Viscous fingering in a soft system

Viscous fingering is a classical hydrodynamic instability that occurs when an invading fluid is injected into a porous medium or Hele-Shaw cell containing a more viscous defending fluid. Viscous fingering is an undesirable feature of many industrial processes, and various strategies have been proposed for suppressing or otherwise controlling it. In a Hele-Shaw cell, for example, viscous fingering can be suppressed by replacing one of the rigid walls with a flexible membrane. The result is a complex fluid-structure interaction problem, the dominant feature of which is inflation of the flow area near the inlet (high pressure) relative to the outlet (low pressure). The resulting negative permeability gradient is ultimately responsible for suppressing the instability. Here, we constrain this problem by working in a system that is deformable but not "inflatable" - A novel Hele-Shaw cell in which one of the rigid walls is coated with soft elastomer. Our system allows for local expansion of the flow area in response to the fluid pressure, but prohibits the generation of a global permeability gradient. We show that the two-way coupling between flow and deformation is nuanced in this constrained system, such that the instability is modified but not necessarily suppressed by softness.