On the fractal dimension of non-Newtonian Hele-Shaw flow subject to Saffman-Taylor instability

Abstract: 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.