Modeling an Elastic Fingering Instability in a Reactive Hele-Shaw Flow

In this paper we develop a mathematical model of Hele-Shaw flows in which two immiscible fluids react and form a gel-like phase at the interface, a phenomenon which has been observed in recent experiments. This phase significantly stiffens the interface. The interface is modeled as an elastic membrane which has a bending rigidity dependent on the local curvature. A dispersion relation is derived using an energy variational method. Several types of instabilities are categorized, and how various physical parameters affect the instability is investigated. Our model is able to account for some of the anomalous fingering instabilities observed in experiments by Podgorski et al. [Phys. Rev. E, 76 (2007), 016202]. Consistent with the experimental observations, the fingering instability in a reactive system is shown to be greater than in a nonreactive system.