Wave evolution on electrified falling films

Document Type

Article

Publication Date

6-10-2006

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. © 2006 Cambridge University Press.

Identifier

33744909139 (Scopus)

Publication Title

Journal of Fluid Mechanics

External Full Text Location

https://doi.org/10.1017/S0022112006009712

e-ISSN

14697645

ISSN

00221120

First Page

361

Last Page

386

Volume

556

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