Modeling sphere suspension microstructure and stress

We develop a model for the microstructure and the stress, in dense suspensions of non-Brownian, perfectly smooth spheres at vanishing particle Reynolds number. These quantities are defined in terms of the second-order moment $\mathit{a}$ of the distribution function of the orientation unit vector between hydrodynamically interacting particles. We show, from first principles, that the evolution equation of $\mathit{a}$ contains a source term that accounts for the association and the dissociation of interacting particle pairs. This term provides a microscopic explanation for typical non-Newtonian behavior, observed in experiments in the literature, including normal stress differences in steady shear flow, as well as time-dependent stress after abruptly reversed shear flow and during oscillating shear flow.

Figure 1

(a) Measured, steady, relative, shear viscosity: ${\zeta }_0={\mathrm{\Sigma }}_{12}/\eta \dot{\gamma }$, as a function of the particle volume fraction $\varphi$. Data are taken from Ref. [8]. The line is Eq. (13a), using ${\varphi }_m=0.58$. (b) Measured, steady, relative, normal stress differences: ${\zeta }_1=({\mathrm{\Sigma }}_{11}-{\mathrm{\Sigma }}_{22})/{\mathrm{\Sigma }}_{12}$ (open markers) and ${\zeta }_2=({\mathrm{\Sigma }}_{22}-{\mathrm{\Sigma }}_{33})/{\mathrm{\Sigma }}_{12}$ (filled markers), as functions of $\varphi$. $▵, ▴$: $a=20\mu \mathrm{m}$ [8]; $▿, ▾$: $a=70\mu \mathrm{m}$ [8]; $\square , \blacksquare$: $a=35\mu \mathrm{m}$ [7]; $◊, ⧫$: $a=70\mu \mathrm{m}$ [7]. The solid line is Eq. (13b) using ${\varphi }_m=0.58$. The dashed line is ${\zeta }_1=0$.

Figure 2

Modeled [Eq. (12)], steady, relative, shear viscosity ${\zeta }_0/\alpha ={\mathrm{\Sigma }}_{12}/\left(\alpha \eta \dot{\gamma }\right)$ and relative, normal stress differences: ${\zeta }_1=({\mathrm{\Sigma }}_{11}-{\mathrm{\Sigma }}_{22})/{\mathrm{\Sigma }}_{12}$ and ${\zeta }_2=({\mathrm{\Sigma }}_{22}-{\mathrm{\Sigma }}_{33})/{\mathrm{\Sigma }}_{12}$ as functions of the microstructure parameter $\beta$.

Figure 3

Modeled [Eq. (12)] time-dependent components of (a) the microstructure $\mathit{a}$ and (b) the particle stress $\mathbf{\Sigma }$ after shear reversal, using $\beta =6$. (c) Modeled time-dependent particle viscosity: $\eta ={\mathrm{\Sigma }}_{12}/\dot{\gamma }$, scaled with the steady value ${\eta }_{\infty }$, together with experimental data, which were taken at a volume fraction of $\varphi =0.4$, and a maximum packing volume fraction [defined in Eq. (13a)] of ${\varphi }_m=0.54$ [12]. (d) Modeled minimum particle viscosity: ${\eta }_{\min }$ [minimum of curve in (c)], after shear reversal, scaled by the steady value ${\eta }_{\infty }$, using various $\beta \in [3,11]$, corresponding to various ${\zeta }_2$ (see Fig. 2), together with experimental data, with varying volume fractions $\varphi \in [0.3,0.5]$ [12], where ${\zeta }_2$ is estimated using Eq. (13b).

Figure 4

Modeled [Eq. (12)] time-dependent components of the microstructure $\mathit{a}$ (a, b) and the particle stress $\mathbf{\Sigma }$ (c, d) in oscillating shear flow, with a strain amplitude of ${\gamma }_0={\dot{\gamma }}_0/\omega ={10}^2$ (a, c) and ${10}^{-2}$ (b, d), using $\beta =6$.

Figure 5

Dynamic particle viscosity [Eq. (17)] in oscillating shear flow, scaled with the steady value, as a function of the strain amplitude. Comparison between model [Eq. (12)] using $\beta =6$ (solid line), and experimental data from Ref. [16] (squares) and from Ref. [18] (triangles), both using $\varphi =0.4$. The dashed line is drawn to guide the eye.

Figure 6

Shear-induced migration per oscillation cycle $\delta r$, scaled with radial position $r$, particle radius $a$, and strain amplitude ${\gamma }_0$. Comparison between model [Eqs. ((12), (20))] using $\beta =6$ (solid line) and experimental data from Ref. [17], using a volume fraction of $\varphi =0.4$ (markers). The dashed line is drawn to guide the eye.