Constraint-Based Approach to Granular Dispersion Rheology

We present a phenomenological model for granular suspension rheology in which particle interactions enter as constraints to relative particle motion. By considering constraints that are formed and released by stress respectively, we derive a range of experimental flow curves in a single treatment and predict singularities in viscosity and yield stress consistent with literature data. Fundamentally, we offer a generic description of suspension flow that is independent of bespoke microphysics.

Figure 1

Two-constraint model: example flow curves. In all panels, ${\varphi }_{\overline{A}B}={\varphi }_{A\overline{B}}=0.86{\varphi }_{AB}$ and ${\varphi }_{\overline{A}\overline{B}}=0.47{\varphi }_{AB}$. (a),(c),(e) $\eta /{\eta }_0$ versus $\sigma /{\sigma }_B$. (b),(d),(f) $f_{\overline{A}}$ (solid lines) and $f_{\overline{B}}$ (dashed lines) versus $\sigma /{\sigma }_B$. Parameters for class 1 (a),(b): ${\sigma }_A/{\sigma }_B={10}^{-6}$, $\alpha =\beta =1$ and $\varphi =0.39{\varphi }_{AB}$; class 3a (c),(d): ${\sigma }_A/{\sigma }_B={10}^2$, $\alpha =\beta =1$ and $\varphi =0.81{\varphi }_{AB}$; class 3b (e),(f): $\alpha =1$, $\beta =0.7$, $\varphi =0.69{\varphi }_{AB}$ at different ${\sigma }_A/{\sigma }_B=0.40$ (purple), 0.45 (green), 0.55 (cyan), 0.65 (orange), and 0.80 (yellow), from top to bottom in (e) and left to right in (f).

Figure 2

Singular behavior for class 1. (a) $\sigma -\varphi$ phase diagram showing jammed (${\varphi }_J\le \varphi$, red) and flowing (${\varphi }_J>\varphi$, white) states. Solid curve, jamming boundary ${\sigma }_{\mathrm{jam}}(\varphi )$. Vertical dashed lines denote different jamming points, as labeled. Symbols, yield stresses from Ref. [5], estimated as the lowest $\sigma$ accessed at each $\varphi$. (b) Symbols, flow curves from Ref. [5] at $\varphi =0.13$, 0.26, 0.35, 0.40, 0.45, 0.47, and 0.50, from bottom to top; ${\eta }_0=0.216\mathrm{Pa}\mathrm{s}$. Lines, model predictions for ${\sigma }_A\to 0$, ${\sigma }_B=1.2\mathrm{Pa}$, $\alpha =1.0$, $\beta =0.5$, and ${\varphi }_{AB}=0.64$, ${\varphi }_{\overline{A}B}={\varphi }_{A\overline{B}}=0.545$, and ${\varphi }_{\overline{A}\overline{B}}=0.20$, at the same $\varphi$.

Figure 3

Singular behavior for class 3a. (a) $\sigma -\varphi$ phase diagram showing jammed (red), stable flowing (white), and unstable flowing (gray) states. Solid line, ${\sigma }_{\mathrm{jam}}(\varphi )$ ($\mathrm{green}=\text{yield stress}$, $\mathrm{blue}=\mathrm{re}\text{−}\mathrm{jamming}\mathrm{stress}$). Dashed curve, boundary of unstable states ${\sigma }_{\mathrm{uns}}(\varphi )$. Symbols, data from Ref. [17]: ($\square$) yield stress (estimated as $\sigma$ at the lowest $\dot{\gamma }$ accessed for each $\varphi$) and ($\circ$) the onset of banded flow. (b) Symbols, flow curves from Ref. [17] at $\varphi =0.40$, 0.42, 0.43, and 0.439, from bottom to top; ${\eta }_0=1\mathrm{mPa}\mathrm{s}$ [33]. Gray points correspond to shear-banded states. Lines, model predictions for ${\sigma }_A=10\mathrm{Pa}$, ${\sigma }_B=0.085\mathrm{Pa}$, $\alpha =0.36$, $\beta =0.38$, and ${\varphi }_{AB}=0.52$, ${\varphi }_{\overline{A}B}=0.38$, ${\varphi }_{A\overline{B}}=0.335$, and ${\varphi }_{\overline{A}\overline{B}}=0.20$. Volume fractions (from bottom to top), $\varphi =0.25$, 0.30, 0.33, 0.35, 0.38, 0.40, 0.42, 0.43, 0.439. Gray parts of the curve are unstable, $d\log \eta /d\log \sigma >1$ states.