1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
|
---
title: "Two-fluid equations"
firstLetter: "T"
publishDate: 2021-10-19
categories:
- Physics
- Plasma physics
date: 2021-10-18T10:12:20+02:00
draft: false
markup: pandoc
---
# Two-fluid equations
The **two-fluid model** describes a plasma as two separate but overlapping fluids,
one for ions and one for electrons.
Instead of tracking individual particles,
it gives the dynamics of fluid elements $\dd{V}$ (i.e. small "blobs").
These blobs are assumed to be much larger than
the [Debye length](/know/concept/debye-length/),
such that electromagnetic interactions between nearby blobs can be ignored.
From Newton's second law, we know that the velocity $\vb{v}$
of a particle with mass $m$ and charge $q$ is as follows,
when subjected only to the [Lorentz force](/know/concept/lorentz-force/):
$$\begin{aligned}
m \dv{\vb{v}}{t}
= q (\vb{E} + \vb{v} \cross \vb{B})
\end{aligned}$$
From here, the derivation is similar to that of the
[Navier-Stokes equations](/know/concept/navier-stokes-equations/).
We replace the time derivative with a
[material derivative](/know/concept/material-derivative/) $\mathrm{D}/\mathrm{D}t$,
and define a blob's velocity $\vb{u}$
as the average velocity of the particles inside it, leading to:
$$\begin{aligned}
m n \frac{\mathrm{D} \vb{u}}{\mathrm{D} t}
= q n (\vb{E} + \vb{u} \cross \vb{B})
\end{aligned}$$
Where we have multiplied by the number density $n$ of the particles.
Due to particle collisions in the fluid,
stresses become important. Therefore, we include
the [Cauchy stress tensor](/know/concept/cauchy-stress-tensor/) $\hat{P}$,
leading to the following two equations:
$$\begin{aligned}
m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t}
&= q_i n_i (\vb{E} + \vb{u}_i \cross \vb{B}) + \nabla \cdot \hat{P}_i{}^\top
\\
m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t}
&= q_e n_e (\vb{E} + \vb{u}_e \cross \vb{B}) + \nabla \cdot \hat{P}_e{}^\top
\end{aligned}$$
Where the subscripts $i$ and $e$ refer to ions and electrons, respectively.
Finally, we also account for momentum transfer between ions and electrons
due to [Rutherford scattering](/know/concept/rutherford-scattering/),
leading to these **two-fluid momentum equations**:
$$\begin{aligned}
\boxed{
\begin{aligned}
m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t}
&= q_i n_i (\vb{E} + \vb{u}_i \cross \vb{B}) + \nabla \cdot \hat{P}_i{}^\top - f_{ie} m_i n_i (\vb{u}_i - \vb{u}_e)
\\
m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t}
&= q_e n_e (\vb{E} + \vb{u}_e \cross \vb{B}) + \nabla \cdot \hat{P}_e{}^\top - f_{ei} m_e n_e (\vb{u}_e - \vb{u}_i)
\end{aligned}
}
\end{aligned}$$
Where $f_{ie}$ is the mean frequency at which an ion collides with electrons,
and vice versa for $f_{ei}$.
For simplicity, we assume that the plasma is isotropic
and that shear stresses are negligible,
in which case the stress term can be replaced
by the gradient $- \nabla p$ of a scalar pressure $p$:
$$\begin{aligned}
m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t}
&= q_i n_i (\vb{E} + \vb{u}_i \cross \vb{B}) - \nabla p_i - f_{ie} m_i n_i (\vb{u}_i - \vb{u}_e)
\\
m_e n_e \frac{\mathrm{D} \vb{u}_e}{\mathrm{D} t}
&= q_e n_e (\vb{E} + \vb{u}_e \cross \vb{B}) - \nabla p_e - f_{ei} m_e n_e (\vb{u}_e - \vb{u}_i)
\end{aligned}$$
Next, we demand that matter is conserved.
In other words, the rate at which particles enter/leave a volume $V$
must be equal to the flux through the enclosing surface $S$:
$$\begin{aligned}
0
&= \pdv{t} \int_V n \dd{V} + \oint_S n \vb{u} \cdot \dd{\vb{S}}
= \int_V \Big( \pdv{n}{t} + \nabla \cdot (n \vb{u}) \Big) \dd{V}
\end{aligned}$$
Where we have used the divergence theorem.
Since $V$ is arbitrary, we can remove the integrals,
leading to the following **continuity equations**:
$$\begin{aligned}
\boxed{
\pdv{n_i}{t} + \nabla \cdot (n_i \vb{u}_i)
= 0
\qquad \quad
\pdv{n_e}{t} + \nabla \cdot (n_e \vb{u}_e)
= 0
}
\end{aligned}$$
Currently, we have 8 equations (2 scalar continuity, 2 vector momentum),
but 16 unknowns $\vb{u}_i$, $\vb{u}_e$, $\vb{E}$, $\vb{B}$, $n_i$, $n_e$, $p_i$ and $p_e$.
We would like to close this system, so we need 8 more.
An obvious choice is [Maxwell's equations](/know/concept/maxwells-equations/),
in particular Faraday's and Ampère's law
(since Gauss' laws are in fact redundant; see the article on Maxwell's equations):
$$\begin{aligned}
\boxed{
\nabla \cross \vb{E} = - \pdv{\vb{B}}{t}
\qquad \quad
\nabla \cross \vb{B} = \mu_0 \Big( n_i q_i \vb{u}_i + n_e q_e \vb{u}_e + \varepsilon_0 \pdv{\vb{E}}{t} \Big)
}
\end{aligned}$$
Now we have 14 equations, so we need 2 more, for the pressures $p_i$ and $p_e$.
This turns out to be the thermodynamic **equation of state**:
for quasistatic, reversible, adiabatic compression
of a gas with constant heat capacities (i.e. a *calorically perfect* gas),
it turns out that:
$$\begin{aligned}
\dv{t} \big( p V^\gamma \big) = 0
\qquad \quad
\gamma
\equiv \frac{C_P}{C_V}
= \frac{N + 2}{N}
\end{aligned}$$
Where $\gamma$ is the *heat capacity ratio*,
and can be calculated from the number of degrees of freedom $N$
of each particle in the gas.
In a fully ionized plasma, $N = 3$.
The density $n \propto 1/V$,
so since $p V^\gamma$ is constant in time,
for some (known) constant $C$:
$$\begin{aligned}
\dv{t} \Big( \frac{p}{n^\gamma} \Big) = 0
\quad \implies \quad
p = C n^\gamma
\end{aligned}$$
In the two-fluid model, we thus have the following two equations of state,
giving us a set of 16 equations for 16 unknowns:
$$\begin{aligned}
\boxed{
p_i = C_i n_i^\gamma
\qquad \quad
p_e = C_e n_e^\gamma
}
\end{aligned}$$
## References
1. F.F. Chen,
*Introduction to plasma physics and controlled fusion*,
3rd edition, Springer.
2. M. Salewski, A.H. Nielsen,
*Plasma physics: lecture notes*,
2021, unpublished.
|
