Control of interfacial instabilities through variable injection rate in a radial Hele-Shaw cell: A nonlinear approach...

In this paper, the nonlinear behavior of immiscible viscous fingering in a circular Hele-Shaw cell under the action of different time-dependent injection flow rate schemes is assessed numerically. Unlike previous studies which addressed the infinite viscosity ratio (inviscid-viscous flow), the problem is tackled by paying special attention to flows with finite viscosity ratio (viscous flow) in which the viscosity of the displacing and the displaced fluids can have any arbitrary value. Systematic numerical simulations based on a complex-variable formulation of Cauchy-Green barycentric coordinates are performed at different mobility ratios and capillary numbers with a focus on the late-time fully nonlinear regime. Additionally, numerical optimization is used to obtain the optimal flow rate schedule through a second-order weakly nonlinear stability analysis in contrast to previous studies in which the optimal flow rate was obtained entirely based on linear stability analysis. It is demonstrated that, irrespective of the values of the mobility ratio and/or the capillary number, for patterns whose constant injection counterpart exhibits linear flow regime, the curvature-driven relaxation time is comparable with the operational time of the time-dependent injection flow rate controlling schemes, and most of the controlling schemes work very well and suppress the fingering phenomenon remarkably with the maximum recovery improvement of 15%. As the nonlinearity of the system increases, the schemes may still perform well, but their effectiveness is more pronounced in patterns with less nonlinearity in their constant injection counterpart than those with higher nonlinearity. As the nonlinearity increases, the curvature-driven relaxation time becomes longer than the operational time of the schemes, leading to a reduction in their effectiveness. Additionally, it is shown that employment of the second-order weakly nonlinear stability analysis to formulate the objective function does not result in any remarkable variation in the obtained optimal flow rate schedule.