41 research outputs found
Mathematical problems arising in interfacial electrohydrodynamics
In this work we consider the nonlinear stability of thin films in the presence of electric fields. We study a perfectly conducting thin film flow down an inclined plane in the presence of an electric field which is uniform in its undisturbed state, and normal to the plate at infinity. In addition, the effect of normal electric fields on films lying above, or hanging from, horizontal substrates is considered. Systematic asymptotic expansions are used to derive fully nonlinear long wave model equations for the scaled interface motion and corresponding flow fields. For the case of an inclined plane, higher order terms are need to be retained to regularize the problem in the sense that the long wave approximation remains valid for long times. For the case of a horizontal plane the fully nonlinear evolution equation which is derived at the leading order, is asymptotically correct and no regularization procedure is required. In both physical situations, the effect of the electric field is to introduce a non-local term which arises from the potential region above the liquid film, and enters through the electric Maxwell stresses at the interface. 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 cut-off, that is, all sufficiently short waves are linearly stable.
For the case of film flow down an inclined plane, the fully nonlinear equation can produce singular solutions (for certain parameter values) after a finite time, even in the absence of an electric field. This difficulty is avoided at smaller amplitudes where the weakly nonlinear evolution is governed by an extension of the Kuramoto-Sivashinsky (KS) equation. Global existence and uniqueness results are proved, and refined estimates of the radius of the absorbing ball in L2 are obtained in terms of the parameters of the equations for a generalized class of modified KS equations. 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. We also carry out a complete study of the nonlinear behavior 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 elucidate parameter regimes 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, which is known to be stable for this class of problems (an analogous statement holds for horizontally supported films also). Our theoretical results indicate that such highly stable flows can be rendered unstable by using electric fields. This opens the way for possible heat and mass transfer applications which can benefit significantly from interfacial oscillations and interfacial turbulence.
For the case of a horizontal plane, a weakly nonlinear theory is not possible due to the absence of the shear flow generated by the gravitational force along the plate when the latter is inclined. We study the fully nonlinear equation, which in this case is asymptotically correct and is obtained at the leading order. The model equation describes both overlying and hanging films - in the former case gravity is stabilizing while in the latter it is destabilizing. The numerical and theoretical analysis of the fully nonlinear evolution is complicated by the fact that the coefficients of the highest order terms (surface tension in this instance) are nonlinear. We implement a fully implicit two level numerical scheme and perform numerical experiments. We also prove global boundedness of positive periodic smooth solutions, using an appropriate energy functional. This global boundedness result is seenin all our numerical results. Through a combination of analysis and extensive numerical experiments we present evidence for 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
Collapsed heteroclinic snaking near a heteroclinic chain in dragged meniscus problems
We study a liquid film that is deposited onto a flat plate that is inclined
at a constant angle to the horizontal and is extracted from a liquid bath at a
constant speed. We additionally assume that there is a constant temperature
gradient along the plate that induces a Marangoni shear stress. We analyse
steady-state solutions of a long-wave evolution equation for the film
thickness. Using centre manifold theory, we first obtain an asymptotic
expansion of solutions in the bath region. The presence of the temperature
gradient significantly changes these expansions and leads to the presence of
logarithmic terms that are absent otherwise. Next, we obtain numerical
solutions of the steady-state equation and analyse the behaviour of the
solutions as the plate velocity is changed. We observe that the bifurcation
curve exhibits snaking behaviour when the plate inclination angle is beyond a
certain critical value. Otherwise, the bifurcation curve is monotonic. The
solutions along these curves are characterised by a foot-like structure that is
formed close to the meniscus and is preceded by a thin precursor film further
up the plate. The length of the foot increases along the bifurcation curve.
Finally, we explain that the snaking behaviour of the bifurcation curves is
caused by the existence of an infinite number of heteroclinic orbits close to a
heteroclinic chain that connects in an appropriate three-dimensional phase
space the fixed point corresponding to the precursor film with the fixed point
corresponding to the foot and then with the fixed point corresponding to the
bath.Comment: Final revised version. 18 pages. To be published in Eur. Phys. J.
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 residuals 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 travelling at the same
velocity. By developing a rigorous coherent-structure theory, we are able to
theoretically predict the pulse-separation distances for which bound states are
formed. Viscous dispersion affects the distances at which bound states are
observed. We show that the theory is in very good agreement with computations
of the second-order model. We also demonstrate that the presence of bound
states allows the film free surface to reach a self-organized state that can be
statistically described in terms of a gas of solitary waves separated by a
typical mean distance and characterized by a typical density
Falling liquid films with blowing and suction
Flow of a thin viscous film down a flat inclined plane becomes unstable to
long wave interfacial fluctuations when the Reynolds number based on the mean
film thickness becomes larger than a critical value (this value decreases as
the angle of inclination with the horizontal increases, and in particular
becomes zero when the plate is vertical). Control of these interfacial
instabilities is relevant to a wide range of industrial applications including
coating processes and heat or mass transfer systems. This study considers the
effect of blowing and suction through the substrate in order to construct from
first principles physically realistic models that can be used for detailed
passive and active control studies of direct relevance to possible experiments.
Two different long-wave, thin-film equations are derived to describe this
system; these include the imposed blowing/suction as well as inertia, surface
tension, gravity and viscosity. The case of spatially periodic blowing and
suction is considered in detail and the bifurcation structure of forced steady
states is explored numerically to predict that steady states cease to exist for
sufficiently large suction speeds since the film locally thins to zero
thickness giving way to dry patches on the substrate. The linear stability of
the resulting nonuniform steady states is investigated for perturbations of
arbitrary wavelengths, and any instabilities are followed into the fully
nonlinear regime using time-dependent computations. The case of small amplitude
blowing/suction is studied analytically both for steady states and their
stability. Finally, the transition between travelling waves and non-uniform
steady states is explored as the suction amplitude increases
Nonlinear waves in counter-current gas-liquid film flow
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
Nonlinear concentric water waves of moderate amplitude
We consider the outward-propagating nonlinear concentric water waves within
the scope of the 2D Boussinesq system. The problem is axisymmetric, and we
derive the slow radius versions of the cylindrical Korteweg - de Vries (cKdV)
and extended cKdV (ecKdV) models. Numerical runs are initially performed using
the full axisymmetric Boussinesq system. At some distance away from the origin,
we use the numerical solution of the Boussinesq system as the "initial
condition" for the derived cKdV and ecKdV models. We then compare the evolution
of the waves as described by both reduced models and the direct numerical
simulations of the axisymmetric Boussinesq system. The main conclusion of the
paper is that the extended cKdV model provides a much more accurate description
of the waves and extends the range of validity of the weakly-nonlinear
modelling to the waves of moderate amplitude.Comment: 29 pages, 17 figure
Kinetic Monte Carlo and hydrodynamic modelling of droplet dynamics on surfaces, including evaporation and condensation
We present a lattice-gas (generalised Ising) model for liquid droplets on
solid surfaces. The time evolution in the model involves two processes: (i)
Single-particle moves which are determined by a kinetic Monte Carlo algorithm.
These incorporate into the model particle diffusion over the surface and within
the droplets and also evaporation and condensation, i.e. the exchange of
particles between droplets and the surrounding vapour. (ii) Larger-scale
collective moves, modelling advective hydrodynamic fluid motion, determined by
considering the dynamics predicted by a thin-film equation. The model enables
us to relate how macroscopic quantities such as the contact angle and the
surface tension depend on the microscopic interaction parameters between the
particles and with the solid surface. We present results for droplets joining,
spreading, sliding under gravity, dewetting, the effects of evaporation, the
interplay of diffusive and advective dynamics, and how all this behaviour
depends on the temperature and other parameters
Two-dimensional pulse dynamics and the formation of bound states on electrified falling films
The flow of an electrified liquid film down an inclined plane wall is investigated with the focus on coherent structures in the form of travelling waves on the film surface, in particular, single-hump solitary pulses and their interactions. The flow structures are analysed first using a long-wave model, which is valid in the presence of weak inertia, and second using the Stokes equations. For obtuse angles, gravity is destabilising and solitary pulses exist even in the absence of an electric field. For acute angles, spatially non-uniform solutions exist only beyond a critical value of the electric field strength; moreover, solitary-pulse solutions are present only at sufficiently high supercritical electric-field strengths. The electric field increases the amplitude of the pulses, can generate recirculation zones in the humps and alters the far-field decay of the pulse tails from exponential to algebraic with a significant impact on pulse interactions. A weak-interaction theory predicts an infinite sequence of bound-state solutions for non-electrified flow, and a finite set for electrified flow. The existence of single-hump pulse solutions and two-pulse bound states is confirmed for the Stokes equations via boundary-element computations. In addition, the electric field is shown to trigger a switch from absolute to convective instability, thereby regularising the dynamics, and this is confirmed by time-dependent simulations of the long-wave model
Absolute and convective instabilities in non-local active-dissipative equations arising in the modelling of thin liquid films
Absolute and convective instabilities in a non-local model that arises in the analysis of thin-film flows over flat or corrugated walls in the presence of an applied electric field are discussed. Electrified liquid films arise, for example, in coating processes where liquid films are deposited onto a target surfaces with a view to producing an evenly coating layer. In practice, the target surface, or substrate, may be irregular in shape and feature corrugations or indentations. This may lead to non-uniformities in the thickness of the coating layer. Attempts to mitigate film-surface irregularities can be made using, for example, electric fields. We analyse the stability of such thin-film flows and show that if the amplitude of the wall corrugations and/or the strength of the applied electric field is increased the convectively unstable flow undergoes a transition to an absolutely unstable flo