Integration through transients for inelastic hard sphere fluids

We compute the rheological properties of inelastic hard spheres in steady shear flow for general shear rates and densities. Starting from the microscopic dynamics we generalize the Integration Through Transients formalism to a fluid of dissipative, randomly driven granular particles. The stress relaxation function is computed approximately within a mode-coupling theory—based on the physical picture that relaxation of shear is dominated by slow structural relaxation, as the glass transition is approached. The transient build-up of stress in steady shear is thus traced back to transient density correlations which are computed self-consistently within mode-coupling theory. The glass transition is signaled by the appearance of a yield stress and a divergence of the Newtonian viscosity, characterizing linear response. For shear rates comparable to the structural relaxation time, the stress becomes independent of shear rate and we observe shear thinning, while for the largest shear rates Bagnold scaling, i.e., a quadratic increase of shear stress with shear rate, is recovered. The rheological properties are qualitatively similar for all values of $\varepsilon$, the coefficient of restitution; however, the magnitude of the stress as well as the range of shear thinning and thickening show significant dependence on the inelasticity.

Figure 1

Protocol H (Shear heating increases the granular temperature): Dynamic state diagrams in the plane spanned by shear rate, $\dot{\gamma }$, and packing fraction, $\phi$, for several values of the coefficient of restitution, $\varepsilon$, as indicated. The effective exponent $R$ quantifying the shear rate dependence of the viscosity, $\eta \left(\dot{\gamma }\right)\sim {\dot{\gamma }}^R$, is color coded. Shear thickening (thinning) corresponds to $R>0\left(R<0\right)$ while $R=0$ indicates Newtonian behavior. The shear rate is normalized by the collision frequency ${\omega }_c(\phi ,T_0)$ of the unsheared system (at granular temperature $T_0)$ with the prescribed volume fraction. The data for all figures are available as Supplemental Material [24].

Figure 2

Protocol H (Shear heating increases the granular temperature). For several values of the coefficient of restitution $\varepsilon$ (columns, as indicated) and several packing fractions (color coded) from $\phi =0.48$ (yellow, bottom) to $\phi =0.60$ (black, top): (First row) Flow curves, shear stress $\sigma$ as a function of shear rate $\dot{\gamma }$, normalized by the density $n$ and the granular temperature at zero shear, $T_0$. (Second row) Viscosity $\eta$ relative to the Boltzmann viscosity ${\eta }_0\left(T_0\right)$, Eq. (66) as a function of shear rate $\dot{\gamma }$. (Third row) Stationary granular temperature $T$, relative to the initial granular temperature, $T_0$, at zero shear as a function of shear rate $\dot{\gamma }$. The dotted lines indicate $T/T_0\equiv 2$. The shear rate is normalized by the collision frequency ${\omega }_c(\phi ,T_0)$ of the unsheared system (at granular temperature $T_0)$ with the prescribed volume fraction.

Figure 3

Protocol T $\left(T\equiv T_0\right)$ or Protocol H $\left(T>T_0\right)$: Dynamic state diagrams in the plane spanned by Péclet number, $Pe$, and packing fraction, $\phi$, for several values of the coefficient of restitution, $\varepsilon$, as indicated. The effective exponent $R$ quantifying the shear rate dependence of the viscosity, $\eta (Pe)\sim {Pe}^R$, is color coded on the same scale as in Fig. 1. The inaccessible regime $Pe>{Pe}_{\infty }$ is left blank. The jagged boundary is due to discretization.

Figure 4

Protocol T $\left(T\equiv T_0\right)$ or Protocol H $\left(T>T_0\right)$: Flow curves, shear stress $\sigma$ as a function of Péclet number $Pe$ (first row), and viscosity $\eta$ relative to the Boltzmann viscosity $\eta \equiv {\eta }_0\left(T\right)$, Eq. (66), (second row) for several values of the coefficient of restitution $\varepsilon$ (columns, as indicated) and several packing fractions (color coded) from $\phi =0.48$ (yellow, bottom) to $\phi =0.60$ (black, top). The filled circles mark the maximum Péclet number ${Pe}_{\infty }$, i.e., the end of the flow curves for the respective densities.

Figure 5

Protocol T $\left(T\equiv T_0\right)$ or Protocol H $\left(T>T_0\right)$: Flow curves, $\sigma (Pe)$, for several values of the coefficient of restitution $\varepsilon$ (as indicated) and several packing fractions (color coded) from $\phi =0.54$ (yellow, bottom) to $\phi =0.60$ (black, top) close to and above the glass transition density. Note the linear scale for the shear stress.

Figure 6

Viscosity $\eta$, normalized by the Boltzmann viscosity ${\eta }_0$, Eq. (66), at small shear rate, $\dot{\gamma }\to 0$. (a) As a function of the coefficient of restitution $\varepsilon$ for a number of packing fractions from $\phi =0.47$ (yellow, bottom) to $\phi =0.51$ (black, top). (b) As a function of packing fraction, $\phi$, for a number of values of the coefficient of restitution, $\varepsilon$, from $\varepsilon =0.1$ (cyan, bottom) to $\varepsilon =0.9$ (magenta, top). Solid lines denote GITT predictions while dashed lines indicate the Enskog low-density expansion by Garzó and Montanero [13]. The arrows indicate the glass transition density, ${\phi }_c\left(\varepsilon \right)$.

Figure 7

(a) Bagnold coefficient $B$ as a function of packing fraction $\phi$. Solid lines are GITT predictions for a number of values of the coefficient of restitution, $\varepsilon$, from $\varepsilon =0.1$ (cyan, bottom) to $\varepsilon =0.9$ (magenta, top). The dashed lines denote Enskog predictions from Savage and Jeffrey [82] (blue), Kumaran [81] (orange), and Mitarai and Nakanishi [80] (green). The last two are for $\varepsilon =0.9$. The arrows indicate the glass transition density, ${\phi }_c\left(\varepsilon \right)$. The dot-dashed lines are the predictions by Jenkins et al. [84, 85] $B_{J1}$ (red) and $B_{J2}$ (violet) for $\varepsilon =0.6$ (cf. Appendix pp6). (b) Yield stress, ${\sigma }_y$, as a function of volume fraction, $\phi$, for a number of values of the coefficient of restitution, $\varepsilon$, from $\varepsilon =0.9$ (magenta, left) to $\varepsilon =0.1$ (cyan, right). Points are calculated and lines are a guide to the eye. The dotted horizontal line indicates the critical yield stress ${\sigma }_y^c$ at the granular glass transition.