Long-scale evolution of thin liquid films

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 close-quote 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. (Abstract Truncated)