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

An algebraic multigrid algorithm is developed based on prolongations by smoothed aggregation. Coarse levels are generated automatically. Guidelines for the selection of method components are presented based on energy considerations. Efficiency of the resulting algorithm is demonstrated by computational results.

Algebraic Multigrid by Smoothed Aggregation for Second and Fourth Order Elliptic Problems

This paper is a review of work on thin fluid films where surface tension is a driving mechanism. Its aim is to highlight the substantial amount of literature dealing with relevant physical models and also analytic work on the resultant equations.
In general the introduction of surface tension into standard lubrication theory leads to a fourth-order nonlinear parabolic equation
$$
\pad{h}{t}+\pad{}{x}\left(C \frac{h^3}{3}\frac{\partial^3 h}{\partial x^3} +f(h,h_x,h_{xx})\right) = 0 ,\label{abeq1}
$$
where $h=h(x,t)$ is the fluid film height. For steady situations this equation may be integrated once and a third-order ordinary differential equation is obtained. Appropriate forms of this equation have been used to model fluid flows in physical situations such as coating, draining of foams, and the movement of contact lenses.
In the introduction a form of the above equation is derived for flow driven by surface tension, surface tension gradients, gravity, and long range molecular forces. Modifications to the equation due to slip, the effect of two free surfaces, two phase fluids, and higher dimensional forms are also discussed. The second section of this paper describes physical situations where surface tension driven lubrication models apply and the governing equations are given. The third section reviews analytical work on the model equations as well as the "generalized lubrication equation" 
$$
\pad{h}{t}+\pad{}{x}\left(h^n h_{xxx}\right) = 0.
$$ 
In particular the discussion focusses on asymptotic results, travelling waves, stability, and similarity solutions. Numerical work is also discussed, while for analytical results the reader is directed to existing literature.

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Thin Films with High Surface Tension

Newton–Krylov-multigrid solvers for large-scale, highly heterogeneous, variably saturated flow problems

Rupture of a thin viscous film on a solid substrate under a balance of destabilizing van der Waals pressure and stabilizing capillary pressure is shown to possess a countably infinite number of similarity solutions in each of which the horizontal lengthscale decreases like (tR−t)2/5 and the film thickness decreases like (tR−t)1/5, where tR−t is the time remaining before rupture. Only the self-similar solution corresponding to the least oscillatory curvature profile is observed in time-dependent numerical simulations of the governing partial differential equation. The numerical strategy employed to obtain the self-similar solutions is developed from far-field asymptotic analysis of the similarity equations.

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Similarity solutions for van der Waals rupture of a thin film on a solid substrate

Thin film rupture

The dynamics of a general three-dimensional thin film subject to van der Waals forces, surface tension, and surfactants is considered. Using an asymptotic analysis based upon the thinness of the film with respect to its lateral extent, evolution equations for the leading-order film thicknesses, tangential velocities, and surfactant concentrations are obtained.

The dynamics of thin films I: general theory

A semicoarsening multigrid algorithm suitable for use on single instruction multiple data (SIMD) architectures has been implemented on the CM-2. The method performs well for strongly anisotropic problems and for problems with coefficients jumping by orders of magnitude across internal interfaces. The parallel efficiency of this method is analyzed, and its actual performance is compared with its performance on some other machines, both parallel and nonparallel.

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A semicoarsening multigrid algorithm for SIMD machines

A system of equations for the dynamics of a general three-dimensional thin film subject to van der Waals forces, surface tension, and surfactants was derived in Part I of this work [M. P. Ida and M. J. Miksis, SIAM J. Appl. Math., 58 (1998), pp. 456--473]. Here we apply these equations to free films with various geometries. A linear and near-linear stability analysis and numerical simulations are done to the specialized equations.

The dynamics of thin films II: applications

The dynamics of a lamella being driven through a capillary tube is studied here. The lamella is modeled as a thin liquid film which makes a finite contact angle with the tube walls. Only two-dimensional solutions are considered. The effect of Van der Waals forces is also included. By considering the thin film limit, we are able to reduce the problem to studying a system of partial differential equations for the film interfaces. This system is studied both numerically and analytically. The stability of the lamella as a function of the physical parameters is determined.

M. P. Ida

Papers

Thin film rupture

The dynamics of thin films I: general theory

A semicoarsening multigrid algorithm for SIMD machines

The dynamics of thin films II: applications

Dynamics of a lamella in a capillary tube