Skip to main content Accessibility help
×
Home

The spreading and stability of a surfactant-laden drop on an inclined prewetted substrate

  • J. V. Goddard (a1) and S. Naire (a1)

Abstract

We consider a viscous drop, loaded with an insoluble surfactant, spreading over an inclined plane that is covered initially with a thin surfactant-free liquid film. Lubrication theory is employed to model the flow using coupled nonlinear evolution equations for the film thickness and surfactant concentration. Exploiting high-resolution numerical simulations, we describe the late-time multi-region asymptotic structure of the spatially one-dimensional spreading flow. A simplified differential–algebraic equation model is derived for key variables characterising the spreading process, using which the late-time spreading and thinning rates are determined. Focusing on the neighbourhood of the drop’s leading-edge effective contact line, we then examine the stability of this region to small-amplitude disturbances with transverse variation. A dispersion relationship is described using long-wavelength asymptotics and numerical simulations, which reveals physical mechanisms and new scaling properties of the instability.

Copyright

Corresponding author

Email address for correspondence: s.naire@keele.ac.uk

References

Hide All
Afsar-Siddiqui, A. B., Luckham, P. F. & Matar, O. K. 2003a The spreading of surfactant solutions on thin liquid films. Adv. Colloid Interface Sci. 106, 183236.
Afsar-Siddiqui, A. B., Luckham, P. F. & Matar, O. K. 2003b Unstable spreading of aqueous anionic surfactant solutions on liquid film. Part 1. Sparingly soluble surfactant. Langmuir 19, 696702.
Afsar-Siddiqui, A. B., Luckham, P. F. & Matar, O. K. 2003c Unstable spreading of aqueous anionic surfactant solutions on liquid films. Part 2. Highly soluble surfactant. Langmuir 19, 703708.
Afsar-Siddiqui, A. B., Luckham, P. F. & Matar, O. K. 2004 Dewetting behavior of aqueous cationic surfactant solutions on liquid films. Langmuir 20, 75757582.
Bertozzi, A. L. & Brenner, M. P. 1997 Linear stability and transient growth in driven contact lines. Phys. Fluids 9 (3), 530539.
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81, 11311198.
Dussan, E. B. & Davis, S. 1974 On the motion of a fluid–fluid interface along a solid surface. J. Fluid Mech. 65, 7195.
Edmonstone, B. D., Matar, O. K. & Craster, R. V. 2004 Flow of surfactant-laden thin films down an inclined plane. J. Engng Maths 50, 141156.
Edmonstone, B. D., Matar, O. K. & Craster, R. V. 2005a Coating of an inclined plane in the presence of insoluble surfactant. J. Colloid Interface Sci. 287, 261272.
Edmonstone, B. D., Matar, O. K. & Craster, R. V. 2005b Surfactant-induced fingering phenomena in thin film flow down an inclined plane. Physica D 209, 6279.
Edmonstone, B. D., Matar, O. K. & Craster, R. V. 2006 A note on the coating of an inclined plane in the presence of soluble surfactant. J. Colloid Sci. 293, 222229.
Eres, M. H., Schwartz, L. W. & Roy, R. V. 2000 Fingering phenomena for driven coating films. Phys. Fluids 12, 12781295.
Grotberg, J. B. 1994 Pulmonary flow and transport phenomena. Annu. Rev. Fluid Mech. 26, 529571.
Grotberg, J. B. 2001 Respiratory fluid mechanics and transport processes. Annu. Rev. Biomed. Engng 3, 421457.
Hocking, L. M. 1990 Spreading and instability of a viscous fluid sheet. J. Fluid Mech. 211, 373392.
Hocking, L. M., Debler, W. R. & Cook, K. E. 1999 The growth of leading-edge distortions on a viscous sheet. Phys. Fluids 11 (2), 307313.
Homsy, G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271311.
Huh, C. & Scriven, L. E. 1971 Hydrodynamical model of steady movement of a solid/liquid/fluid contact line. J. Colloid Interface Sci. 35, 85101.
Huppert, H. E. 1982 The flow and instability of viscous gravity currents down a slope. Nature 300, 427429.
Jensen, O. 1994 Self-similar, surfactant driven flows. Phys. Fluids A 6, 10841094.
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.
Jensen, O. E. & Halpern, D. 1998 The stress singularity in surfactant-driven thin-film flows. Part 1. Viscous elements. J. Fluid Mech. 372, 273300.
Jensen, O. E. & Naire, S. 2006 The spreading and stability of a surfactant-laden drop on a prewetted substrate. J. Fluid Mech. 554, 524.
Jerrett, J. M. & de Bruyn, J. R. 1992 Fingering instability of a gravitationally driven contact line. Phys. Fluids A 4 (2), 234242.
Kondic, L. E. & Diez, J. A. 2001 Pattern formation in the flow of thin films down an incline: constant flux configuration. Phys. Fluids 13 (11), 31643184.
Kondic, L. E. & Diez, J. A. 2002 Computing three-dimensional thin film flows using contact lines. J. Comput. Phys. 183, 274306.
Levy, R. & Shearer, M. 2006 The motion of a thin film driven by surfactant and gravity. SIAM J. Appl. Maths 66, 15881609.
Levy, R., Shearer, M. & Witelski, T. P. 2007 Gravity-driven thin liquid films with insoluble surfactant: smooth travelling waves. Eur. J. Appl. Maths 8 (6), 679708.
Marmur, A. & Lelah, M. D. 1981 The spreading of aqueous surfactant solutions on glass. Chem. Engng Commun. 13, 133143.
Matar, O. K. & Craster, R. V. 2009 Dynamics of surfactant-assisted spreading. Soft Matt. 5, 38013809.
Mavromoustaki, A.2011 Long-wave dynamics of single- and two-layer flows. PhD thesis, Imperial College of Science, Technology and Medicine, London.
Mavromoustaki, A., Matar, O. K. & Craster, R. V. 2012a Dynamics of a climbing surfactant-laden film – I: base-state flow. J. Colloid Interface Sci. 371, 107120.
Mavromoustaki, A., Matar, O. K. & Craster, R. V. 2012b Dynamics of a climbing surfactant-laden film – II: stability. J. Colloid Interface Sci. 371, 121135.
Rosen, M. J. 2004 Surfactants and Interfacial Phenomena, 3rd edn. Wiley.
Schwartz, L. W. 1989 Viscous flows down an inclined plane: instability and finger formation. Phys. Fluids A 1, 443445.
Silvi, N. & Dussan, E. B. 1985 On the rewetting of an inclined solid surface by a liquid. Phys. Fluids 28 (1), 57.
Spaid, M. A. & Homsy, G. M. 1996 Stability of Newtonian and viscoelastic dynamic contact lines. Phys. Fluids 8, 460478.
Troian, S. M., Herbolzheimer, E. & Safran, S. A. 1990 Model for the fingering instability of the spreading surfactant drops. Phys. Rev. Lett. 65, 333336.
Troian, S. M., Herbolzheimer, E., Safran, S. A. & Joanny, J. F. 1989 Fingering instabilities of driven spreading films. Europhys. Lett. 10 (1), 2530.
Warner, M. R. E., Craster, R. V. & Matar, O. K. 2004 Fingering phenomena associated with insoluble surfactant spreading on thin liquid films. J. Fluid Mech. 510, 169200.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

The spreading and stability of a surfactant-laden drop on an inclined prewetted substrate

  • J. V. Goddard (a1) and S. Naire (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed