Compression-driven viscous fingering in a Hele-Shaw cell

The gas-driven displacement of a viscous liquid from a Hele-Shaw cell is a classical problem in interfacial fluid dynamics. This problem has received significant attention because the gas-liquid interface is hydrodynamically unstable, forming striking finger-like patterns that have attracted research interest for decades. Mathematical models for this problem almost universally take both fluids to be incompressible and identify the capillary number as the key parameter that sets the intensity of the instability. Recently, much research effort has been targeted at controlling the fingering instability by manipulating the injection rate, the geometry of the Hele-Shaw cell, or the properties of the fluids; however, the practicality of these strategies can be limited. In contrast, compression of the invading gas will naturally, passively, and unavoidably lead to an unsteady injection rate unless the injection pressure is held constant. Here, we study the impact of gas compression on viscous fingering using an axisymmetric model, linear stability analysis, fully nonlinear numerical simulations, and laboratory experiments. We identify a second key parameter, the compressibility number, that plays a strong role in the development of the fingering pattern and we conduct experiments and simulations over a wide range of capillary and compressibility numbers. We show that increasing the compressibility number systematically delays the onset of fingering and decreases the ultimate severity of the fingering pattern at high capillary number. Our results provide an unprecedented comparison of experiments with simulations for viscous fingering, a comprehensive understanding of the role of compression in unstable gas-liquid displacement flows, and insight into a new mechanism for controlling the development of fingering patterns.