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We study the stability of multi-layer radial flows in porous media within theHele-Shaw model. We perform a linear stability analysis for radial flowsconsisting of an arbitrary number of fluid layers with interfaces separatingfluids of constant viscosity and with positive viscosity jump at each interfacein the direction of flow. Several different time-dependent injection strategiesare analyzed including the maximal injection rate that maintains a stable flow.We find numerically that flows with more fluid layers can be stable with fastertime-dependent injection rates than comparable flows with fewer fluid layers.In particular, the injection rate for a stable flow increases at a rate that isproportional to the number of interfaces to the two-thirds power for largetimes. Additionally, we show that in any multi-layer radial Hele-Shaw flow, ifall of the interfaces are circular except for one perturbed interface thenthere exists a time-dependent injection rate such that the circular interfacesremain circular as they propagate and the disturbance on the perturbedinterface decays.  The motion of the interfaces within linear theory is also investigatednumerically for the case of constant injection rates. It is found that: (i) Adisturbance of one interface can be transferred to the other interface(s); (ii)The disturbances on the interfaces can develop either in phase or out of phasefrom any arbitrary initial disturbance; and (iii) The dynamics of the flow canchange dramatically with the addition of more interfaces.

Time-dependent injection strategies and interfacial stability in multi-layer Hele-Shaw and porous media flows