Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-22T22:06:25.776Z Has data issue: false hasContentIssue false

The interface dynamics of a surfactant drop on a thin viscous film

Published online by Cambridge University Press:  08 November 2016

MARINA CHUGUNOVA
Affiliation:
Institute of Mathematical Sciences, Claremont Graduate University, 150 E. 10th St., Claremont, California 91711, USA email: [email protected]
JOHN R. KING
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK email: [email protected]
ROMAN M. TARANETS
Affiliation:
Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine, Donetsk, 83114, Ukraine UCLA Department of Mathematics, Los Angeles, CA 90095, USA email: [email protected]

Abstract

We study a system of two coupled parabolic equations that models the spreading of a drop of an insoluble surfactant on a thin liquid film. Unlike the previously known results, the surface diffusion coefficient is not assumed constant and depends on the surfactant concentration. We obtain sufficient conditions for finite speed of support propagation and for waiting-time phenomenon by application of an extension of Stampacchia's lemma for a system of functional equations.

Type
Papers
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The research of Roman Taranets leading to these results has received funding from the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement No PIIF-GA-2009-254521 - [TFE]. This work was partially supported by a grant from the Simons Foundation (#275088 to Marina Chugunova).

References

[1] Francisco, B. & Friedman, A. (1990) Higher order nonlinear degenerate parabolic equations. J. Differ. Equ. 83 (1), 179206.Google Scholar
[2] Francisco, B. (1996) Finite speed of propagation and continuity of the interface for thin viscous flows. Adv. Differ. Equ. 1 (3), 337368.Google Scholar
[3] Francisco, B. (1996) Finite speed of propagation for thin viscous flows when 2 ⩽ n < 3. C. R. Acad. Sci. Paris Sér. I Math. 322 (12), 11691174.Google Scholar
[4] Andrea, L. B. & Pugh, M. C. (2000) Finite-time blow-up of solutions of some long-wave unstable thin film equations. Indiana Univ. Math. J. 49 (4), 13231366.Google Scholar
[5] Bertozzi, A. L. & Pugh, M. C. (1998) Long-wave instabilities and saturation in thin film equations. Commun. Pure Appl. Math. 51 (6), 625661.Google Scholar
[6] Bertozzi, A. L. & Pugh, M. C. (2000) Finite-time blow-up of solutions of some long-wave unstable thin film equations. Indiana Univ. Math. J. 49 (4), 13231366.Google Scholar
[7] Blowey, J. F., King, J. R. & Langdon, S. (2007) Small- and waiting-time behavior of the thin-film equation. SIAM J. Appl. Math. 67 (6), 17761807.Google Scholar
[8] Michael, S. B. & Grotberg, J. B. (1988) Monolayer flow on a thin film. J. Fluid Mech. 193, 151170.Google Scholar
[9] Chugunova, M., Pugh, M. & Taranets, R. (2010) Nonnegative solutions for a long-wave unstable thin film equation with convection. SIAM J. Math. Anal. 42 (4), 18261853.Google Scholar
[10] Chugunova, M. & Taranets, R. (2013) Nonnegative weak solutions for a degenerate system modelling the spreading of surfactant on thin films. Appl. Math. Res. Exp. AMRX 2013 (1), 102126.Google Scholar
[11] Chugunova, M. & Taranets, R. (2012) Qualitative analysis of coating flows on a rotating horizontal cylinder. Int. J. Differ. Equ. 2012 (1), art.id 570283:1–30.Google Scholar
[12] Dal Passo, R., Garcke, H. & Grün, G. (1998) On a fourth-order degenerate parabolic equation: Global entropy estimates, existence, and qualitative behavior of solutions. SIAM J. Math. Anal. 29 (2), 321342.Google Scholar
[13] Dal Passo, R., Giacomelli, L. & Grün, G. (2001) A waiting time phenomenon for thin film equations. Ann. Scuola Norm. Sup. Pisa. 30, 437463.Google Scholar
[14] Dal Passo, R., Giacomelli, L. & Shishkov, A. (2001) The thin film equation with nonlinear diffusion. Commun. Partial Differ. Equ. 26 (9–10), 15091557.Google Scholar
[15] Èĭdel'man, S. D. (1969) Parabolic Systems. Translated from the Russian by Scripta Technica, London. North-Holland Publishing Co., Amsterdam.Google Scholar
[16] Escher, J., Hillairet, M., Laurençot, Ph. & Walker, Ch. (2011) Global weak solutions for a degenerate parabolic system modeling the spreading of insoluble surfactant. Indiana Univ. Math. J. 60, 19752020.Google Scholar
[17] Escher, J., Hillairet, M., Laurençot, Ph. & Walker, Ch. (2012) Thin film equations with soluble surfactant and gravity: Modeling and stability of steady states. Math. Nachr. 285 (2–3), 210222.Google Scholar
[18] Fallest, D. W., Lichtenberger, A. M., Fox, C. J. & Daniels, K. E. (2010) Fluorescent visualization of a spreading surfactant. New J. Phys. 12, 073029.Google Scholar
[19] Fischer, J. (2013) Optimal lower bounds on asymptotic support propagation rates for the thin-film equation. J. Differ. Equ. 255 (10), 31273149.Google Scholar
[20] Fischer, J. (2014) Upper bounds on waiting times for the thin-film equation: The case of weak slippage. Arch. Rational Mech. Anal. 211 (3), 771818.Google Scholar
[21] Jensen, O. E. & Grotberg, J. B. (1992) Insoluble surfactant spreading on a thin viscous film: Shock evolution and film rupture. J. Fluid Mech. 240, 259288.Google Scholar
[22] Harald, G. & Wieland, S. (2006) Surfactant spreading on thin viscous films: Nonnegative solutions of a coupled degenerate system. SIAM J. Math. Anal. 37 (6), 20252048.Google Scholar
[23] Giacomelli, L. & Shishkov, A. (2005) Propagation of support in one-dimensional convected thin-film flow. Indiana Univ. Math. J. 54 (4), 11811215.Google Scholar
[24] Giacomelli, L. & Grün, G. (2006) Lower bounds on waiting times for degenerate parabolic equations and systems. Interfaces Free Boundaries 8 (1), 111129.Google Scholar
[25] Grün, G. (2003) Droplet spreading under weak slippage: A basic result on finite speed of propagation. SIAM J. Math. Anal. 34 (4), 9921006.Google Scholar
[26] Grün, G. (2004) Droplet spreading under weak slippage: The waiting time phenomenon. Ann. I. H. Poincare, Analyse Non Lineaire. 21 (2), 255269.Google Scholar
[27] Grün, G. (2004) Droplet spreading under weak slippage–existence for the cauchy problem. Comm. Partial Differ. Equ. 29 (11–12), 16971744.Google Scholar
[28] Kamin, S. & Vazquez, J. L. (1991) Asymptotic behaviour of solutions of the porous medium equation with changing sign. SIAM J. Math. Anal. 22 (1), 3445.CrossRefGoogle Scholar
[29] Chan, K. Y. & Borhan, A. (2005) Surfactant-assisted spreading of a liquid drop on a smooth solid surface. J. Colloid Interface Sci. 287 (1), 233248.Google Scholar
[30] Lucassen, J. & Hansen, R. S. (1967) Damping of waves on monolayer-covered surfaces II. Influence of bulk-to-surface diffusional interchange on ripple characteristics. J. Colloid Interface Sci. 23, 319328.Google Scholar
[31] Yu, N. & Taranets, R. (2011) Backward motion and waiting time phenomena for degenerate parabolic equations with nonlinear gradient absorption. Manuscr. Math. 136 (3–4), 475500.Google Scholar
[32] Peterson, E. R. & Shearer, M. (2012) Simulation of spreading surfactant on a thin liquid film. Appl. Math. Comput. 218, 51575167.Google Scholar
[33] Shishkov, A. E. & Taranets, R. M. (2004) On the equation of the flow of thin films with nonlinear convection in multidimensional domains. Ukr. Mat. Visn. 1 (3), 402444.Google Scholar
[34] Sinz, D. K. N., Hanyak, M., Zeegers, J. C. H. & Darhuber, A. A. (2011) Insolubale surfactant spreading along thin liquid films confined by chemical surface patterns. Phys. Chem. Chem. Phys. 13, 97689777.Google Scholar
[35] Solonnikov, V. A. (1965) On boundary value problems for linear parabolic systems of differential equations of general form. Trudy Mat. Inst. Steklov 83, 3163.Google Scholar
[36] Starov, V. M., Ryck, A. & Velardet, M. G. (1997) On the spreading of an insoluble surfactant over a thin viscous liquid layer. J. Colloid Interface Sci. 190, 104113.Google Scholar
[37] Swanson, E. R., Strickland, S. L., Shearer, M. & Daniels, K. E. (2014) Surfactant spreading on a thin liquid film: Reconciling models and experiments. J. Eng. Math. doi: 10.1007/s10665-014-9735-0.Google Scholar
[38] Taranets, R. (2006) Propagation of perturbations in thin capillary film equations with nonlinear diffusion and convection. Siberian Math. J. 47, 914931.CrossRefGoogle Scholar
[39] Taranets, R. & Shishkov, A. E. (2003) Effect of time delay of support propagation in equations of thin films. Ukrainian Math. J. 55, 11311152.Google Scholar
[40] Taranets, R. (2003) Propagation of perturbations in the equations of thin capillary films with nonlinear absorption. Proc. Inst. Appl. Math. Mech. 8, 180194.Google Scholar
[41] Taranets, R. (2002) Solvability and global behavior of solutions of the equation of thin films with nonlinear dissipation and absorption. Proc. Inst. Appl. Math. Mech. 7, 192209.Google Scholar
[42] Touhami, Y., Rana, D., Neale, G. H. & Hornof, V. (2001) Study of polymer-surfactant interactions via surface tension measurements. Colloid Polymer Sci. 279 (3), 297300.Google Scholar
[43] Vázquez, J. L. (2007) The Porous Medium Equation: Mathematical Theory. Oxford University Press, Cary, NC, USA.Google Scholar
[44] Venables, J. A. (2000) Introduction to Surface and Thin Film Processes. Cambridge University Press, Cambridge, UK.Google Scholar