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/bernoullis-theorem/index.md | 26 ++++++++++++-------------
 1 file changed, 13 insertions(+), 13 deletions(-)

(limited to 'source/know/concept/bernoullis-theorem')

diff --git a/source/know/concept/bernoullis-theorem/index.md b/source/know/concept/bernoullis-theorem/index.md
index 12bd0ca..6b933d2 100644
--- a/source/know/concept/bernoullis-theorem/index.md
+++ b/source/know/concept/bernoullis-theorem/index.md
@@ -10,8 +10,8 @@ layout: "concept"
 ---
 
 For inviscid fluids, **Bernuilli's theorem** states
-that an increase in flow velocity $\va{v}$ is paired
-with a decrease in pressure $p$ and/or potential energy.
+that an increase in flow velocity $$\va{v}$$ is paired
+with a decrease in pressure $$p$$ and/or potential energy.
 For a qualitative argument, look no further than
 one of the [Euler equations](/know/concept/euler-equations/),
 with a [material derivative](/know/concept/material-derivative/):
@@ -22,16 +22,16 @@ $$\begin{aligned}
     = \va{g} - \frac{\nabla p}{\rho}
 \end{aligned}$$
 
-Assuming that $\va{v}$ is constant in $t$,
-it becomes clear that a higher $\va{v}$ requires a lower $p$.
+Assuming that $$\va{v}$$ is constant in $$t$$,
+it becomes clear that a higher $$\va{v}$$ requires a lower $$p$$.
 
 
 ## Simple form
 
 For an incompressible fluid
-with a time-independent velocity field $\va{v}$ (i.e. **steady flow**),
+with a time-independent velocity field $$\va{v}$$ (i.e. **steady flow**),
 Bernoulli's theorem formally states that the
-**Bernoulli head** $H$ is constant along a streamline:
+**Bernoulli head** $$H$$ is constant along a streamline:
 
 $$\begin{aligned}
     \boxed{
@@ -40,8 +40,8 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Where $\Phi$ is the gravitational potential, such that $\va{g} = - \nabla \Phi$.
-To prove this theorem, we take the material derivative of $H$:
+Where $$\Phi$$ is the gravitational potential, such that $$\va{g} = - \nabla \Phi$$.
+To prove this theorem, we take the material derivative of $$H$$:
 
 $$\begin{aligned}
     \frac{\mathrm{D} H}{\mathrm{D} t}
@@ -63,7 +63,7 @@ $$\begin{aligned}
     + \va{v} \cdot \big( \va{g} + \nabla \Phi \big) + \va{v} \cdot \Big( \frac{\nabla p}{\rho} - \frac{\nabla p}{\rho} \Big)
 \end{aligned}$$
 
-Using the fact that $\va{g} = - \nabla \Phi$,
+Using the fact that $$\va{g} = - \nabla \Phi$$,
 we are left with the following equation:
 
 $$\begin{aligned}
@@ -72,12 +72,12 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Assuming that the flow is steady, both derivatives vanish,
-leading us to the conclusion that $H$ is conserved along the streamline.
+leading us to the conclusion that $$H$$ is conserved along the streamline.
 
 In fact, there exists **Bernoulli's stronger theorem**,
-which states that $H$ is constant *everywhere* in regions with
-zero [vorticity](/know/concept/vorticity/) $\va{\omega} = 0$.
-For a proof, see the derivation of $\va{\omega}$'s equation of motion.
+which states that $$H$$ is constant *everywhere* in regions with
+zero [vorticity](/know/concept/vorticity/) $$\va{\omega} = 0$$.
+For a proof, see the derivation of $$\va{\omega}$$'s equation of motion.
 
 
 ## References
-- 
cgit v1.2.3