WIAS Preprint No. 1670, (2011)

Stationary solutions for two-layer lubrication equations


  • Jachalski, Sebastian
  • Huth, Robert
  • Kitavtsev, Georgy
  • Peschka, Dirk
    ORCID: 0000-0002-3047-1140
  • Wagner, Barbara
    ORCID: 0000-0001-8306-3645

2010 Mathematics Subject Classification

  • 76Dxx 76Txx 35B40 35C20 49Jxx


  • thin films, gamma-convergence, matched asymptotics, free boundaries, bilayer




We investigate stationary solutions of flows of thin liquid bilayers in an energetic formulation which is motivated by the gradient flow structure of its lubrication approximation. The corresponding energy favors the liquid substrate to be only partially covered by the upper liquid. This is expressed by a negative spreading coefficient which arises from an intermolecular potential combining attractive and repulsive forces and leads to an ultra-thin layer of thickness ε. For the corresponding lubrication models existence of stationary solutions is proven. In the limit ε to 0 matched asymptotic analysis is applied to derive sharp-interface models and the corresponding contact angles, i.e. the Neumann triangle. In addition we use Γ-convergence and derive the equivalent sharp-interface models rigorously in this limit. For the resulting model existence and uniqueness of energetic minimizers are proven. The minimizers agree with solutions obtained by matched asymptotics.

Appeared in

  • SIAM J. Appl. Math., 73 (2013) pp. 1183--1202.

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