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THE EFFECT OF MICROSTRUCTURE ON THE TWO DIMENSIONAL FLOW OF A BINGHAM FLUID IN POROUS MEDIA

Abstract

Bingham fluids exhibit a yield stress which means that they exhibit a nonzero rate of strain only when the shearing stress is larger than the yield stress. The first analysis of the flow of a Bingham fluid which might be regarded as being of relevance to porous media is that through a circular pipe, and it is what might be termed Hagen-Poiseuille-Bingham flow. The resulting variation of the mean flow with the applied pressure gradient is known as the Buckingham-Reiner relation. In recent years attention has focussed on flows through a random network of such circular capillaries in order to model realistic porous media, and the main aim has been to acquire information about the breakthough pressure gradient and the overall magnitude of the resulting one-dimensional flow as a function of the applied pressure gradient.
The present work extends these one dimensional (1D) analyses beginning with a presentation of some novel analytical expressions for different types of 1D media. The main focus, however, is on two dimensional (2D) networks of channels.  We consider how the detailed microstructure affects how not only the amplitude of the mean flow but also its direction are dependent on the magnitude and direction of the applied pressure gradient.
First we consider various networks which tesselate the plane, namely those with square, triangular and hexagonal patterns. In each case we find that the network yields anisotropic responses to the direction of the applied pressure gradient, and this is especially so near the threshold gradient where flow may not arise in certain ranges of orientation of the applied pressure gradient but will in others. Thus we have a yield-stress induced anisotropy because all three of these networks are isotropic when filled with a Newtonian fluid. These networks become isotropic as the pressure gradient increases. 
Then we consider random networks in the sense that the nodes are perturbed randomly away from those corresponding either to a square network or a triangular network. In both cases we assume that the network remains periodic overall, but that each periodic unit is itself randomly composed. We find that randomly perturbed square networks remain anisotropic even when averaged over many cases, but that triangular networks approach isotropy much more readily as the number of nodes within the periodic unit increases.

Biography

Andrew Rees is a mathematician who has been lecturing in the Department of Mechanical Engineering at the University of Bath, UK, for over 25 years. His BSc was from Imperial College, London in 1980 and his PhD, which was entitled, "Convection in Porous Media", was awarded by the University of Bristol in 1986. After a brief time at the University of Exeter as a fixed-term lecturer in Mathematics, he joined the University of Bath.
His main area of research is the analysis and simulation of thermoconvective instabilities and these analytical and numerical tools are applied primarily to flows in porous media. His most recent research interest is on the modelling and simulation of flow and convection of Bingham fluids in porous media.
He is the author of over 150 journal papers and is on the editorial boards of Transport in Porous Media, the International Journal for
Numerical Methods in Heat and Fluid Flow, and Computational Thermal Sciences.
Although he has pursued an academic career, he still finds time to engage in music making with two orchestras, two operatic societies, his local church choir and many other occasional groups. He was awarded a diploma of the Trinity College of Music, London in 1977 for violin performance.