Dmitri Tseluiko

Wave evolution on electrified falling films

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.

Electrified viscous thin film flow over topography

The gravity-driven flow of a liquid film down an inclined wall with periodic indentations in the presence of a normal electric field is investigated. The film is assumed to be a perfect conductor, and the bounding region of air above the film is taken to be a perfect dielectric. In particular, the interaction between the electric field and the topography is examined by predicting the shape of the film surface under steady conditions. A nonlinear, non-local evolution equation for the thickness of the liquid film is derived using a long-wave asymptotic analysis. Steady solutions are computed for flow into a rectangular trench and over a rectangular mound, whose shapes are approximated with smooth functions. The limiting behaviour of the film profile as the steepness of the wall geometry is increased is discussed. Using substantial numerical evidence, it is established that as the topography steepness increases towards rectangular steps, trenches, or mounds, the interfacial slope remains bounded, and the film does not touch the wall. In the absence of an electric field, the film develops a capillary ridge above a downward step and a slight depression in front of an upward step. It is demonstrated how an electric field may be used to completely eliminate the capillary ridge at a downward step. In contrast, imposing an electric field leads to the creation of a free-surface ridge at an upward step. The effect of the electric field on film flow into relatively narrow trenches, over relatively narrow mounds, and down slightly inclined substrates is also considered.

Rigorous coherent-structure theory for falling liquid films: Viscous dispersion effects on bound-state formation and self-organization

We examine the interaction of two-dimensional solitary pulses on falling liquid films. We make use of the second-order model derived by Ruyer-Quil and Manneville [Eur. Phys. J. B 6, 277 (1998); Eur. Phys. J. B 15, 357 (2000); Phys. Fluids 14, 170 (2002)] by combining the long-wave approximation with a weighted residual technique. The model includes (second-order) viscous dispersion effects which originate from the streamwise momentum equation and tangential stress balance. These effects play a dispersive role that primarily influences the shape of the capillary ripples in front of the solitary pulses. We show that different physical parameters, such as surface tension and viscosity, play a crucial role in the interaction between solitary pulses giving rise eventually to the formation of bound states consisting of two or more pulses separated by well-defined distances and traveling at the same velocity. By developing a rigorous coherent-structure theory, we are able to theoretically predict the pulse-separ...

Noise induced state transitions, intermittency and universality in the noisy Kuramoto-Sivashinksy equation

Consider the effect of pure additive noise on the long-time dynamics of the noisy Kuramoto-Sivashinsky (KS) equation close to the instability onset. When the noise acts only on the first stable mode (highly degenerate), the KS solution undergoes several state transitions, including critical on-off intermittency and stabilized states, as the noise strength increases. Similar results are obtained with the Burgers equation. Such noise-induced transitions are completely characterized through critical exponents, obtaining the same universality class for both equations, and rigorously explained using multiscale techniques.

Effect of an electric field on film flow down a corrugated wall at zero Reynolds number

The effect of an electric field on a liquid layer flowing down an inclined, corrugated wall at zero Reynolds number is investigated. The layer is taken to be either a perfect conductor or a perfect dielectric. The region above the layer is assumed to be a perfect dielectric. Steady flow down a wall with small-amplitude sinusoidal corrugations is considered, and it is shown how the electric field can be used to control the amplitude of the free-surface deflection and the phase shift between the free surface and the wall profile. Steady flow over walls with large amplitude sinusoidal corrugations or other-shaped indentations is studied by using the boundary-element method. Results for flow into a wide rectangular trench are compared to previous model predictions based on the lubrication approximation. For a perfect-conductor film, the results confirm that the height of the capillary ridge, which appears above a downward step, monotonically decreases as the electric field strength increases. Solutions for a ...

A global attracting set for nonlocal Kuramoto–Sivashinsky equations arising in interfacial electrohydrodynamics

We study a generalized class of nonlocal evolution equations which includes those arising in the modelling of electrified film flow down an inclined plane, with applications in enhanced heat or mass transfer through interfacial turbulence. Global existence and uniqueness results are proved and refined estimates of the radius of the absorbing ball in

Nonlinear waves in counter-current gas–liquid film flow

L^2

Nonlinear Dynamics of Electrified Thin Liquid Films

are obtained in terms of the parameters of the equations (the length of the system and the dimensionless electric field-measuring parameter multiplying the nonlocal term). The established estimates are compared with numerical solutions of the equations which in turn suggest an optimal upper bound for the radius of the absorbing ball. A scaling argument is used to explain this and a general conjecture is made based on extensive computations.

Additive noise effects in active nonlinear spatially extended systems

We investigate the dynamics of a thin laminar liquid film flowing under gravity down the lower wall of an inclined channel when turbulent gas flows above the film. The solution of the full system of equations describing the gas―liquid flow faces serious technical difficulties. However, a number of assumptions allow isolating the gas problem and solving it independently by treating the interface as a solid wall. This permits finding the perturbations to pressure and tangential stresses at the interface imposed by the turbulent gas in closed form. We then analyse the liquid film flow under the influence of these perturbations and derive a hierarchy of model equations describing the dynamics of the interface, i.e. boundary-layer equations, a long-wave model and a weakly nonlinear model, which turns out to be the Kuramoto― Sivashinsky equation with an additional term due to the presence of the turbulent gas. This additional term is dispersive and destabilising (for the counter-current case; stabilizing in the co-current case). We also combine the long-wave approximation with a weighted-residual technique to obtain an integral-boundary-layer approximation that is valid for moderately large values of the Reynolds number. This model is then used for a systematic investigation of the flooding phenomenon observed in various experiments: as the gas flow rate is increased, the initially downward-falling film starts to travel upwards while just before the wave reversal the amplitude of the waves grows rapidly. We confirm the existence of large-amplitude stationary waves by computing periodic travelling waves for the integral-boundary-layer approximation and we corroborate our travelling-wave results by time-dependent computations.

Continuous and discontinuous dynamic unbinding transitions in drawn film flow.

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 c...