European Journal of Applied Mathematics



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Comparison of two dynamic contact line models for driven thin liquid films


RACHEL LEVY a1 and MICHAEL SHEARER a1
a1 Department of Mathematics and Center for Research in Scientific Computation, North Carolina State University. Raleigh, NC 27695, USA email: shearer@math.ncsu.edu

Article author query
levy r   [Google Scholar] 
shearer m   [Google Scholar] 
 

Abstract

The modeling of the motion of a contact line, the triple point at which solid, liquid and air meet, is a major outstanding problem in the fluid mechanics of thin films [2, 9]. In this paper, we compare two well-known models in the specific context of Marangoni driven films. The precursor model replaces the contact line by a sharp transition between the bulk fluid and a thin layer of fluid, effectively pre-wetting the solid; the Navier slip model replaces the usual no-slip boundary condition by a singular slip condition that is effective only very near the contact line. We restrict attention to traveling wave solutions of the thin film PDE for a film driven up an inclined planar solid surface by a thermally induced surface tension gradient. This involves analyzing third order ODE that depend on several parameters. The two models considered here have subtle differences in their description, requiring a careful treatment when comparing traveling waves and effective contact angles. Numerical results exhibit broad agreement between the two models, but the closest comparison can be done only for a rather restricted range of parameters. The driven film context gives contact angle results quite different from the case of a film moving under the action of gravity alone. The numerical technique for exploring phase portraits for the third order ODE is also used to tabulate the kinetic relation and nucleation condition, information that can be used with the underlying hyperbolic conservation law to explain the rich combination of wave structures observed in simulations of the PDE and in experiments [3, 15].

(Published Online May 3 2005)
(Received November 27 2003)
(Revised April 29 2004)