It's really interesting from an applied mathematical point of view that this same model applies to a lot of different physical systems, including lava flow and petroleum processing. In all these cases, there are two key components that lead to interesting behavior. First, the effective viscosity of the fluid is nonlinear with respect to the concentrations of the two fluids. Second, when a branch in the network splits into two daughter branches, the distribution of fluid phases in the daughter branches, known as skimming, can be a complicated function depending on the concentrations of the phases, the flow rate in the branch, and the geometry of the junction.
Recent work (by John, Brian and m has developed an empirical skimming model for a tabletop experimental setup. This should allow us to