Comment on "low-dimensional models for vertically falling viscous films"

International audienceA Comment on the Letter by Mohan K. R. Panga and Vemuri Balakotaiah, Phys. Rev. Lett. 90, 154501 (2003). The authors of the Letter offer a Repl

On the speed of solitary waves running down a vertical wall

International audienceSolitary-wave solutions to surface equations or two-equation models of film flows are investigated within the framework of dynamical system theory. The limiting behaviour of one-humped solitary waves (homoclinic orbits) at large Reynolds numbers is considered. Their predicted speed is in good agreement with numerical findings. The theory also explains the absence of solitary-wave solutions to the Benney equation in the same limit. © 2005 Cambridge University Press

Selection of solitary waves in vertically falling liquid films

Two-dimensional solitary waves at the surface of a film flow down a vertical plane are considered. When the system is subjected to inlet white noise, solitary waves are formed after an inception region and interact with each other. Using open-domain simulations of reduced equation models, we investigate numerically their late time process dynamics. Close to the instability threshold, the waves synchronize themselves into bound states. For higher values of the Reynolds number, the separation distance between the waves increases and the synchronization process at work is weaker. Performing statistics, we show that the mean characteristics of the waves correspond to the minimal value of the mean film thickness along the traveling-wave branch of solutions. In this regime, synchronization occurs through the waves tails which is associated with a change of scaling of the waves features. A similar behavior is observed performing simulations in periodic domains: the selected waves maximize the mean flow rate. (C) 2016 Elsevier Ltd. All rights reserved

Analytical approach to viscous fingering in a cylindrical Hele-Shaw cell

We report analytical results for the development of the viscous fingering instability in a cylindrical Hele-Shaw cell of radius a and thickness b. We derive a generalized version of Darcy's law in such cylindrical background, and find it recovers the usual Darcy's law for flow in flat, rectangular cells, with corrections of higher order in b/a. We focus our interest on the influence of cell's radius of curvature on the instability characteristics. Linear and slightly nonlinear flow regimes are studied through a mode-coupling analysis. Our analytical results reveal that linear growth rates and finger competition are inhibited for increasingly larger radius of curvature. The absence of tip-splitting events in cylindrical cells is also discussed.Comment: 14 pages, 3 ps figures, Revte

Improved modeling of flows down inclined planes

New models of film flows down inclined planes have been derived by combining a gradient expansion at first or second order to weighted residual techniques with polynomials as test functions. The two-dimensional formulation has been extended to account for three-dimensional flows as well. The full second-order two-dimensional model can be expressed as a set of four coupled evolution equations for four slowly varying fields, the thickness h, the flow rate q and two other quantities measuring the departure from the flat-film semi-parabolic velocity profile. A simplified model has been obtained in terms of h and q only. Including viscous dispersion effects properly, it closely sticks to the asymptotic expansion in the appropriate limit. Our new models improve over previous ones in that they remain valid deep into the strongly nonlinear regime, as shown by the comparison of our results relative to travelling-wave and solitary-wave solutions with those of both direct numerical simulations and experiments

Modeling film flows down inclined planes

A new model of film flow down an inclined plane is derived by a method combining results of the classical long wavelength expansion to a weighted-residuals technique. It can be expressed as a set of three coupled evolution equations for three slowly varying fields, the thickness h, the flow-rate q, and a new variable τ that measures the departure of the wall shear from the shear predicted by a parabolic velocity profile. Results of a preliminary study are in good agreement with theoretical asymptotic properties close to the instability threshold, laboratory experiments beyond threshold and numerical simulations of the full Navier-Stokes equations

Pulse dynamics in a power-law falling film

We examine the stability, dynamics and interactions of solitary waves in a two-dimensional vertically falling thin liquid film that exhibits shear-thinning effects. We use a low-dimensional two-field model that describes the evolution of both the local flow rate and the film thickness and is consistent up to second-order terms in the long-wave expansion. The shear-thinning behaviour is modelled via a power-law formulation with a Newtonian plateau in the limit of small strain rates. our results show the emergence of a hysteresis behaviour as the control parameter (the Reynolds number) is increased which is directly related to the shear-thinning character of the liquid an can be quantified with both linear analysis arguments and a physical interpretation. We also study pulse interactions, observing that two pulses may attract or repel each other either monotonically or in an oscillatory manner. In large domains we find that for a given Reynolds number the final state depends on the initial condition, a consequence of the presence of multiple solutions

Falling Liquid Films

This research monograph gives a detailed review of the state-of-the-art theoretical methodologies for the analysis of dissipative wave dynamics and pattern formation on the surface of a film falling down a planar, inclined substrate. This prototype is an open-flow hydrodynamic instability representing an excellent paradigm for the study of complexity in active nonlinear media with energy supply, dissipation and dispersion. Whenever possible, the link between theory and experiments is illustrated and the development of order-of-magnitude estimates and scaling arguments is used to facilitate th

Three-dimensional instabilities of quasi-solitary waves in a falling liquid film

Abstract The stability of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\gamma _2$ travelling waves at the surface of a film flow down an inclined plane is considered experimentally and numerically. These waves are fast, one-humped and quasi-solitary. They undergo a three-dimensional secondary instability if the flow rate (or Reynolds number) is sufficiently high. Rugged or scallop wave patterns are generated by the interplay between a short-wave and a long-wave instability mode. The short-wave mode arises in the capillary region of the wave, with a mechanism of capillary origin which is similar to the Rayleigh–Plateau instability, whereas the long-wave mode deforms the entire wave and is triggered by a Rayleigh–Taylor instability. Rugged waves are observed at relatively small inclination angles. At larger angles, the long-wave mode predominates and scallop waves are observed. For a water film the transition between rugged and scallop waves occurs for an inclination angle around 12°