Influence of inertia on viscous fingering patterns: rectangular and radial flows.

Recently, there has been a growing interest in the impact of inertial effects on the development of the Saffman-Taylor instability. Experiments and theory indicate that inertia may have a significant influence on the system's behavior. We employ a perturbative-mode-coupling method to examine how the stability and morphology of the viscosity-driven fingering patterns are affected by inertia. Both rectangular and radial Hele-Shaw flow geometries are considered. In the rectangular configuration useful results can be deduced analytically, and in closed form. In particular, we have found that inertia has a stabilizing role at the linear stage, and tends to widen the fingers at the weakly nonlinear regime. These analytical results are consistent with existing experimental findings. The analysis of the system is not as simple in radial flow geometry, but it still allows the capture of inertially induced, enhanced finger tip splitting events at the onset of nonlinearities.