Suppressing viscous fingering with rotation: Linear predictions and nonlinear simulations

In this work, we investigate the possibility of suppressing injection-driven, viscous fingering instabilities in a radial Hele-Shaw cell, via the action of centrifugal forces. We consider the situation in which an inviscid fluid of negligible density is injected into a viscous and denser one, while the entire cell is rotated with constant angular velocity. Linear stability theory indicates that injection is destabilizing, while rotation tends to stabilize the expanding interface. Two linear stability criteria are proposed to determine a critical angular velocity required to stabilize the interface. The first criterion is based on a standard wavelength selection procedure, and maximizes the linear growth. A second criterion relies on the maximum of the interfacial perturbation amplitudes. Our linear stability findings indicate that the perturbation amplitude-based criterion provides a more robust condition for thorough interface stabilization. To explore the complete nonlinear dynamics of the system, and to verify the efficacy of our angular velocity-based controlling criteria, we perform fully numerical simulations, based on the level set method. Our numerical simulations show that the conventional growth-rate-based criterion is unsuccessful to predict the critical stabilizing angular velocity. On the other hand, our numerical findings validate the maximum-amplitude criterion as a proper and accurate controlling mechanism for suppressing viscous fingering. In this way, our fully nonlinear numerical results do substantiate our analytical linear stability predictions.