)) yielding three rheometric parameters: consistency K (cognate with viscosity); flow index n (a measure of shear-thinning); yield stress τ 0 . The consistency K of suspensions of particles of arbitrary aspect ratio can be accurately predicted by the model of Maron & Pierce (Maron & Pierce 1956 J. Colloid Sci. 11, 80-95 (doi:10.101690023-X)) with the maximum packing fraction φ m as the only fitted parameter. We derive empirical relationships for φ m and n as a function of average particle aspect ratio r p and for τ 0 as a function of φ m and a fitting parameter τ * . These relationships can be used to predict the rheology of suspensions of prolate particles from measured φ and r p . By recasting our data in terms of the Einstein coefficient, we relate our rheological observations to the underlying particle motions via Jeffery's (Jeffery 1922 Proc. R. Soc. Lond. A 102, 161-179 (doi:10.1098/rspa.1922.0078)) theory. We extend Jeffery's work to calculate, numerically, the Einstein coefficient for a suspension of many, initially randomly oriented particles. This provides a physical, microstructural explanation of our observations, including transient oscillations seen during run start-up and changes of rheological regime as φ increases.

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A semiempirical constitutive model for the visco-elastic rheology of bubble suspensions with gas volume fractions φ < 0.5 and small deformations (Ca 1) is developed. The model has its theoretical foundation in a physical analysis of dilute emulsions. The constitutive equation takes the form of a linear Jeffreys model involving observable material parameters: the viscosity of the continuous phase, gas volume fraction, the relaxation time, bubble size distribution and an empirically determined dimensionless constant. The model is validated against observations of the deformation of suspensions of nitrogen bubbles in a Newtonian liquid (golden syrup) subjected to forced oscillations. The effect of φ and frequency of oscillation f on the elastic and viscous components of the deformation are investigated. At low f , increasing φ leads to an increase in viscosity, whereas, at high f , viscosity decreases as φ increases. This behaviour can be understood in terms of bubble deformation rates and we propose a dimensionless quantity, the dynamic capillary number Cd, as the parameter which controls the behaviour of the system. Previously published constitutive equations and observations of the rheology of bubble suspensions are reviewed. Hitherto apparently contradictory findings can be explained as a result of Cd regime. A method for dealing with polydisperse bubble size distributions is also presented.

1] The rheology of crystal-bearing magma and lava depends on both the shape and volume fraction of the suspended crystals. We present the results of analogue rheometric experiments on monodisperse suspensions of solid particles in a Newtonian liquid, in which particle volume fraction and aspect ratio r p are varied systematically. We find that the effect of on viscosity is well captured by the Maron-Pierce model, and that this model is valid across the range of particle aspect ratios investigated (0.04 ≤ r p ≤ 22, i.e., strongly-oblate to strongly-prolate) when the maximum packing fraction m is treated as a fitting parameter. The value of m derived from fitting to our experimental data depends strongly and systematically on particle aspect ratio; hence, m represents an effective proxy for the influence of particle shape on suspension rheology. We present a simple relationship for m (r p ) which allows the viscosity of a suspension to be calculated as a function of and r p only. We investigate the impact of accounting for crystal shape when modelling volcanic flows by simulating the eruption of magma carrying crystals of different aspect ratio, and conclude that the effect of crystal shape should not be neglected. Citation: Mueller, S., E. W. Llewellin, and H. M. Mader (2011), The effect of particle shape on suspension viscosity and implications for magmatic flows, Geophys. Res. Lett., 38, L13316,