A phase-field model for capillary bulldozing

The invasion of non-wetting gas into a horizontal, liquid-filled tube or Hele-Shaw cell is a classical problem in fluid mechanics that has been studied extensively from a variety of perspectives. However, the addition of a sedimented granular material to the defending liquid phase can fundamentally change the mechanics of the problem by introducing friction, leading to a class of ``multiphase frictional flows'' that remain relatively poorly understood. For example, recent experiments [Dumazer et al., 2016, PRL] show that, in a capillary tube, the motion of the gas-liquid interface will bulldoze the granular material, accumulating a pile of grains on the liquid side of the interface that will grow until it forms a plug and clogs the tube. Here, we present a thermodynamically consistent phase-field model for capillary bulldozing. The model involves three phases -- gas, liquid, and liquid-solid mixture -- and takes the form of a coupled pair of nonlinear conservation laws and a linear elliptic equation for the velocity of liquid-solid mixture. We solve our model numerically for a variety of different scenarios to develop insight into the roles of sliding friction, rearrangement, capillarity, viscosity, and plug formation during capillary bulldozing.