Pressure and energy of compressional shocks in two-dimensional Yukawa systems

The propagation of compressional shocks in two-dimensional (2D) dusty plasmas is investigated using MD simulations under various conditions. The shock Hugoniot curves of the relationship between the shock front speed $D$ and the mean particle speed $\overline{v}$ after shocks are obtained and analytically fit to parabolic expressions. As the screening parameter increases, the weaker Yukawa interparticle interaction cause the shock Hugoniot curves to be more linear. Combining the obtained shock Hugoniot curves with the Rankine-Hugoniot jump relations, analytic expressions of pressure and energy after the shocks in 2D Yukawa systems are obtained, which are functions of the observable quantities, like the shock front speed $D$ or the mean particle speed $\overline{v}$ or the specific volume.

Figure 1

The sketch of compressional shocks in our simulations. A piston of a reflecting boundary moves from left to right at a constant speed $v_p$, to generate shocks in 2D Yukawa system with the conditions of ${\mathrm{\Gamma }}_0$ and ${\kappa }_0$. After the shock, the simulated particles would collectively move at a speed $v$ along the piston moving direction. The generated shock front would propagate at a constant speed $D$. The periodic boundary conditions are applied only in the $y$ direction, as in Ref. [9].

Figure 2

The spatiotemporal evolution of the number density $n_d$ of particles in our simulation. The simulation box is divided into 480 narrow rectangular bins with the width of $a_0$ in the $x$ direction, and the counted particle number in each bin is just the plotted number density here. The dashed line corresponds to the propagation of the shock front. Here the shock front speed is $D=1.64a_0{\omega }_{pd}$, while the piston moving speed is $v_p=0.707a_0{\omega }_{pd}$. For comparison, we also draw a solid line, whose slope represents the longitudinal sound speed $C_l=0.961a_0{\omega }_{pd}$ for the 2D Yukawa system with the condition of ${\kappa }_0=0.75$ and ${\mathrm{\Gamma }}_0=800$ here.

Figure 3

The obtained one-particle distribution function $f_1(\zeta ,v_x)$. Here, $\zeta$ is the Lagrange coordinate ($\zeta =x-Dt$), so that the simulated box has been divided to three regions of the postshock, shock front, and preshock, as marked in the $\zeta$ axis. To prepare $f_1(\zeta ,v_x)$, we divide the $\zeta -v_x$ plane into bins in both $\zeta$ and $v_x$ axes, and count the particle numbers in each cell (one element of the 2D array $f_1[\zeta ,v_x)]$. The width of the bin in the $\zeta$ axis is $0.5a_0$, while in the $v_x$ axis, the width is $0.01a_0{\omega }_{pd}$. Then, we divide the counted number in each cell by the total particle number in all cells, to obtain the one-particle distribution function $f_1(\zeta ,v_x)$. For each value of $\zeta$, we average $f_1(\zeta ,v_x)$ with the corresponding weight, then we obtain the mean particle speed ${\overline{v}}_x\left(\zeta \right)$, as the solid curve shown here. Note that the piston speed and the initial conditions of the Yukawa system here are the same as those in Fig. 2.

Figure 4

The obtained $D-\overline{v}$ relations, i.e., the shock Hugoniot curves, for 2D Yukawa systems. Various symbols come from different values of ${\kappa }_0$ from our shock simulations. The parabolic expression of $D=C_0+B\overline{v}+A{\overline{v}}^2$ is chosen to fit to these symbols, as the curves shown, and the corresponding fitting coefficients are shown in the legend. Clearly, with the decrease of ${\kappa }_0$, the stronger repulsion between particles would increase the shock front speed, or lift the shock Hugoniot curves. For comparison, the analytical $D-\overline{v}$ relation for the 2D ideal gas is also presented here.

Figure 5

The obtained pressure of 2D Yukawa systems in the postshock region. From our derivation of Eq. (5), the pressure in the postshock region can be analytically expressed by the ratio of $\frac{\tau }{{\tau }_0}$ (a), or postshock mean particle speed $\overline{v}$ (b), or shock front speed $D$ (c), respectively. For comparison, the pressure of 2D ideal gas condition ($T_0=300K$) after shocks is also presented.

Figure 6

The obtained energy of 2D Yukawa systems in the postshock region. From the Hugoniot equation of Eq. (6), the energy increase $e-e_0$ after shocks equals $\frac{1}{2}(P+P_0)({\tau }_0-\tau )$, corresponding to the area of the shadow in panel (a). The energy after shocks can been easily obtained by substituting the pressure expression $P$ in Eq. (5a) into the Hugoniot equation of Eq. (6). We plot the analytic results of the energy in the postshock region $e$ with the specific volume $\frac{\tau }{{\tau }_0}$ in panel (b).