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Diffstat (limited to 'content/know/concept/two-fluid-equations')
| -rw-r--r-- | content/know/concept/two-fluid-equations/index.pdc | 101 |
1 files changed, 93 insertions, 8 deletions
diff --git a/content/know/concept/two-fluid-equations/index.pdc b/content/know/concept/two-fluid-equations/index.pdc index cd77f5e..df45e73 100644 --- a/content/know/concept/two-fluid-equations/index.pdc +++ b/content/know/concept/two-fluid-equations/index.pdc @@ -32,10 +32,9 @@ $$\begin{aligned} From here, the derivation is similar to that of the [Navier-Stokes equations](/know/concept/navier-stokes-equations/). -We replace the time derivative with a +We replace $\dv*{t}$ with a [material derivative](/know/concept/material-derivative/) $\mathrm{D}/\mathrm{D}t$, -and define a blob's velocity $\vb{u}$ -as the average velocity of the particles inside it, leading to: +and define $\vb{u}$ as the blob's center-of-mass velocity: $$\begin{aligned} m n \frac{\mathrm{D} \vb{u}}{\mathrm{D} t} @@ -134,7 +133,7 @@ of a gas with constant heat capacities (i.e. a *calorically perfect* gas), it turns out that: $$\begin{aligned} - \dv{t} \big( p V^\gamma \big) = 0 + \frac{\mathrm{D}}{\mathrm{D} t} \big( p V^\gamma \big) = 0 \qquad \quad \gamma \equiv \frac{C_P}{C_V} @@ -148,10 +147,10 @@ In a fully ionized plasma, $N = 3$. The density $n \propto 1/V$, so since $p V^\gamma$ is constant in time, -for some (known) constant $C$: +for some constant $C$: $$\begin{aligned} - \dv{t} \Big( \frac{p}{n^\gamma} \Big) = 0 + \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p}{n^\gamma} \Big) = 0 \quad \implies \quad p = C n^\gamma \end{aligned}$$ @@ -161,13 +160,99 @@ giving us a set of 16 equations for 16 unknowns: $$\begin{aligned} \boxed{ - p_i = C_i n_i^\gamma + \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p_i}{n_i^\gamma} \Big) + = 0 \qquad \quad - p_e = C_e n_e^\gamma + \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p_e}{n_e^\gamma} \Big) + = 0 } \end{aligned}$$ +## Fluid drifts + +The momentum equations reduce to the following +if we assume the flow is steady $\pdv*{\vb{u}}{t} = 0$, +and neglect electron-ion momentum transfer on the right: + +$$\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, |