From 075683cdf4588fe16f41d9f7b46b9720b42b2553 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Wed, 17 Jul 2024 10:01:43 +0200
Subject: Improve knowledge base

---
 source/know/concept/two-fluid-equations/index.md | 47 ++++++++++++++----------
 1 file changed, 28 insertions(+), 19 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 e224e3e..a00a2f9 100644
--- a/source/know/concept/two-fluid-equations/index.md
+++ b/source/know/concept/two-fluid-equations/index.md
@@ -98,15 +98,17 @@ leading to the following **continuity equations**:
 
 $$\begin{aligned}
     \boxed{
-        \pdv{n_i}{t} + \nabla \cdot (n_i \vb{u}_i)
-        = 0
-        \qquad \quad
-        \pdv{n_e}{t} + \nabla \cdot (n_e \vb{u}_e)
-        = 0
+        \begin{aligned}
+            0
+            &= \pdv{n_i}{t} + \nabla \cdot (n_i \vb{u}_i)
+            \\
+            0
+            &= \pdv{n_e}{t} + \nabla \cdot (n_e \vb{u}_e)
+        \end{aligned}
     }
 \end{aligned}$$
 
-These are 8 equations (2 scalar continuity, 2 vector momentum),
+These are 8 equations (2 scalars for continuity, 2 vectors for 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/),
@@ -115,9 +117,13 @@ in particular Faraday's and Ampère's law
 
 $$\begin{aligned}
     \boxed{
-        \nabla \cross \vb{E} = - \pdv{\vb{B}}{t}
-        \qquad \quad
-        \nabla \cross \vb{B} = \mu_0 \Big( n_i q_i \vb{u}_i + n_e q_e \vb{u}_e + \varepsilon_0 \pdv{\vb{E}}{t} \Big)
+        \begin{aligned}
+            \nabla \cross \vb{E}
+            &= - \pdv{\vb{B}}{t}
+            \\
+            \nabla \cross \vb{B}
+            &= \mu_0 \Big( n_i q_i \vb{u}_i + n_e q_e \vb{u}_e + \varepsilon_0 \pdv{\vb{E}}{t} \Big)
+        \end{aligned}
     }
 \end{aligned}$$
 
@@ -129,7 +135,7 @@ it turns out that:
 
 $$\begin{aligned}
     \frac{\mathrm{D}}{\mathrm{D} t} \big( p V^\gamma \big) = 0
-    \qquad \quad
+    \qquad \qquad
     \gamma
     \equiv \frac{C_P}{C_V}
     = \frac{N + 2}{N}
@@ -146,7 +152,7 @@ for some constant $$C$$:
 
 $$\begin{aligned}
     \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p}{n^\gamma} \Big) = 0
-    \quad \implies \quad
+    \qquad \implies \qquad
     p = C n^\gamma
 \end{aligned}$$
 
@@ -155,11 +161,13 @@ giving us a set of 16 equations for 16 unknowns:
 
 $$\begin{aligned}
     \boxed{
-        \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p_i}{n_i^\gamma} \Big)
-        = 0
-        \qquad \quad
-        \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p_e}{n_e^\gamma} \Big)
-        = 0
+        \begin{aligned}
+            0
+            &= \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p_i}{n_i^\gamma} \Big)
+            \\
+            0
+            &= \frac{\mathrm{D}}{\mathrm{D} t} \Big( \frac{p_e}{n_e^\gamma} \Big)
+        \end{aligned}
     }
 \end{aligned}$$
 
@@ -169,15 +177,16 @@ using simple differentiation and the ideal gas law:
 
 $$\begin{aligned}
     p = C n^\gamma
-    \quad \implies \quad
+    \qquad \implies \qquad
     \nabla p
     = \gamma \frac{C n^{\gamma}}{n} \nabla n
     = \gamma p \frac{\nabla n}{n}
     = \gamma k_B T \nabla n
 \end{aligned}$$
 
-Note that the ideal gas law was not used immediately,
-to allow for $$\gamma \neq 1$$.
+Note that we waited until now to use the ideal gas law,
+in order to include the case $$\gamma \neq 1$$.
+
 
 
 ## Fluid drifts
-- 
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