We present analyses to provide a generalized rheological equation for suspensions and emulsions of non-Brownian particles. These multiparticle systems are subjected to a steady straining flow at low Reynolds number. We first consider the effect of a single deformable fluid particle on the ambient velocity and stress fields to constrain the rheological behavior of dilute mixtures. In the homogenization process, we introduce a first volume correction by considering a finite domain for the incompressible matrix. We then extend the solution for the rheology of concentrated system using an incremental differential method operating in a fixed and finite volume, where we account for the effective volume of particles through a crowding factor. This approach provides a self-consistent method to approximate hydrodynamic interactions between bubbles, droplets, or solid particles in concentrated systems. The resultant non-linear model predicts the relative viscosity over particle volume fractions ranging from dilute to the the random close packing in the limit of small deformation (capillary or Weissenberg numbers) for any viscosity ratio between the dispersed and continuous phases. The predictions from our model are tested against published datasets and other constitutive equations over different ranges of viscosity ratio, volume fraction, and shear rate. These comparisons show that our model, is in excellent agreement with published datasets. Moreover, comparisons with experimental data show that the model performs very well when extrapolated to high capillary numbers (C
 a≫1). We also predict the existence of two dimensionless numbers; a critical viscosity ratio and critical capillary numbers that characterize transitions in the macroscopic rheological behavior of emulsions. Finally, we present a regime diagram in terms of the viscosity ratio and capillary number that constrains conditions where emulsions behave like Newtonian or Non-Newtonian fluids.

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Volatile dynamics in crystal-rich magma bodies, perspectives from laboratory experiments and theory

A generalized equation for rheology of emulsions and suspensions of deformable particles subjected to simple shear at low Reynolds number

Although the relative velocity of a single crystal or bubble in a quiescent fluid (melt) is well characterized, the interplay of crystals/bubbles in multiparticle systems and its effect on their settling/rising velocity is poorly quantified. We propose a theoretical model for the hindered velocity of non-Brownian emulsions and suspensions of nondeformable fluid and solid particles in the creeping flow regime. The model is based on three sets of correction: two on the drag coefficient experienced by each particle to account for both return flow and Smoluchowski effects and a correction on the rheology to account for nonlocal interactions introduced as a mean-field effective viscosity. Our model is tested against new and published experimental data over a wide range of particle volume fraction and viscosity ratio between the fluids. We find an excellent agreement between our model and experiments. The model is then applied to show that hindered settling can increase mineral residence time by up to an order of magnitude in convecting magma chambers.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1002/2014GL062570

Volatile dynamics in crystal-rich magma bodies, perspectives from laboratory experiments and theory

Citations

A generalized equation for rheology of emulsions and suspensions of deformable particles subjected to simple shear at low Reynolds number

Unifying the relative hindered velocity in suspensions and emulsions of nondeformable particles

Rheological state variables: A framework for viscosity parametrization in crystal-rich magmas

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