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/cavitation/index.md | 36 ++++++++++++++++-----------------
 1 file changed, 18 insertions(+), 18 deletions(-)

(limited to 'source/know/concept/cavitation')

diff --git a/source/know/concept/cavitation/index.md b/source/know/concept/cavitation/index.md
index 346edc4..679e66f 100644
--- a/source/know/concept/cavitation/index.md
+++ b/source/know/concept/cavitation/index.md
@@ -20,11 +20,11 @@ the [Rayleigh-Plesset equation](/know/concept/rayleigh-plesset-equation/)
 for an inviscid liquid without surface tension.
 Note that the RP equation assumes incompressibility.
 
-We assume that the whole liquid is at a constant pressure $p_\infty$,
-and the bubble is empty, such that the interface pressure $P = 0$,
-meaning $\Delta p = - p_\infty$.
-At first, the radius is stationary $R'(0) = 0$,
-and given by a constant $R(0) = a$.
+We assume that the whole liquid is at a constant pressure $$p_\infty$$,
+and the bubble is empty, such that the interface pressure $$P = 0$$,
+meaning $$\Delta p = - p_\infty$$.
+At first, the radius is stationary $$R'(0) = 0$$,
+and given by a constant $$R(0) = a$$.
 The simple Rayleigh-Plesset equation is then:
 
 $$\begin{aligned}
@@ -32,7 +32,7 @@ $$\begin{aligned}
     = - \frac{p_\infty}{\rho}
 \end{aligned}$$
 
-To solve it, we multiply both sides by $R^2 R'$
+To solve it, we multiply both sides by $$R^2 R'$$
 and rewrite it in the following way:
 
 $$\begin{aligned}
@@ -44,7 +44,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 It is then straightforward to integrate both sides
-with respect to time $\tau$, from $0$ to $t$:
+with respect to time $$\tau$$, from $$0$$ to $$t$$:
 
 $$\begin{aligned}
     - \frac{2 p_\infty}{3 \rho} \int_0^t \dv{}{\tau}\Big( R^3 \Big) \dd{\tau}
@@ -58,7 +58,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Rearranging this equation yields the following expression
-for the derivative $R'$:
+for the derivative $$R'$$:
 
 $$\begin{aligned}
     (R')^2
@@ -66,10 +66,10 @@ $$\begin{aligned}
 \end{aligned}$$
 
 This equation is nasty to integrate.
-The trick is to invert $R(t)$ into $t(R)$,
+The trick is to invert $$R(t)$$ into $$t(R)$$,
 and, because we are only interested in collapse,
-we just need to consider the case $R' < 0$.
-The time of a given radius $R$ is then as follows,
+we just need to consider the case $$R' < 0$$.
+The time of a given radius $$R$$ is then as follows,
 where we are using slightly sloppy notation:
 
 $$\begin{aligned}
@@ -79,8 +79,8 @@ $$\begin{aligned}
     = \int_{R}^{a} \frac{\dd{R}}{R'}
 \end{aligned}$$
 
-The minus comes from the constraint that $R' < 0$, but $t \ge 0$.
-We insert the expression for $R'$:
+The minus comes from the constraint that $$R' < 0$$, but $$t \ge 0$$.
+We insert the expression for $$R'$$:
 
 $$\begin{aligned}
     t
@@ -89,9 +89,9 @@ $$\begin{aligned}
 \end{aligned}$$
 
 This integral needs to be looked up,
-and involves the hypergeometric function ${}_2 F_1$.
-However, we only care about *collapse*, which is when $R = 0$.
-The time $t_0$ at which this occurs is:
+and involves the hypergeometric function $${}_2 F_1$$.
+However, we only care about *collapse*, which is when $$R = 0$$.
+The time $$t_0$$ at which this occurs is:
 
 $$\begin{aligned}
     t_0
@@ -101,9 +101,9 @@ $$\begin{aligned}
 
 With our assumptions, a bubble will always collapse.
 However, unsurprisingly, reality turns out to be more complicated:
-as $R \to 0$, the interface velocity $R' \to \infty$.
+as $$R \to 0$$, the interface velocity $$R' \to \infty$$.
 By looking at the derivation of the Rayleigh-Plesset equation,
-it can be shown that the pressure just outside the bubble diverges due to $R'$.
+it can be shown that the pressure just outside the bubble diverges due to $$R'$$.
 This drastically changes the liquid's properties, and breaks our assumptions.
 
 
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