Nonlinear Dynamics of Electrified Thin Liquid Films

Abstract

We study a nonlinear nonlocal evolution equation describing the hydrodynamics of thin films in the presence of normal electric fields. The liquid film is assumed to be perfectly conducting and to completely wet the upper or lower surface of a horizontal flat plate. The flat plate is held at constant voltage, and a vertical electric field is generated by a second parallel electrode kept at a different constant voltage and placed at a large vertical distance from the bottom plate. The fluid is viscous, and gravity and surface tension act. The equation is derived using lubrication theory and contains an additional nonlinear nonlocal term representing the electric field. The electric field is linearly destabilizing and is particularly important in producing nontrivial dynamics in the case when the film rests on the upper side of the plate. We give rigorous results on the global boundedness of positive periodic smooth solutions, using an appropriate energy functional. We also implement a fully implicit numerical scheme and perform extensive numerical experiments. Through a combination of analysis and numerical experiments we present evidence for the global existence of positive smooth solutions. This means, in turn, that the film does not touch the wall in finite time but asymptotically at infinite time. Numerical solutions are presented to support such phenomena, which are also observed in hanging films when electric fields are absent.

Electric field stabilization of viscous liquid layers coating the underside of a surface

“…We describe our numerical solution of (41) in the Appendix. Note that in the absence of the lower dielectric slab (region 0), the nonlocal expression (40) simplifies to a form that shares strong similarities with the corresponding term in the evolution equation previously derived by Tseluiko and Papageorgiou [20]. In the respective case the sign is shifted due to their destabilizing vertical electric field configuration, while the prefactor is also different, as here we include the influence of the upper solid region 3 through T .…”

“…An additional coupling to the fields in the slab regions 0 and 3 is present due to the voltage continuity conditions at y = 0,1 + and the Gauss laws (18) and (20). Following [19], we use the fact that the complex functions ∂ x φ j − i∂ y φ j with j = 0,3 are analytic in their respective domains and apply Cauchy's theorem in regions 0 and 3 in turn.…”

“…Implementing our upscaling method is straightforward with Fourier interpolation; we take our Fourier transformed solutionĥ M on a grid with M grid points, pad M higher wave numbers with zero, and then transform back to obtain a solution h 2M defined on 2M grid points. We also control the time step by employing a local error indicator e m , which approximates [20,25]:…”

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 new equation is interesting both physically and mathematically because when β = 0 solutions are found to terminate (under fairly general initial conditions) in touchdown singularities in finite time, as opposed to the absence of touchdown in the Hammond equation which supports intricate multi-scale dynamics over extremely large time-scales (for rigorous results see Tseluiko & Papageorgiou (2007) and for extensive numerical work see Lister et al (2006)). We also note that (4.8) emerges in the zero electrokinetic limit of the more general electrolytic fluids version derived by Conroy et al (2010); in addition, it provides the structures near touchdown of the electrokinetic model, if the latter terminates in a singularity.…”

“…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. Also of interest are the boundary integral computations of Newhouse & Pozrikidis (1992) of the capillary driven dynamics of threads in core-annular arrangements in perfectly cylindrical tubes, who show that the dynamics are ultimately attracted to pinching or wall touchdown (strictly speaking the latter is precluded by the mathematical properties of the Hammond equation and some recent very large time computations studies of it -see Tseluiko & Papageorgiou (2007) and Lister et al (2006), respectively), depending on whether the undisturbed annulus thickness is above or below a threshold value. Our numerical work accurately predicts this boundary in the solution space and we also quantify the effect of the electric field showing that relatively thick annuli that lead to pinching in the absence of a field, can be driven to wall touchdown in its presence, thus indicating a physical mechanism to suppress breakup into droplets.…”