Traveling shock front in quasi-two-dimensional granular flows

While the density profile of a granular shock front can be obtained by the conventional treatment of supersonic fluids, its temperature profile is very different from that in ordinary shocks. We study the density and temperature profiles of a traveling granular shock generated by piling up metal spheres in a closed bottom quasi-two-dimensional channel. We successfully account for the temperature profile in the granular shock using a simple kinetic theory in terms of energy transfer from the mean flow direction to the transverse direction. Contrary to ordinary fluids and previous granular shock experiments, the granular shock width is found to increase with the inflow rate.

Figure 1

Schematic diagram of the experimental setup. The photo at the right is a typical image.

Figure 2

(Color online) Mean density profile measured during the interval $0.518\text{s}<t<0.528\text{s}$ at inflow rate $\gamma =1367{\text{s}}^{-1}$. The dash line is the running average of the mean profile. The inset shows the trajectories of the particle during this time interval.

Figure 3

(Color online) Mean density profiles in the laboratory frame, from right to left: $t=0.2$, 0.4, 0.52, 0.7, 0.8, and 0.9 s. The inset shows the corresponding positions $y_o$ of the shock front.

Figure 4

(Color online) Density profiles at different inflow rates from left to right: $\gamma =650$, 814, 1054, 1367, 1510, and $1890{\text{s}}^{-1}$. The inset shows the variation of density profile width $\lambda$ with $\gamma$.

Figure 5

(Color online) (a) Density $n$, (b) mean kinetic energy $⟨E_k⟩$, (c) mean flow $\frac{m}{2}{⟨\mathbf{v}⟩}^2$, and (d) thermal kinetic energy profile $\frac{m}{2}⟨\delta {\mathbf{v}}^{\mathbf{2}}⟩$ in the comoving frame. The inset in (a) shows the probability functions $P\left(v_y\right)$ and $P\left(v_x\right)$ at an upstream location.

Figure 6

(Color online) (a) Density profiles $\left(n_1,n_2\right)$ for type-1 and type-2 particles. The upper inset shows the probability function $P\left(v_y\right)$ at different locations: $y=5$, 15, 25, 35, and 45 mm and the lower inset shows the $T_{2}x$ and $T_{2}y$ profiles; profiles of the thermal kinetic energy (b) $T_y$ and (c) $T_x$.