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/reynolds-number/index.md | 38 ++++++++++++++--------------
 1 file changed, 19 insertions(+), 19 deletions(-)

(limited to 'source/know/concept/reynolds-number/index.md')

diff --git a/source/know/concept/reynolds-number/index.md b/source/know/concept/reynolds-number/index.md
index aa9559c..9ae4f4b 100644
--- a/source/know/concept/reynolds-number/index.md
+++ b/source/know/concept/reynolds-number/index.md
@@ -12,7 +12,7 @@ layout: "concept"
 The [Navier-Stokes equations](/know/concept/navier-stokes-equations/)
 are infamously tricky to solve,
 so we would like a way to qualitatively predict
-the behaviour of a fluid without needing the flow $\va{v}$.
+the behaviour of a fluid without needing the flow $$\va{v}$$.
 Consider the main equation:
 
 $$\begin{aligned}
@@ -20,14 +20,14 @@ $$\begin{aligned}
     = - \frac{\nabla p}{\rho} + \nu \nabla^2 \va{v}
 \end{aligned}$$
 
-In this case, the gravity term $\va{g}$
+In this case, the gravity term $$\va{g}$$
 has been absorbed into the pressure term:
-$p \to p\!+\!\rho \Phi$,
-where $\Phi$ is the gravitational scalar potential,
-i.e. $\va{g} = - \nabla \Phi$.
+$$p \to p\!+\!\rho \Phi$$,
+where $$\Phi$$ is the gravitational scalar potential,
+i.e. $$\va{g} = - \nabla \Phi$$.
 
-Let us introduce the dimensionless variables $\va{v}'$, $\va{r}'$, $t'$ and $p'$,
-where $U$ and $L$ are respectively a characteristic velocity and length
+Let us introduce the dimensionless variables $$\va{v}'$$, $$\va{r}'$$, $$t'$$ and $$p'$$,
+where $$U$$ and $$L$$ are respectively a characteristic velocity and length
 of the system at hand:
 
 $$\begin{aligned}
@@ -57,7 +57,7 @@ $$\begin{aligned}
     = - \frac{U^2}{L} \nabla' p' + \frac{U \nu}{L^2} \nabla'^2 \va{v}'
 \end{aligned}$$
 
-After dividing out $U^2/L$,
+After dividing out $$U^2/L$$,
 we arrive at the form of the original equation again:
 
 $$\begin{aligned}
@@ -66,7 +66,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 The constant factor of the last term
-leads to the definition of the **Reynolds number** $\mathrm{Re}$:
+leads to the definition of the **Reynolds number** $$\mathrm{Re}$$:
 
 $$\begin{aligned}
     \boxed{
@@ -75,17 +75,17 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-If we choose $U$ and $L$ appropriately for a given system,
+If we choose $$U$$ and $$L$$ appropriately for a given system,
 the Reynolds number allows us to predict the general trends.
 It can be regarded as the inverse of an "effective viscosity":
-when $\mathrm{Re}$ is large, viscosity only has a minor role,
-but when $\mathrm{Re}$ is small, it dominates the dynamics.
+when $$\mathrm{Re}$$ is large, viscosity only has a minor role,
+but when $$\mathrm{Re}$$ is small, it dominates the dynamics.
 
 Another way is thus to see the Reynolds number
 as the characteristic ratio between the advective term
 (see [material derivative](/know/concept/material-derivative/))
 to the [viscosity](/know/concept/viscosity/) term,
-since $\va{v} \sim U$:
+since $$\va{v} \sim U$$:
 
 $$\begin{aligned}
      \mathrm{Re}
@@ -94,7 +94,7 @@ $$\begin{aligned}
      = \frac{U L}{\nu}
 \end{aligned}$$
 
-In other words, $\mathrm{Re}$
+In other words, $$\mathrm{Re}$$
 describes the relative strength of intertial and viscous forces.
 Returning to the dimensionless Navier-Stokes equation:
 
@@ -103,7 +103,7 @@ $$\begin{aligned}
     = - \nabla' p' + \frac{1}{\mathrm{Re}} \nabla'^2 \va{v}'
 \end{aligned}$$
 
-For large $\mathrm{Re} \gg 1$,
+For large $$\mathrm{Re} \gg 1$$,
 we can neglect the latter term,
 such that redimensionalizing yields:
 
@@ -120,10 +120,10 @@ for an ideal fluid, i.e. a fluid without viscosity.
 ## Stokes flow
 
 A notable case is so-called **Stokes flow** or **creeping flow**,
-meaning flow at $\mathrm{Re} \ll 1$.
+meaning flow at $$\mathrm{Re} \ll 1$$.
 In this limit, the Navier-Stokes equations can be linearized:
-since $\mathrm{Re}$ is the advective-to-viscous ratio,
-$\mathrm{Re} \ll 1$ implies that we can ignore the advective term, leaving:
+since $$\mathrm{Re}$$ is the advective-to-viscous ratio,
+$$\mathrm{Re} \ll 1$$ implies that we can ignore the advective term, leaving:
 
 $$\begin{aligned}
     \boxed{
@@ -135,7 +135,7 @@ $$\begin{aligned}
 This equation is called the **unsteady Stokes equation**.
 Usually, however, such flows are assumed to be steady
 (i.e. time-invariant), leading to the **steady Stokes equation**,
-with $\eta = \rho \nu$:
+with $$\eta = \rho \nu$$:
 
 $$\begin{aligned}
     \boxed{
-- 
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