Inertial effects of miscible viscous fingering in a Hele-Shaw cell

Abstract

The effects of inertia on miscible displacements in a Hele-Shaw cell are analyzed. Both linear stability analysis and nonlinear simulations are performed. The results reveal that inertia tends to attenuate viscous fingering. An analysis based on the quasi-steady-state approximation (QSSA) revealed that the growth rate at initial time decreases monotonically as a modified Reynolds number Re* increases, while the most dangerous wavenumber tends to shift toward smaller values. It was also found that the cutoff wavenumber is not affected by inertia and that inertial effects are very limited for longwave instabilities. Initial value calculations extended the QSSA and confirmed the decrease of the growth rate with Re* for long times. Full nonlinear simulations based on a spectral method showed that inertial effects tend to delay the initial development of the instability and result in wider fingers at later breakthrough times. Furthermore, a quantitative analysis of the mixing area shows that inertia can extend the dispersive regime. A re-scaling of the mixing area and time is proposed and the curves at any Peclet and modified Reynolds numbers were superposed onto a single universal curve. With this, the mixing area of any flow with different inertial effects can be predicted.