The effect of dust size distribution on shock wave in quantum dusty plasma

Abstract

Present paper studies the contribution of both the quantum effect and the dust size distribution effect on the shock wave characters. It is concluded that the quantum effect of dust particles is negligible, while the quantum effects of both electrons and ions on the shock wave of a quantum dusty plasma cannot be neglected in certain cases. It is found that the speed and the amplitude of the shock wave, considering the dust size distribution, are larger than that of the quantum dusty plasma with the average dust size of a monosized dust plasma. Furthermore, the speed and the amplitude of the shock wave increase, while the width of the shock wave decreases as the ratios of the maximum dust size to the minimum one increases. The quantum effects may affect both the amplitude and width of the shock wave, while it has no effect on the shock wave speed. The width of the shock wave increases, while its amplitude decreases as $H_e$ and $H_i$ increase.

Appendices

Appendix A

We have taken the following parameters [27, 32, 33, 58]: $Z_d=10^{3},$ $m_d=2\times 10^{-17}$ kg, $m_e=0.91\times 10^{-30}$ kg, $m_i=2 \times 10^{-26}$ kg, $n_d=1.5\times 10^{27}$ m$^{-3}$, $n_i=2\times 10^{30}$ m$^{-3}$, $\hbar =1.05\times 10^{-34}$J$\cdot$s, $T_{\mathrm {eff}}=3.1\times 10^{-13}$ J, $\mu _{i}=1.3$, $\mu _{e}=0.3$. We have

$$\begin{aligned} H_{e}= & {} \sqrt{\frac{\hbar ^{2}\overline{Z_{d0}}\omega _{pd}^{2}}{m_{e}c_{d}^{4}}}\approx 3.4 \end{aligned}$$

(A.1)

$$\begin{aligned} H_{i}= & {} \sqrt{\frac{\hbar ^{2}\overline{Z_{d0}}\omega _{pd}^{2}}{m_{i}c_{d}^{4}}}\approx 0.02 \end{aligned}$$

(A.2)

$$\begin{aligned} H_{d}= & {} \sqrt{\frac{\hbar ^{2}\overline{Z_{d0}}\omega _{pd}^{2}}{\overline{m}_{d}c_{d}^{4}}}\approx 7\times 10^{-7}. \end{aligned}$$

(A.3)

Hence, the contributions of dust and ion are much lower than the electron.

Appendix B

We have model fluid equations

$$\begin{aligned}&{\partial n_{dj} \over \partial t} +\nabla \cdot (n_{dj} \mathbf {u_{dj}} )=0 \end{aligned}$$

(B.1)

$$\begin{aligned}&{\partial \mathbf {u_{dj}} \over \partial t} +\mathbf {u_{dj}}(\nabla \cdot \mathbf {u_{dj}})=\frac{Z_{dj}}{m_{dj}}\nabla \phi \nonumber \\&-\frac{\sigma _d }{m_{dj}}n_{dj}\frac{Z_{dj}}{m_{dj}}\Omega (\mathbf {u_{dj}}\times \mathbf {x}) +\frac{\eta }{m_{dj}}\nabla ^{2}\mathbf {u_{dj}}\nonumber \\&+\frac{H_{d}^{2}}{2m_{dj}^{2}}\nabla \left( \frac{\nabla ^{2}\sqrt{n_{dj}}}{\sqrt{n_{dj}}}\right) \end{aligned}$$

(B.2)

$$\begin{aligned}&\nabla ^{2}\phi =\mu _{e}n_e-\mu _i n_i+\sum _{j=1}^{N} Z_{dj}n_{dj} \end{aligned}$$

(B.3)

$$\begin{aligned}&\begin{aligned} 0=\nabla \phi -\sigma _{e}n_{e}\nabla n_{e} +\frac{H_{e}^{2}}{2}\nabla \left( \frac{\nabla ^{2}\sqrt{n_{e}}}{\sqrt{n_{e}}}\right) \end{aligned} \end{aligned}$$

(B.4)

$$\begin{aligned}&\begin{aligned} 0=-\nabla \phi -\sigma _{i}n_{i}\nabla n_{i} +\frac{H_{i}^{2}}{2}\nabla \left( \frac{\nabla ^{2}\sqrt{n_{i}}}{\sqrt{n_{i}}}\right) . \end{aligned} \end{aligned}$$

(B.5)

We introduce the stretched coordinates

$$\begin{aligned}&X=\epsilon (x-\lambda t),\quad Y=\epsilon y, \quad =\epsilon z \end{aligned}$$

(B.6)

$$\begin{aligned}&T=\epsilon ^{3}t \end{aligned}$$

(B.7)

and

$$\begin{aligned}&{\partial \over \partial x}=\epsilon {\partial \over \partial X} \end{aligned}$$

(B.8)

$$\begin{aligned}&{\partial \over \partial y}=\epsilon {\partial \over \partial Y} \end{aligned}$$

(B.9)

$$\begin{aligned}&{\partial \over \partial z}=\epsilon {\partial \over \partial Z} \end{aligned}$$

(B.10)

$$\begin{aligned}&{\partial ^{2} \over \partial x^{2}}=\epsilon ^{2}{\partial ^{2} \over \partial X^{2}} \end{aligned}$$

(B.11)

$$\begin{aligned}&{\partial \over \partial t}=\epsilon ^{3}{\partial \over \partial T}-\epsilon \lambda {\partial \over \partial X}. \end{aligned}$$

(B.12)

The asymptotic expansions of the perturbed quantities are given as

$$\begin{aligned}&n_{dj}=n_{dj0}+\epsilon ^{2}n_{dj1}+\epsilon ^{4}n_{dj2}+\cdots \nonumber \\&n_{e}=1+\epsilon ^{2}n_{e1}+\epsilon ^{4}n_{e2}+\cdots \nonumber \\&n_{i}=1+\epsilon ^{2}n_{i1}+\epsilon ^{4}n_{i2}+\cdots \nonumber \\&u_{djx}=\epsilon ^{2}u_{djx1}+\epsilon ^{4}u_{djx2}+\cdots \nonumber \\&u_{djy}=\epsilon ^{3}u_{djy1}+\epsilon ^{4}u_{djy2}+\cdots \nonumber \\&u_{djz}=\epsilon ^{3}u_{djz1}+\epsilon ^{4}u_{djz2}+\cdots \nonumber \\&\phi =\epsilon ^{2}\phi _1+\epsilon ^{4}\phi _2+\cdots . \end{aligned}$$

(B.13)

Substituting eqs (B.8)–(B.13) into eqs (B.1)–(B.5), and collecting the powers of $\epsilon ^{3}$, we assume $\eta =\epsilon \eta '$

$$\begin{aligned}&u_{djx1}=\frac{\lambda }{n_{dj0}}n_{dj1} \end{aligned}$$

(B.14)

$$\begin{aligned}&n_{dj1}=\frac{Z_{dj}n_{dj0}}{\sigma _{d}n_{dj0}^{2}-\lambda ^{2}m_{dj}}\phi _{1} \end{aligned}$$

(B.15)

$$\begin{aligned}&u_{djz1}=\left( 1+\frac{\sigma _{d}n_{dj0}^{2}}{\lambda ^{2}m_{dj}-\sigma _{d}n_{dj0}^{2}}\right) \frac{1}{\Omega }{\partial \phi _{1} \over \partial Y} \end{aligned}$$

(B.16)

$$\begin{aligned}&u_{djy1}=-\left( 1+\frac{\sigma _{d}n_{dj0}^{2}}{\lambda ^{2}m_{dj}-\sigma _{d}n_{dj0}^{2}}\right) \frac{1}{\Omega }{\partial \phi _{1} \over \partial Y} \end{aligned}$$

(B.17)

$$\begin{aligned}&0=\mu _e n_{e1}-\mu _in_{i1}+\sum _{j=1}^{N}Z_{di}n_{dj1} \end{aligned}$$

(B.18)

$$\begin{aligned}&n_{e1}=\frac{\phi _1}{\sigma _e} \end{aligned}$$

(B.19)

$$\begin{aligned}&n_{i1}=-\frac{\phi _1}{\sigma _i}. \end{aligned}$$

(B.20)

Collecting terms of the same powers of $\epsilon ^{4}$, the y- and z-components of the second-order perturbed velocity are calculated as

$$\begin{aligned} u_{djz2}= & {} \frac{\lambda m_{d}}{Z_{dj}\Omega }{\partial u_{djy1} \over \partial X} \end{aligned}$$

(B.21)

$$\begin{aligned} u_{djy2}= & {} -\frac{\lambda m_{d}}{Z_{dj}\Omega }{\partial u_{djz1} \over \partial X}. \end{aligned}$$

(B.22)

From eq. (B.15) and (B.18)–(B.20), we get

$$\begin{aligned} \sum _{j=1}^{N}\frac{n_{dj0}Z_{dj}^{2}}{\lambda ^{2}m_{dj}-\sigma _{d}n_{dj0}^{2}}=1. \end{aligned}$$

(B.23)

We put $\sigma _d=0$ leading to a simplified linear dispersion relation

$$\begin{aligned} \lambda ^{2}=\sum _{j=1}^{N}\frac{n_{dj0}Z_{dj}^{2}}{m_{dj}}. \end{aligned}$$

(B.24)

Collecting the powers of $\epsilon ^{5}$, we have

$$\begin{aligned}&{\partial n_{dj1} \over \partial T}-\lambda {\partial n_{dj2} \over \partial X}+{\partial (n_{dj1}u_{djx1}) \over \partial X}+n_{dj0}{\partial u_{djx2} \over \partial X}\nonumber \\&-n_{dj0}\frac{\lambda m_{dj}}{Z_{dj}\Omega ^{2}}\left( 1+\frac{\sigma _{d}n_{dj0}^{2}}{\lambda ^{2}m_{dj}-\sigma _{d}n_{dj0}^{2}}\right) {\partial \over \partial X}{\partial ^{2} \phi _{1} \over \partial Y^{2}}\nonumber \\&-n_{dj0}\frac{\lambda m_{dj}}{Z_{dj}\Omega ^{2}}\left( 1+\frac{\sigma _{d}Z_{dj}n_{dj0}^{2}}{\lambda ^{2}m_{dj}-\sigma _{d}n_{dj0}^{2}}\right) {\partial \over \partial X}{\partial ^{2} \phi _{1} \over \partial Z^{2}}\nonumber \\&=0 \end{aligned}$$

(B.25)

$$\begin{aligned}&{\partial u_{djx1} \over \partial T}-\lambda {\partial u_{djx2} \over \partial X}+u_{djx1}{\partial u_{djx1} \over \partial X}=\frac{Z_{dj}}{m_{dj}}{\partial \phi _{2} \over \partial X}\nonumber \\&-\frac{\sigma _d}{m_{dj}}n_{dj0}{\partial n_{dj2} \over \partial X}-\frac{\sigma _d}{m_{dj}}n_{dj1}{\partial n_{dj1} \over \partial X} \nonumber \\&+\frac{\eta '}{m_{dj}}{\partial ^{2} u_{djx1} \over \partial X^{2}}+\frac{H_{d}^{2}}{4n_{dj0}m_{dj}^{2}}{\partial (\nabla ^{2}n_{dj1}) \over \partial X} \end{aligned}$$

(B.26)

$$\begin{aligned}&\frac{\lambda ^{2}m_{dj}}{Z_{dj}\Omega ^{2}}\left( 1+\frac{\sigma _{d}n_{dj0}^{2}}{\lambda ^{2}m_{dj}-\sigma _{d}n_{dj0}^{2}}\right) {\partial \over \partial Y}{\partial ^{2} \phi _{1} \over \partial X^{2}}\nonumber \\&=\frac{Z_{dj}}{m_{dj}}{\partial \phi _{2} \over \partial Y}-\frac{\sigma _d}{m_{dj}}n_{dj0}{\partial n_{dj2} \over \partial Y}\nonumber \\&-\frac{\sigma _d}{m_{dj}}n_{dj1}{\partial n_{dj1} \over \partial Y}+\frac{H_{d}^{2}}{4n_{dj0}m_{dj}^{2}}{\partial (\nabla ^{2}n_{dj1}) \over \partial Y} \end{aligned}$$

(B.27)

$$\begin{aligned}&\frac{\lambda ^{2}m_{dj}}{Z_{dj}\Omega ^{2}}\left( 1+\frac{\sigma _{d}n_{dj0}^{2}}{\lambda ^{2}m_{dj}-\sigma _{d}n_{dj0}^{2}}\right) {\partial \over \partial Z}{\partial ^{2} \phi _{1} \over \partial X^{2}}\nonumber \\&=\frac{Z_{dj}}{m_{dj}}{\partial \phi _{2} \over \partial Z}-\frac{\sigma _d}{m_{dj}}n_{dj0}{\partial n_{dj2} \over \partial Z}\nonumber \\&-\frac{\sigma _d}{m_{dj}}n_{dj1}{\partial n_{dj1} \over \partial Z} +\frac{H_{d}^{2}}{4n_{dj0}m_{dj}^{2}}{\partial (\nabla ^{2}n_{dj1}) \over \partial Z} \end{aligned}$$

(B.28)

$$\begin{aligned}&\nabla ^{2}\phi _{1}=\mu _{e}n_{e2}-\mu _{i}n_{i2}+\sum _{j=1}^{N}Z_{dj}n_{dj2} \end{aligned}$$

(B.29)

$$\begin{aligned}&0=\nabla \phi _{2}-\sigma _{e}n_{e1}\nabla n_{e1}-\sigma _{e}\nabla n_{e2}+\frac{H_{e}^{2}}{4}\nabla (\nabla ^{2}n_{e1}) \nonumber \\\end{aligned}$$

(B.30)

$$\begin{aligned}&0=-\nabla \phi _{2}-\sigma _{i}n_{i1}\nabla n_{i1}-\sigma _{i}\nabla n_{i2}+\frac{H_{i}^{2}}{4}\nabla (\nabla ^{2}n_{i1}).\nonumber \\ \end{aligned}$$

(B.31)

Solving eqs (B.25)–(B.31) with the aid of eqs (B.14)–(B.24), we get ZK-Burgers equation

$$\begin{aligned}&{\partial \phi _1 \over \partial T}+A\phi _1{\partial \phi _1 \over \partial X}+B{\partial ^{3} \phi _1 \over \partial X^{3}}\nonumber \\&+C{\partial \over \partial X}\left( {\partial ^{2} \phi _1 \over \partial Y^{2}}+{\partial ^{2} \phi _1 \over \partial Z^{2}}\right) -D{\partial ^{2} \phi _1 \over \partial X^{2}}=0, \end{aligned}$$

(B.32)

where the values of A, B, C and D are consistent with eqs (16)–(19).