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Three-dimensional wave evolution on electrified falling films
- R. J. Tomlin, D. T. Papageorgiou, G. A. Pavliotis
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- Journal:
- Journal of Fluid Mechanics / Volume 822 / 10 July 2017
- Published online by Cambridge University Press:
- 06 June 2017, pp. 54-79
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We consider the full three-dimensional dynamics of a thin falling liquid film on a flat plate inclined at some non-zero angle to the horizontal. In addition to gravitational effects, the flow is driven by an electric field which is normal to the substrate far from the flow. This extends the work of Tseluiko & Papageorgiou (J. Fluid Mech., vol. 556, 2006b, pp. 361–386) by including transverse dynamics. We study both the cases of overlying and hanging films, where the liquid lies above or below the substrate, respectively. Starting with the Navier–Stokes equations coupled with electrostatics, a fully nonlinear two-dimensional Benney equation for the interfacial dynamics is derived, valid for waves that are long compared to the film thickness. The weakly nonlinear evolution is governed by a Kuramoto–Sivashinsky equation with a non-local term due to the electric field effect. The electric field term is linearly destabilising and produces growth rates proportional to $|\unicode[STIX]{x1D743}|^{3}$, where $\unicode[STIX]{x1D743}$ is the wavenumber vector of the perturbations. It is found that transverse gravitational instabilities are always present for hanging films, and this leads to unboundedness of nonlinear solutions even in the absence of electric fields – this is due to the anisotropy of the nonlinearity. For overlying films and a restriction on the strength of the electric field, the equation is well-posed in the sense that it possesses bounded solutions. This two-dimensional equation is studied numerically for the case of periodic boundary conditions in order to assess the effects of inertia, electric field strength and the size of the periodic domain. Rich dynamical behaviours are observed and reported. For subcritical Reynolds number flows, a sufficiently strong electric field can promote non-trivial dynamics for some choices of domain size, leading to fully two-dimensional evolutions of the interface. We also observe two-dimensional spatiotemporal chaos on sufficiently large domains. For supercritical flows, such two-dimensional chaotic dynamics emerges in the absence of a field, and its presence enhances the amplitude of the fluctuations and broadens their spectrum.
Nonlinear interfacial dynamics in stratified multilayer channel flows
- E. S. Papaefthymiou, D. T. Papageorgiou, G. A. Pavliotis
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- Journal:
- Journal of Fluid Mechanics / Volume 734 / 10 November 2013
- Published online by Cambridge University Press:
- 08 October 2013, pp. 114-143
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The dynamics of viscous immiscible pressure-driven multilayer flows in channels are investigated using a combination of modelling, analysis and numerical computations. More specifically, the particular system of three stratified layers with two internal fluid–fluid interfaces is considered in detail in order to identify the nonlinear mechanisms involved due to multiple fluid surface interactions. The approach adopted is analytical/asymptotic and is valid for interfacial waves that are long compared with the channel height or individual undisturbed liquid layer thicknesses. This leads to a coupled system of fully nonlinear partial differential equations of Benney type that contain a small slenderness parameter that cannot be scaled out of the problem. This system is in turn used to develop a consistent coupled system of weakly nonlinear evolution equations, and it is shown that this is possible only if the underlying base-flow and fluid parameters satisfy certain conditions that enable a synchronous Galilean transformation to be performed at leading order. Two distinct canonical cases (all terms in the equations are of the same order) are identified in the absence and presence of inertia, respectively. The resulting systems incorporate all of the active physical mechanisms at Reynolds numbers that are not large, namely, nonlinearities, inertia-induced instabilities (at non-zero Reynolds number) and surface tension stabilization of sufficiently short waves. The coupled system supports several instabilities that are not found in single long-wave equations including, transitional instabilities due to a change of type of the flux nonlinearity from hyperbolic to elliptic, kinematic instabilities due to the presence of complex eigenvalues in the linearized advection matrix leading to a resonance between the interfaces, and the possibility of long-wave instabilities induced by an interaction between the flux function of the system and the surface tension terms. All of these instabilities are followed into the nonlinear regime by carrying out extensive numerical simulations using spectral methods on periodic domains. It is established that instabilities leading to coherent structures in the form of nonlinear travelling waves are possible even at zero Reynolds number, in contrast to single interface (two-layer) systems; in addition, even in parameter regimes where the flow is linearly stable, the coupling of the flux functions and their hyperbolic–elliptic transitions lead to coherent structures for initial disturbances above a threshold value. When inertia is present an additional short-wave instability enters and the systems become general coupled Kuramoto–Sivashinsky-type equations. Extensive numerical experiments indicate a rich landscape of dynamical behaviour including nonlinear travelling waves, time-periodic travelling states and chaotic dynamics. It is also established that it is possible to regularize the chaotic dynamics into travelling wave pulses by enhancing the inertialess instabilities through the advective terms. Such phenomena may be of importance in mixing, mass and heat-transfer applications.