From 78b0c780b1d2c3d31cc092d52e4dfc38928e7c6b Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Thu, 21 Oct 2021 15:46:15 +0200
Subject: Expand knowledge base

---
 content/know/concept/two-fluid-equations/index.pdc | 101 +++++++++++++++++++--
 1 file changed, 93 insertions(+), 8 deletions(-)

(limited to 'content/know/concept/two-fluid-equations')

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,
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