Nonlinear simulations of elastic fingering in a Hele-Shaw cell

This work is motivated by the recent experiments of two reacting fluids in a Hele-Shaw cell (Podgorski et al., 2007) and associated linear stability analysis of a curvature weakening model (He et al., 2012). Unlike the classical Hele-Shaw problem posed for moving interfaces with surface tension, the curvature weakening model is concerned with a newly-produced gel-like phase that stiffens the interface, thus the interface is modeled as an elastic membrane with curvature dependent rigidity that reflects geometrically induced breaking of intermolecular bonds. Here we are interested in exploring the long-time interface dynamics in the nonlinear regime. We perform simulations using a spectrally accurate boundary integral method, together with a rescaling scheme to dramatically speed up the intrinsically slow evolution of the interface. We find curvature weakening inhibits tip-splitting and promotes side-branching morphology. At long times, numerical results reveal that there exist nonlinear, stable, self-similarly evolving morphologies.