## Abstract

A thin liquid layer flowing due to gravity and a surface tension gradient is taken into account. On the liquid/gas interface one of the boundary conditions reduces to the fact that the normal stress equals the atmospheric pressure. This is the main difference between our study those where the same boundary condition expresses the fact that the normal stress is proportional to the curvature. In theses, by using the standard lubrication theory, a fourth-order nonlinear parabolic equation for the fluid film height is obtained. In ours, by using the same theory, a nonlinear conservation law with a noncovenx flux function is deduced for the same variable. For this equation a similarity solution is carried out. It shows that the behavior of the liquid layer depends essentially upon the gradient of surface tension and is quite insensitive to the viscosity of the liquid. “Viscous” and weak formulations for the conservation law are also carried out. An entropy condition to pick out physically relevant weak solutions is used.

## Authors

**Gheorghiu**

Tiberiu Popoviciu Institute of Numerical Analysis

## Keywords

Marangoni flow; normal stress; atmospheric pressure; nonlinear conservation law; non-convex flux; similarity solution; entropy condition;

### References

See the expanding block below.

## Cite this paper as

C.I. Gheorghiu, *On the behavior of a thin liquid layer flowing due to gravity and a surface tension gradient, I. Mathematical aspects*, Studia Univ. Babeş-Bolyai Math., **XLI** (1996) 47-54.

?

## About this paper

##### Publisher Name

Babes-Bolyai University

##### Paper on journal website

##### Print ISSN

0252-1938

##### Online ISSN

2065-961x

## MR

?

## ZBL

?

## Google Scholar

?

## References

## Paper in html format

## References

*Elementary Fluid Dynamics,*Oxford Univ. Press. 1990.

[2] Chifu, E., Gheorghiu, C. I., Stan, I., *Surface Mobility of Surfactant Solutions XI. Numerical Analysis for the Marangoni and Gravity Flow in a Thin Liquid Layer of Triangular Section, * Rev. Roumaine Chim., 29, pp. 31-42, 1984.

[3] Jensen, O.E., Grotberg, J. B., *Insoluble surfactant spreading on a thin viscous film: shock evolution and film rupture, *J. Fluid Mech. 240, pp. 259-288, 1992.

[4] Le Veque, R. J., *Numerical Methods for Conservation Laws, Brikhauser, 1990.
*

[5] Myers, T.G., * Thin films with high surface tension, * submitted to SIAM review, Oct. 1995.

[6] Myers, T.G., *Surface tension driven thin films flows*, ECMI Newsletter, 19, pp. 23-24, 1996.

[7] Renardy, M., *A singularly perturbed problem related to surfactant spreading on thin films, * Nonlin. Anal. (to appear).

[8] Renardy, M., *On an equation describing the spreading of surfactants on thim films, * Nonlin Anal. (to appear).

[9] Williams, B.M., Davis, S.H., * Nonlinear Theory of Film Rupture, * J. Colloid Interface Sci., 90, pp. 220-228, 1982.