Rigorous lubrication approximation

We rigorously carry out a lubrication approximation for a liquid thin film which spreads on a solid, driven by surface tension. We consider a two-dimensional Darcy liquid as simple model case. Of particular interest to us is the codimension-two free boundary, i.e. the triple junctions where solid, liquid and vapor meet. In the considered regime of complete wetting, the contact angle vanishes throughout the evolution. We show in particular that this contact-angle condition is preserved in the lubrication approximation.

[1]  P. Gennes.  Wetting: statics and dynamics , 1985 .

[2]  Andrea L. Bertozzi,et al.  Stable and unstable singularities in the unforced Hele‐Shaw cell , 1996 .

[3]  Droplet spreading under weak slippage: the optimal asymptotic propagation rate in the multi-dimensional case , 2002 .

[4]  S. Bankoff,et al.  Long-scale evolution of thin liquid films , 1997 .

[5]  Francisco Bernis,et al.  Finite speed of propagation and continuity of the interface for thin viscous flows , 1996, Advances in Differential Equations.

[6]  Felix Otto,et al.  Lubrication approximation with prescribed nonzero contact anggle , 1998 .

[7]  J. Duchon,et al.  Évolution d’une interface par capillarité et diffusion de volume I. Existence locale en temps , 1984 .

[8]  Mary C. Pugh,et al.  Global solutions for small data to the Hele-Shaw problem , 1993 .

[9]  Lorenzo Giacomelli,et al.  Droplet spreading: intermediate scaling law by PDE methods , 2002 .

[10]  Robert Almgren,et al.  Singularity formation in Hele–Shaw bubbles , 1996 .

[11]  Joachim Escher,et al.  Classical solutions for Hele-Shaw models with surface tension , 1997, Advances in Differential Equations.

[12]  Mary C. Pugh,et al.  The lubrication approximation for thin viscous films: Regularity and long-time behavior of weak solutions , 1996 .

[13]  Lorenzo Giacomelli,et al.  Variatonal formulation for the lubrication approximation of the Hele-Shaw flow , 2001 .

[14]  M. Bertsch,et al.  Nonnegative solutions of a fourth-order nonlinear degenerate parabolic equation , 1995 .

[15]  G. Prokert.  Existence results for Hele–Shaw flow driven by surface tension , 1998, European Journal of Applied Mathematics.