Local shear stress and its correlation with local volume fraction in concentrated non-Brownian suspensions: Lattice Boltzmann simulation

The local shear stress of non-Brownian suspensions was investigated using the lattice Boltzmann method coupled with the smoothed profile method. Previous studies have only focused on the bulk rheology of complex fluids because the local rheology of complex fluids was not accessible due to technical limitations. In this study, the local shear stress of two-dimensional solid particle suspensions in Couette flow was investigated with the method of planes to correlate non-Newtonian fluid behavior with the structural evolution of concentrated particle suspensions. Shear thickening was successfully captured for highly concentrated suspensions at high particle Reynolds number, and both the local rheology and local structure of the suspensions were analyzed. It was also found that the linear correlation between the local particle stress and local particle volume fraction was dramatically reduced during shear thickening. These results clearly show how the change in local structure of suspensions influences the local and bulk rheology of the suspensions.

Figure 1

(a) Distribution of shear stress around a single particle and (b) stress distribution as a function of channel height, $H$.

Figure 2

(a) Streamlines around a single particle in Couette flow for $H/R=5$. (b) Normalized angular velocity for different ratio of channel height to particle radius, H/R.

Figure 3

(a) Lubrication force between two identical spheres (3D) for different particle radius, $R$. The dashed line was obtained from the lubrication theory, Eq. (27). (b) Lubrication force between two identical cylinders (2D) for different particle radius, $R$.

Figure 4

$x$ component of velocity and shear stress (of the solvent) at time step 300 000 $(\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=0.01)$. (a)–(c) denote particle configurations with $x$ component of velocity in color for $\varphi =0.1$, $\varphi =0.3$, and $\varphi =0.5$. (d)–(f) show the distribution of shear stress (of the solvent) for $\varphi =0.1$, $\varphi =0.3$, and $\varphi =0.5$.

Figure 5

Relative shear viscosity at different volume fraction of solid particles ($\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=0.01$). (a) WSS and (b) MOP.

Figure 6

Relative shear viscosity in terms of volume fraction of solid particles $(\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=0.01)$. Error bar was obtained by averaging the data from 200 000 to 300 000 time steps (for five simulation sets). Both algorithms (WSS and MOP) predict the Krieger-Dougherty equation very well.

Figure 7

Relative viscosity measured by wall shear stress (WSS) and by the method of planes (MOP): (a) $\varphi =0.4$, $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=0.01$; (b) $\varphi =0.4$, $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=1.0$; (c) $\varphi =0.5$, $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=0.01$; (d) $\varphi =0.5$, $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=1.0$; (e) $\varphi =0.6$, $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=0.01$; and (f) $\varphi =0.6$, $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=1.0$.

Figure 8

Shear viscosity as a function of particle Reynolds number. Increase of relative shear viscosity was observed at high particle Reynolds number $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}$. The results from WSS and MOP are matched well with each other (error bar was obtained by time average from 200 000 to 300 000 time steps for five simulation sets which have different initial particle configurations).

Figure 9

Time averaged pair-distribution function for $\varphi =0.6$. (a) $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=0.01$, (b) $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=0.01$, and (c) $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=1.0$. Abscissa denotes flow direction and ordinate denotes gradient direction. Darker region represents higher probability to find particles.

Figure 10

Particle configuration at time step 300 000. (a) $\varphi =0.5$, $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=0.01$; (b) $\varphi =0.5$, $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=0.1$; (c) $\varphi =0.5$, $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=1.0$; (d) $\varphi =0.6$, $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=0.01$; (e) $\varphi =0.6$, $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=0.1$; and (f) $\varphi =0.6$, $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=1.0$. The particles in the same cluster have the same color. Nonclustering particles are black, and the particles in the larger cluster are close to white.

Figure 11

Time averaged probability of the number of solid particles in clusters. (a) $\varphi =0.4$ and (b) $\varphi =0.6$. Larger clusters were formed with an increase in particle Reynolds number at higher volume fraction.

Figure 12

Time averaged angular distribution of clusters. (a) $\varphi =0.4$ and (b) $\varphi =0.6$. Higher angular distribution was observed over 90°, which means that the clusters are aligned to the compressive axis.

Figure 13

Time averaged (from 200 000 to 300 000) local shear stress measured by the method of planes (MOP) to the gradient direction for $\varphi =0.6$: (a) $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=0.01$ and (b) $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=1.0$. Time averaged particle volume (area) fraction: (c) $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=0.01$ and (d) $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=1.0$.

Figure 14

Local particle stress and local particle volume fractions at $t=300,000$. (a) $\varphi =0.4$, $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=0.01$; (b) $\varphi =0.4$, $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=1.0$; (c) $\varphi =0.5$, $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=0.01$; (d) $\varphi =0.5$, $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=1.0$; (e) $\varphi =0.6$, $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=0.01$; and (f) $\varphi =0.6$, $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}=1.0$. Dashed line (medium dash, light gray) above the local stress line denotes time averaged local stress (total) which was normalized by the shear viscosity of pure solvent multiplied by wall shear rate.

Figure 15

Pearson's correlation coefficient between local particle stress and local particle volume fractions as a function of time. (a) $\varphi =0.4$, (b) $\varphi =0.5$ and (c) $\varphi =0.6$.

Figure 16

Time averaged Pearson's correlation coefficient between local particle stress and local particle volume fraction. The correlation decreases with the increase in local particle volume fraction and particle Reynolds number, $\mathrm{R}{\mathrm{e}}_{\mathrm{p}}$.