 1997 The thin liquid lining of a weakly curved cylindrical tube. J. Fluid Mech. 331, 373–403.CrossRefGoogle Scholar
Johnson, M., Kamm, R. D., Ho, L. W., Shapiro, A. & Pedley, T. J. 1991 The nonlinear growth of surface-tension-driven instabilities of a thin annular film. J. Fluid Mech. 233, 141–156.CrossRefGoogle Scholar
Jones, A. F. & Wilson, S. D. R. 1978 The film drainage problem in droplet coalescence. J. Fluid Mech. 87, 263–288.CrossRefGoogle Scholar
Joseph, D. D., Bai, R., Chen, K. P. & Renardy, Y. Y. 1997 Core-annular flows. Annu. Rev. Fluid Mech. 29, 65–90.CrossRefGoogle Scholar
Kalliadasis, S. & Chang, H.-C. 1994 Drop formation during the coating of vertical fibres. J. Fluid Mech. 261, 135–168.CrossRefGoogle Scholar
Kas-Danouche, S., Papageorgiou, D. T. & Siegel, M. 2009 Nonlinear dynamics of core-annular flows in the presence of surfactant. J. Fluid Mech. 626, 415–448.CrossRefGoogle Scholar
Keast, P. & Muir, K. H. 1991 Algorithm 688 EPDCOL: a more efficient PDECOL code. ACM Trans. Math. Software 17, 153–166.CrossRefGoogle Scholar
Kerchman, V. 1995 Strongly nonlinear interfacial dynamics in core-annular flows. J. Fluid Mech. 290, 131–166.CrossRefGoogle Scholar
Kliakhandler, I. L., Davis, S. H. & Bankoff, S. G. 2001 Viscous beads on vertical fibre. J. Fluid Mech. 429, 381–390.CrossRefGoogle Scholar
Lister, J. R., Rallison, J. M., King, A. A., Cummings, L. J. & Jensen, O. E. 2006 Capillary drainage of an annular film: the dynamics of collars and lobes. J. Fluid Mech. 552, 311–343.CrossRefGoogle Scholar
Maitland, G. C. 2000 Oil and gas production. Curr. Opinion Colloid Interface Sci. 5, 301–311.CrossRefGoogle Scholar
Moffatt, H. K. 1977 Behaviour of a viscous film on the outer surface of a rotating cylinder. J. Mec. 16, 651–673.Google Scholar
Newhouse, L. A. & Pozrikidis, C. 1992 The Rayleigh–Taylor instability of a viscous-liquid layer resting on a plane wall. J. Fluid Mech. 217, 615–638.CrossRefGoogle Scholar
Olbricht, W. L. 1996 Pore-scale prototypes of multiphase flow in porous media. Annu. Rev. Fluid Mech. 28, 187–213.CrossRefGoogle Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980.CrossRefGoogle Scholar
Papageorgiou, D. T., Maldarelli, C. & Rumschitzki, D. S. 1990 Nonlinear interfacial stability of core-annular film flow. Phys. Fluids A 2, 340–352.CrossRefGoogle Scholar
Quéré, D. 1990 Thin-films flowing on vertical fibers. Europhys. Lett. 13, 721–726.CrossRefGoogle Scholar
Quéré, D. 1999 Fluid coating on a fiber. Annu. Rev. Fluid Mech. 31, 347–384.CrossRefGoogle Scholar
Ruyer-Quil, C., Treveleyan, P., Giorgiutti-Dauphiné, F., Duprat, C. & Kalliadasis, S. 2008 Modelling film flows down a fibre. J. Fluid Mech. 603, 431–462.CrossRefGoogle Scholar
Saville, D. A. 1997 Electrohydrodynamics: the Taylor–Melcher leaky-dielectric model. Annu. Rev. Fluid Mech. 29, 27–64.CrossRefGoogle Scholar
Squires, T. M. & Bazant, M. Z. 2004 Induced-charge electro-osmosis. J. Fluid Mech. 509, 217–252.CrossRefGoogle Scholar
Squires, T. M. & Quake, S. R. 2005 Micro-fluidics: fluid physics at the nanoliter scale. Rev. Mod. Phys. 77, 977–1024.CrossRefGoogle Scholar
Thorodsen, S. T. & Mahadevan, L. 1997 Experimental study of coating flows in a partially-filled horizontally rotating cylinder. Exp. Fluids 23, 1–13.CrossRefGoogle Scholar
Tsvelodub, O. Y. & Trifonov, Y. Y. 1992 Nonlinear waves on the surface of a falling liquid film. Part 2. Bifurcations of the first-family waves and other types of nonlinear waves. J. Fluid Mech. 244, 149–169.CrossRefGoogle Scholar
Vaynblat, D., Lister, J. R. & Witelski, T. P. 2001 Rupture of thin viscous films by van der Waals forces: evolution and self-similarity. Phys. Fluids A 13, 1130–1140.CrossRefGoogle Scholar
Wang, Q. 2009 Nonlinear evolution of annular layers and liquid threads in electric fields. PhD thesis, New Jersey Institute of Technology, Newark, New Jersey.Google Scholar
Wei, H.-H. & Rumschitzki, D. 2002 The linear stability of a core-annular flow in an asymptotically corrugated tube. J. Fluid Mech. 466, 113–147.CrossRefGoogle Scholar
Weidner, D. E., Schwartz, L. W. & Eres, M. H. 1997 Simulation of coating layer evolution and drop formation on horizontal cylinders. J. Colloid Interface Sci. 187, 243–258.CrossRefGoogle ScholarPubMed
Zhang, W. W. & Lister, J. R. 1999 Similarity solutions for van der Waals rupture of a thin film on a solid substrate. Phys. Fluids 11, 2454–2462.CrossRefGoogle Scholar