  The finger-like branching pattern that occurs when a less viscous fluid
displaces a more viscous one confined between two parallel plates has been
widely studied as a classical example of a mathematically-tractable
hydrodynamic instability since the time of Saffman and Taylor. Fingering in
such Hele-Shaw geometries have been generated not only with Newtonian fluids
but also with various non-Newtonian fluids including fine granular material
displaced by gas, liquid, or large grains. Here we study a granular Hele-Shaw
system to explore whether the absence of cohesive forces in dry granular
material can produce an ideal venue for studying the hitherto-unrealizable
singular hydrodynamics predicted in the zero-surface-tension limit. We
demonstrate that the grain-gas interface does indeed exhibit fractal structure
and sharp cusps associated with finite-time singularities. Above the yield
stress, the scaling for the finger width is distinct from that for ordinary
fluids, reflecting unique granular properties such as friction-induced
dissipation as opposed to viscous damping. Despite such differences, the
dimension of the global fractal structure and the shape of the singular cusps
on the interface agree with the theories based on simple Laplacian growth of
conventional fluid fingering in the zero-surface-tension limit. Our study
provides new insights not only on the dynamics of two-phase dense granular
flows and granular pattern formation, but also on the fluid dynamics in the
zero-surface-tension limit.
