Long-scale evolution of thin liquid films

Abstract

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 f...

Validity domain of the Benney equation including the Marangoni effect for closed and open flows

“…It even faithfully captures all physical mechanisms at small Reynolds numbers and its widespread use still confirms its interest (Oron et al 1997). For this reason, the knowledge of its validity in the parameter space -i.e.…”

“…In any case, at low Reynolds number, viscosity ensures the in-depth coherence of the flow which allows to reduce the full set of equations to a single evolution equation for the film thickness, as initially performed by Benney (1966). The equation is usually truncated at first or second order in ε and still referred to as the Benney equation (see the review by Oron et al (1997)). Now, the incompatibility between the two estimates ε L and ε N can be understood within the framework of the Benney expansion considering the assumption (1.1), where the genuine small parameter is εRe instead of ε only.…”

Deformation and dewetting of thin liquid films induced by moving gas jets

“…In the absence of steep gradients, the height profile h(x, y, t) of a liquid film is accurately described by the lubrication equation [36] …”

A minimal model for solvent evaporation and absorption in thin films

“…Detailed models based on thermodynamic principles have been proposed as a means of accurately describing the transport of heat and mass during the drying process [23,24,25]. Finally, there is a wide body of literature concerning the application of hydrodynamic models to the problem of film drying [26,27]. These models explicitly account for the motion of the liquid phase and have been used to understand flow-driven solute deposition, e.g., the coffee-stain effect [28,29,30,31]; skin formation [32,33]; and the onset of Marangoni instabilities caused by composition gradients at the evaporating surface [34,35,36,37,38].…”