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.

A model for the flow of a thin film, with and without solidification, on an arbitrary three-dimensional substrate is presented. The problem is reduced to two simultaneous partial differential equations for the film and solid layer thicknesses. The flow model (with the solidification rate set to zero) is the first such model to describe thin film flow on an arbitrary three-dimensional surface. Various limits are investigated to recover previous models for flow on flat, cylindrical and two-dimensional curved surfaces. With solidification a previous model for accretion on a flat substrate is retrieved. It is shown how the model may be reduced to standard forms, such as solidification on a flat surface, circular and non-circular cylinders, aerofoils and spheres. Numerical solutions are obtained by combining an ADI scheme with a shock capturing method. Results are presented for flow and accretion on a flat surface, aerofoil and sphere.