Modeling the Microstructure and Stress in Dense Suspensions under Inhomogeneous Flow

Under inhomogeneous flow, dense suspensions exhibit behavior that violates the conventional homogeneous rheology. Specifically, one finds flowing regions with a macroscopic friction coefficient below the yielding criterion, and volume fraction above the jamming criterion. We demonstrate the underlying physics by incorporating shear rate fluctuations into a recently proposed tensor model for the microstructure and stress, and applying the model to an inhomogeneous flow problem. The model predictions agree qualitatively with particle-based simulations.

Figure 1

(a) Snapshot of particle-based simulation of Kolmogorov flow with relative domain size $L/a=278$. The gray level indicates the particle pressure (black and white represent negative and positive, respectively) and the driving force $f_x(y)$ is sketched with the black line. (b),(c) Sketches of the $xy$ component of $\mathit{E}$ and ${\mathit{E}}_c$ respectively, showing raw signals (gray) and filtered signals (black). The latter is zero at $y/L=0.25$, 0.75 for $⟨\mathit{E}⟩$ but not for $⟨{\mathit{E}}_c⟩$. The dashed lines indicate the abscissas. (d) Macroscopic friction coefficient $\mu$ as a function of the nondimensional shear rate $S$ predicted by simulation in homogeneous shear flow (filled circles) and in Kolmogorov flow for $L/a=278$ (open squares), 139 (open downward triangles), and 56 (open rightward triangles) and predicted by constitutive model for $S_{\mathrm{rms}}=0$ (solid line) $6\times {10}^{-3}$ (dashed line), $1.5\times {10}^{-2}$ (dash-dotted line) and $4.2\times {10}^{-2}$ (dotted line). (e) Nondimensional shear rate fluctuations $S_{\mathrm{rms}}$ as a function of $S$ predicted by simulation. The markers are as in (d).

Figure 2

Kolmogorov flow profiles as a function of the normalized distance to the nearest center line $y^{\prime }/L$, predicted by constitutive model with $S_{\mathrm{rms}}=0$ (gray lines), $S_{\mathrm{rms}}={10}^{-2}$ (solid black lines), and $S_{\mathrm{rms}}={10}^{-1}$ (dashed black lines) and by simulations with $L/a=278$ (filled circles) and 56 (open circles). (a) Volume fraction $\varphi$; (b) normalized velocity $u_x/(\widehat{f}{L}^2/{\eta }_s)$; (c) anisotropy of particle contacts $A$; (d) normalized coordination number $Z/Z_J$ in simulation and normalized jamming coordinate $\xi /{\xi }_J$ in constitutive model.