Macroscopic thin liquid films are entities that are important in biophysics, physics, and engineering, as well as in natural settings. They can be composed of common liquids such as water or oil, rheologically complex materials such as polymers solutions or melts, or complex mixtures of phases or components. When the films are subjected to the action of various mechanical, thermal, or structural factors, they display interesting dynamic phenomena such as wave propagation, wave steepening, and development of chaotic responses. Such films can display rupture phenomena creating holes, spreading of fronts, and the development of fingers. In this review a unified mathematical theory is presented that takes advantage of the disparity of the length scales and is based on the asymptotic procedure of reduction of the full set of governing equations and boundary conditions to a simplified, highly nonlinear, evolution equation or to a set of equations. As a result of this long-wave theory, a mathematical system is obtained that does not have the mathematical complexity of the original free-boundary problem but does preserve many of the important features of its physics. The basics of the long-wave theory are explained. If, in addition, the Reynolds number of the flow is not too large, the analogy with Reynolds's theory of lubrication can be drawn. A general nonlinear evolution equation or equations are then derived and various particular cases are considered. Each case contains a discussion of the linear stability properties of the base-state solutions and of the nonlinear spatiotemporal evolution of the interface (and other scalar variables, such as temperature or solute concentration). The cases reducing to a single highly nonlinear evolution equation are first examined. These include: (a) films with constant interfacial shear stress and constant surface tension, (b) films with constant surface tension and gravity only, (c) films with van der Waals (long-range molecular) forces and constant surface tension only, (d) films with thermocapillarity, surface tension, and body force only, (e) films with temperature-dependent physical properties, (f) evaporating/condensing films, (g) films on a thick substrate, (h) films on a horizontal cylinder, and (i) films on a rotating disc. The dynamics of the films with a spatial dependence of the base-state solution are then studied. These include the examples of nonuniform temperature or heat flux at liquid-solid boundaries. Problems which reduce to a set of nonlinear evolution equations are considered next. Those include (a) the dynamics of free liquid films, (b) bounded films with interfacial viscosity, and (c) dynamics of soluble and insoluble surfactants in bounded and free films. The spreading of drops on a solid surface and moving contact lines, including effects of heat and mass transport and van der Waals attractions, are then addressed. Several related topics such as falling films and sheets and Hele-Shaw flows are also briefly discussed. The results discussed give motivation for the development of careful experiments which can be used to test the theories and exhibit new phenomena.

Long-scale evolution of thin liquid films

The dynamics and stability of thin liquid films have fascinated scientists over many decades: the observations of regular wave patterns in film flows down a windowpane or along guttering, the patterning of dewetting droplets, and the fingering of viscous flows down a slope are all examples that are familiar in daily life. Thin film flows occur over a wide range of length scales and are central to numerous areas of engineering, geophysics, and biophysics; these include nanofluidics and microfluidics, coating flows, intensive processing, lava flows, dynamics of continental ice sheets, tear-film rupture, and surfactant replacement therapy. These flows have attracted considerable attention in the literature, which have resulted in many significant developments in experimental, analytical, and numerical research in this area. These include advances in understanding dewetting, thermocapillary- and surfactant-driven films, falling films and films flowing over structured, compliant, and rapidly rotating substrates, and evaporating films as well as those manipulated via use of electric fields to produce nanoscale patterns. These developments are reviewed in this paper and open problems and exciting research avenues in this thriving area of fluid mechanics are also highlighted.

Dynamics and stability of thin liquid films

Since the pioneering experiment by the father-son team of the Kapitza family during their house arrest in the late forties (Kapitza & Kapitza 1949), wave evolution on a falling film has intrigued many researchers. One of its main attractions is its simplicity-it is an open-flow hydrodynamic instability that occurs at very low flow rates. It can hence be studied with the simplest experimental apparatus, an obviously important factor for the Kapitzas. Yet, it yields a rich spectrum of fascinating wave dynamics, including a very unique and experimentally well-characterized sequence of nonlinear secondary transitions that begins with a selected monochromatic disturbance and leads eventually to nonstationary and broad-banded (in both frequency and wave number) "turbulent" wave dynamics. (Turbu­ lence here is used interchangeably with irregular spatio-temporal fluc­ tuations.) While this transition to "interfacial turbulence" or "spatio­ temporal chaos" seems to be quite analogous to other classical instabilities at first glance, there are subtle but important differences that have recently come to light. The pertinent nonlinear mechanisms behind these secondary transitions are the focus of the present review. We shall be mostly concerned with transitions on a free-falling vertical film. Wave dynamics on an inclined plane is quite analogous to the vertical limit and most experiments and theories have focused on the latter. For the vertical film, the problem is defined by two independent dimensionless parameters and we prefer the Russian convention of using the Reynolds

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Wave evolution on a falling film

We present novel measurements of the primary instabilities of thin liquid films flowing down an incline. A fluorescence imaging method allows accurate measurements of film thickness h(x, y, t) in real time with a sensitivity of several microns, and laser beam deflection yields local measurements with a sensitivity of less than one micron. We locate the instability with good accuracy despite the fact that it occurs (asymptotically) at zero wavenumber, and determine the critical Reynolds number R, for the onset of waves as a function of angle ,8. The measurements of R,(/3) are found to be in good agreement with calculations, as are the growth rates and wave velocities. We show experimentally that the initial instability is convective and that the waves are noisesustained. This means that the waveform and its amplitude are strongly affected by external noise at the source. We investigate the role of noise by varying the level of periodic external forcing. The nonlinear evolution of the waves depends strongly on the initial wavenumber (or the frequency f). A new phase boundary e(R) is measured, which separates the regimes of saturated finite amplitude waves (at high f) from multipeaked solitary waves (at low f). This boundary probably corresponds approximately to the sign reversal of the third Landau coefficient in weakly nonlinear theory. Finally, we show that periodic waves are unstable over a wide frequency band with respect to a convective subharmonic instability. This instability leads to disordered two-dimensional waves.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/6E877A58289CDBA17C4426664B7F2C24/S0022112093001387a.pdf/div-class-title-measurements-of-the-primary-instabilities-of-film-flows-div.pdf

Measurements of the primary instabilities of film flows

A numerical and analytical study of the Kuramoto–Sivashinsky partial differential equation (PDE) in one spatial dimension with periodic boundary conditions is presented. The structure, stability, and bifurcation characteristics of steady state and time-dependent solutions of the PDE for values of the parameter $\alpha $ less than 40 are examined. The numerically observed primary and secondary bifurcations of steady states, as well as bifurcations to constant speed traveling waves (limit cycles), are analytically verified. Persistent homoclinic and heteroclinic saddle connections are observed and explained via the system symmetries and fixed point subspaces of appropriate isotropy subgroups of $O( 2 )$. Their effect on the system dynamics is discussed, and several tertiary bifurcations, observed numerically, are presented.

Back in the saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation

The method of integral relations is used to derive a nonlinear two-wave equation for long waves on the surface of vertical falling liquid films. This equation is valid within a range of moderate Reynolds numbers and and be reduced in some cases to other well-known equations.

The theoretical results for the fastest growing waves are compared with the experimental results concerning velocities, wave numbers, and growth rates of the waves in the inception region. The validity of the theoretical assumptions is also confirmed by direct measurements of instantaneous velocity profiles in a wave liquid film.

The results of the experimental investigation concerning nonlinear stationary waves and the evolution of initial solitary disturbances are presented.

S. V. Alekseenko

Papers

Wave formation on a vertical falling liquid film