Finger tip behavior in small gap gradient Hele-Shaw flows

Finger tip splitting and finger tip narrowing phenomena are not commonly observed during early flow stages in parallel-plate rectangular Hele-Shaw cells. However, an interesting experimental study performed by Zhao [Phys. Rev. A 45, 2455 (1992)] showed that finger tip behavior can be significantly affected by the addition of a small gradient to the gap of the cell, so that the plates are no longer exactly parallel. It was found that pattern forming events at the finger tip are dictated by the sign of the gap gradient, where positive (negative) sign produces wider (narrower) fingers. We approach the nonparallel-plate situation analytically through a perturbative mode coupling theory and show that these important finger tip features can be predicted at early nonlinear stages of the dynamics.