Time-dependent injection strategies for multilayer Hele-Shaw and porous media flows

We use linear stability analysis to demonstrate how to stabilize multilayer radial Hele-Shaw and porous media flows with a time-dependent injection rate. Sufficient conditions for an injection rate that maintains a stable flow are analytically derived for flows with an arbitrary number of fluid layers. We show numerically that the maximum injection rate for a stable flow decreases proportional to t(-1/3) for t >> 1 regardless of the number of fluid layers. However, the constant of proportionality depends on the number of layers and increases at a rate that is proportional to the number of interfaces to the two-thirds power. Therefore, flows with more fluid layers can be stable with faster time-dependent injection rates than comparable flows with fewer fluid layers, even when the additional layers are very thin. We also show that for unstable flows, which may be required to inject a given amount of fluid in a fixed amount of time, an increasing injection rate is less unstable than a constant or decreasing injection rate, and that the inclusion of more fluid layers can overcome poor injection strategies.