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/euler-equations/index.md | 46 ++++++++++++++--------------
 1 file changed, 23 insertions(+), 23 deletions(-)

(limited to 'source/know/concept/euler-equations')

diff --git a/source/know/concept/euler-equations/index.md b/source/know/concept/euler-equations/index.md
index ddaa27a..3730ea3 100644
--- a/source/know/concept/euler-equations/index.md
+++ b/source/know/concept/euler-equations/index.md
@@ -18,10 +18,10 @@ the surrounding assumptions about the fluid.
 
 ## Incompressible fluid
 
-In a fluid moving according to the velocity vield $\va{v}(\va{r}, t)$,
+In a fluid moving according to the velocity vield $$\va{v}(\va{r}, t)$$,
 the acceleration felt by a particle is given by
-the **material acceleration field** $\va{w}(\va{r}, t)$,
-which is the [material derivative](/know/concept/material-derivative/) of $\va{v}$:
+the **material acceleration field** $$\va{w}(\va{r}, t)$$,
+which is the [material derivative](/know/concept/material-derivative/) of $$\va{v}$$:
 
 $$\begin{aligned}
     \va{w}
@@ -38,20 +38,20 @@ $$\begin{aligned}
     = \va{f^*} \dd{V}
 \end{aligned}$$
 
-Where $\dd{m}$ and $\dd{V}$ are the particle's mass volume,
-and $\rho$ is the fluid density, which we assume, in this case, to be constant in space and time.
-Then the **effective force density** $\va{f^*}$ represents the net force-per-particle.
-By dividing the law by $\dd{V}$, we find:
+Where $$\dd{m}$$ and $$\dd{V}$$ are the particle's mass volume,
+and $$\rho$$ is the fluid density, which we assume, in this case, to be constant in space and time.
+Then the **effective force density** $$\va{f^*}$$ represents the net force-per-particle.
+By dividing the law by $$\dd{V}$$, we find:
 
 $$\begin{aligned}
     \rho \va{w}
     = \va{f^*}
 \end{aligned}$$
 
-Next, we want to find another expression for $\va{f^*}$.
-We know that the overall force $\va{F}$ on an arbitrary volume $V$ of the fluid
-is the sum of the gravity body force $\va{F}_g$,
-and the pressure contact force $\va{F}_p$ on the enclosing surface $S$.
+Next, we want to find another expression for $$\va{f^*}$$.
+We know that the overall force $$\va{F}$$ on an arbitrary volume $$V$$ of the fluid
+is the sum of the gravity body force $$\va{F}_g$$,
+and the pressure contact force $$\va{F}_p$$ on the enclosing surface $$S$$.
 Using the divergence theorem, we then find:
 
 $$\begin{aligned}
@@ -62,8 +62,8 @@ $$\begin{aligned}
     = \int_V \va{f^*} \dd{V}
 \end{aligned}$$
 
-Where $p(\va{r}, t)$ is the pressure field,
-and $\va{g}(\va{r}, t)$ is the gravitational acceleration field.
+Where $$p(\va{r}, t)$$ is the pressure field,
+and $$\va{g}(\va{r}, t)$$ is the gravitational acceleration field.
 Combining this with Newton's law, we find the following equation for the force density:
 
 $$\begin{aligned}
@@ -72,7 +72,7 @@ $$\begin{aligned}
     = \rho \va{g} - \nabla p
 \end{aligned}$$
 
-Dividing this by $\rho$,
+Dividing this by $$\rho$$,
 we get the first of the system of Euler equations:
 
 $$\begin{aligned}
@@ -85,7 +85,7 @@ $$\begin{aligned}
 
 The last ingredient is **incompressibility**:
 the same volume must simultaneously
-be flowing in and out of an arbitrary enclosure $S$.
+be flowing in and out of an arbitrary enclosure $$S$$.
 Then, by the divergence theorem:
 
 $$\begin{aligned}
@@ -94,7 +94,7 @@ $$\begin{aligned}
     = \int_V \nabla \cdot \va{v} \dd{V}
 \end{aligned}$$
 
-Since $S$ and $V$ are arbitrary,
+Since $$S$$ and $$V$$ are arbitrary,
 the integrand must vanish by itself everywhere:
 
 $$\begin{aligned}
@@ -103,10 +103,10 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Combining this with the equation for $\va{w}$,
+Combining this with the equation for $$\va{w}$$,
 we get a system of two coupled differential equations:
 these are the Euler equations for an incompressible fluid
-with spatially uniform density $\rho$:
+with spatially uniform density $$\rho$$:
 
 $$\begin{aligned}
     \boxed{
@@ -119,13 +119,13 @@ $$\begin{aligned}
 \end{aligned}$$
 
 The above form is straightforward to generalize to incompressible fluids
-with non-uniform spatial densities $\rho(\va{r}, t)$.
+with non-uniform spatial densities $$\rho(\va{r}, t)$$.
 In other words, these fluids are "lumpy" (variable density),
 but the size of their lumps does not change (incompressibility).
 
 To update the equations, we demand conservation of mass:
-the mass evolution of a volume $V$
-is equal to the mass flow through its boundary $S$.
+the mass evolution of a volume $$V$$
+is equal to the mass flow through its boundary $$S$$.
 Applying the divergence theorem again:
 
 $$\begin{aligned}
@@ -134,7 +134,7 @@ $$\begin{aligned}
     = \int_V \dv{\rho}{t} + \nabla \cdot (\rho \va{v}) \dd{V}
 \end{aligned}$$
 
-Since $V$ is arbitrary, the integrand must be zero.
+Since $$V$$ is arbitrary, the integrand must be zero.
 This leads to the following **continuity equation**,
 to which we apply a vector identity:
 
@@ -172,7 +172,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Usually, however, when discussing incompressible fluids,
-$\rho$ is assumed to be spatially uniform,
+$$\rho$$ is assumed to be spatially uniform,
 in which case the latter equation is trivially satisfied.
 
 
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