  The Saffman-Taylor viscous fingering problem is investigated for the
displacement of a non-Newtonian fluid by a Newtonian one in a radial Hele-Shaw
cell. We execute a mode-coupling approach to the problem and examine the
morphology of the fluid-fluid interface in the weak shear limit. A differential
equation describing the early nonlinear evolution of the interface modes is
derived in detail. Owing to vorticity arising from our modified Darcy's law, we
introduce a vector potential for the velocity in contrast to the conventional
scalar potential. Our analytical results address how mode-coupling dynamics
relates to tip-splitting and side branching in both shear thinning and shear
thickening cases. The development of non-Newtonian interfacial patterns in
rectangular Hele-Shaw cells is also analyzed.
