A shallow water type model to describe the dynamic of thin partially wetting films for the simulation of anti-icing systems

Abstract

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. 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. 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. We use an extended 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.