arXiv2004
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.