# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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

$\frac{\partial h}{\partial t}+\frac{\partial }{\partial x}\left(C\frac{{h}^{3}}{3}\frac{{\partial }^{3}h}{\partial {x}^{3}}+f\left(h,{h}_{x},{h}_{xx}\right)\right)=0,$

where $h=h\left(x,t\right)$ 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.

##### MSC:
 35K55 Nonlinear parabolic equations 76A20 Thin fluid films (fluid mechanics) 76D08 Lubrication theory 35B40 Asymptotic behavior of solutions of PDE 35K65 Parabolic equations of degenerate type 76D45 Capillarity (surface tension) 35Q35 PDEs in connection with fluid mechanics 35C07 Traveling wave solutions of PDE 76A20 Thin fluid films (fluid mechanics)