Modeling of thin liquid films with arbitrary many layers

Abstract

We propose the generalization of the thin film equation (TFE) to arbitrarily many immiscible liquid layers. Then, we provide different pathways for deriving the hydrodynamic pressure within the individual layers, showing how to understand the equation as a Cahn-Hilliard-type conservation equation and providing an algorithm to derive the associated Onsager Matrix. Furthermore, we employ a numerical solver based on the multilayer shallow water-lattice Boltzmann method (LBM) for two and three liquid layers in pseudo two and three dimensions to gain insights into the dynamics of the system and to validate the model. We perform a linear stability analysis and assess droplet equilibrium shapes. Furthermore, we compare the dynamics of the proposed thin film equation to full Navier-Stokes simulations and show the possible equilibrium states of the multilayer liquid thin film system.