Journal of Fluid Mechanics



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


O. E. JENSEN a1 and S. NAIRE a2
a1 School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK
a2 Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, UK

Article author query
jensen oe   [Google Scholar] 
naire s   [Google Scholar] 
 

Abstract

We consider a viscous drop, loaded with an insoluble surfactant, spreading over a flat plane that is covered initially with a thin liquid film. Lubrication theory allows the flow to be modelled using coupled nonlinear evolution equations for the film thickness and surfactant concentration. Exploiting high-resolution numerical simulations, we describe the multi-region asymptotic structure of the spatially one-dimensional spreading flow and derive a simplified ODE model that captures its dominant features at large times. The model includes a version of Tanner's law accounting for a Marangoni flux through the drop's effective contact line, the magnitude of which is influenced by a rarefaction wave in the film ahead of the contact line. Focusing on the neighbourhood of the contact line, we then examine the stability of small-amplitude disturbances with spanwise variation, using long-wavelength asymptotics and numerical simulations to describe the growth-rate/wavenumber relationship. In addition to revealing physical mechanisms and new scaling properties, our analysis shows how initial conditions and transient dynamics have a long-lived influence on late-time flow structures, spreading rates and contact-line stability.

(Published Online April 24 2006)
(Received February 17 2005)
(Revised August 23 2005)



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