Migrate from 'jekyll-katex' to 'kramdown-math-sskatex'

1 files changed, 22 insertions, 22 deletions

diff --git a/source/know/concept/rayleigh-plesset-equation/index.md b/source/know/concept/rayleigh-plesset-equation/index.md
index 4718683..b3ec8f5 100644
--- a/source/know/concept/rayleigh-plesset-equation/index.md
+++ b/source/know/concept/rayleigh-plesset-equation/index.md
@@ -16,7 +16,7 @@ Notably, it leads to [cavitation](/know/concept/cavitation/).
 
 Consider the main
 [Navier-Stokes equation](/know/concept/navier-stokes-equations/)
-for the velocity field $\va{v}$:
+for the velocity field $$\va{v}$$:
 
 $$\begin{aligned}
     \frac{\mathrm{D} \va{v}}{\mathrm{D} t}
@@ -24,9 +24,9 @@ $$\begin{aligned}
     = - \frac{\nabla p}{\rho} + \nu \nabla^2 \va{v}
 \end{aligned}$$
 
-We make the ansatz $\va{v} = v(r, t) \vu{e}_r$,
-where $\vu{e}_r$ is the basis vector;
-in other words, we demand that the only spatial variation of the flow is in $r$.
+We make the ansatz $$\va{v} = v(r, t) \vu{e}_r$$,
+where $$\vu{e}_r$$ is the basis vector;
+in other words, we demand that the only spatial variation of the flow is in $$r$$.
 The above equation then becomes:
 
 $$\begin{aligned}
@@ -44,17 +44,17 @@ $$\begin{aligned}
     = 0
 \end{aligned}$$
 
-This is only satisfied if $r^2 v$ is constant with respect to $r$,
-leading us to a solution $v(r)$ given by:
+This is only satisfied if $$r^2 v$$ is constant with respect to $$r$$,
+leading us to a solution $$v(r)$$ given by:
 
 $$\begin{aligned}
     v(r)
     = \frac{C(t)}{r^2}
 \end{aligned}$$
 
-Where $C(t)$ is an unknown function that does not depend on $r$.
+Where $$C(t)$$ is an unknown function that does not depend on $$r$$.
 We then insert this result in the main Navier-Stokes equation,
-and isolate it for $\ipdv{p}{r}$, yielding:
+and isolate it for $$\ipdv{p}{r}$$, yielding:
 
 $$\begin{aligned}
     \pdv{p}{r}
@@ -63,8 +63,8 @@ $$\begin{aligned}
     = - \rho \bigg( \frac{1}{r^2} C' - \frac{2}{r^5} C^2 \bigg)
 \end{aligned}$$
 
-Integrating this with respect to $r$ yields the following expression for $p$,
-where $p_\infty(t)$ is the (possibly time-dependent) pressure at $r = \infty$:
+Integrating this with respect to $$r$$ yields the following expression for $$p$$,
+where $$p_\infty(t)$$ is the (possibly time-dependent) pressure at $$r = \infty$$:
 
 $$\begin{aligned}
     p(r)
@@ -73,7 +73,7 @@ $$\begin{aligned}
 
 From the definition of [viscosity](/know/concept/viscosity/),
 we know that the normal [stress](/know/concept/cauchy-stress-tensor/)
-$\sigma_{rr}$ in the liquid is given by:
+$$\sigma_{rr}$$ in the liquid is given by:
 
 $$\begin{aligned}
     \sigma_{rr}(r)
@@ -81,11 +81,11 @@ $$\begin{aligned}
 \end{aligned}$$
 
 We now consider a spherical bubble
-with radius $R(t)$ and interior pressure $P(t)$ along its surface.
-Since we know the liquid pressure $p(r)$,
-we can find $P$ from $\sigma_{rr}(r)$.
+with radius $$R(t)$$ and interior pressure $$P(t)$$ along its surface.
+Since we know the liquid pressure $$p(r)$$,
+we can find $$P$$ from $$\sigma_{rr}(r)$$.
 Furthermore, to include the effects of surface tension, we simply add
-the [Young-Laplace law](/know/concept/young-laplace-law/) to $P$:
+the [Young-Laplace law](/know/concept/young-laplace-law/) to $$P$$:
 
 $$\begin{aligned}
     P
@@ -93,18 +93,18 @@ $$\begin{aligned}
     = p(R) - 2 \rho \nu \Big( \frac{-2}{R^3} C \Big) + \alpha \frac{2}{R}
 \end{aligned}$$
 
-We isolate this for $p(R)$, and equate it to
-our expression for $p(r)$
-at the surface $r\!=\!R$:
+We isolate this for $$p(R)$$, and equate it to
+our expression for $$p(r)$$
+at the surface $$r\!=\!R$$:
 
 $$\begin{aligned}
     P - \rho \nu \frac{4}{R^3} C - \alpha \frac{2}{R}
     = p_\infty + \rho \bigg( \frac{1}{R} C' - \frac{1}{2 R^4} C^2 \bigg)
 \end{aligned}$$
 
-Isolating for $P$,
-and inserting the fact that $R'(t) = v(t)$,
-such that $C = r^2 v = R^2 R'$,
+Isolating for $$P$$,
+and inserting the fact that $$R'(t) = v(t)$$,
+such that $$C = r^2 v = R^2 R'$$,
 yields:
 
 $$\begin{aligned}
@@ -115,7 +115,7 @@ $$\begin{aligned}
     &= p_\infty + \rho \bigg( 2 (R')^2 + R R'' - \frac{1}{2} (R')^2 + \nu \frac{4}{R} R' \bigg) + \alpha \frac{2}{R}
 \end{aligned}$$
 
-Rearranging this and defining $\Delta p \equiv P - p_\infty$
+Rearranging this and defining $$\Delta p \equiv P - p_\infty$$
 leads to the Rayleigh-Plesset equation:
 
 $$\begin{aligned}