Dynamics and stability of an annular electrolyte film

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.

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

“…The analysis is an electrical modification of Hammond (1983) and an electrostatic version of the electrokinetic study of Conroy et al (2010). We introduce the small parameter ǫ = d−1 and assume that the viscosity ratio is of order one, so that in particular ǫλ ≪ 1.…”

“…Recent and ongoing analytical studies concentrate on describing the pinching process asymptotically by utilizing the separation of radial and axial scales and mapping the dynamics to a class of self-similar solutions which are universal when inertia is present; notable studies include the work of Eggers (1993), Eggers & Dupont (1994), Papageorgiou (1995), Brenner et al (1996) for jets surrounded by a passive medium; Craster et al (2002), Craster et al (2003), Craster et al (2005), for surfactant-covered or compound jets; Conroy et al (2010), for core-annular arrangements in the presence of electrokinetic effects. Significant computational work has also been carried out with the aim of simulating the phenomena and evaluating the asymptotic theories (the latter are considerably less demanding numerically) -see Newhouse & Pozrikidis (1992), Pozrikidis (1999), Lister & Stone (1998), Sierou & Lister (2003), who simulate Stokes flows using boundary integral methods, and Ambravaneswaran et al (2002), Chen et al (2002), Notz et al (2001), Notz & Basaran (2004), Collins et al (2007), Hameed et al (2008) who compute the flow at arbitrary Reynolds number and in some instances include the effects of surfactants and electric fields -the extensions and novel aspects of the present work are outlined later.…”

“…One motivation for this study is the theoretical understanding of electrified core-annular flows as could arise in multiphase flows in porous media, for example. In a recent study, Conroy et al (2010), we considered the dynamics of an annular electrolyte film thus extending the study of Hammond (1983) to electrolytic multiphase systems, and showed that the film can either touch the wall in finite time (in contrast to the dynamics of the Hammond equation) or develop into non-uniform stationary states depending on the electric double layer interaction. In this article we simulate such flows numerically for arbitrary undisturbed annular layer thicknesses in order to establish the range of validity of the asymptotic models and provide results at arbitrary parameter values beyond this range.…”

Thin viscous ferrofluid film in a magnetic field

“…We have checked that mass is conserved in the numerical solutions and that the results are consistent with linear theory during the initial stages of the film evolution. Versions of the code have also been used previously (see Conroy et al 17,18 ) for the evolution of electrified films and viscous threads. We first investigate the numerical solution for perturbations of the film at a zero degree incline to compare the results with previous work.…”

“…For ferrofluids with a linear relationship between magnetization and magnetic field, the dynamics are similar to electrified films. 10 There are many papers on the dynamics of electrified films [17][18][19][20][21] that can be used to understand ferrofluid dynamics; however, the two problems diverge for cases where the magnetic field is sufficiently large to warrant a non-linear relationship between magnetization and magnetic field.…”