Instability and fingering in a rotating Hele–Shaw cell or porous medium

The simplified mathematical model for two-phase displacement in a Hele–Shaw cell, i.e., slow flow between closely spaced parallel plates, has been extended to include effects arising from a constant rate of rotation of the system. A linear stability analysis for an initially concentric circular drop reveals that it is unstable both to translation and, depending on the rotation rate, to a number of fingering modes. The nonlinear evolution with time of two of these modes is traced using a boundary-integral numerical scheme. Ultimately, the liquid mass will break up into a number of drops. Apart from small distortion due to the Coriolis force, the patterns formed are essentially independent of both the liquid viscosity and the cell plate spacing or porous-medium permeability. Potential application areas include certain aspects of coating flows, mold filling, and a class of liquid-filled spinning projectiles with permeable packing.