  The dynamics of the interface between two immiscible fluids in a rotating
Hele-Shaw cell are studied experimentally, theoretically and by phase-field
simulations of the H-S equations. As the central, denser fluid is centrifuged,
it forms fingering patterns with long, thin radial filaments ended by a
droplet, alternating with incoming fingers of the outer fluid. Simulations show
the length (width) of the filaments to grow (decay) roughly exponentially, and
the incoming finger tips to asymptotically approach a finite radius for n-fold
symmetric initial conditions; these thus tend to a stationary-shape, which is
calculated. The filament width decays with a time constant which depends only
on the viscosity contrast, whereas its length exhibits a completely universal
growth rate, related to the run away of an isolated droplet, for which we give
an exact solution. The exponential behavior is clear for high, but not low
viscosity contrasts A. Both experiments and simulations show systematic
pinch-off of the droplets at the tips of the filaments for low and not for high
A. A lubrication approximation is derived and successfully accounts for the
filament thinning; it explains why pure exponential thinning is not observed
for low A, and it could clarify the presence or absence of finite-time
pinch-off, since the (morphological) agreement of experiments and simulations
suggests that this phenomenon is contained in the Hele-Shaw equations. For low
A, the experimental time constant appears to be different from that predicted
by standard Hele-Shaw boundary conditions and observed in simulations. An
effective slip condition for the Poiseuille flow of inner liquid across the
cell gap in the case of two liquids gives a possible explanation of this
discrepancy.
