Two-Phase Displacement Instabilities in Variable-Gap Hele-Shaw Cells

Recently, there has been interest in stabilizing mechanisms for the classical viscous fingering instabilities in Hele-Shaw cells. Here, we critically review the stability criterion and derive it more generally, going beyond the single-finger criterion and analytically deriving critical slowing-down times in the slow-flow regime. We rederive the stability analysis for the linear opening scenario, obtaining a multiple-finger instability phase diagram, where we determine that the linear opening profile does not increase the stability range in the (U, W) space unless velocity corrections are considered. Furthermore, we found a crossing in the stability ranges at critical values of (Uc,Wc), indicating an enlarged stability for Hele-Shaw cells with a linear profile, only for wide cells and low flow speeds. We further analyze other opening profiles amenable to analytical treatment, including the power-law and exponential profiles.