Shear viscosity of a model for confined granular media

The shear viscosity in the dilute regime of a model for confined granular matter is studied by simulations and kinetic theory. The model consists on projecting into two dimensions the motion of vibrofluidized granular matter in shallow boxes by modifying the collision rule: besides the restitution coefficient that accounts for the energy dissipation, there is a separation velocity that is added in each collision in the normal direction. The two mechanisms balance on average, producing stationary homogeneous states. Molecular dynamics simulations show that in the steady state the distribution function departs from a Maxwellian, with cumulants that remain small in the whole range of inelasticities. The shear viscosity normalized with stationary temperature presents a clear dependence with the inelasticity, taking smaller values compared to the elastic case. A Boltzmann-like equation is built and analyzed using linear response theory. It is found that the predictions show an excellent agreement with the simulations when the correct stationary distribution is used but a Maxwellian approximation fails in predicting the inelasticity dependence of the viscosity. These results confirm that transport coefficients depend strongly on the mechanisms that drive them to stationary states.

Figure 1

Dimensionless correction of the stationary temperature $\widehat{T}$ and the normalized cumulants $a_{2,3,4}$ as a function of the inelasticity coefficient $q$. Results from simulations (solid circles) obtained as an extrapolation to vanishing density for $\widehat{T}$ as a polynomial fit in density and at $n{\sigma }^2=0.005$ for the cumulants. The error bars for $\widehat{T}$ result from the fitting procedure and for the cumulants the error bars are estimated from the deviation between the simulations at the four studied densities. Theoretical predictions with $K=0$ (thin solid line) $K=1$ (dotted), $K=2$ (dashed), and $K=3$ (thick solid line) terms in the polynomial expansion of the distribution function (25).

Figure 2

Dimensionless shear viscosity $\widehat{\eta }=\eta /{\eta }_0$, where ${\eta }_0=1/\left(2\sigma \right)\sqrt{mT/\pi }$ [28], as a function of the inelasticity $q$. Average of viscosity over the four studied densities (solid circles, with error bars estimated from the deviation between the simulations at the four studied densities) and theoretical predictions for $M=2$ with $K=0$ (dotted), $K=1$ (dashed), $K=2$ (solid thick line). The $K=2$ and $K=3$ cases are indistinguishable.

Figure 3

Normalized decay rate ${\lambda }_{\perp }/\left(k^2{\nu }_0\right)$ against $k/\sigma n$. Simulation results for densities $n{\sigma }^2=0.005$ (solid circles), $n{\sigma }^2=0.010$ (solid squares), $n{\sigma }^2=0.015$ (empty triangles), and $n{\sigma }^2=0.020$ (empty circles) in the case of inelasticity $q=0.2$. Other inelasticities give similar results. The solid line corresponds to the fit ${\lambda }_{\perp }/k^2=\nu [1+c{(k/\sigma n)}^2]$ with $\nu =(0.2660\pm 0.0009){\eta }_0/mn$ and $c=-0.0895\pm 0.0092$.

Figure 4

Schematic representation of the direct (left) and inverse (right) collisions. The postcollisional velocities are related to the precollisional ones via the relations ${\mathbf{c}}_1^{\prime }={\mathbf{h}}_1({\mathbf{c}}_1,{\mathbf{c}}_2,\widehat{\sigma }),{\mathbf{c}}_2^{\prime }={\mathbf{h}}_2({\mathbf{c}}_1,{\mathbf{c}}_2,\widehat{\sigma }),{\mathbf{c}}_1={\mathbf{h}}_1({\mathbf{c}}_1^{*},{\mathbf{c}}_2^{*},-\widehat{\sigma })$, and ${\mathbf{c}}_2={\mathbf{h}}_2({\mathbf{c}}_1^{*},{\mathbf{c}}_2^{*},-\widehat{\sigma })$. Note that in the inverse collision, the sign of the unit vector is reversed to guarantee that ${\mathbf{c}}_{12}·\widehat{\sigma }>0$.