Controlling fingering instabilities in nonflat Hele-Shaw geometries.

In analogy to the viscous fingering problem between parallel flat plates, injection-driven flow in nonflat (conical and spherical) Hele-Shaw cells gives rise to ramified interfacial structures. The multiple occurrences of fingertip-splitting events makes the resulting pattern difficult to predict and to control. We calculate expressions for time-dependent injection rates that suppress the development of bifurcated shapes on curved confined environments. Implementation of these injection procedures leads to interfacial shapes with a robust n-fold symmetry. A weakly nonlinear approach is used to extract important analytical information about the influence of the geometric and topological features of the nonflat cells on the efficiency of the proposed fingering control processes.