Suppression of viscous fingering in nonflat Hele-Shaw cells

Viscous fingering formation in flat Hele-Shaw cells is a classical and widely studied fluid mechanical problem. Recently, instead of focusing on the development of the fingering instability, researchers have devised different strategies aiming to suppress its appearance. In this work, we study a protocol that intends to inhibit the occurrence of fingering instabilities in nonflat (spherical and conical) Hele-Shaw cell geometries. By using a mode-coupling theory to describe interfacial evolution, plus a variational controlling technique, we show that viscous fingering phenomena can be minimized in such a confined, curved environment by properly manipulating a time-dependent injection flow rate Q(t). Explicit expressions for Q(t) are derived for the specific cases of spherical and conical cells. The suitability of the controlling method is verified for linear and weakly nonlinear stages of the flow.