From 16555851b6514a736c5c9d8e73de7da7fc9b6288 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 20 Oct 2022 18:25:31 +0200
Subject: Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

---
 source/know/concept/two-fluid-equations/index.md | 84 ++++++++++++------------
 1 file changed, 42 insertions(+), 42 deletions(-)

(limited to 'source/know/concept/two-fluid-equations/index.md')

diff --git a/source/know/concept/two-fluid-equations/index.md b/source/know/concept/two-fluid-equations/index.md
index 425d50d..e224e3e 100644
--- a/source/know/concept/two-fluid-equations/index.md
+++ b/source/know/concept/two-fluid-equations/index.md
@@ -11,13 +11,13 @@ 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").
+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,
+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/):
 
 $$\begin{aligned}
@@ -27,19 +27,19 @@ $$\begin{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:
+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.
+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/) $\hat{P}$,
+the [Cauchy stress tensor](/know/concept/cauchy-stress-tensor/) $$\hat{P}$$,
 leading to the following two equations:
 
 $$\begin{aligned}
@@ -50,7 +50,7 @@ $$\begin{aligned}
         &= 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.
+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**:
@@ -67,12 +67,12 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where $f_{ie}$ is the mean frequency at which an ion collides with electrons,
-and vice versa for $f_{ei}$.
+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 $p$:
+by the gradient $$- \nabla p$$ of a scalar pressure $$p$$:
 
 $$\begin{aligned}
     m_i n_i \frac{\mathrm{D} \vb{u}_i}{\mathrm{D} t}
@@ -83,8 +83,8 @@ $$\begin{aligned}
 \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 $S$:
+In other words, the rate at which particles enter/leave a volume $$V$$
+must be equal to the flux through the enclosing surface $$S$$:
 
 $$\begin{aligned}
     0
@@ -93,7 +93,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Where we have used the divergence theorem.
-Since $V$ is arbitrary, we can remove the integrals,
+Since $$V$$ is arbitrary, we can remove the integrals,
 leading to the following **continuity equations**:
 
 $$\begin{aligned}
@@ -107,7 +107,7 @@ $$\begin{aligned}
 \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 $p_e$.
+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
@@ -121,7 +121,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Now we have 14 equations, so we need 2 more, for the pressures $p_i$ and $p_e$.
+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),
@@ -135,14 +135,14 @@ $$\begin{aligned}
     = \frac{N + 2}{N}
 \end{aligned}$$
 
-Where $\gamma$ is the *heat capacity ratio*,
-and can be calculated from the number of degrees of freedom $N$
+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, $N = 3$.
+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 $C$:
+The density $$n \propto 1/V$$,
+so since $$p V^\gamma$$ is constant in time,
+for some constant $$C$$:
 
 $$\begin{aligned}
     \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p}{n^\gamma} \Big) = 0
@@ -163,8 +163,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Note that from the relation $p = C n^\gamma$,
-we can calculate the $\nabla p$ term in the momentum equation,
+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:
 
 $$\begin{aligned}
@@ -177,13 +177,13 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Note that the ideal gas law was not used immediately,
-to allow for $\gamma \neq 1$.
+to allow for $$\gamma \neq 1$$.
 
 
 ## Fluid drifts
 
 The momentum equations reduce to the following
-if we assume the flow is steady $\ipdv{\vb{u}}{t} = 0$,
+if we assume the flow is steady $$\ipdv{\vb{u}}{t} = 0$$,
 and neglect electron-ion momentum transfer on the right:
 
 $$\begin{aligned}
@@ -194,9 +194,9 @@ $$\begin{aligned}
     &\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:
+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:
 
 $$\begin{aligned}
     0
@@ -205,9 +205,9 @@ $$\begin{aligned}
     &= 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 $\vb{u}_\perp$:
+Isolating for $$\vb{u}_\perp$$ tells us
+that the fluids drifts perpendicularly to $$\vb{B}$$,
+with velocity $$\vb{u}_\perp$$:
 
 $$\begin{aligned}
     \vb{u}_\perp
@@ -216,11 +216,11 @@ $$\begin{aligned}
 \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 $\vb{v}_E$
+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** $\vb{v}_D$:
+and the second term is called the **diamagnetic drift** $$\vb{v}_D$$:
 
 $$\begin{aligned}
     \boxed{
@@ -236,9 +236,9 @@ $$\begin{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:
+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:
 
 $$\begin{aligned}
     \vb{J}
@@ -247,7 +247,7 @@ $$\begin{aligned}
     = \frac{\vb{B} \cross \nabla (p_i + p_e)}{B^2}
 \end{aligned}$$
 
-Using the ideal gas law $p = k_B T n$,
+Using the ideal gas law $$p = k_B T n$$,
 this can be rewritten as follows:
 
 $$\begin{aligned}
@@ -255,7 +255,7 @@ $$\begin{aligned}
     = 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,
+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,
-- 
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