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Insoluble surfactant spreading on a thin viscous film: shock evolution and film rupture

Published online by Cambridge University Press:  26 April 2006

O. E. Jensen  and
J. B. Grotberg
O. E. Jensen
Affiliation:
Department of Biomedical Engineering, Northwestern University, Evanston. IL 60208, USA
J. B. Grotberg
Affiliation:
Department of Biomedical Engineering, Northwestern University, Evanston. IL 60208, USA Department of Anesthesia, Northwestern University Medical School, Chicago, IL 60611, USA
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Abstract

Lubrication theory and similarity methods are used to determine the spreading rate of a localized monolayer of insoluble surfactant on the surface of a thin viscous film, in the limit of weak capillarity and weak surface diffusion. If the total mass of surfactant increases as tα, then at early times, when spreading is driven predominantly by Marangoni forces, a planar (axisymmetric) region of surfactant is shown to spread as t(1 + α)/3 (t(1 + α)/4). A shock exists at the leading edge of the monolayer; asymptotic methods are used to show that a wavetrain due to capillary forces exists ahead of the shock at small times, but that after a finite time it is swamped by diffusive effects. For α < ½ (α < 1), the diffusive lengthscale at the shock grows faster than the length of the monolayer, ultimately destroying the shock; subsequently, spreading is driven by diffusion, and proceeds as t½. The asymptotic results are shown to be good approximations of numerical solutions of the governing partial differential equations in the appropriate limits. Additional forces are also considered: weak vertical gravity can also destroy the shock in finite time, while effects usually neglected from lubrication theory are important only early in spreading. Experiments have shown that the severe thinning of the film behind the shock can cause it to rupture: the dryout process is modelled by introducing van der Waals forces.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 240 , July 1992 , pp. 259 - 288
Copyright
© 1992 Cambridge University Press

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References

Ahmad, J. & Hansen, R. S. 1972 A simple quantitative treatment of the spreading of monolayers on thin liquid films. J. Colloid Interface Sci. 38, 601–604.Google Scholar
Borgas, M. S. & Grotberg, J. B. 1988 Monolayer flow on a thin film J. Fluid Mech. 193, 151–170 (herein called BG).
Bretherton, F. P. 1961 The motion of long bubbles in tubes. J. Fluid Mech. 10, 166–188.Google Scholar
Dipietro, N. D., Huh, C. & Cox, R. G. 1978 The hydrodynamics of the spreading of one liquid on the surface of another. J. Fluid Mech. 84, 529–549.Google Scholar
Espinosa, F. F. 1991 Spreading of surfactant in a small pulmonary airway. MS Thesis, Massachusetts Institute of Technology.
Fraaije, J. G. E. M. & Cazabat, A. M. 1989 Dynamics of spreading on a liquid substrate. J. Colloid Interface Sci. 133, 452–460.Google Scholar
Gaver, D. P. Iii & Grotberg, J. B. 1990 The dynamics of a localized surfactant on a thin film. J. Fluid Mech. 213, 127–148 (herein called GG1).Google Scholar
Gaver, D. P. Iii & Grotberg, J. B. 1992 Droplet spreading on a thin viscous film. J. Fluid Mech. 235, 399–414 (herein called GG2).Google Scholar
Halpern, D. & Grotberg, J. B. 1992 Dynamics and transport of a localized soluble surfactant on a thin film. J. Fluid Mech. 237, 1–11.Google Scholar
Robertson, B. 1984 Pathology and pathophysiology of neonatal surfactant deficiency (‘Respiratory Distress Syndrome’, ‘Hyaline Membrane Disease’). In Pulmonary Surfactant (ed. B. Robertson, L. M. G. van Golde & J. J. Batenburg) Elsevier
Ruckenstein, E. & Jain, R. K. 1974 Spontaneous rupture of thin liquid films. Chem. Soc. Faraday Trans. 2 70, 132–147.Google Scholar
Scriven, L. E. 1960 Dynamics of a fluid interface: equation of motion for Newtonian surface fluids. Chem. Engng Sci. 12, 98–108.Google Scholar
Shapiro, D. L. 1989 Development of surfactant replacement therapy. In Surfactant Replacement Therapy (ed. D. L. Shapiro & R. H. Notter) New York: A. R. Liss
Sharma, A. & Ruckenstein, E. 1986 An analytical nonlinear theory of thin film rupture and its application to wetting films. J. Colloid Interface Sci. 113, 456–479.Google Scholar
Stone, H. A. 1989 A simple derivation of the time-dependent convective-diffusion equation for surfactant transport along a deforming interface. Phys. Fluids A2, 111–112.Google Scholar
Troian, S. M., Herbolzheimer, E. & Safran, S. A. 1990 Model for the fingering instability of spreading surfactant drops. Phys. Rev. Lett. 65, 333–336.Google Scholar
Troian, S. M., Wu, X. L. & Safran, S. A. 1989 Fingering instability in thin wetting films. Phys. Rev. Lett. 62, 1496–1499.Google Scholar
Williams, M. B. & Davis, S. H. 1982 Nonlinear theory of film rupture. J. Colloid Interface Sci. 90, 220–228.Google Scholar