Simulation of granular jets: Is granular flow really a perfect fluid?

We perform three-dimensional simulations of the impact of a granular jet for both frictional and frictionless grains. Small shear stress observed in the experiment [X. Cheng et al., Phys. Rev. Lett. 99, 188001 (2007)] is reproduced through our simulation. However, the fluid state after the impact is far from a perfect fluid, and thus the similarity between granular jets and quark gluon plasma is superficial because the observed viscosity is finite and its value is consistent with the prediction of the kinetic theory.

Figure 1

Snapshot of a three-dimensional simulation. Blue particles are grains and red ones are wall particles. Grains consisting of a regular lattice with missing particles collide on a bumpy wall, where grains are scattered randomly, and the jet is ejected along with the wall.

Figure 2

The dependence of ${\psi }_0$ (rad) on $X$ for the frictional case with ${\tilde{\phi }}_0=0.90$, $e=0.75$, and $D_{\mathrm{jet}}/d=4.5$. ${\psi }_0$ does not depend on the restitution coefficient. The dotted lines are asymptotic ones, ${\psi }_0=C_0$ for $X\gg 1$ and ${\psi }_0=C_{1}X+C_2$ for $X\ll 1$, and the solid line is the interpolation function. The asymptotic behavior which is observed in experiments [10] is obtained.

Figure 3

The dependence of $v_2$ on $\varepsilon$ for the frictional case with ${\tilde{\phi }}_0=0.90$ and $e=0.75$, where the initial cross section of the jet is fixed to $16d^2$. $v_2$ does not depend linearly on $\varepsilon$. It should be noted that the observed $v_2$ is much smaller than the value reported in Ref. [10].

Figure 4

A schematic picture of the calculation region. The cylinder with radius $R_{\mathrm{tar}}$ and height $5d$ is divided into cylindrical mesh, where physical quantities are estimated.

Figure 5

The profile of the stress tensor ${\sigma }_{\alpha \beta }(m{u}_0^2/d^3)$ as functions of distance from the jet axis $r$ with $R_{\mathrm{tar}}$ for frictional grains with $\widetilde{{\phi }_0}=0.90$. The off-diagonal components of the stress tensor ${\sigma }_{rz}$ and ${\sigma }_{zr}$ are much smaller than the diagonal components as $|{\sigma }_{rz}/{\sigma }_{zz}|\simeq 1.7\times {10}^{-2}$ at $r/R_{\mathrm{tar}}=0.1$.

Figure 6

The comparison between the theoretical pressure in Eq. (2) and the observed pressure for several $\widetilde{{\phi }_0}$, where the vertical axis is $P$ divided by the density $n$ and the granular temperature $T_g$ for the frictionless case. The inset contains a comparison between those for the frictional case. Black solid lines in each figure denote Eq. (2) for the frictionless and the frictional case, respectively.

Figure 7

The diagonal components of the stress tensor for the frictional case with several $\widetilde{{\phi }_0}$ divided by $n{T}_{\alpha }$, where $T_{\alpha }$ is the temperature for the $\alpha$ direction, $T_{\alpha }\equiv {\sum }_{i}m{u}_{i\alpha }^2/N$. Red empty points, blue filled points, purple half-filled points, and the solid black line denote $P_z/n{T}_z,P_r/n{T}_r,P_{\theta }/n{T}_{\theta }$, and Eq. (2), respectively.

Figure 8

Scaled VDF $\tilde{f}\left(c_{\alpha }\right)$ for $\widetilde{{\phi }_0}=0.90$. Empty points and filled points denote the frictionless case and the frictional case, respectively. The black solid line is $\tilde{f}\left(c_{\alpha }\right)=\exp (-c_{\alpha }^2/2)/\sqrt{2\pi }$ and the dashed line is $\tilde{f}\left(c_{\alpha }\right)=\exp (-\sqrt{2}|c_{\alpha }\left|\right)/\sqrt{2}$.

Figure 9

Nondimensional shear viscosity ${\eta }^{*}$, which is defined in Eq. (5), for several ${\phi }_0/{\phi }_{\mathrm{fcc}}$ in $0.2R_{\mathrm{tar}}<r<R_{\mathrm{tar}}$. Black solid lines denote theoretical curves.