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Heterogeneous dispersions as microcontinuum fluids

Published online by Cambridge University Press:  11 February 2020

Benjamin E. Dolata Open the ORCID record for Benjamin E. Dolata [Opens in a new window]  and
Roseanna N. Zia Open the ORCID record for Roseanna N. Zia [Opens in a new window]
Benjamin E. Dolata
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, CA 94305, USA Robert Frederick Smith School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14853, USA
Roseanna N. Zia*
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, CA 94305, USA
*
†Email address for correspondence: rzia@stanford.edu
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Abstract

We present a non-local ‘microcontinuum’ constitutive equation describing particle suspensions undergoing heterogeneous flows in the low Stokes and particle Reynolds number regime. Most prior models rely on mesoscale averages of structural dynamics that smear out non-local effects or otherwise place strong restrictions on flow type. We relieve this restriction here by ensemble averaging the exact equations that govern the suspension at the microscale, thereby obtaining explicit structure-property connections. The result is a pointwise, heterogeneous ensemble average of the suspension stress valid for suspended particles of arbitrary shape, size, composition and concentration, as well as arbitrary sources of heterogeneity, including non-uniform shear fields, heterogeneous force fields or spatially varying volume fractions. This non-local model accounts for spatial and temporal variations in the flow structure over arbitrary length and time scales. We express the microcontinuum constitutive equation as a superposition of gradient operators that automatically accounts for flow heterogeneity over any length scale larger than the particle size. Batchelor’s result for a homogeneous suspension is recovered in the limit of zero gradients; non-zero gradient terms provide non-local corrections to the average stress and account for statistical heterogeneity in the suspension. We utilize energy methods to compute the influence of these gradient operators on the pointwise-averaged stress tensor, revealing a deep connection to the microhydrodynamics formalism. We apply this general framework to a dilute suspension of spherical particles and show that the resultant constitutive equations are consistent with experimental observations.

JFM classification

Complex Fluids: Colloids Mathematical Foundations: General fluid mechanics Micro-/Nano-fluid dynamics: Non-continuum effects
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 888 , 10 April 2020 , A28
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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