Geometric approach to stationary shapes in rotating Hele-Shaw flows

We consider the geometry of a special family of curves associated with the motion of a two-fluid interface in a rotating Hele-Shaw cell. This family of stationary exact solutions with surface tension consists of interface shapes which balance exactly the competing capillary and centrifugal forces. The result is the formation of patterned structures presenting fingers that eventually assume a teardrop-like shape, and tend to be detached from the main body of the rotating fluid (occurrence of a pinch-off event). By using the vortex-sheet formalism, we approach the problem analytically, and show that the curvature of these particular curves can be very simply expressed as a function of the radial distance to the rotation axis. Motivated by this fact, and through a simple geometric interpretation, we show that the exact solutions for the rotating Hele-Shaw problem satisfy a first-order ordinary differential equation which is readily solved, so that the shape solutions are determined up to quadratures. The ability to probe a number of key morphological features of such solutions analytically is demonstrated, and a gallery of plots is provided to highlight these findings. The possibility of accessing a criterion to predict the occurrence of pinch-off is also discussed. Finally, we prove that the general solutions are dense, and use this fact to guarantee the very existence of the neat, highly symmetric physical patterns.