  We introduce a discrete numerical method based on the diffusion-limited
aggregation (DLA) approach to simulate two-fluid Hele-Shaw flow subject to the
Saffman-Taylor interfacial instability, in the case where the displaced fluid
is non-Newtonian. Focusing on fluids for which the most relevant non-Newtonian
aspect of the thin-gap flow is shear-thinning, we introduce a history-dependent
aspect into the algorithm, modeling shear-rate-dependent fluid viscosity. The
main finding is that the morphology of the emerging patterns, characterized by
the fractal dimension, is modified in a nontrivial manner by the shear-thinning
nature of the displaced fluid. In particular, we consistently find that
shear-thinning leads to the formation of patterns characterized by a smaller
fractal dimension, compared to the corresponding Newtonian fluid.
