141 research outputs found
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.
Absolute and convective instabilities in non-local active-dissipative equations arising in the modelling of thin liquid films
This paper was presented at the 4th Micro and Nano Flows Conference (MNF2014), which was held at University College, London, UK. The conference was organised by Brunel University and supported by the Italian Union of Thermofluiddynamics, IPEM, the Process Intensification Network, the Institution of Mechanical Engineers, the Heat Transfer Society, HEXAG - the Heat Exchange Action Group, and the Energy Institute, ASME Press, LCN London Centre for Nanotechnology, UCL University College London, UCL Engineering, the International NanoScience Community, www.nanopaprika.eu.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 flow
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
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
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
Deformation of a liquid film by an impinging gas jet: Modelling and experiments
© 2019, Avestia Publishing. We consider liquid in a cylindrical beaker and study the deformation of its surface under the influence of an impinging gas jet. Analyzing such a system not only is of fundamental theoretical interest, but also of industrial importance, e.g., in metallurgical applications. The solution of the full set of governing equations is computationally expensive. Therefore, to obtain initial insight into relevant regimes and timescales of the system, we first derive a reduced-order model (a thin-film equation) based on the long-wave assumption and on appropriate decoupling the gas problem from that for the liquid and taking into account a disjoining pressure. We also perform direct numerical simulations (DNS) of the full governing equations using two different approaches, the Computational Fluid Dynamics (CFD) package in COMSOL and the volume-of-fluid Gerris package. The DNS are used to validate the results for the thinfilm equation and also to investigate the regimes that are beyond the range of validity of this equation. We additionally compare the computational results with experiments and find good agreement
Additive noise effects in active nonlinear spatially extended systems
We examine the effects of pure additive noise on spatially extended systems
with quadratic nonlinearities. We develop a general multiscale theory for such
systems and apply it to the Kuramoto-Sivashinsky equation as a case study. We
first focus on a regime close to the instability onset (primary bifurcation),
where the system can be described by a single dominant mode. We show
analytically that the resulting noise in the equation describing the amplitude
of the dominant mode largely depends on the nature of the stochastic forcing.
For a highly degenerate noise, in the sense that it is acting on the first
stable mode only, the amplitude equation is dominated by a pure multiplicative
noise, which in turn induces the dominant mode to undergo several critical
state transitions and complex phenomena, including intermittency and
stabilisation, as the noise strength is increased. The intermittent behaviour
is characterised by a power-law probability density and the corresponding
critical exponent is calculated rigorously by making use of the first-passage
properties of the amplitude equation. On the other hand, when the noise is
acting on the whole subspace of stable modes, the multiplicative noise is
corrected by an additive-like term, with the eventual loss of any stabilised
state. We also show that the stochastic forcing has no effect on the dominant
mode dynamics when it is acting on the second stable mode. Finally, in a regime
which is relatively far from the instability onset, so that there are two
unstable modes, we observe numerically that when the noise is acting on the
first stable mode, both dominant modes show noise-induced complex phenomena
similar to the single-mode case
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Dynamic unbinding transitions and deposition patterns in dragged meniscus problems
This paper was presented at the 4th Micro and Nano Flows Conference (MNF2014), which was held at University College, London, UK. The conference was organised by Brunel University and supported by the Italian Union of Thermofluiddynamics, IPEM, the Process Intensification Network, the Institution of Mechanical Engineers, the Heat Transfer Society, HEXAG - the Heat Exchange Action Group, and the Energy Institute, ASME Press, LCN London Centre for Nanotechnology, UCL University College London, UCL Engineering, the International NanoScience Community, www.nanopaprika.eu.We sketch main results of our recent work on the transfer of a thin liquid film onto a flat plate
that is extracted from a bath of pure non-volatile liquid. Employing a long-wave hydrodynamic model, that
incorporates wettability via a Derjaguin (disjoining) pressure, we analyse steady-state meniscus profiles as the
plate velocity is changed. We identify four qualitatively different dynamic transitions between microscopic
and macroscopic coatings that are out-of-equilibrium equivalents of equilibrium unbinding transitions. The
conclusion briefly discusses how the gradient dynamics formulation of the problem allows one to systematically
extend the employed one-component model into thermodynamically consistent two-component models as used
to describe, e.g., the formation of line patterns during the Langmuir-Blodgett transfer of a surfactant layer
Recommended from our members
Absolute and convective instabilities in non-local active-dissipative equations arising in the modelling of thin liquid films
This paper was presented at the 4th Micro and Nano Flows Conference (MNF2014), which was held at University College, London, UK. The conference was organised by Brunel University and supported by the Italian Union of Thermofluiddynamics, IPEM, the Process Intensification Network, the Institution of Mechanical Engineers, the Heat Transfer Society, HEXAG - the Heat Exchange Action Group, and the Energy Institute, ASME Press, LCN London Centre for Nanotechnology, UCL University College London, UCL Engineering, the International NanoScience Community, www.nanopaprika.eu.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 flow
Dynamics of a thin film flowing down a heated wall with finite thermal diffusivity
Consider the dynamics of a thin film flowing down a heated substrate. The substrate heating generates a temperature distribution on the free surface, which in turn induces surface-tension gradients and corresponding thermocapillary stresses that affect the free surface and therefore the fluid flow. We study here the effect of finite substrate thermal diffusivity on the film dynamics. Linear stability analysis of the full Navier-Stokes and heat transport equations indicates if the substrate diffusivity is sufficiently small, the film becomes unstable at a finite wavelength and at a Reynolds number smaller than that predicted in the long-wavelength limit. This property is captured in a reduced-order system of equations derived using a weighted-residual integral-boundary-layer method. This reduced-order model is also used to compute the bifurcation diagrams of solution branches connecting the trivial flat film to traveling waves including solitary pulses. The effect of finite diffusivity is to separate a simultaneous Hopf-transcritical bifurcation into its individual component bifurcations. The appropriate Hopf bifurcation then connects only to the solution branch of negative-hump pulses, with wave speed less than the linear wave speed, while the branch of positive-single-hump pulses merges with the branch of positive-two-hump pulses at a supercritical Reynolds number. In the regime where finite-wavelength instability occurs, there exists a Hopf-bifurcation pair connected by a branch of periodic solutions, whose period cannot be increased indefinitely. Numerical simulation of the reduced-order system shows the development of a train of coherent structures, each of which resembles a stationary positive-hump pulse, and, in the regime of finite-wavelength instability, wavelength selection and saturation to periodic traveling waves
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