Thin films with high surface tension. (English) Zbl 0908.35057

Summary: 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 \[ \frac{\partial h}{\partial t}+\frac{\partial}{\partial x}\left(C \frac{h^3}{3}\frac{\partial^3 h}{\partial x^3} +f(h,h_x,h_{xx})\right) = 0, \] 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” \[ \frac{\partial h}{\partial t}+\frac{\partial}{\partial x} \left(h^n h_{xxx}\right) = 0. \] 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.


35K55 Nonlinear parabolic equations
76A20 Thin fluid films
76D08 Lubrication theory
35B40 Asymptotic behavior of solutions to PDEs
35K65 Degenerate parabolic equations
76D45 Capillarity (surface tension) for incompressible viscous fluids
35Q35 PDEs in connection with fluid mechanics
35C07 Traveling wave solutions
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