Capillary and geometrically driven fingering instability in nonflat Hele-Shaw cells

The usual viscous fingering instability arises when a fluid displaces another of higher viscosity in a flat Hele-Shaw cell, under sufficiently large capillary number conditions. In this traditional framing, the reverse flow case (more viscous fluid displacing a less viscous one) and the viscosity-matched situation (fluids of equal viscosities) are stable. We revisit this classical fluid dynamic problem, now considering flow in a nonflat Hele-Shaw cell. For a specific nonflat environment, we show that both the reverse and the viscosity-matched flows can become unstable, even at low capillary number. This peculiar fluid fingering instability is driven by the combined action of capillary effects and geometric properties of the nonflat Hele-Shaw cell. Our theoretical results indicate that the Hele-Shaw cell geometry significantly impacts the linear stability and nonlinear pattern-forming dynamics of the system. This suggests that the geometry of the medium plays an important role in favoring the occurrence of fingering patterns in nonflat, confined fluid flows.