Shear Thickening in Non-Brownian Suspensions: An Excluded Volume Effect

Shear thickening appears as an increase of the viscosity of a dense suspension with the shear rate, sometimes sudden and violent at high volume fraction. Its origin for noncolloidal suspension with non-negligible inertial effects is still debated. Here we consider a simple shear flow and demonstrate that fluid inertia causes a strong microstructure anisotropy that results in the formation of a shadow region with no relative flux of particles. We show that shear thickening at finite inertia can be explained as an increase of the effective volume fraction when considering the dynamically excluded volume due to these shadow regions.

Figure 1

Normalized effective viscosity $\mu /{\mu }_0$ versus $\varphi$ for four particle Reynolds numbers Re. Symbols: (open circles) $\mathrm{Re}=0.1$, (crosses) $\mathrm{Re}=1$, (asterisk) $\mathrm{Re}=5$, and (open squares) $\mathrm{Re}=10$; dash-dotted line, Eilers fit (1) with ${\varphi }_m=0.6$ and $B=1.7$. Inset: $\mu /{\mu }_0$ versus Re. Red solid line $\varphi =0.11$; long-dashed green line $\varphi =0.21$; dashed green line $\varphi =0.26$; dotted magenta line $\varphi =0.315$.

Figure 2

Projection of the normalized angle-dependent pair-distribution function $g(r\simeq 2a,\widehat{\mathbf{r}})$ on the wall-parallel plane (with mean flow from right to left) $x\mathrm{\text{−}}z$ plane for $\mathrm{Re}=0.1$ (a) and $\mathrm{Re}=10$ (b) for $\varphi =0.315$.

Figure 3

Contour plot of particle pair relative flux $\mid \mathbf{q}\mid$, Eq. (2) in the shear plane $x\mathrm{\text{−}}y$ for $\varphi =0.315$, $\mathrm{Re}=0.1$ (a) and $\mathrm{Re}=10$ (b). Local mean flow is from right to left in the horizontal direction. The direction of $\mathbf{q}$ in the plane is shown by arrows. The black contour corresponds to $\mid \mathbf{q}\mid =q_{\mathrm{th}}$.

Figure 4

Effective viscosity versus effective volume fraction ${\varphi }_e$: (red pluses) $\varphi =0.11$; (green crosses) $\varphi =0.21$; (blue asterisk) $\varphi =0.26$; and (magenta open squares) $\varphi =0.315$. Lines: Eilers fit (1): dash-dotted, best fit of present data ${\varphi }_m=0.6$ and $B=1.7$; dotted, fitting parameters in 15, 16, 17 ${\varphi }_m=0.58$ and $B=1.5$. Inset: Relative increment of the volume fraction as a function of Re.

Figure 5

(a) The pair-distribution function averaged over unit sphere, $G(r)\equiv (1/4\pi )\int g(r,\widehat{r})d\Omega$ versus $r/a-2$ for $\varphi =0.315$ and several values of Re: (red pluses) $\mathrm{Re}=0.1$; (green crosses) $\mathrm{Re}=1.0$; (blue asterisk) $\mathrm{Re}=5.0$; and (magenta open squares) $\mathrm{Re}=10.0$. Inset: Second-order structure function of longitudinal velocity differences of the spheres versus $r/a-2$. (b) Probability distribution function of the number of clusters formed by $N$ spheres.

Figure 6

Wall-normal profile of (a) the average local volume fraction and (b) the average particle velocity $V(y)$ for $\varphi =0.315$. The inset shows $K\equiv v_{\mathrm{rms}}^2$ versus the wall distance $y/H$. Symbols are as in Fig. 5.