Shock wave propagation in vibrofluidized granular materials

Shock wave formation and propagation in vertically vibrated quasi-two-dimensional granular materials are studied by digital high speed photography. Steep density and temperature wave fronts form at the bottom of the granular layer when the layer collides with vibrating plate. Then the fronts propagate upwards through the layer. The temperature front is always in the transition region between the upward and downward granular flows. The effects of driving parameters and particle number on the shock are also explored.

Figure 1

(a) Time-space profiles of packing fraction ${\nu }_r$ from $j=1$ to 20. Solid circles represent peak-value points of density at each strip, dashed line is the polynomial fitting of peak values with time. $fk\Delta t=0$ corresponds to the time when the plate reaches its minimum height; (b) background packing fraction ${\nu }_b$ versus height $j$. Parameters are $f=15\mathrm{Hz}$, $\Gamma =5$, and $N=150$. The height of peak-value point at each strip is set at 90% of total height of that strip.

Figure 2

Two dimensional background velocity field (left column), Mach number $\mathit{Ma}$ (asterisk), and scaled temperature (open circle) (right column) as a function of height (right) at four different times. At time $fk\Delta t=0.5$ the plate is at its highest position. Parameters here are the same as those in Fig. 1.

Figure 3

The peak value of Mach number and the normalized shock width $W_s∕\zeta$ as a function of time. The mean free path of particles is $\zeta (j,k)=\sqrt{\pi ∕8}∕\left[n(j,k)\pi d^2\right]$, where $n(j,k)=\nu (j,k)∕V_p$ is the number density. The shock begins to form at $fk\Delta t=0.1$ and disappear at $fk\Delta t=0.5$. Shock width $W_s∕\zeta$ reaches its maximum value 16.8 at $fk\Delta t=0.34$. Parameters here are the same as those in Fig. 1.

Figure 4

(Color online) Shock location as a function of time with different $\Gamma$ (a) and $N$ (b). Shock speed as functions of $\Gamma$ and $N$ are shown separately in the insets of (a) and (b). $f$ and $N$ are fixed at $20\mathrm{Hz}$ and 150 in (a) and $f$ and $\Gamma$ are fixed at $20\mathrm{Hz}$ and 8 in (b). Solid line represents the vibration of the plate.