diff options
Diffstat (limited to 'source/know/concept/two-fluid-equations/index.md')
| -rw-r--r-- | source/know/concept/two-fluid-equations/index.md | 273 |
1 files changed, 273 insertions, 0 deletions
diff --git a/source/know/concept/two-fluid-equations/index.md b/source/know/concept/two-fluid-equations/index.md new file mode 100644 index 0000000..b9f1e94 --- /dev/null +++ b/source/know/concept/two-fluid-equations/index.md @@ -0,0 +1,273 @@ +--- +title: "Two-fluid equations" +date: 2021-10-19 +categories: +- Physics +- Plasma physics +layout: "concept" +--- + +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 $\idv{}{t}$ with a +[material derivative](/know/concept/material-derivative/) $\mathrm{D}/\mathrm{D}t$, +and define $\vb{u}$ as the blob's center-of-mass velocity: + +$$\begin{aligned} + m n \frac{\mathrm{D} \vb{u}}{\mathrm{D} t} + = q n (\vb{E} + \vb{u} \cross \vb{B}) +\end{aligned}$$ + +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**: + +$$\begin{aligned} + \boxed{ + \pdv{n_i}{t} + \nabla \cdot (n_i \vb{u}_i) + = 0 + \qquad \quad + \pdv{n_e}{t} + \nabla \cdot (n_e \vb{u}_e) + = 0 + } +\end{aligned}$$ + +These are 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 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 capacity (i.e. a *calorically perfect* gas), +it turns out that: + +$$\begin{aligned} + \frac{\mathrm{D}}{\mathrm{D} t} \big( p V^\gamma \big) = 0 + \qquad \quad + \gamma + \equiv \frac{C_P}{C_V} + = \frac{N + 2}{N} +\end{aligned}$$ + +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 constant $C$: + +$$\begin{aligned} + \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p}{n^\gamma} \Big) = 0 + \quad \implies \quad + p = C n^\gamma +\end{aligned}$$ + +In the two-fluid model, we thus have the following two equations of state, +giving us a set of 16 equations for 16 unknowns: + +$$\begin{aligned} + \boxed{ + \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p_i}{n_i^\gamma} \Big) + = 0 + \qquad \quad + \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p_e}{n_e^\gamma} \Big) + = 0 + } +\end{aligned}$$ + +Note that from the relation $p = C n^\gamma$, +we can calculate the $\nabla p$ term in the momentum equation, +using simple differentiation and the ideal gas law: + +$$\begin{aligned} + p = C n^\gamma + \quad \implies \quad + \nabla p + = \gamma \frac{C n^{\gamma}}{n} \nabla n + = \gamma p \frac{\nabla n}{n} + = \gamma k_B T \nabla n +\end{aligned}$$ + +Note that the ideal gas law was not used immediately, +to allow for $\gamma \neq 1$. + + +## Fluid drifts + +The momentum equations reduce to the following +if we assume the flow is steady $\ipdv{\vb{u}}{t} = 0$, +and neglect electron-ion momentum transfer on the right: + +$$\begin{aligned} + m_i n_i (\vb{u}_i \cdot \nabla) \vb{u}_i + &\approx q_i n_i (\vb{E} + \vb{u}_i \cross \vb{B}) - \nabla p_i + \\ + m_e n_e (\vb{u}_e \cdot \nabla) \vb{u}_e + &\approx q_e n_e (\vb{E} + \vb{u}_e \cross \vb{B}) - \nabla p_e +\end{aligned}$$ + +We take the cross product with $\vb{B}$, +which leaves only the component $\vb{u}_\perp$ of $\vb{u}$ +perpendicular to $\vb{B}$ in the Lorentz term: + +$$\begin{aligned} + 0 + &= q n (\vb{E} + \vb{u}_\perp \cross \vb{B}) \cross \vb{B} - \nabla p \cross \vb{B} - m n \big( (\vb{u} \cdot \nabla) \vb{u} \big) \cross \vb{B} + \\ + &= q n (\vb{E} \cross \vb{B} - \vb{u}_\perp B^2) - \nabla p \cross \vb{B} - m n \big( (\vb{u} \cdot \nabla) \vb{u} \big) \cross \vb{B} +\end{aligned}$$ + +Isolating for $\vb{u}_\perp$ tells us +that the fluids drifts perpendicularly to $\vb{B}$, +with velocity $\vb{u}_\perp$: + +$$\begin{aligned} + \vb{u}_\perp + = \frac{\vb{E} \cross \vb{B}}{B^2} - \frac{\nabla p \cross \vb{B}}{q n B^2} + - \frac{m \big( (\vb{u} \cdot \nabla) \vb{u} \big) \cross \vb{B}}{q B^2} +\end{aligned}$$ + +The last term is often neglected, +which turns out to be a valid approximation if $\vb{E} = 0$, +or if $\vb{E}$ is parallel to $\nabla p$. +The first term is the familiar $\vb{E} \cross \vb{B}$ drift $\vb{v}_E$ +from [guiding center theory](/know/concept/guiding-center-theory/), +and the second term is called the **diamagnetic drift** $\vb{v}_D$: + +$$\begin{aligned} + \boxed{ + \vb{v}_E + = \frac{\vb{E} \cross \vb{B}}{B^2} + } + \qquad \quad + \boxed{ + \vb{v}_D + = - \frac{\nabla p \cross \vb{B}}{q n B^2} + } +\end{aligned}$$ + +It is called *diamagnetic* because +it creates a current that induces +a magnetic field opposite to the original $\vb{B}$. +In a quasi-neutral plasma $q_e n_e = - q_i n_i$, +the current density $\vb{J}$ is given by: + +$$\begin{aligned} + \vb{J} + = q_e n_e (\vb{v}_{De} - \vb{v}_{Di}) + = q_e n_e \Big( \frac{\nabla p_i \cross \vb{B}}{q_i n_i B^2} - \frac{\nabla p_e \cross \vb{B}}{q_e n_e B^2} \Big) + = \frac{\vb{B} \cross \nabla (p_i + p_e)}{B^2} +\end{aligned}$$ + +Using the ideal gas law $p = k_B T n$, +this can be rewritten as follows: + +$$\begin{aligned} + \vb{J} + = k_B \frac{\vb{B} \cross \nabla (T_i n_i + T_e n_e)}{B^2} +\end{aligned}$$ + +Curiously, $\vb{v}_D$ does not involve any net movement of particles, +because a pressure gradient does not necessarily cause particles to move. +Instead, there is a higher density of gyration paths +in the high-pressure region, +so that the particle flux through a reference plane is higher. +This causes the fluid elements to drift, +but not the guiding centers. + + + +## 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. |