2.1. The Viscosity Coefficient

In the hydrodynamic regime, electronic dynamics is dominated by viscosity, rather than impurity scatting [Reference Cohen and Goldstein56]. The electron flow in which numerous target nuclei are suspended (forming a suspension-like) may be regarded as a homogeneous medium. Such a medium has an effective viscosity η [Reference Landau and Lifshitz65]. In the present work, we neglect the compressibility, the thermal conductivity effects, and magnetic and temperature effects. Viscosity is solely characterized by momentum relaxation in fluid. It should be noted that in general, the electron viscosity is not necessarily related to electron-electron collisions. Any process providing the relaxation of the second moment of the electron distribution function leads to viscosity. Thus, the viscosity coefficient η is proportional to the second-order relaxation time τ ₂ [Reference Alekseev55].

(1) $\begin{array}{c}\eta =\frac{1}{4}{v}_F^2{\tau }_2,\frac{1}{{\tau }_2\left(T\right)}=\frac{1}{{\tau }_{2,ee}\left(T\right)}+\frac{1}{{\tau }_{\mathrm{2,0}}},\end{array}$

where v_F is the Fermi velocity. Additionally, $1/{\tau }_{2,ee}\left(T\right)$ is the interelectron relaxation rate, while $1/{\tau }_{\mathrm{2,0}}$ , being the “residual” relaxation rate of the shear stress at T ⟶ 0 owing to scattering of electrons on disorder, is not related to the electron-electron collisions [Reference Alekseev55, Reference Alekseev and Dmitriev66]. The temperature effect is not considered here, i.e., ${\tau }_2\approx {\tau }_{\mathrm{2,0}}$ . Furthermore, the character of the viscous flow strongly depends on the geometry and probe configurations of the sample in general. For instance, the velocity field in a curved pipe flow becomes more nonuniform than in a straight pipe, which may enhance the viscosity effect [Reference Gusev, Levin, Levinson and Bakarov53]. In principle, a range of viscosity values is to be expected in different electron flows [Reference Moll, Kushwaha, Nandi, Schmidt and Mackenzie67]. For a typical system of the viscous flow, the high-quality GaAs quantum wells with the electron density $n\simeq 9.1\times {10}^{11}{\text{cm}}^{-2}$ , 2.9 × 10¹¹cm⁻², or 6.0 × 10¹¹cm⁻², the following fitting parameters, ${\tau }_{\mathrm{2,0}}=0.8\times {10}^{-11}\text{s}$ , 1.1×10⁻¹¹s, or 0.69 × 10⁻¹¹s, are used in references [Reference Levin, Gusev, Levinson, Kvon and Bakarov51, Reference Gusev, Levin, Levinson and Bakarov53, Reference Alekseev55]. The viscosity coefficient η values of two well-known 2D electron flows, GaAs quantum wells at 1.4 K of two different configurations and graphene, are 1200cm²s⁻¹, 3000cm²s⁻¹, and 1000cm²s⁻¹ [Reference Levin, Gusev, Levinson, Kvon and Bakarov51–Reference Gusev, Levin, Levinson and Bakarov53], respectively, an order of magnitude higher than that of honey. For comparison, liquid honey has typical viscosities of 20–50cm²s⁻¹ [Reference Bandurin, Torre and Krishna Kumar52]. The τ ₂ is related to the temperature and density of the material. $r_s={\left(2\pi n_0{a}_B^2\right)}^{-1/2}$ is also the function of density dependent on the material character. Therefore, when the temperature takes a certain value, the value of τ ₂ for the corresponding range can be obtained by giving the value of the density function r_s, as reported in [Reference Alekseev and Dmitriev66].

Consequently, as for $r_s=2$ , we can set the values of τ _2,0 in the range between ${\tau }_2\approx {\tau }_{\mathrm{2,0}}=1.1\times {10}^{-16}\text{s}$ and 1.1 × 10⁻¹³s at T ⟶ 0, since the characteristic time (inverse of the electron plasma frequency $1/{\omega }_p={\left(2\pi n_0{e}^2/m_e{a}_B\right)}^{-1/2}$ ) is about 10⁻¹⁶ − 10⁻¹³ s. According to reference [Reference Alekseev and Dmitriev66], the shear stress relaxation rate $1/{\tau }_2$ as the function of temperature for the GaAs quantum well was presented for the density n = 1.6 × 10¹¹cm⁻², which increases from the order of 10¹¹s⁻¹ to 10¹²s⁻¹ with increasing temperature T. Therefore, as for τ ₂, the order of 10⁻¹³s–10⁻¹⁶s can be related with the graphene, GaN, and thin metal film for corresponding carrier concentrations n of the order of 10¹²cm⁻², 10¹³cm⁻², and 10¹²–10¹⁶cm⁻² [Reference Zhang, Gao and Guo47, Reference Bandurin, Torre and Krishna Kumar52, Reference Liang, Wei, Wang and Li68], the carrier concentrations of which can be changed by doping and other methods.

As a result, the corresponding viscosity η in the case assumed by this work can take values of 0.33cm²s⁻¹, 3.3cm²s⁻¹, 33cm²s⁻¹, and 330cm²s⁻¹. As follows, the influence of the viscous effect (1) is considered for the modified expressions of the perturbed density, the stopping power, and the velocity vector field of the perturbed electron gas.

2.2. Derivation of the Fluid Model

We consider an idealized 2DEG with an equilibrium density, n ₀ = n _i0 = n _e0, which is composed of free electrons and motionless ions. The region z > 0 is the vacuum. A particle of charge $Z_{1}e$ moves with constant velocity v parallel to 2DEG along the x axis with density ${\rho }_{\text{ext}}=Z_{1}e\delta \left(r-vt\right)\delta \left(z-z_0\right)$ , where $v=v{e}_x$ , $r=r\left(x,y\right)$ , and z ₀ is the distance from the plane, described in a Cartesian coordinate system with $R=\left(x,y,z\right)$ , as shown in Figure 1. Therefore, the homogeneous 2DEG will be perturbed by the charged particle and can be regarded as a charged viscous fluid with velocity field $u_e\left(r,t\right)$ and the electron gas density (per unit area) $n_e\left(r,t\right)$ .

Figure 1: Schematic illustration of the interaction system: a particle of charge $Z_{1}e$ moving with v above 2DEG described in Cartesian coordinate, $R=\left(x,y,z\right)$ , along the x axis at a distance z ₀ above the 2DEG plane.

By employing the linearized quantum hydrodynamic model of the incompressible viscous fluid, the electronic excitation on the 2DEG surface can be described by the continuity equation as

(2) $\begin{array}{}\frac{\partial n_e}{\partial t}+{\nabla }_{\Vert }\cdot \left(n_e{u}_e\right)=0,\end{array}$

the momentum-balance equation as

(3) $\begin{array}{}{m}_e\left(\frac{\partial u_e}{\partial t}+\left(u_e\cdot {\nabla }_{\Vert }\right)u_e\right)={e{\nabla }_{\Vert }\Phi |}_{z=0}-{\nabla }_{\Vert }{w}_e+\frac{{\hslash }^2}{2m_e}{\nabla }_{\Vert }\left(\frac{1}{\sqrt{n_e}}{\nabla }_{\Vert }^2\sqrt{n_e}\right)-\gamma m_e{u}_e+m_e\eta {\Delta }_{\Vert }{u}_e,\end{array}$

and Poisson’s equation as

(4) $\begin{array}{}{\nabla }^2\Phi =4\pi e\left(n_e\delta \left(z\right)-n_0\delta \left(z\right)-Z_1\delta \left(r-vt\right)\delta \left(z-z_0\right)\right),\end{array}$

where m_e is the electron mass, e is the elementary charge, and ℏ is the Planck constant divided by 2π. It is worth noting that, in (2) and (3), ${\nabla }_{\Vert }=\left(\partial /\partial x\right){\hat{e}}_x+\left(\partial /\partial y\right){\hat{e}}_y$ , which is used to describe the quantum electron gas surface, is different from $\nabla =\left(\partial /\partial x\right){\hat{e}}_x+\left(\partial /\partial y\right){\hat{e}}_y+\left(\partial /\partial z\right){\hat{e}}_z$ unrestricted in (4) and ${\Delta }_{\Vert }={\nabla }_{\Vert }^2=\left(\left({\partial }^2/\partial x^2\right)+\left({\partial }^2/\partial y^2\right)\right)$ . In addition, the viscosity is taken into account through the coefficient of viscosity η. Φ is the total potential which consisted of the external potential Φ₀ of the moving charged particle and the induced potential, Φ_ind, generated by the perturbation of the electron gas density on the 2DEG surface. In the right-hand side of (3), the first term is the force on electrons of the electron gas surface given by the tangential component of the electric field. The second and third terms are regarded as the quantum effects due to the internal interactions in the electron species and the quantum pressure, where $w_e=\pi {\hslash }^2{n}_e/m_e$ , being the Fermi energy of 2DEG. The fourth term is the frictional force on electrons, owing to the positive charge background, with γ being the frictional coefficient, while the last term, $m_e\eta {\Delta }_{\Vert }{u}_e$ , is the viscous force result from the electron gas at different speeds, where η is the viscosity coefficient.

There is a weak perturbation in 2DEG by the moving charged particles. Hence, we can linearize the above equations by assuming the density, velocity field, and the potential written as $n_e\left(r,t\right)=n_0+n_{e1}\left(r,t\right)\left(\left(n_{e1}\left(r,t\right)\right)\ll n_0\right)$ , $u_e\left(r,t\right)=u_{e1}\left(r,t\right)$ , $\Phi \left(r,z,t\right)={\Phi }_1\left(r,z,t\right)$ , and $\overline{\Phi }\left(r,0,t\right)={\overline{\Phi }}_1\left(r,0,t\right)$ , where $n_{e1}\left(r,t\right)$ , $u_{e1}\left(r,t\right)$ and ${\Phi }_1\left(r,z,t\right)$ represent the first-order perturbed values of the density, velocity, and potential. The linearized equations of the electronic excitation on the electron gas surface are obtained as follows:

(5) $\begin{array}{}\frac{\partial n_{e1}}{\partial t}+n_0{\nabla }_{\Vert }\cdot u_{e1}=0,\end{array}$

(6) $\begin{array}{}{m}_e\frac{\partial u_{e1}}{\partial t}={e{\nabla }_{\Vert }{\overline{\Phi }}_1|}_{z=0}-\frac{\pi {\hslash }^2}{m_e}{\nabla }_{\Vert }{n}_{e1}+\frac{{\hslash }^2}{4m_e{n}_0}{\nabla }_{\Vert }\left({\nabla }_{\Vert }^2{n}_{e1}\right)-\gamma m_e{u}_{e1}+m_e\eta {\Delta }_{\Vert }{u}_{e1},\end{array}$

(7) $\begin{array}{}{\nabla }^2{\Phi }_1=4\pi e\left(n_{e1}\delta \left(z\right)-Z_1\delta \left(r-vt\right)\delta \left(z-z_0\right)\right).\end{array}$

We adopt the time-space Fourier transform:

(8) $\begin{array}{}F\left(R,t\right)=\iint \frac{dQd\omega }{{\left(2\pi \right)}^4}f\left(Q,\omega \right)e^{i\left(Q\cdot R-\omega t\right)},\end{array}$

where F(R,t) stands for any of the above-listed perturbed quantities, and $Q=\left(k_x,k_y,k_z\right)$ is the wave vector. By using the Fourier transform in (5), (6), and (7), we can readily obtain the perturbed $n_{e1}\left(r,t\right)$ , the induced potential ${\Phi }_{\text{ind}}\left(r,z,t\right)$ , and the velocity field $u_{e1}\left(r,t\right)$ of perturbed electron gas:

(9) $\begin{array}{}{n}_{e1}\left(r,t\right)=-n_0\frac{Z_1{e}^2}{2\pi m_e}\int {\text{d}k\text{ke}}^{-k{z}_0}{D}^{-1}\left(k,\omega \right)e^{ik\cdot \left(r-vt\right)},\end{array}$

(10) $\begin{array}{}{\Phi }_{\text{ind}}\left(r,z,t\right)=\frac{Z_1{e}^3{n}_0}{m_e}\int dk{e}^{-k\left(z+z_0\right)}{D}^{-1}\left(k,\omega \right)e^{ik\cdot \left(r-vt\right)},\end{array}$

$\begin{array}{}\text{(11a)}& u_{e1x}\left(r,t\right)=-\frac{Z_1{e}^2}{2\pi m_e}\int dk\cdot \frac{\omega }{k}\cdot \frac{k_x}{D\left(k,w\right)}\cdot e^{-k{z}_0}\cdot e^{ik\left(r-vt\right)},\end{array}$

$\begin{array}{}\text{(11b)}& u_{e1y}\left(r,t\right)=-\frac{Z_1{e}^2}{2\pi m_e}\int dk\cdot \frac{\omega }{k}\cdot \frac{k_y}{D\left(k,w\right)}\cdot e^{-k{z}_0}\cdot e^{ik\left(r-vt\right)},\end{array}$

where $D\left(k,\omega \right)=\left(\left(\omega \tilde{\omega }-{\left(k{v}_F\right)}^2\left(1+k^2/2k_F^2\right)\right)/\left(2-{\omega }_p^2\left(k{a}_B\right)+i\eta k^{2}w\right)\right)$ , with $\tilde{\omega }=\omega +i\gamma$ , Bohr radius $a_B={\hslash }^2/m_e{e}^2$ , Fermi velocity $v_F=\hslash k_F/m_e$ , Fermi wave number $k_F=\sqrt{2\pi n_0}$ , Fermi wave length ${\lambda }_f=1/k_F$ , electron plasma frequency ${\omega }_p={\left(2\pi n_0{e}^2/m_e{a}_B\right)}^{1/2}$ , and the 2D wave vector $k=\left(k_x,k_y\right)$ . In addition, the direction of the projectile velocity v is along with the x axis, for which $\omega =k\cdot v$ , so that we can obtain $\omega =k_{x}v$ .

For convenience, we introduce the dimensionless variables: $u=\omega /{\omega }_p$ , $q_y=k_y/k_F$ , $q=k/k_F$ , $\tilde{v}=v/v_B$ , $\tilde{l}=l/{\lambda }_f$ , $\tilde{x}=\left(\left(x-vt\right)/{\lambda }_f\right)$ , $\tilde{\gamma }=\gamma /{\omega }_p$ , and $\tilde{\eta }=\eta$ /w_p/ $k_{\text{F}}^2$ , where l stands for any quantity of length. By using the dimensionless variables above, (9), (10), (11a) and (11b) can be reduced to

(12) $\begin{array}{}\frac{n_{e1}\left(r,t\right)}{n_0}=A_n{\int }_{-\infty }^{+\infty }\text{d}{q}_y{\int }_{-\infty }^{+\infty }du\cdot \frac{q\cdot e^{-q\widetilde{z_0}}}{{\left(D_0\right)}^2}\cos \left(\tilde{y}{q}_y\right)\times \left(\mathrm{Re}\left(D_0\right)\cos \left(\frac{\tilde{x}u}{\tilde{v}}\right)+\mathrm{Im}\left(D_0\right)\sin \left(\frac{\tilde{x}u}{\tilde{v}}\right)\right),\end{array}$

(13) $\begin{array}{}\frac{{\Phi }_{ind}\left(r,z,t\right)}{{\Phi }_n}=A_{\varphi }{\int }_{-\infty }^{+\infty }d{q}_y{\int }_{-\infty }^{+\infty }du\cdot \frac{e^{-q\left(\widetilde{z_0},+,\tilde{z}\right)}}{{\left(D_0\right)}^2}\cos \left(\tilde{y}{q}_y\right)\times \left(\mathrm{Re}\left(D_0\right)\cos \left(\frac{\tilde{x}u}{\tilde{v}}\right)+\mathrm{Im}\left(D_0\right)\sin \left(\frac{\tilde{x}u}{\tilde{v}}\right)\right),\end{array}$

$\begin{array}{}\text{(14a)}& \frac{u_{e1x}\left(r,t\right)}{v_B}=A_{ux}{\int }_{-\infty }^{+\infty }d{q}_y{\int }_{-\infty }^{+\infty }du\cdot \frac{u^2\cdot e^{-q\overline{z}}}{q\cdot {\left(D_0\right)}^2}\cos \left(\tilde{y}{q}_y\right)\times \left(\mathrm{Re}\left(D_0\right)\cos \left(\frac{\tilde{x}u}{\tilde{v}}\right)+\mathrm{Im}\left(D_0\right)\sin \left(\frac{\tilde{x}u}{\tilde{v}}\right)\right),\end{array}$

$\begin{array}{}\text{(14b)}& \frac{u_{e1y}\left(r,t\right)}{v_B}=A_{uy}{\int }_{-\infty }^{+\infty }d{q}_y{\int }_{-\infty }^{+\infty }du\cdot \frac{u\cdot q_y\cdot e^{-q\overline{z}}}{q\cdot {\left(D_0\right)}^2}\sin \left(\tilde{y}{q}_y\right)\times \left(\mathrm{Re}\left(D_0\right)\sin \left(\frac{\tilde{x}u}{\tilde{v}}\right)-\mathrm{Im}\left(D_0\right)\cos \left(\frac{\tilde{x}u}{\tilde{v}}\right)\right),\end{array}$

where $A_n=\left(-Z_1\right)/2\pi r_s\tilde{v}$ , $A_{\Phi }=Z_1/2\pi \tilde{v}{r}_s^2$ , ${\Phi }_n=e/a_B$ , $A_{ux}=\left(\left(-Z_1\right)/2\pi {\tilde{v}}^2{r}_s\right)$ , and $A_{\text{uy}}=Z_1/2\pi \tilde{v}{r}_s$ , and

(15) $\begin{array}{}{D}_0\left(q,u\right)=u\left(u+i\tilde{\gamma }\right)-q^2\left(\frac{\left(1+\left(q^2/2\right)\right)}{2r_s^2}\right)-\frac{q}{r_s}+{\text{iuq}}^2\tilde{\eta },\end{array}$

is the dimensionless $D\left(k,\omega \right)$ . Besides, $r_s={\left(2\pi n_0{a}_B^2\right)}^{-1/2}$ is the so-called Wigner radius as the function of density dependent on the material.

The stopping power $S\left(v\right)=-\left(\text{d}E/\text{d}x\right)$ originates from the induced potential:

(16) $\begin{array}{}S\left(v\right)=e{Z}_1\frac{\partial {\Phi }_{\text{ind}}}{\partial x}{|}_{r=vt}^{z=z_0}.\end{array}$

Thus, we can obtain the expression of the dimensionless stopping power:

(17) $\begin{array}{}\frac{S\left(v\right)}{S_0}=A_s\iint \text{d}u\text{d}{q}_y{e}^{-2q{\tilde{z}}_0}u\frac{I_m\left(D_0\right)}{{\left(D_o\right)}^2},\end{array}$

where $A_s=Z_1^2/2\pi \tilde{v}{r}_s^3$ , $S_0={\left(e/a_B\right)}^2$ . The form of dimensionless equations (12), (13), (14a), (14b), and (17) obtained above are similar to reference [Reference Li, Song and Wang64], but the internal forms are different, as reflected in the fact that the imaginary part of D ₀(q, u) takes into account the viscous effect $u{q}^2\tilde{\eta }$ .

In the next section, the results of the perturbed electrons gas density, the velocity vector field of the perturbed electron gas, and the stopping power are obtained numerically according to (12), (14a), (14b), and (17), respectively, where the MATLAB solver function (integral2 function) is used to solve the double integral. Referring to [Reference Li, Song and Wang64] for parameter selection, we take the charge number of the incident particle is a proton Z ₁ = 1, the friction coefficient $\gamma =0.02{\omega }_p$ , the height $z_0=3.0{\lambda }_f$ , and r_s = 2 kept fixed throughout the study. The QHD method is suitable to calculate the stopping power for higher particle velocities greater than the Bohr velocity, as discussed in our previous work [Reference Zhang, Song and Wang69]. Thus, for the velocity of the incident particle, we start from v = v_B.

3.1. The Perturbed Electron Density

We have solved (12) numerically for the values of the plasma configuration parameters $v=2v_B$ and r_s = 2 in the case of with and without viscosity. The results are shown in Figure 2, showing the distribution of perturbed electron gas density n _e1 (normalized by n ₀). Then, how the perturbed density of the electron gas is impacted by viscosity will be discussed. Equation (1) shows that the viscosity coefficient η is a function of the relaxation time τ ₂, linearly increasing as τ ₂ increases. To examine the effects of the viscosity on the perturbation of the electron gas, we plot Figure 2 for four relaxation time values (τ ₂ = 1.1 × 10⁻¹⁶, 1.1 × 10⁻¹⁵, 1.1 × 10⁻¹⁴, and 1.1 × 10⁻¹³) with r_s = 2 and $v=2v_B$ . Note that, as τ ₂ increases (namely, η increases), the oscillation amplitudes of the perturbed density decrease, and then, finally, the peak is suppressed and disappears gradually. We can also see in Figure 2 that the perturbed region becomes smaller when τ ₂ increases. This is because when the incident particles move above 2DEG, part energy of the incident particles is dissipated by the viscous effect, leading to a decrease in the energy that makes the electron gas movement. It can be clearly seen that electron density perturbation is significantly suppressed at high viscosity.

Figure 2: The perturbed electron gas density (normalized by n ₀) with and without viscosity for different relaxation times τ ₂ (i.e., different viscosity coefficients $\eta =\left(1/4\right)v_F^2{\tau }_2$ and the dimensionless variables $\overline{\eta }=\eta$ /w_p/ $k_{\text{F}}^2$ ): (a) ${\tau }_2=1.1\times {10}^{-16}$ , (b) ${\tau }_2=1.1\times {10}^{-15}$ , (c) ${\tau }_2=1.1\times {10}^{-14}$ , and (d) ${\tau }_2=1.1\times {10}^{-13}$ with r_s = 2 and $v=2v_B$ .

The comparison of the perturbed electron gas density for the different moving speeds ( $v=v_B$ , $v=1.5v_B$ , $v=2v_B$ , $v=3v_B$ , and $v=4v_B$ ) is shown in Figure 3 with three relaxation time values (τ ₂ = 1.1 × 10⁻¹⁶, 1.1 × 10⁻¹⁵, and 1.1 × ¹⁰⁻¹⁴). In the case of low viscosity shown in Figure 3(a), one can find that the curves of the perturbed electron gas density demonstrate an obvious difference for spatial change, which indicates that the projectile speed has a strong influence on the wake-field. The number of wake-field oscillations excited behind the incident particles decreases with increasing velocity. The maximum values decrease in magnitude with the increasing v, and its position shifts toward lower values. The trend of this change shows good agreement with the conclusion in reference [Reference Li, Song and Wang45], which also showed the decrease in the maximum values and the number of wake-field oscillations. Figures 3(b) and 3(c) show the curves of the perturbed electron gas density for different viscosities with ${\tau }_2=1.1\times {10}^{-15}$ and 1.1 × 10⁻¹⁴, showing similar change tendencies with those in Figure 3(a) for fixed τ ₂. However, as shown in Figures 3(b) and 3(c), an interesting phenomenon appears when the incident particle moves along x direction with a certain speed in different values of the viscosity, in which the strength of the perturbed electron gas density decreases and the perturbed region become smaller as viscosity increases, which is in agreement with that shown in Figure 2. At a fixed incident particle velocity, the curve exhibits multiple wake-field oscillations behind the particle with low viscosity, while the oscillations vanish with high viscosity.

Figure 3: Comparisons of the perturbed electron gas density (normalized by n ₀) in the different moving speeds of the incident particles in the case of considering different relaxation times τ ₂ (i.e., different viscosity coefficients $\eta =\left(1/4\right)v_F^2{\tau }_2$ and the dimensionless variables $\overline{\eta }=\eta$ /w_p/ $k_{\text{F}}^2$ ): (a) ${\tau }_2=1.1\times {10}^{-16}$ , (b) ${\tau }_2=1.1\times {10}^{-15}$ , and (c) ${\tau }_2=1.1\times {10}^{-14}$ with r_s = 2.

3.2. The Spatial Distribution of the Velocity Vector Field

Figures 4 and 5 show the spatial distribution of the velocity vector field of perturbed electron gas for different incident particle speeds to further understand the influence of viscosity. We first show in Figures 4(a) and 4(b) the spatial distribution of the velocity vector field with $v=2v_B$ , r_s = 2, and ${\tau }_2=1.1\times {10}^{-15}$ . One can see from Figures 4(a) and 4(b) that the wake-field is almost axially symmetric about the y axis, compared with those in the case of considering the viscosity shown in Figure 4(b), in which the V-shaped cones structures lag slightly behind the incident particles, along with fewer oscillatory lateral wakes. Besides, we also observe that these structures are composed of multiple cones, which become fewer in consideration of the viscosity. However, the magnitude of the velocity vector field $\left(u_{e1}\right)$ can be also determined by the viscosity, which can be inferred from (14a) and (14b) in the imaginary part of D ₀. Thus, the magnitude of the velocity field of the electrons $\left(u_{e1}\right)$ in the wake-field region is shown in Figures 4(c) and 4(d). The main features observed in Figures 4(a) and 4(b) are shown in Figures 4(c) and 4(d). Moreover, the maximum value of $\left(u_{e1}\right)$ is suppressed when the viscosity is taken into account, indicating the viscosity makes the electron gas more difficult to be disturbed. These characteristics closely resemble the spatial distribution of perturbed electron gas density n _e1.

Figure 4: The spatial distribution and magnitude of the velocity vector field (normalized by v_B): (a) the spatial distribution of the velocity vector field without viscosity, (b) the spatial distribution of the velocity vector field with viscosity, (c) the magnitude of the velocity vector field without viscosity, and (d) the magnitude of the velocity vector field with viscosity, where $v=2v_B$ , r_s = 2, and τ₂ = 1.1 × 10⁻¹⁵.

Figure 5: The spatial distribution of the velocity vector field of perturbed electron gas u_e1 (normalized by v_B) in the case of viscosity for different relaxation times (i.e., different viscosity coefficients $\eta =\left(1/4\right)v_F^2{\tau }_2$ and the dimensionless variables $\overline{\eta }=\eta$ /w_p/ $k_{\text{F}}^2$ ): τ₂ = (a) 1.1 × 10⁻¹⁶, (b) 1.1 × 10⁻¹⁵, and (c) 1.1 × 10⁻¹⁴, with $v=3v_B$ .

As demonstrated above, viscosity affects the spatial distribution and affects the maximum value of the velocity vector field. Thus, such a viscosity effect on the velocity vector of the perturbed electron gas u_e1 is shown in Figures 5(a)–5(c) with viscosity for different relaxation times ${\tau }_2=1.1\times {10}^{-16}$ , ${\tau }_2=1.1\times {10}^{-15}$ , and ${\tau }_2=1.1\times {10}^{-14}$ with $v=3v_B$ . From Figure 5(a), we can see that in the low-viscosity case $\left({\tau }_2=1.1\times {10}^{-16}\right)$ , the characteristics closely resemble the case of without considering the viscosity in Figure 4(a). The similar cone structures of the velocity field and invariably symmetrical spatial distributions concerning the y axis are still reproduced in the wake-field regions, showing the evident dynamic polarization of the perturbed electron gas. The same pattern has also been found in reference [Reference Hu, Song, Hu and Wang19], in which the dynamic polarization of perturbed electrons becomes obvious with increasing magnetic field. However, as shown in Figures 5(b) and 5(c) with the increasing viscosity, the opening cone angles of the V-shaped wake-field decrease, and the oscillation tails disappear gradually, showing the wakening of the dynamic polarization, which is completely different from the velocity distribution under the influence of magnetic field that the dynamic polarization is enhanced as the magnetic field increases [Reference Hu, Song, Hu and Wang19]. It can be concluded that the movement of the perturbed electron gas is increasingly restricted under the influence of viscosity. In this case, the electrons cannot fully respond to the disturbance from the incident particles, giving rise to the reduction in the wake region.

3.3. Stopping Power

The viscosity significantly impacts the density and velocity vector field of the perturbed electron gas. As a result, such an influence can also be seen in the stopping power, as shown in Figure 6, showing how the electronic stopping character is impacted by viscosity. The influence of viscosity is reflected clearly in Figure 6 where the stopping power is plotted as a function of velocity for the same condition shown in Figure 2. The main features observed in the perturbed density are reproduced in the stopping power. With the increasing τ ₂ values shown in Figure 6, at the high-velocity region, the two curves for the stopping power show agreement with each other, while at the low-velocity region, the peak values of the stopping power get weaker, showing less energy loss from projectiles. Besides, note that the peak position of the stopping power shifts mainly to the lower-velocity region with the viscosity. Finally, the peak is limited to become a broad plateau, indicating that the energy loss of the particle is hardly influenced by the viscosity when the viscosity reaches a considerable value. The above results are entirely different from those in reference [Reference Hu, Song, Hu and Wang19], in which the significant enhancement of the stopping power has been observed with the increasing magnetic field. Therefore, one can expect that the differences of the results between the two situations of with and without viscosity are due to the action of the viscosity on the perturbed electron gas density and velocity vector field in the plane at their distribution and magnitude, which can be seen in parts of the perturbed electron density and the spatial distribution of the velocity vector field.

Figure 6: The stopping power S (normalized by $S_0={\left(e/a_B\right)}^2$ ) versus the moving speed with and without viscosity for different relaxation times τ ₂ (i.e., different viscosity coefficients η and the dimensionless variables $\overline{\eta }=\eta$ /w_p/ $k_{\text{F}}^2$ ): (a) ${\tau }_2=1.1\times {10}^{-16}$ , (b) ${\tau }_2=1.1\times {10}^{-15}$ , (c) ${\tau }_2=1.1\times {10}^{-14}$ , and (d) ${\tau }_2=1.1\times {10}^{-13}$ with r_s = 2.