Liquid films moving onto a solid substrate are present in a lot of natural and industrial processes and have been the object of a lot of research studies for several decades. In the context of deicing, when a thermal protection system is activated, the supercooled water droplets impacting an aircraft surface don't freeze instantaneously and can coalesce and form a thin liquid film as a result of aerodynamic forces. Experimental studies show that this liquid film isn't always stable and can split into rivulets that may refreeze on unprotected surfaces. The modeling of rivulet flows and thus the accurate prediction of wet and dry surfaces is still an unsolved problem.

The objective of this work is to model the motion and the instabilities of partially wetting thin liquid films to derive models for the formation of wet and dry surfaces. This has a direct influence on the estimation of the wall heat and mass fluxes such as evaporation or exchanges with the boundary layer. For thin films, capillary forces generally play an important role and could strongly influence both the motion of the contact line and the development of longitudinal (surface waves) and transversal instabilities (dewetting and rivulets formation). To predict such flows, shallow water models are generally preferred to the full Navier-Stokes equations. The derivation of such models is based on closure assumptions on the normal film velocity profile which can be justified either by asymptotic analysis or by empirical arguments.

The main idea of the work consists in reformulating the shallow water equations by introducing a "disjoining pressure" to model the effects of a partial wetting. This new term appears like a regularization of the discontinuous forces at the contact line. Emphasis is put on the numerical treatment of the capillary forces, especially those acting in the vicinity of the contact line, since they can strongly influence the development of instabilities. 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 the classical shallow water systems with two equations. A conservative formulation of the system and the associated energy are derived.

One-dimensional numerical simulations using a first order implicit finite volume scheme have been performed. Droplet's stationnary shape, spreading length and time on an horizontal substrate is well recovered for all contact angle. Moreover, based on a linear stability analysis, unstable dewetting regimes of an infinite film of uniform thickness are identified and simulated. The add of a dynamic contact angle, the hysteresis effects and the 3D extension of the method are also in progress.