Head-on collision of compressional shocks in two-dimensional Yukawa systems

The head-on collision of compressional shocks in two-dimensional dusty plasmas is investigated using both molecular dynamical and Langevin simulations. Two compressional shocks are generated from the inward compressional boundaries in simulations. It is found that, during the collision of shocks, there is a generally existing time delay of shocks $\tau$, which diminishes monotonically with the increasing compressional speed of boundaries, corresponding to the time resolution of the studied system. Dispersive shock waves (DSWs) are generated around the shock front for some conditions. It is also found that the period of the DSW decreases monotonically with the inward compressional speed of boundaries, more substantially than the time delay of shocks $\tau$. When the inward compressional speed of boundaries increases further, the DSWs gradually vanish. We speculate that, for these high compressional speeds of boundaries, the period of the DSW might be reduced to a comparable timescale of the time delay of shocks $\tau$, i.e., the time resolution of our studied system, or even shorter, thus the DSW reasonably vanishes.

Figure 1

Snapshot of particle positions of two-dimensional Yukawa solids before the head-on collision of two compressional shocks in our simulations. Two pistons on the left and right boundaries both move inward with a constant speed of $v_p$, generating two compressional shocks with the inward propagating speed of $D$ [40]. At the shock front region, a few ripples of the particle density appear, called the dispersive shock wave (DSW). Here, we focus on the properties of the DSW and the dynamics of the latter head-on collisions of these two compressional shocks. Note that, in our simulations ($N=16384$) reported here, the periodic boundary conditions are applied in the $y$ direction.

Figure 2

The spatiotemporal evolution of the number density $n$ of particles for the head-on collision of two compressional shocks when $M_p=0.441$ for the 2D Yukawa conditions of $\mathrm{\Gamma }=800$ and $\kappa =0.75$. The simulation box is divided into 480 narrow rectangular bins with the width of $a$ in the $x$ direction, and the counted particle number in each bin is just the plotted number density here. Two shock fronts can clearly be observed from this number density evolution in (a) as two distinctive lines with the constant slope, corresponding to the shock front speed $D$, which we marked as the dashed line. These two distinctive lines merge together when $t{w}_{pd}=181$, which means that these two shock fronts collide with each other. By magnifying the collision procedure in (a), as in (b), we can see that, after the head-on collision of these two shock fronts (marked as the dashed line), there is a clear time delay of $\tau {\omega }_{pd}=2.5$ before two new shocks (marked as the dotted line) propagate away. The variation property of this time delay of shocks $\tau$ as a function of the Yukawa conditions and the shock parameters is what we will study here. Note that, in both panels, before the head-on collision of two shocks, we observe a few stripes with the same slope of the shock front, reflecting the structure of the DSW. Also, after the collision, these stripes form a grid, further reflecting the interaction of the DSW.

Figure 3

The time delay of shocks $\tau$ as the function of the Mach number of the inward compressional speed $M_p$ for the 2D Yukawa conditions of $\mathrm{\Gamma }=800$ and $\kappa =0.75$. We obtain the time delay of shocks $\tau$ in different Mach numbers with and without frictional gas drag from the spatiotemporal evolution plots of the number density $n$, when the Mach number of the inward compressional speed $M_p=v_p/C_l$ is varied from 0.147 to 1.324. Here, the time delay of shocks $\tau$ decreases monotonically with the compressional speed of $v_p$ for both conditions with and without frictional gas drag, and it seems to reach a saturation level of $\tau {\omega }_{pd}\approx 1$ at the high Mach numbers from our simulation. We speculate that the observed timescale of $\tau {\omega }_{pd}\approx 1$ is related to the timescale of individual particle motion.

Figure 4

The obtained one-particle distribution function $f_1(\zeta ,v_x)$, where $\zeta$ is the Lagrange coordinate $\zeta =(x-Dt)$, from our frictionless MD simulations of $\mathrm{\Gamma }=800$ and $\kappa =0.75$. To calculate $f_1(\zeta ,v_x)$, we divide the $\zeta -v_x$ plane into bins in both the $\zeta$ and $v_x$ directions, and we count the particle number in each cell. Then, we divide the counted number in each cell by the total particle number in all cells to obtain the one-particle distribution function $f_1(\zeta ,v_x)$, as in [7]. In our simulations, the only varied condition is the inward compressional speed of two boundaries in the $x$ direction, which are (a) $M_p=0.22$, (b) $M_p=0.441$, (c) $M_p=0.662$, (d) $M_p=0.809$, (e) $M_p=0.956$, and (f) $M_p=1.324$, respectively. Here, we refer to the distance between the first and second peaks of $f_1(\zeta ,v_x)$ in the shock front as the first wavelength of the oscillation ${\lambda }_1$. Similarly, the distance between the second and third peaks is called the second wavelength ${\lambda }_2$, as marked in (a). Clearly, from (a)–(f), the wavelengths ${\lambda }_1$ and ${\lambda }_2$ gradually diminish with the Mach number, until the DSW completely vanishes. Note that we also find that the first wavelength ${\lambda }_1$ seems to be always slightly larger than the second one ${\lambda }_2$.

Figure 5

The time delay of shocks $\tau$ (triangle), and the first $T_1$ (square) and second $T_2$ (circle) periods of the DSW, as functions of the inward compressional speed of boundary $v_p$ for the 2D Yukawa conditions of $\mathrm{\Gamma }=800$ and $\kappa =0.75$. Here, the time delay $\tau$ is obtained from Fig. 3, while the first and second periods $T_1$ and $T_2$ of shock waves are obtained from $T=\lambda /D$, where ${\lambda }_1$ and ${\lambda }_2$ are measured from the calculated $f_1(\zeta ,v_x)$ as in Fig. 4. When the compressional speed of the boundaries $v_p$ increases, the three timescales (the time delay of shocks $\tau$, and the first $T_1$ and second $T_2$ periods of dispersive shock waves) all decrease monotonically. We can find that the first and second periods $T_1$ and $T_2$ of shock waves are always larger than the time delay of shock $\tau$, although $T_1$ and $T_2$ diminish much more substantially than $\tau$. And the time delay of shocks $\tau$ seems to reach a saturation level of $\tau {\omega }_{pd}\approx 1$ at the high Mach numbers from our simulation data. We speculate that the unity timescale of $\tau {\omega }_{pd}\approx 1$ seems to be related to the motion of individual particles around the shock front. We believe that the observed timescale of $\tau {\omega }_{pd}\approx 1$ is just the time resolution of our studied shock system, which determines the property of the DSW. We speculate that, for a high Mach number of $v_p$, the period of the DSW might be reduced to the time resolution of our studied system, or even shorter. As a result, the DSW cannot be detected anymore with this time resolution, and the DSW reasonably vanishes around the shock front. Note that the legends of (a)–(e) correspond here to the results of the panels in Fig. 4. We also plot the results of the obtained time delay of shocks and the first period of shock wave from the Langevin simulations, as two types of hollow symbols shown. Clearly, although the frictional gas damping expands both the time delay of shocks $\tau$ and the first period of DSW $T_1$, the whole variation trends of $\tau$ and $T_1$ as a function of $v_p$ are still the same as the case without gas damping.