Mapping the local viscosity of non-Newtonian fluids flowing through disordered porous structures.

Flow of non-Newtonian fluids through topologically complex structures is ubiquitous in most biological, industrial and environmental settings. The interplay between local hydrodynamics and the fluid's constitutive law determines the distribution of flow paths. Consequently the spatial heterogeneity of the viscous resistance controls mass and solute transport from the micron to the meter scale. Examples range from oil recovery and groundwater engineering to drug delivery, filters and catalysts. Here we present a new methodology to map the spatial variation of the local viscosity of a non-Newtonian fluid flowing through a complex pore geometry. We use high resolution image velocimetry to determine local shear rates. Knowing the local shear rate in combination with a separate measurement of the fluid's constitutive law allows to quantitatively map the local viscosity at the pore scale. Our experimental results-which closely match with three-dimensional numerical simulations-demonstrate that the exponential decay of the longitudinal velocity distributions, previously observed for Newtonian fluids, is a function of the spatial heterogeneity of the local viscosity. This work sheds light on the relationship between hydraulic properties and the viscosity at the pore scale, which is of fundamental importance for predicting transport properties, mixing, and chemical reactions in many porous systems.

Laplacian networks: Growth, local symmetry, and shape optimization.

Inspired by river networks and other structures formed by Laplacian growth, we use the Loewner equation to investigate the growth of a network of thin fingers in a diffusion field. We first review previous contributions to illustrate how this formalism reduces the network's expansion to three rules, which respectively govern the velocity, the direction, and the nucleation of its growing branches. This framework allows us to establish the mathematical equivalence between three formulations of the direction rule, namely geodesic growth, growth that maintains local symmetry, and growth that maximizes flux into tips for a given amount of growth. Surprisingly, we find that this growth rule may result in a network different from the static configuration that optimizes flux into tips.