Wave evolution on electrified falling films

Abstract

The nonlinear stability of falling film flow down an inclined flat plane is investigated when an electric field acts normal to the plane. A systematic asymptotic expansion is used to derive a fully nonlinear long-wave model equation for the scaled interface, where higher-order terms must be retained to make the long-wave approximation valid for long times. The effect of the electric field is to introduce a non-local term which comes from the potential region above the liquid film. This term is always linearly destabilizing and produces growth rates proportional to the cubic power of the wavenumber – surface tension is included and provides a short wavelength cutoff. Even in the absence of an electric field, the fully nonlinear equation can produce singular solutions after a finite time. This difficulty is avoided at smaller amplitudes where the weakly nonlinear evolution is governed by an extension of the Kuramoto–Sivashinsky equation. This equation has solutions which exist for all time and allows for a complete study of the nonlinear behaviour of competing physical mechanisms: long-wave instability above a critical Reynolds number, short-wave damping due to surface tension and intermediate growth due to the electric field. Through a combination of analysis and extensive numerical experiments, we find parameter ranges that support non-uniform travelling waves, time-periodic travelling waves and complex nonlinear dynamics including chaotic interfacial oscillations. It is established that a sufficiently high electric field will drive the system to chaotic oscillations, even when the Reynolds number is smaller than the critical value below which the non-electrified problem is linearly stable. A particular case of this is Stokes flow.

Coherent Structures in Nonlocal Dispersive Active-Dissipative Systems

“…Such nonlocal extensions of the KS equation were derived, e.g., in [53,54] to describe wave dynamics on electrified falling films, and the global existence and uniqueness of the solutions and the existence of the global attractor are given in [52]. The analyticity properties of solutions in a more general setting have been recently analyzed in [3].…”

Thin viscous ferrofluid film in a magnetic field

“…In this way, the film, from the outer field, looks like a line (or sheet in three dimensions) of magnetic material. In the surrounding air, the region (2 and 3) is not slender so we re-scale as z =zδ −1 (see Tseluiko and papageorgiou 19 for a similar approach for a voltage potential). In addition, we take ξ ′ =ξ/δ, F =Fδ −1 and H ∞ = 1 + δH 1 (x).…”

“…This equation is similar to the models for a perfectly conducting fluid film in a normal electric field. 19,20,26 The main difference in our model is the presence of a non-linear relationship for the magnetization that has been retained by keeping the next term in the asymptotic expansion.…”

“…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.…”

Liquid layer flow over convex corners in the presence of an electric field

“…These flows have been the subject of much attention recently in the literature, and the theory and many applications are discussed extensively in [2]. In recent work the behaviour of liquid layer flows under the influence of an electric field has also been studied, see for example Tseluiko & Papageorgiou [3]. But in many of these papers it is assumed that the Reynolds number is small.…”