Viscous potential flow analysis of radial fingering in a Hele-Shaw cell

The problem of radial fingering in two phase gas/liquid flow in a Hele-Shaw cell under injection of gas is studied here. The fingers arise as an instability of a time-dependent flow. The instability is analyzed as a viscous potential flow, in which potential flow analysis of Paterson [L. Paterson, J. Fluid Mech. 113, 513 (1981)] and others is augmented to account for the effects of viscosity on the normal stress at the gas/liquid interface. The addition of these new effects brings our theory into a much better agreement with experiments of Maxworthy [T. Maxworthy, Phys. Rev. A 39, 5863 (1989)] than other theories.