Pattern Formation in Non-Newtonian Hele-Shaw Flow

We study theoretically the Saffman-Taylor instability of an air bubble expanding into a non-Newtonian fluid in a Hele-Shaw cell, with the motivation of understanding suppression of tip-splitting and the formation of dendritic structures observed in the flow of complex fluids, such as polymeric liquids or liquid crystals. A standard visco-elastic flow model is simplified in the case of flow in a thin gap, and it is found that there is a distinguished limit where shear thinning and normal stress differences are apparent, but elastic response is negligible. This observation allows formulation of a generalized Darcy\u27s law, where the pressure satisfies a nonlinear elliptic boundary value problem. Numerical simulation shows that shear-thinning alone modifies considerably the pattern formation and can produce fingers whose tip-splitting is suppressed, in agreement with experimental results. These fingers grow in an oscillating fashion, shedding "side-branches" from their tips, closely resembling solidification patterns. A careful analysis of the parametric dependencies of the system provides an understanding of the conditions required to suppress tip-splitting, and an interpretation of experimental observations, such as emerging length-scales. (C) 2001 American Institute of Physics.</p

Science in the sandbox: Fluctuations, friction and instabilities

The study of granular materials is a novel and rapidly growing field. These materials are interest for a number of reasons, both practical and theoretical. They exhibit a rich of novel dyanamical states, and they exhibit 'phases'-solid, liquid, and gas-that resemble conventional thermodynamic phases. However, the presence of strong dissipation through friction and inelasticity places these systems well outside the usual class of systems that can be explained by equilibrium thermodynamics. Thus, there are important challenges to create new kinds of statistical physics and new analytical descriptions for the mean and fluctuating behavior of these materials. We explore recent work that focuses on several important issues. These include force propagation and fluctuations in static and driven systems. It is well known that forces propagate through granular structures along networks-force chains, whose structure is a function of history. It is much less clear how to describe this process, and even what kind of structures evolve in physical experiments. After a brief overview of the field, we consider models of force propagation and recent experiments to test these models. Among the latter are experiments that probe force profiles at the base of sandpiles and experiments that determine the Green's function response to point perturbations in granular systems. We also explore the nature of force fluctuations in slowly evolving systems, particulary sheared granular systems. These can be very strong-with rms fluctuations in the force that are as strong as the mean force. Finally, we pursue the analogy between conventional phases of matter, where we particularly focus on the transition between fluid and solid granular states in the presence of sustained horizontal shaking

Velocity-jump instabilities in Hele-Shaw flow of associating polymer solutions

We study fracturelike flow instabilities that arise when water is injected into a Hele-Shaw cell filled with aqueous solutions of associating polymers. We explore various polymer architectures, molecular weights, and solution concentrations. Simultaneous measurements of the finger tip velocity and of the pressure at the injection point allow us to describe the dynamics of the finger in terms of the finger mobility, which relates the velocity to the pressure gradient. The flow discontinuities, characterized by jumps in the finger tip velocity, which are observed in experiments with some of the polymer solutions, can be modeled by using a nonmonotonic dependence between a characteristic shear stress and the shear rate at the tip of the finger. A simple model, which is based on a viscosity function containing both a Newtonian and a non-Newtonian component, and which predicts nonmonotonic regions when the non-Newtonian component of the viscosity dominates, is shown to agree with the experimental data