Suppose that the top half of space (\(y\ge 0\)), confined by an unlimited flat plate (\(\text{y}=0\)), is complete with an RGP of electrons and ions. Because the ratio among electron and ion masses in ionized RGP is too slight (\(\frac{{m}_{e}}{{m}_{i}}\ll 1\)), the ions will be treated as a motionless neutralizing background. Firstly, the RGP is incomplete ES and the wall rest. The plate then begins to move abruptly in its plane, with velocity\({U}_{0}{e}^{-\alpha t}\) is along the x-axis (\({U}_{0}\)and \(\alpha\)are constants). The temperature of the entire system (electrons + ions + surface) is maintained at a constant temperature. The nomenclature defines all physical parameters.
Where \(\overrightarrow{F}\) operating upon every electron be calculated as follows [19–22]:
\(\overrightarrow{F}=-e\overrightarrow{E}+\frac{-e}{{c}_{o}}(\overrightarrow{c}\wedge \overrightarrow{B})\) . By assuming
(1) \(\overrightarrow{u}\equiv ( {V}_{x},\text{0,0}),\overrightarrow{J}\equiv (qn{V}_{x},\text{0,0}),\overrightarrow{E}\equiv ({E}_{x},\text{0,0})\text{ and }\overrightarrow{B}\equiv (\text{0,0},{B}_{z})\).
E x , Bz, Jx, and Vx, are considered (y, t) functions. Equations of Maxwell's are satisfied by this selection.
In the RGP, the EVDF is \(f\left(y,\overrightarrow{c},t \right), that\)can be calculated from the BE [21–23] which can be composed in the BGK type [24–26] as:
(2) \(\frac{\partial f}{\partial t}+\overrightarrow{c}\cdot \frac{\partial f}{\partial \stackrel{⃑}{r}}+\frac{\overrightarrow{F}}{m}\cdot \frac{\partial f}{\partial \stackrel{⃑}{c}}=\frac{1}{\tau }({f}_{0}-f)\), where\({f}_{0}=n(2\pi RT{)}^{-\frac{3}{2}}{exp}\left(\frac{-{\left(\overrightarrow{c}-\overrightarrow{u}\right)}^{2}}{2RT}\right)\)
By substitutions from (1) into (2), we get:
(3) \(\frac{\partial f}{\partial t}+{c}_{y}\frac{\partial f}{\partial y}-\frac{e{B}_{z}}{m{c}_{0}}({c}_{y}\frac{\partial f}{\partial {c}_{x}}-{c}_{x}\frac{\partial f}{\partial {c}_{y}})+\frac{e{E}_{x}}{m}\frac{\partial f}{\partial {c}_{x}}=\frac{1}{\tau }({f}_{0}-f)\).
Considering the solution of Eq. (4) as in [9]:
where \({V}_{x1}\)and \({V}_{x2}\) are two undetermined functions of variables t and y.
Appling Gard's moment method [1, 4] multiplying Eq. (4) by \({Q}_{i}\left(\overrightarrow{c}\right)\) and integrating overall values of \(\overrightarrow{c}\), the kinetic transfer equations are obtained as:
(5)\(\frac{\partial }{\partial t}\int {Q}_{i}fd\stackrel{̱}{c}+\frac{\partial }{\partial y}\int {c}_{y}{Q}_{i}fd\stackrel{̱}{c}+\frac{e{E}_{x}}{m}\int f\frac{\partial {Q}_{i}}{\partial {c}_{x}}d\stackrel{̱}{c}+\)
$$+\frac{q{B}_{z}}{m{c}_{0}}\int ({c}_{x}\frac{\partial {Q}_{i}}{\partial {c}_{y}}-{c}_{y}\frac{\partial {Q}_{i}}{\partial {c}_{x}})d\stackrel{̱}{c}=\frac{1}{\tau }\int ({f}_{0}-f){Q}_{j}d\stackrel{̱}{c}$$
.
The integrals over velocity are computed using the formula [2],
(6) \(\int {Q}_{i}\left(\overrightarrow{c}\right)fd\stackrel{̱}{c}={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{0}{\int }_{-\infty }^{\infty }{Q}_{i}{f}_{1}d\stackrel{̱}{c}+{\int }_{-\infty }^{\infty }{\int }_{0}^{-\infty }{\int }_{-\infty }^{\infty }{Q}_{i}{f}_{2}d\stackrel{̱}{c}\), where \({Q}_{i}={Q}_{i}\left(\overrightarrow{c}\right)\), \(i=\text{1,2}\) and\(d\stackrel{̱}{c}=d{c}_{x}d{c}_{y}d{c}_{z}\)
where \({c}_{x},{c}_{y}\text{ and }{c}_{z}\)correspondingly, three components of the electron speed along x, y, and z axes. Moreover, the component of both \(E\) and \(B\) can be calculated from Maxwell's equations:
(7)\(\frac{\partial {E}_{x}}{\partial y}-\frac{1}{{c}_{0}}\frac{\partial {B}_{z}}{\partial t}=0\)
(8)\(\frac{\partial {B}_{z}}{\partial y}-\frac{1}{{c}_{0}}\frac{\partial {E}_{x}}{\partial t}-\frac{4\pi en}{{c}_{0}}{V}_{x}=0\)
where \(n=\int fd\stackrel{̱}{c}\text{, n}{\text{V}}_{x}=\int {c}_{x}fd\stackrel{̱}{c},\) with the boundary and initial conditions
(9)\(\left.\begin{array}{c}{E}_{x}(y,0)={B}_{z}(y,0)=0\text{ }\text{,}\\ {E}_{x}(y,t)\text{ }\text{a}\text{n}\text{d}\text{ }{B}_{z}(y,t)\text{ }\text{ }\text{a}\text{r}\text{e}\text{ }\text{f}\text{i}\text{n}\text{i}\text{t}\text{e}\text{ }\text{a}\text{s}\text{ }\text{y}\to \infty .\end{array}\right\}\)
The dimensionless variables are introduced as,
(10)\(\left.\begin{array}{c}t={t}^{\text{'}}\tau ,y=\frac{{y}^{\text{'}}\tau }{\sqrt{2\pi }}{V}_{Th} ,{V}_{x}={V}_{x}^{\text{'}}{V}_{Th} ,{\tau }_{xy}={\tau }_{xy}^{\text{'}}{V}_{Th} ,M=\frac{c}{{V}_{Th}},{M}_{p}=\frac{{U}_{0}}{{V}_{Th}} \\ {B}_{z}={B}_{z}^{\text{'}}\frac{m{c}_{0} {V}_{Th}}{e\tau }\left(\frac{\sqrt{2\pi }}{{V}_{Th}}\right),{E}_{x}^{\text{'}}={E}_{x}^{\text{'}}\frac{m{V}_{Th}}{e\tau }\text{ }\text{ },\rho =n{m}_{e} ,{V}_{Th}=\sqrt{\frac{2K{T}_{0}}{{m}_{e}}.}\end{array}\right\}\)
For \({M}^{2}\ll 1\) (small Mach number), we could consider that changes in temperature and density are negligible, i.e., \(T=1+O\left({M}^{2}\right)\) and \(n=1+O\left({M}^{2}\right)\). Let
(11) \({V}_{x}=\frac{1}{2}({V}_{x1}+{V}_{x2})\), \({\tau }_{xy}=\frac{{P}_{xy}}{\rho {U}_{0}\sqrt{R{T}_{e}/2\pi }}=({V}_{x2}-{V}_{x1})\).
By applying the non-dimension variable Eq. (5) for \({Q}_{1}={c}_{x}\) and \({Q}_{2}={c}_{x}{c}_{y}\)becomes
(12) \(\frac{\partial {V}_{x}^{\text{'}}}{\partial {t}^{\text{'}}}+\frac{\partial {\tau }_{xy}^{\text{'}}}{\partial {y}^{\text{'}}}-{E}_{x}^{\text{'}}=0\),
(13) \(\frac{\partial {\tau }_{xy}^{\text{'}}}{\partial {t}^{\text{'}}}+2\pi \frac{\partial {V}_{x}^{\text{'}}}{\partial {y}^{\text{'}}}+{\tau }_{xy}^{\text{'}}=0\),
with the boundary and initial conditions
(14)\(\left.\begin{array}{c}{V}_{x}^{\text{'}}({y}^{\text{'}},0)={\tau }_{xy}^{\text{'}}({y}^{\text{'}},0)=0,\text{ }\\ 2{V}_{x}^{\text{'}}(0,{t}^{\text{'}})+{\tau }_{xy}^{\text{'}}(0,{t}^{\text{'}})=2{M}_{p}{e}^{-{\alpha }_{1}{t}^{\text{'}}}\text{ }\\ {V}_{x}^{\text{'}}\text{ and }{\tau }_{xy}^{\text{'}}\text{ are finite as y}\to \infty ,{\alpha }_{1}=\alpha \tau .\end{array}\right\}\)
In equations (7)-(9) and (12)- (14), we remove the dash over the dimensionless variables for brevity's purpose. Thus, we have the following equations system (disregard current of displacement) [27]:
(15) \(\frac{\partial {V}_{x}}{\partial t}+\frac{\partial {\tau }_{xy}}{\partial y}-{E}_{x}=0\),
(16) \(\frac{\partial {\tau }_{xy}}{\partial t}+2\pi \frac{\partial {V}_{x}}{\partial y}+{\tau }_{xy}=0\),
(17) \(\frac{\partial {E}_{x}}{\partial y}-\frac{\partial {B}_{z}}{\partial t}=0\),
(18) \(\frac{\partial {B}_{z}}{\partial y}-{\alpha }_{0}{V}_{x}=0\), where \({\alpha }_{0}=\frac{{{V}_{Th}}^{2}{\tau }^{2}{e}^{2}{n}_{e}}{{m}_{e}{c}_{0}^{2}}\),
with the boundary and initial conditions
(19)\(\left.\begin{array}{c}{V}_{x}(y,0)={\tau }_{xy}(y,0)={E}_{x}(y,0)={B}_{z}(y,0)=0 ,\\ 2{V}_{x}(0,t)+{\tau }_{xy}(0,t)=2{M}_{p}{e}^{-{\alpha }_{1}t},\text{ for }t>0\text{ };\text{ }\\ {V}_{x}\text{ , }{\tau }_{xy}\text{, }{E}_{x}\text{ and }{B}_{z}\text{ are finite as y}\to \infty .\text{ }\end{array}\right\}\)
We can reduce our basic Eqs. (15-18), to one equation:
(20) \(\frac{{\partial }^{4}{V}_{x}(y,t)}{\partial {t}^{2}\partial {y}^{2}}-\alpha \frac{{\partial }^{2}{V}_{x}(y,t)}{\partial {t}^{2}}-\frac{{\partial }^{3}{V}_{x}(y,t)}{\partial t\partial {y}^{2}}-\alpha \frac{\partial {V}_{x}(y,t)}{\partial t}-2\pi \frac{{\partial }^{4}{V}_{x}(y,t)}{\partial {t}^{4}}=0\).