Multifluid tensorial equations for the flow of semi-dilute monodisperse suspensions

TL;DR

This paper derives tensorial two-fluid equations for semi-dilute monodisperse suspensions using volume averaging, incorporating microstructure and particle interactions, applicable in arbitrary geometries.

Contribution

It introduces a novel set of tensorial two-fluid equations with closure models that account for microstructure anisotropy and multi-particle hydrodynamic interactions in semi-dilute suspensions.

Findings

Derived tensorial two-fluid equations for suspensions.

Developed closure models for particle stress and microstructure.

Performed magnitude analysis to simplify equations for laminar flow.

Abstract

Using the volume averaging technique of Jackson (1997), we derive a set of two-fluid equations that describe the dynamics of a mono-disperse non-Brownian colloidal suspension in the semi-dilute regime. The equations are tensorial and can be applied in arbitrary geometries. Closure models are developed that represent the stress surrounding each particle as a sum of stresses due to fluid movement through a fixed bed of particles, and those due to interactions between particles. Emphasising pragmatism, the developed closure models are consistent with current knowledge of particle interactions in these systems but employ parameters that can be tuned to represent the microstructure of specific particle suspensions. Within the interaction model, a model for the particle distribution around each particle is used that depends on the strain rate field, allowing anisotropy of the microstructure (and hence normal suspension stresses) to develop within the suspension in response to arbitrary strain fields. Force moments acting on particles during particle interactions are calculating by summing hydrodynamic contributions between particle pairs, but adjusted to recognise that multi-particle interactions can increase the effective stress generated during each interaction. Finally, an order of magnitude analysis is performed on the derived momentum equations to determine what terms are significant, leaving only terms in the final equations that are required to predict behaviour during laminar flow within the targeted semi-dilute regime. In a companion paper (referred to as paper II, Noori et. al., 2024) we chose sets of microstructure parameters that predict experimentally measured bulk suspension behaviour, creating a link between particle properties (i.e. roughness), suspension microstructure and shear-induced particle migration in arbitrary flow fields.