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Dealing with contact line forces in shallow water models
Julien Lallement (ONERA)
The objective of this work is to model motion and instabilities of partially wetting thin liquid films. Emphasis is put on the numerical treatment of capillary forces, especially those acting in the vicinity of the contact line, since they can strongly influence the development of instabilities. The main idea of the work consists in reformulating the shallow water equations by introducing a "disjoining pressure" to model partial wetting effects. This new term allows smoothing the singular force acting at the contact line by replacing it by a distributed force.
Based on the work of Noble & Vila, we use an augmented conservative system that consists in reducing the order of the shallow water system by adding one evolution equation. This model is suited for numerical purposes since the surface tension term only involves second order derivatives instead of third order derivatives. In addition to that, it is possible to write energy balance equation corresponding to this model that implies the conservation of the energy. The existence of this energy balance equation was used as a criterion to build our system with partial wetting effects.
Onedimensional numerical simulations using a first order implicit finite volume scheme have been performed. Droplet's stationnary shape and spreading length on an horizontal substrate is well recovered. Moreover, based on a linear stability analysis, stable and unstable dewetting regimes of a finite film of uniform thickness can be identified and simulated.

