  We study the stability of multi-layer radial flows in porous media within the
Hele-Shaw model. We perform a linear stability analysis for radial flows
consisting of an arbitrary number of fluid layers with interfaces separating
fluids of constant viscosity and with positive viscosity jump at each interface
in the direction of flow. Several different time-dependent injection strategies
are analyzed including the maximal injection rate that maintains a stable flow.
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, the injection rate for a stable flow increases at a rate that is
proportional to the number of interfaces to the two-thirds power for large
times. Additionally, we show that in any multi-layer radial Hele-Shaw flow, if
all of the interfaces are circular except for one perturbed interface then
there exists a time-dependent injection rate such that the circular interfaces
remain circular as they propagate and the disturbance on the perturbed
interface decays.
  The motion of the interfaces within linear theory is also investigated
numerically for the case of constant injection rates. It is found that: (i) A
disturbance 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 phase
from any arbitrary initial disturbance; and (iii) The dynamics of the flow can
change dramatically with the addition of more interfaces.
