Diffusion Limited Aggregation and Saffman-Taylor Instability in Non-Newtonian Hele-Shaw Flow

Supported by NSF Grant No. DMS-1211713

Two-phase flow in quasi 2D geometry is relevant to a number of applications, in particular related to the flow in porous media.    This relevance serves as one important motivation for considering fluid instabillities in the so-called Hele-Shaw geometry (flow between two glass plates). The participating students have carried out experimental, theoretical, and computational study of flow stability in such geometry, focusing in particular on the setup where a less viscous fluid is injected into a more viscous one, in a setup known to lead to instability carrying the name of Saffman-Taylor.  The aim of this project has been to study the instability when the more viscous fluid is non-Newtonian, with viscosity that depends on shear rate. To characterize the properties of the emerging patterns, the students have used several methods to calculate the fractal dimension based on the data collected from experimental trials and extensive simulations of diffusion limited aggregation type.  Both experimental and computational results suggest that the fractal dimensions between Newtonian and non-Newtonian setups differ.  This results, if confirmed, will be of relevance to further work in this field.



Figure 1: Experimental realizations of Saffman-Taylor instability: Newtonian (top/left) and non-Newtonian (bottom/right)


 



Figure 2: Diffusion limited aggregation method for simulating  Saffman-Taylor instability: Newtonian (top/left) and non-Newtonian (bottom/right)