Recent Computational Advances in Dense Suspension Mechanics

Abstract

Dense suspensions of particles dispersed in liquids are central to industrial and geophysical processes and serve as model systems for out-of-equilibrium soft matter. At high particle concentrations, they exhibit stress-dependent rheology, including discontinuous shear thickening and shear jamming, arising from frictional contacts. Nonlinear physics arises from the interplay among direct contacts, interfacial chemistry, and fluid-mediated hydrodynamics. The relative importance of these mechanisms depends on particle properties and flow conditions, making predictive modeling inherently multi-scale and, therefore, computationally challenging. Recent advances in computational methods have transformed our ability to simulate the physics of dense suspensions across scales. In this Perspective, we discuss state-of-the-art simulation frameworks that integrate the mechanics of dry granular materials, mediated by contact friction, with suspension hydrodynamics to provide predictive models of dense suspension rheology. We highlight recent computational developments for simulating dense suspensions at varying levels of fidelity, ranging from particle-resolved to continuum models, as well as models that investigate their mesoscale organization during flow. Together, these approaches reveal a hierarchical structure in which microscale constraints give rise to mesoscale frictional networks that ultimately govern macroscopic flow. By synthesizing developments across computational mechanics and soft matter physics, this Perspective highlights emerging directions toward a predictive, multi-scale modeling framework of dense suspensions in realistic geometries and complex flow environments.

Figures (6)

• Figure 1: Multi-scale nature of suspension flow: The channel flow through a contraction offering a paradigmatic exampleTop Left: Role of constitutive equation in macro-scale fluid mechanics governed by constitutive equation. The constitutive equation is governed by colloidal dynamics, which is in turn governed by forces and constraints at play: attraction-repulsion interactions, near-field hydrodynamics, Brownian motion, rolling and sliding friction constraints, and force chain networks.
• Figure 2: Definitions of flow regimesTop: Regimes defined based on particle size $a$: At the smallest length scale, Brownian motion and hydrodynamics dominate, while gravity and friction govern the physics in the granular regime; both friction and hydrodynamics are crucial for the non-colloidal suspensions, and particle inertia can be ignored. Bottom Flow regimes based on scaled particle concentration in rheology of colloidal and non-colloidal suspensions: From dilute (yellow shading) to non-dilute (blue) and dense (red) limits based on the relative importance of interactions; full long-ranged hydrodynamic interactions are crucial in dilute limit; near-field lubrication interaction are important in semi-dilute limit; frictional contact interactions dominate the physics in dense limit. Bottom panel is adapted from Reference guazzelli_2018.
• Figure 3: Connecting Jamming and Particle Features. Left: Different types of constraints on relative particle motion: Hard sphere, frictionless $\{\mu_s,\mu_r\} = \{0,0\}$ having isostatic condition $Z_{\mathrm{iso}}^{\{0,0\}}=2d$ in $d-$ dimension leading to $\phi_J^{\{0,0\}} \approx 0.65$ in 3-$d$; infinite sliding $\{\mu_s,\mu_r\} = \{\infty,0\}$ having isostatic condition $Z_{\mathrm{iso}}^{\{\infty,0\}}=d+1$ in $d-$ dimension leading to $\phi_J^{\{0,0\}} \approx 0.57$ in 3-$d$; infinite sliding and rolling $\{\mu_s,\mu_r\} = \{\infty,\infty\}$ having isostatic condition $Z_{\mathrm{iso}}^{\{\infty,\infty\}}=d(d+1)/(2d-1)$ in $d-$ dimension leading to $\phi_J^{\{0,0\}} \approx 0.365$ in 3-$d$. Middle: Jamming phase diagram: Jamming volume fraction $\phi_J^{\{\mu_s,\mu_r\}}$ plotted as a function of $\mu_r$ for several values of $\mu_s$. Right: Control knob of jamming by tuning particle properties: (top) particle shape, (middle) roughness, and (bottom) interfacial chemistry. The shading on each panel corresponds to the range of jamming volume fraction (middle panel). The middle panel is adapted from Ref. Singh_2022.
• Figure 4: Connecting interparticle forces with rheology.Center: Representative shear-thickening response of a dense suspension, showing relative viscosity $\eta_r$ as a function of dimensionless shear stress $\sigma/\sigma_0$. The background shading indicates the dominant force scale at a given stress: hydrodynamic and conservative interactions (low--intermediate stress) and frictional contact forces (high stress). Different symbols denote simulation models incorporating hydrodynamics and friction with varying conservative interactions. Circles (black solid line) correspond to the critical load model (CLM). Squares represent electrostatic repulsion, with darker shades indicating increasing Debye length $\kappa^{-1}$. Triangles denote simulations including both repulsive and attractive forces, with darker shades corresponding to a larger Hamaker constant $A$.
• Figure 5: From microscopic constraints to mesoscale frictional contact networkTop: Sliding constraints only: Considering sliding constraints only leads to a highly branched frictional contact network (FCN), with local frictional coordination number $Z_\mu^{\mathrm{local}} \ge 3$ (green colored) dominating, with a few particles with $Z_\mu^{\mathrm{local}} <2$ and minimal (yet finite) rattlers ($Z_\mu^{\mathrm{local}} =0$). Strong force chains in the primary (compressive) axis need orthogonal support to maintain mechanical stability. Bottom: Sliding and rolling constraints: A suspension in which particles interact with both sliding and rolling constraints, similar rheology (viscosity $\eta_r$) has a very distinct FCN. The FCN is primarily composed of particles with ($Z_\mu^{\mathrm{local}} \le 2$) and a very few particles with ($Z_\mu^{\mathrm{local}} \ge 3$), and is composed of many rattlers. The mechanical stability for particles interacting with both sliding and rolling constraints does not require orthogonal support. Figure is adapted from Ref. Sharma_2025.