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/capillary-action/index.md | 42 +++++++++++++--------------
 1 file changed, 21 insertions(+), 21 deletions(-)

(limited to 'source/know/concept/capillary-action')

diff --git a/source/know/concept/capillary-action/index.md b/source/know/concept/capillary-action/index.md
index 1ee3cd4..4b9e76c 100644
--- a/source/know/concept/capillary-action/index.md
+++ b/source/know/concept/capillary-action/index.md
@@ -16,7 +16,7 @@ It occurs when the [Laplace pressure](/know/concept/young-laplace-law/)
 from surface tension is much larger in magnitude than the
 [hydrostatic pressure](/know/concept/hydrostatic-pressure/) from gravity.
 
-Consider a spherical droplet of liquid with radius $R$.
+Consider a spherical droplet of liquid with radius $$R$$.
 The hydrostatic pressure difference
 between the top and bottom of the drop
 is much smaller than the Laplace pressure:
@@ -25,17 +25,17 @@ $$\begin{aligned}
     2 R \rho g \ll 2 \frac{\alpha}{R}
 \end{aligned}$$
 
-Where $\rho$ is the density of the liquid,
-$g$ is the acceleration due to gravity,
-and $\alpha$ is the energy cost per unit surface area.
+Where $$\rho$$ is the density of the liquid,
+$$g$$ is the acceleration due to gravity,
+and $$\alpha$$ is the energy cost per unit surface area.
 Rearranging the inequality yields:
 
 $$\begin{aligned}
     R^2 \ll \frac{\alpha}{\rho g}
 \end{aligned}$$
 
-From the right-hand side we define the **capillary length** $L_c$,
-so gravity is negligible if $R \ll L_c$:
+From the right-hand side we define the **capillary length** $$L_c$$,
+so gravity is negligible if $$R \ll L_c$$:
 
 $$\begin{aligned}
     \boxed{
@@ -44,10 +44,10 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-In general, for a system with characteristic length $L$,
+In general, for a system with characteristic length $$L$$,
 the relative strength of gravity compared to surface tension
-is described by the **Bond number** $\mathrm{Bo}$
-or **Eötvös number** $\mathrm{Eo}$:
+is described by the **Bond number** $$\mathrm{Bo}$$
+or **Eötvös number** $$\mathrm{Eo}$$:
 
 $$\begin{aligned}
     \boxed{
@@ -58,16 +58,16 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-The right-most side gives an alternative way of understanding $\mathrm{Bo}$:
-$m$ is the mass of a cube with side $L$, such that the numerator is the weight force,
+The right-most side gives an alternative way of understanding $$\mathrm{Bo}$$:
+$$m$$ is the mass of a cube with side $$L$$, such that the numerator is the weight force,
 and the denominator is the tension force of the surface.
-In any case, capillary action can be observed when $\mathrm{Bo \ll 1}$.
+In any case, capillary action can be observed when $$\mathrm{Bo \ll 1}$$.
 
 The most famous example of capillary action is **capillary rise**,
-where a liquid "climbs" upwards in a narrow vertical tube with radius $R$,
+where a liquid "climbs" upwards in a narrow vertical tube with radius $$R$$,
 apparently defying gravity.
 Assuming the liquid-air interface is a spherical cap
-with constant [curvature](/know/concept/curvature/) radius $R_c$,
+with constant [curvature](/know/concept/curvature/) radius $$R_c$$,
 then we know that the liquid is at rest
 when the hydrostatic pressure equals the Laplace pressure:
 
@@ -77,12 +77,12 @@ $$\begin{aligned}
     = 2 \alpha \frac{\cos\theta}{R}
 \end{aligned}$$
 
-Where $\theta$ is the liquid-tube contact angle,
-and we are neglecting variations of the height $h$ due to the curvature
+Where $$\theta$$ is the liquid-tube contact angle,
+and we are neglecting variations of the height $$h$$ due to the curvature
 (i.e. the [meniscus](/know/concept/meniscus/)).
-By isolating the above equation for $h$,
+By isolating the above equation for $$h$$,
 we arrive at **Jurin's law**,
-which predicts the height climbed by a liquid in a tube with radius $R$:
+which predicts the height climbed by a liquid in a tube with radius $$R$$:
 
 $$\begin{aligned}
     \boxed{
@@ -91,7 +91,7 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Depending on $\theta$, $h$ can be negative,
+Depending on $$\theta$$, $$h$$ can be negative,
 i.e. the liquid might descend below the ambient level.
 
 
@@ -106,7 +106,7 @@ $$\begin{aligned}
      \approx 2 \pi R (\alpha_{sg} - \alpha_{sl})
 \end{aligned}$$
 
-Where $\alpha_{sg}$ and $\alpha_{sl}$ are the energy costs
+Where $$\alpha_{sg}$$ and $$\alpha_{sl}$$ are the energy costs
 of the solid-gas and solid-liquid interfaces.
 Thanks to the [Young-Dupré relation](/know/concept/young-dupre-relation/),
 we can rewrite this as follows:
@@ -116,7 +116,7 @@ $$\begin{aligned}
      = 2 \alpha \cos\theta
 \end{aligned}$$
 
-Isolating this for $h$ simply yields Jurin's law again, as expected.
+Isolating this for $$h$$ simply yields Jurin's law again, as expected.
 
 
 
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