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Understanding the rheological properties of suspensions is a vital step in designing efficient equipment for numerous industrial applications and also for modeling many important natural phenomena which involve fluid-particle flows. Although extensive research has been carried out on the macroscopic rheological properties of suspensions the effective bulk viscosity has largely been neglected, probably because of the non-obvious way in which such effects are manifested. For a pure fluid the bulk viscosity relates the excess pressure in a fluid undergoing volume change to the rate of expansion or compression of the fluid. The bulk viscosity measures the energy dissipated in causing a change in the fluid's density. For a suspension the effective bulk viscosity is defined analogously to that of a pure fluid as the constant of proportionality relating the deviation of the trace of the average macroscopic stress from its equilibrium value to the average rate of expansion of the suspension. The effective bulk viscosity is then a measure of the energy dissipated in changing the number density of the particulate phase. Although the fluid and the particles may be not be compressible individually, when considered as a phase both the fluid and particle phases are compressible. When the particles are forced closer or pulled apart the fluid between them gets squeezed out or in respectively, and this squeezing motion generates an isotropic stress proportional to the rate of expansion, i.e., a bulk viscosity effect. Consequently the bulk viscosity would play an important role in the modeling of suspensions, especially when there is a rapid and sharp variation in particle concentration - shocks in particle volume fraction - which occur frequently in suspension flows.

Exact expressions were derived for computing the bulk viscosity for all particle concentrations and all expansion rates, and were shown to be completely analogous to the well known formulae used to compute the shear viscosity from the deviatoric macroscopic stres. The single particle correction to the suspension bulk viscosity similar to Einstein's correcion to the shear viscosity, was derived and was shown to be proportional to the fluid shear viscosity. The finite size of a rigid particle in expansion flow causes incompressible disturbance flows around it which contribute to the total stress. Therefore the bulk viscosity may not be negligible even for very dilute suspensions depending on the magnitude of the fluid shear viscosity. The effect of rigid particles on the bulk viscosity was determined to second order in volume fraction and to first order in the Peclet number via theoretical analysis. The Peclet number (Pe) is deÞned as the expansion rate made dimensionless with the Brownian time scale. The bulk viscosity for dilute suspensions was determined numerically for all rates of compression and was found to undergo compression thinning at small Pe and compression thickening for large compression rates, eventually reaching a plateau in the limit of infinite Pe. The temporal response of the suspension in oscillatory expansion/compression was also studied analytically to determine the frequency-dependent bulk viscosity and the high-frequency bulk modulus of hard spheres with hydrodynamic interactions. An analytical expression for the pressure autocorrelation function for Brownian hard spheres was also derived from the temporal response analysis, and it was found to decay as t^(-3/2) in the long-time limit.

At high particle concentrations the many-body interactions between particles become important and lubrication interactions between nearly touching particles comprise the dominant contribution to the bulk stress, necessiating the use of numerical simulations to calculate the total stress in the suspension. The Stokesian Dynamics (SD) technique developed by Brady and Bossis (1988) was adapted to allow for a uniform linear expansion flow and to compute the trace of the stress tensor for determining the particle-phase pressure. The Accelerated Stokesian Dynamics (ASD) (Sierou and Brady, 2001) and the ASDB-nf (Banchio and Brady, 2003) techniques developed more recently to speed up the SD method were also adapted for expansion flows. The updated simulation methods have enabled the study of a larger variety of suspension flows where the particle phase may undergo expansion or compression either by changing the number density of particle, or having the particles themselves expand or contract in addition to any other imposed forcing. Brownian Dynamics (BD) simulations were also performed to isolate the effect of hydrodynamic interactions on the microstructure.

Dynamic simulations of Brownian suspensions at equilibrium were performed using BD, SD and ASD to calculate the bulk viscosity from the pressure autocorrelation function. In the absence of hydrodynamic interactions the pressure autocorrelation function was found to scale as the pair-distribution function at contact and with hydrodynamics it also scales as inverse of the short-time self-diffusivity. The scaled pressure autocorrelation data from our simulations matches very well with the analytical curve, thereby validating the theoretical work. The rate of decay of the stress autocorrelation functions was found to scale as the single particle Stokes-Einstein diffusivity at small concentrations and with the long-time self-diffusivity at volume fractions greater than 35%. With this scaling the dilute theory result for the bulk viscosity was extrapolated to predict the Brownian contribution to the bulk viscosity correctly for all concentrations. The direct hydrodynamic contribution to the bulk viscosity was also determined for all concentrations by averaging over an ensemble of particle configurations. Dynamic compression simulations were performed with a range of Peclet numbers to study the effect of compression on the microstructure and the bulk viscosity, and reproduced the compression thinning and compression thickening behavior predicted from theoretical analysis.

Finally macroscopic equations for the modeling of suspension flows were derived with the bulk viscosity term and used to simulate simple one dimensional compression of a suspension. The bulk viscosity adds an additional diffusive term to the momentum balance for the particle phase, and therefore affects the temporal evolution of the concentration profile on a macroscopic scale.