Enhanced time dependent injection rate for multilayer stable Hele-Shaw flows

There are classical asymptotic results on the time dependent injection rate for a stable radial displacement of one viscous fluid by a less viscous one in a Hele-Shaw cell. However, the injection rate can be faster in a multi-layer setting. In particular, we find numerically that flows with more fluid layers can be stable with faster time-dependent injection rates than comparable flows with fewer fluid layers. In particular, we show that for large times, the time-dependent injection rate for a stable radial Hele-Shaw flow increases at a rate that is proportional to the number of interfaces to the two-thirds power.