Effect of normal contact forces on the stress in shear rate invariant particle suspensions

We present a tensorial theory for the microstructure and the stress in shear rate invariant particle suspensions that includes hydrodynamic and normal but not tangential hard sphere interaction forces. The theory predicts that hydrodynamic forces produce a negligible first normal stress difference, while contact forces produce a positive first normal stress difference. The theory thereby provides a rationale for seemingly contradicting experimental observations in the literature. In addition, the theory captures the experimentally observed time dependence of the shear stress after shear reversal.

Figure 1

Measured, steady, relative first and second normal stress differences ${\zeta }_1=({\mathrm{\Sigma }}_{11}-{\mathrm{\Sigma }}_{22})/{\mathrm{\Sigma }}_{12}$ (open markers) and ${\zeta }_2=({\mathrm{\Sigma }}_{22}-{\mathrm{\Sigma }}_{33})/{\mathrm{\Sigma }}_{12}$ (solid markers), under shear rate invariant conditions and as functions of the particle volume fraction $\varphi$. $▵,a=20\mu \mathrm{m}$; $▿,a=70\mu \mathrm{m}$ (polystyrene in water, UCON oil and zinc bromide [2]); $\square ,a=35\mu \mathrm{m}$; $◊,a=70\mu \mathrm{m}$ (polystyrene in poly(ethylene glycol-ran-propylene glycol) monobutylether [3]); $⊲,a=5\mu \mathrm{m}$ (poly (methyl methacrylate) (PMMA) in Triton X-100, anhydrous zinc chloride, and water (TZW) [4]); $\oplus ,a=98\mu \mathrm{m}$ (PMMA in TZW [5]); $◯,a=22\mu \mathrm{m}$ (glass in corn syrup and glycerin [6]); and ☆, $a=20\mu \mathrm{m}$ (polystyrene in silicone fluid [7]). The lines are drawn to guide the eye, and the lower line represents the empirical relation [Eq. (16)].

Figure 2

(a) Modeled [Eqs. (9), (10)] microstructure $\mathit{a}$ as a function of the pair association rate $\beta$. (b) Volume fraction $\varphi$ dependence of the planar microstructure ${\mathit{a}}^{2\mathrm{D}}$, measured in Ref. [22] [Eq. (14), markers] and modeled [Eqs. (13), (15), and (16), lines].

Figure 3

Modeled [Eqs. (9), (10), and (12)] relative first normal stress difference ${\zeta }_1=({\mathrm{\Sigma }}_{11}-{\mathrm{\Sigma }}_{22})/{\mathrm{\Sigma }}_{12}$ (dashed lines) and second normal stress difference ${\zeta }_2=({\mathrm{\Sigma }}_{22}-{\mathrm{\Sigma }}_{33})/{\mathrm{\Sigma }}_{12}$ (solid lines), as functions of the pair association rate $\beta$ for systems dominated by (a) hydrodynamic forces $(\alpha ,\chi )=(1,0)$ and (b) contact forces $(\alpha ,\chi )=(0,1)$.

Figure 4

(a) Modeled [Eqs. (9), (10), and (12)] suspension shear stress, scaled with the steady value ${\mathrm{\Sigma }}_{12}/{\mathrm{\Sigma }}_{12,\infty }$, as a function of the strain $\dot{\gamma }t$, after shear reversal, using various $\beta ,\alpha$, and $\chi$. (b) Measured ${\mathrm{\Sigma }}_{12}/{\mathrm{\Sigma }}_{12,\infty }$ as a function of $\dot{\gamma }t$, after shear reversal, using various volume fractions $\varphi$ [12].