Controlling viscous fingering in tapered Hele-Shaw cells

We present a theoretical study of a variant of the classical viscous fingering instability, which occurs when a high viscosity fluid is displaced by a low viscosity fluid in a Hele-Shaw cell. In our system, the Hele-Shaw cell is tapered in the direction of fluid displacement. We consider two tapered Hele-Shaw geometries (rectilinear and radial), which have a constant depth gradient in the flow direction. We find that the presence of a depth gradient can alter the stability of the interface offering opportunities to control and tune fingering instabilities. In particular, the stability of the interface is now determined by both the viscosity contrast of the fluids and the ratio of the depth gradient to the capillary number of the system. We also demonstrate several applications of our analysis, including the inhibition of viscous fingering by controlling the injection flow rate in a radially tapered Hele-Shaw cell.