Nonlinear stability of a charged electrified viscous liquid sheet under the action of a horizontal electric field

Abstract

In a recent paper [D. T. Papageorgiou and P. G. Petropoulos, J. Eng. Math. 50, 223 (2004)] we considered the linear stability of a two-dimensional incompressible leaky dielectric viscous liquid sheet surrounded by a hydrodynamically passive conducting medium, when an electric field is applied parallel to the initially flat bounding fluid interfaces. It was established that for order-one Reynolds numbers and when the dielectric permittivity ratio, εp=εin∕εout, and the electric conductivity ratio, σR=σout∕σin, satisfy εpσR>1, the flow is linearly stable in the absence of an electric field. When a field is present a band of unstable long waves emerges whose size increases as the field increases (the band remains finite, that is short waves are stable, for large fields). In the present study we consider the nonlinear dynamics in the vicinity of the zero electric field bifurcation. The scalings determined from the linear stability calculations are used to derive canonical strongly nonlinear evolution equations for the leading order shape of the sheet and the corresponding horizontal velocity. Numerical simulations indicate that for a wide class of initial conditions, a quasisteady state is reached in the long time when the layer organizes into a number of lobes connected by slowly draining threads whose height vanishes asymptotically in time. The number of lobes and their volumes depend on initial conditions. Using this insight, we construct an ordinary differential equation which describes the shape of the sheet in the limit t→∞.

Accurate low-order modeling of electrified falling films at moderate Reynolds number

“…3, where h i is the interfacial shape of each low-order model, and h DNS 337 is the output of the direct numerical simulations. Previous studies [19,30] have shown that the 338 critical governing parameters for the stability of the system are the ratios of the permittivities and we will investigate it in more detail in Sec. V. We find that in fact the simplified models provide better 346 accuracy than the regularized models.…”

“…This has included work on the full leaky dielectric formulation [17,18] 40 as well as the simpler situations where both regions have large conductivities [19], or indeed where 41 one region is a perfect conductor [20]. Notably, given a permittivity ratio R and a conductivity ratio…”

Electrified coating flows on vertical fibres: enhancement or suppression of interfacial dynamics

“…is also important, this being the cylindrical analogue of the term (σ * /σ − 1) noted by Ozen et al (2006b). The growth rate, Re(s), is a quadratic in k 2 with a double root at k 2 = 0.…”

“…In a cylindrical geometry there are fewer leaky dielectric analyses, consideration mainly being restricted to situations where one fluid is a perfect insulator or conductor (Wang, Mählmann & Papageorgiou 2009;Wang & Papageorgiou 2011), or a jet rather than an annular region (Mestel 1994;Gañán-Calvo 1997;Shin et al 2001;Gañán-Calvo & Montanero 2009). Notably, given a permittivity ratio R and a conductivity ratio σ R between the two regions, the two dimensionless groupings (1 − σ R / R ) and (1 − σ 2 R / R ) have previously been shown to be critical to determining whether the electric field is linearly stabilizing or destabilizing (Ozen et al 2006a;Ozen, Papageorgiou & Petropoulos 2006b). In addition, extensive consideration has been given to electrified jets, albeit this time in the presence of electrokinetic effects (Conroy et al 2011a,b).…”

Gravity-capillary waves on the free surface of a liquid dielectric in a tangential electric field

“…The applied interest in studying the liquid surface dynamics in external electric field is related to the possibility of controlling the behavior of liquid surfaces and suppressing hydrodynamic instabilities [7][8][9][10]. The features of the nonlinear evolution of capillary waves at the liquid interfaces in the presence of the horizontal field were analyzed in [11][12][13]. It has been shown in [14][15][16] that, in the case of a strong electric field (the effects of gravitational and capillary forces are negligibly small), nonlinear waves on the surface of a liquid with high dielectric constant can propagate without distortions along (or against) the field direction.…”