A Cahn-Hilliard framework for thin-film flows in the partial-wetting regime
Abstract:Traditional mathematical descriptions of multiphase flow in porous media rely on a multiphase extension of Darcy’s law, and lead to nonlinear second-order (advection-diffusion) partial differential equations for fluid saturations. Here, we study horizontal redistribution of immiscible fluids. The traditional Darcy-flow model predicts that the spreading of a finite amount of liquid in a horizontal porous medium never stops; a prediction that is not substantiated by observation. To help guide the development of new models of multiphase flow in porous media , we draw an analogy with the flow of thin films. The flow of thin films over flat surfaces has been the subject of much theoretical, experimental and computational research . Under the lubrication approximation, the classical mathematical model for these flows takes the form of a nonlinear fourth-order PDE, where the fourth-order term models the effect of surface tension . This classical model, however, effectively assumes that the film is perfectly wetting to the substrate and, therefore, does not capture the partial wetting regime. Partial wetting is responsible for stopping the spread of a liquid puddle.
Here, we present experiments of (large-volume) liquid spreading over a flat horizontal substrate in the partial wetting regime, and characterize the spreading regimes that we observe. We extend our previous theoretical work of two-phase flow in a capillary tube , and develop a macroscopic phase-field model of thin-film flows with partial wetting. Our model naturally accounts for the dynamic contact angle at the contact line, and therefore permits modeling thin-film flows without invoking a precursor film, leading to compactly-supported solutions that reproduce the spreading dynamics and the static equilibrium configuration observed in the experiments. We anticipate that this modeling approach will provide a natural mathematical framework to describe spreading and redistribution of immiscible fluids in porous media.