On the behavior of a thin liquid layer flowing due to gravity and a surface tension gradient

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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 and those in which the same boundary condition expresses the fact that the normal stress is proportional to the curvature.  In these, 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 non-convex 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

C.I. 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.

Paper coordinates

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

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Publisher Name

Babes-Bolyai University

Print ISSN

0252-1938

Online ISSN

2065-961x

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References

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References

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[5] Myers, T. G.,  Thin films with high surface,  submitted to  SIAM review, Oct. 1995.
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[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 thin films. Nonlin. Anal. (to appear).

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

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