LONG-SCALE EVOLUTION OF THIN LIQUID-FILMS

Macroscopic thin liquid films are entities that are important in bioph ysics, physics, and engineering, as well as in natural settings. They can be composed of common liquids such as water or oil, theologically complex materials such as polymers solutions or melts, or complex mixt ures of phases or components. When the films are subjected to the acti on of various mechanical, thermal, or structural factors, they display interesting dynamic phenomena such as wave propagation, wave steepeni ng, and development of chaotic responses. Such films can display ruptu re 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 base d on the asymptotic procedure of reduction of the full set of governin g 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 doe s preserve many of the important features of its physics. The basics o f the long-wave theory are explained. If, in addition, the Reynolds nu mber 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 considere d. Each case contains a discussion of the linear stability properties of the base-state solutions and of the nonlinear spatiotemporal evolut ion of the interface (and other scalar variables, such as temperature or solute concentration). The cases reducing to a single highly nonlin ear evolution equation are first examined. These include: (a) films wi th 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 tensi on only, (d) films with thermocapillarity, surface tension, and body f orce only, (e) films with temperature-dependent physical properties, ( f) evaporating/condensing films, (g) films on a thick substrate, (h) f ilms on a horizontal cylinder, and (i) films on a rotating disc. The d ynamics of the films with a spatial dependence of the base-state solut ion are then studied. These include the tramples of nonuniform tempera ture or heat flux at liquid-solid boundaries. Problems which reduce to a set of nonlinear evolution equations are considered next. Those inc lude (a) the dynamics of free liquid films, (b) bounded films with int erfacial viscosity, and (c) dynamics of soluble and insoluble surfacta nts in bounded and free films. The spreading of drops on a solid surfa ce and moving contact lines, including effects of heat and mass transp ort and van er Waals attractions, are then addressed. Several related topics such as falling films and sheets and Hele-Shaw flows are also b riefly discussed. The results discussed give motivation for the develo pment of careful experiments which can be used to test the theories an d exhibit new phenomena.