A continuum model for patterns and flow in frictional fluid dynamics

The injection of non-wetting gas into a liquid-filled Hele-Shaw cell is a classical problem in a Hele-Shaw cell. This problem has received significant attention because of the distinct interfacial patterns, often referred to as viscous fingers, that form due to hydraulic instabilities. A somewhat popular modification on this problem involves the air injection into a confined granular suspension such that the liquid phase is mixed with density-matched beads. However, if these beads are left to settle on the bottom of the geometry, frictional forces between both adjacent beads as well as the walls of the geometry are introduced. The addition of these grains leads to a class of so-called ``multiphase frictional flows’’, which have implications in a variety of natural and engineering scenarios. Despite their ubiquity and importance, these frictional flows remain relatively understood, particularly from a theoretical perspective. Here, we present a reduced order continuum model for the capillary bulldozing of a sedimented granular material. The model involves three phases – gas, liquid, and solid – here the granular pile and the overlying fluid layer evolves as coupled thin films. By performing a numerical investigation of our model, we present a unified description of the emerging patterns whose morphology varies as the dynamics transition from a friction-dominated to a viscous-dominated regime.