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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: [email protected]
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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

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