Unifying homogeneous and inhomogeneous rheology of dense suspensions

TL;DR

This paper introduces a unified rheological framework for dense suspensions that accounts for both homogeneous and inhomogeneous flows by incorporating a new dimensionless number, suspension temperature, to describe local particle fluctuations.

Contribution

The study develops a new constitutive law, $mbda(J,Theta_s)$, unifying the description of dense suspension rheology across flow regimes, including non-local effects.

Findings

The suspension temperature $Theta_s$ effectively captures flow heterogeneity.

The proposed model predicts spatial variations of key rheological parameters.

Scaling laws are identified for different particle shapes and interactions.

Abstract

The rheology of dense suspensions lacks a universal description due to the involvement of a wide variety of parameters, ranging from the physical properties of solid particles to the nature of the external deformation or applied stress. While the former controls microscopic interactions, spatial variations in the latter induce heterogeneity in the flow, making it difficult to find suitable constitutive laws to describe the rheology in a unified way. For homogeneous driving with a spatially uniform strain rate, the rheology of non-Brownian dense suspensions is well described by the conventional $\mu(J)$ rheology. However, this rheology fails in the inhomogeneous case due to non-local effects, where the flow in one region is influenced by the flow in another. Here, motivated by observations from simulation data, we introduce a new dimensionless number, the suspension temperature $\Theta_s$, which contains information on local particle velocity fluctuations. We find that $\mu(J,\Theta_s)$ provides a unified description for both homogeneous and inhomogeneous flows. By employing scaling theory, we identify a set of constitutive laws for dense suspensions of frictional spherical particles and frictionless rod-shaped particles. Combining these scaling relations with the momentum balance equation for our model system, we predict the spatial variation of the relevant dimensionless numbers, the volume fraction $\phi$, the viscous number $J$, the macroscopic friction coefficient $\mu$, and $\Theta_s$ solely from the nature of the imposed external driving.