WIAS Preprint No. 1009, (2005)

Dynamics of a surface-gradient-driven liquid film rising from a reservoir onto a substrate


  • Evans, Peter
  • Münch, Andreas

2010 Mathematics Subject Classification

  • 76D08 37N10 76D45 76B45 76A20 34E10 34B60 76D27

2008 Physics and Astronomy Classification Scheme

  • 68.15.+e 47.20.Ky 47.15.Gf 68.03.Kn 68.03.Cd


  • Lubrication theory, Gravity and surface tension driven liquid flows, undercompressive waves, Landau-Levich drag-out problem, coating flows, capillary meniscus, rarefaction fans


On a tilted heated substrate, surface tension gradients can draw liquid up out of a reservoir. The resulting film thickness profile is controlled by the tilt of the substrate, the imposed temperature gradient, and the thickness of a postulated thin precursor layer. We study the evolution of this film in time, using a lubrication model. A number of distinct behaviours are possible as the substrate tilt angle and other parameters are varied. We use recent results for the multiple stationary profiles possible near the meniscus and examine how these can interact with the advancing front. We show that it is in fact possible to systematically determine the evolution of the entire film profile from the meniscus to the apparent contact line. This allows a categorisation of the range of behaviours for a transversely-uniform profile, in a two-dimensional parameter space. In addition to combinations of meniscus profiles involving capillary fronts and double shock structures, we describe a new combination of a Type I meniscus with a rarefaction fan, and either undercompressive or classical waves for the advancing front, that arises for certain ranges of large substrate tilt and of precursor thickness.

Appeared in

  • SIAM J. Appl. Math., 66 (2006) pp. 1610-1631.

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