Reviving the Suspension Balance Model

TL;DR

The paper addresses the contested foundation of the Suspension Balance Model by clarifying how hydrodynamic forces should be partitioned between the particle and fluid phases. It introduces a perturbative configuration-based partition that includes both direct and indirect hydrodynamic contributions, removing divergences at contact and yielding a particle-phase stress that matches the equilibrium Brownian (thermodynamic) stress. Through Stokesian Dynamics simulations, the authors show that the properly partitioned particle-phase stress $n\langle{\boldsymbol{\sigma}}^{\mathrm{p}}\rangle$ agrees with the suspension stress minus the isolated single-particle Einstein correction $\boldsymbol{\Sigma}^{(\mathrm{p})}_0$, i.e. $\boldsymbol{\Sigma}^{(\mathrm{p})}-\boldsymbol{\Sigma}^{(\mathrm{p})}_0$, across concentrations. This agreement validates the SBM, demonstrates thermodynamic consistency, and confirms that the measured suspension stress can be used in the particle-phase momentum balance for accurate predictions of shear-induced migration and related phenomena. The approach strengthens the SBM’s physical foundation and its practical use in predicting complex suspension flows. All mathematical notation is presented with explicit $...$ delimiters to ensure precision in formal contexts.

Abstract

The Suspension Balance Model (SBM) [J. Fluid Mech. \textbf{275}, 157 (1994)] for two-phase flows uses the momentum balance of the particle phase as a closure for the particle flux, showing that particle migration is driven by the divergence of the particle-phase stress. The underlying basis of this model was challenged by Nott~et~al.\ [Phys. Fluids \textbf{23}, 043304 (2011)] where the authors argued that the hydrodynamic contributions to the suspension stress should not appear in the particle-phase momentum balance, being replaced by a different particle-phase stress. The particle-phase stress proposed by Nott~et~al., while mathematically correct, involves the partitioning of the (non-pairwise-additive) hydrodynamic forces, and care is needed to understand how the force on a chosen particle is affected by a second particle. We show by a simple two-particle calculation what is the proper partitioning, and show that it is consistent thermodynamically and gives the correct equilibrium osmotic pressure of Brownian colloids. Using Stokesian Dyanmics suspension rheology, we quantitatively demonstrate that the hydrodynamic contribution to the suspension stress is virtually identical to particle-phase stress; the only difference is that the isolated single-particle hydrodynamic stress contribution -- the Einstein viscosity correction -- must be removed from the suspension stress when used to predict particle flux. Our results validate a key assumption of the SBM and therefore revive its physical foundation.

Figures (4)

• Figure 1: Schematic illustration of computing the pairwise hydrodynamic force $\widetilde{\boldsymbol{f}}_{12}^\mathrm{h}$ for a $3$-particle system in a quiescent fluid ($\langle{\boldsymbol{u}}\rangle=0$). On the left, the hydrodynamic force on particle $1$, $\boldsymbol{f}_1^\mathrm{h}$, is calculated with all the particles present. On the right, $\boldsymbol{f}_1^\mathrm{h}{}'(2)$ is computed with particle $2$ removed (dashed line). In both cases, the particle velocities are unchanged.
• Figure 2: The suspension osmotic pressures $\Pi$, and the particle-phase osmotic pressures from the resistance and the perturbative approaches, $\Pi^\mathrm{p}$ and $\widetilde{\Pi}^\mathrm{p}$ respectively, as functions of the volume fraction $\phi$ for hard-sphere Brownian suspensions at $\mathrm{Pe} = 0$. The dashed line shows the Carnahan-Starling equation of state.
• Figure 3: The suspension and particle-phase viscosities: (a)--(c): the Brownian viscosities $\eta_\mathrm{B}$, ${\eta}_\mathrm{B}^\mathrm{p}$, and $\widetilde{\eta}_\mathrm{B}^\mathrm{p}$; (d)--(f): the flow viscosities $\eta_\mathrm{E}-\tfrac{5}{2}\phi\eta_0$, ${\eta}_\mathrm{E}^\mathrm{p}$, and $\widetilde{\eta}_\mathrm{E}^\mathrm{p}$ as functions of the Péclet number $\mathrm{Pe}$ at volume fractions (a,d): $\phi=0.3$, (b,e): $\phi=0.4$, and (c,f): $\phi=0.5$. At $\phi=0.4$ the SD results of sd-brownian-susp_brady_jfm2000 are also shown.
• Figure 4: (The suspension friction coefficient $\mu$ and the particle-phase friction coefficients $\mu^\mathrm{p}$ and $\widetilde{\mu}^\mathrm{p}$, as functions of Péclet number, Pe, at volume fractions (a): $\phi=0.3$, (b): $\phi=0.4$, and (c): $\phi=0.5$. Here, the friction coefficients are defined without the solvent contribution.