Delayed instability and smoothed gap profile under translational oscillatory shear in viscous fingering

In confined geometries, the interface between two fluids becomes unstable when a less viscous fluid displaces a more viscous one. This leads to the formation of finger-like protrusions, called the viscous fingering instability. This study uses a radial Hele-Shaw cell as the confining geometry and investigates the role of the profile of the fluid across the gap on this instability. We perturb the profile by introducing uniaxial translational oscillatory shear to the plates while injecting the less viscous, miscible fluid from the center. In the direction parallel to the shear axis, the gradient of the gap-averaged viscosity profile at the interface tip initially remains constant, and then drops to a lower constant value. The new value becomes smaller as the shear amplitude increases and the radius at which this gradient decays varies inversely with the shear speed. We find that together with the reduction of the gap-averaged viscosity gradient, increasing shear speed or shear amplitude also delays the onset of fingering and reduces the growth rate of fingers. The uniaxial shear breaks the symmetry between the directions parallel and perpendicular to the shear axis. The finger growth rate differs between these two directions and there is a larger onset radius and finger width in the perpendicular direction compared to the parallel one.