Toward the zero surface tension limit: The granular fingering instability

TL;DR

This paper investigates granular fingering patterns in a Hele-Shaw cell, revealing fractal structures and singularities that align with classical fluid fingering theories in the zero-surface-tension limit, despite granular-specific differences.

Contribution

It demonstrates that dry granular materials can produce fingering instabilities with properties similar to idealized fluid models, offering new insights into granular flow dynamics and zero-surface-tension hydrodynamics.

Findings

Granular interfaces exhibit fractal structures and cusps similar to fluid fingering.

Fingering scaling laws differ above the yield stress due to granular friction.

Interface structures agree with Laplacian growth theories in the zero-surface-tension limit.

Abstract

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.