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Gravity-driven thin liquid films with insoluble surfactant: smooth traveling waves

Published online by Cambridge University Press:  01 December 2007

RACHEL LEVY ,
MICHAEL SHEARER  and
THOMAS P. WITELSKI
RACHEL LEVY
Affiliation:
Department of Mathematics, Duke University, Durham NC 27708, USA Department of Mathematics, Harvey Mudd College, Claremont, CA 91711, USA
MICHAEL SHEARER
Affiliation:
Department of Mathematics, Duke University, Durham NC 27708, USA Department of Mathematics and Center for Research in Scientific Computation, N.C. State University Raleigh, NC 27695, USA
THOMAS P. WITELSKI
Affiliation:
Department of Mathematics, Duke University, Durham NC 27708, USA
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Abstract

The flow of a thin layer of fluid down an inclined plane is modified by the presence of insoluble surfactant. For any finite surfactant mass, traveling waves are constructed for a system of lubrication equations describing the evolution of the free-surface fluid height and the surfactant concentration. The one-parameter family of solutions is investigated using perturbation theory with three small parameters: the coefficient of surface tension, the surfactant diffusivity, and the coefficient of the gravity-driven diffusive spreading of the fluid. When all three parameters are zero, the nonlinear PDE system is hyperbolic/degenerate-parabolic, and admits traveling wave solutions in which the free-surface height is piecewise constant, and the surfactant concentration is piecewise linear and continuous. The jumps and corners in the traveling waves are regularized when the small parameters are nonzero; their structure is revealed through a combination of analysis and numerical simulation.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 18 , Issue 6 , December 2007 , pp. 679 - 708
Copyright
Copyright © Cambridge University Press 2007

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References

[1]Beretta, E.Hulshof, J. & Peletier, L. A. (1996) On an ODE from forced coating flow. J. Diff. Eq., 130 (1), 247–265.CrossRefGoogle Scholar
[2]Bernis, F. & Peletier, L. A. (1996) Two problems from draining flows involving third-order ordinary differential equations. SIAM J. Math. Anal., 27 (2), 515–527.CrossRefGoogle Scholar
[3]Bertozzi, A. L. & Brenner, M. P. (1997) Linear stability and transient growth in driven contact lines. Phys. Fluids, 9 (3): 530–539.CrossRefGoogle Scholar
[4]Bertozzi, A. L., Münch, A. & Shearer, M. (1999) Undercompressive shocks in thin film flows. Physica D, 134 (4), 431–464.CrossRefGoogle Scholar
[5]Bowen, M., Sur, J., Bertozzi, A. L. & Behringer, R. P. (2005) Nonlinear dynamics of two-dimensional undercompressive shocks. Physica D, 209 (1–4), 36–48.CrossRefGoogle Scholar
[6]Edmonstone, B. D., Matar, O. K. & Craster, R. V. (2004) Flow of surfactant-laden thin films down an inclined plane. J. Eng. Math., 50 (2–3), 141–156.CrossRefGoogle Scholar
[7]Edmonstone, B. D., Matar, O. K. & Craster, R. V. (2005) Surfactant-induced fingering phenomena in thin film flow down an inclined plane. Phys. D, 209 (1-4), 62–79.CrossRefGoogle Scholar
[8]Galaktionov, V. A. (2006) On higher-order viscosity approximations of odd-order nonlinear PDEs. Preprint.CrossRefGoogle Scholar
[9]Huppert, H. E. (1982) Flow and instability of a viscous current down a slope. Nature, 300, 427–429.CrossRefGoogle Scholar
[10]Jensen, O. E. & Grotberg, J. B. (1992) Insoluble surfactant spreading on a thin viscous film: shock evolution and film rupture. J. Fluid Mech., 240, 259–288.CrossRefGoogle Scholar
[11]Jensen, O. E. & Grotberg, J. B. (1993) The spreading of heat or soluble surfactant along a thin liquid film. Phys. Fluids A, 5 (1), 58–68.CrossRefGoogle Scholar
[12]King, J. R. (1988) Approximate solutions to a nonlinear diffusion equation. J. Eng. Math., 22 (1), 53–72.CrossRefGoogle Scholar
[13]King, J. R. & Please, C. P. (1986) Diffusion of dopant in crystalline silicon: an asymptotic analysis. IMA J. Appl. Math., 37 (3), 185–197.CrossRefGoogle Scholar
[14]Levy, R. & Shearer, M. (2006) The motion of a thin film driven by surfactant and gravity. SIAM J. Appl. Maths, 66 (5), 1588–1609.CrossRefGoogle Scholar
[15]Dal Passo, R.Bertsch, M. & Nitsch, C. (2005) A system of degenerate parabolic nonlinear pde's: a new free boundary problem. Interfaces Free Bound., 7, 255–276.Google Scholar
[16]Marangoni, C. (1865) On the expansion of a drop of liquid floating in the surface of another liquid. PhD thesis, University of Pavia.Google Scholar
[17]Matar, O. K. (2002) Nonlinear evolution of thin free viscous films in the presence of soluble surfactant. Phys. Fluids, 14, 4216–4234.CrossRefGoogle Scholar
[18]Mock, M. S. (1976) On fourth-order dissipation and single conservation laws. Commun. Pure Appl. Math., 29, 383–388.CrossRefGoogle Scholar
[19]Moriarty, J. A., Schwartz, L. W. & Tuck, E. O. (1991) Unsteady spreading of thin liquid films with small surface tension. Phys. Fluids A, 3 (5), 733–742.CrossRefGoogle Scholar
[20]Münch, A. (2003) Pinch-off transition in Marangoni-driven thin films. Phys. Rev. Lett., 91 (1), 016105.CrossRefGoogle ScholarPubMed
[21]Münch, A. & Wagner, B. A. (1999) Numerical and asymptotic results on the linear stability of a thin film spreading down a slope of small inclination. Eur. J. Appl. Math., 10 (3), 297–318.CrossRefGoogle Scholar
[22]Myers, T. G. (1998) Thin films with high surface tension. SIAM Rev., 40 (3), 441–462.CrossRefGoogle Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. (1997) Long-scale evolution of thin liquid films. Rev. Modern Phys., 69 (3), 931–980.CrossRefGoogle Scholar
[24]Renardy, M. (1996) A singularly perturbed problem related to surfactant spreading on thin films. Nonlinear Anal., 27 (3), 287–296.CrossRefGoogle Scholar
[25]Renardy, M. (1997) A degenerate parabolic-hyperbolic system modeling the spreading of surfactants. SIAM J. Math. Anal., 28 (5), 1048–1063.CrossRefGoogle Scholar
[26]Schaeffer, D. G. & Shearer, M. (1987) Riemann problems for nonstrictly hyperbolic systems of conservation laws. Trans. AMS, 301, 267–306.Google Scholar
[27]Stone, H. A. (1990) A simple derivation of the time-dependent convective-diffusion equation for surfactant transport along a deforming interface. Phys. Fluids A, 2 (1), 111–112.CrossRefGoogle Scholar
[28]Troy, W. C. (1993) Solutions of third-order differential equations relevant to draining and coating flows. SIAM J. Math. Anal., 24 (1), 155–171.CrossRefGoogle Scholar
[29]Tuck, E. O. & Schwartz, L. W. (1990) A numerical and asymptotic study of some third-order ordinary differential equations relevant to draining and coating flows. SIAM Rev., 32, 453–469.CrossRefGoogle Scholar
[30]Warner, M. R. E. & Craster, R.V. (2004) Fingering phenomena created by a soluble surfactant deposition on a thin liquid film. Phys. Fluids, 16, 2933–2951.CrossRefGoogle Scholar
[31]Witelski, T. P., Shearer, M. & Levy, R. (2006) Growing surfactant waves in thin liquid films driven by gravity. Appl. Math. Res. Exp., 2006 (15487), 1–21.Google Scholar