WIAS Preprint No. 941, (2004)

Marangoni-driven liquid films rising out of a meniscus onto a nearly horizontal substrate


  • Münch, Andreas
  • Evans, P. L.

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


We revisit here the situation of a thin liquid film driven up an inclined substrate by a thermally induced Marangoni shear stress against the counter-acting parallel component of gravity. In contrast to previous studies, we focus here on the meniscus region, in the case where the substrate is nearly horizontal, so there is a significant contribution from the normal component of gravity. Our numerical simulations show that the time-dependent lubrication model for the film profile can reach a steady state in the meniscus region that is unlike the monotonic solutions found in [Münch, SIAM J. Appl. Math., 62(6):2045-2063, 2002]. A systematic investigation of the steady states of the lubrication model is carried out by studying the phase space of the corresponding third order ODE system. We find a rich structure of the phase space including multiple non-monotonic solutions with the same far-field film thickness.

Appeared in

  • Phys. D, 209 (2005), pp. 164--177.

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