The Hele-Shaw cell involves two immiscible fluids separated by an interface. Possible topology changes in the interface are investigated. In particular, we ask whether a thin neck between two masses of the fluid can develop, get thinner, and finally break. To study this, we employ the lubrication approximation, which implies for a symmetrical neck that the neck thickness h obeys h, +(hh ")"=0.The question is whether, starting with smooth positive initial data for h, one can achieve h =0, and hence a possible broken neck within a finite time. One possibility is that, instead of breaking, the neck gets continually thinner and finally goes to zero thickness only at infinite time. Here, we investigate one set of initial data and argue that in this case the system does indeed realize this infinite-time breakage scenario.
The motion of the interface between two Auids in a quasi-two-dimensional geometry is studied via simulations. We consider the case in which a zero-viscosity Quid displaces one with finite viscosity and compare the interfaces that arise with zero surface tension with those that occur when the surface tension is not zero. The interface dynamics can be analyzed in terms of a complex analytic function that maps the unit circle into the interface between the Auids. The physical region of the domain is the exterior of the circle, which then maps into the region occupied by the more viscous Auid. In this physical region, the mapping is analytic and its derivative is never zero. This paper focuses upon the determination of the nature of the interface and the positions of the singularities of the derivative of the mapping function g. Two kinds of initial conditions are considered: case A, in which the singularities closest to the unit circle are poles; and case 8, in which the t =0 interface is described by a function g with only zeros inside the unit circle. In either case, different behaviors are found for relatively smaller and larger surface tensions. In case A, when the surface tension is relatively small, the problem is qualitatively similar with and without surface tension: the singularities move outward and asymptotically approach the unit circle. For relatively large surface tension, the singularities, still polelike, move towards the center of the unit circle instead. In case B, for zero surface tension, the zeros move outward and hit the unit circle after a finite time, whereupon the solution breaks down. For finite but relatively small surface tension, each initial zero disappears and is replaced by a pair of polelike excitations that seem to approach the unit circle asymptotically, while for a relatively large surface tension, each initial zero is replaced by a polelike singularity that then moves towards the unit circle.
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