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'

---
 .../know/concept/navier-stokes-equations/index.md  | 30 +++++++++++-----------
 1 file changed, 15 insertions(+), 15 deletions(-)

(limited to 'source/know/concept/navier-stokes-equations')

diff --git a/source/know/concept/navier-stokes-equations/index.md b/source/know/concept/navier-stokes-equations/index.md
index fd26860..964acda 100644
--- a/source/know/concept/navier-stokes-equations/index.md
+++ b/source/know/concept/navier-stokes-equations/index.md
@@ -33,10 +33,10 @@ $$\begin{aligned}
     = \va{f^*}
 \end{aligned}$$
 
-$\mathrm{D}/\mathrm{D}t$ is the [material derivative](/know/concept/material-derivative/),
-$\rho$ is the density, and $\va{f^*}$ is the effective force density,
-expressed in terms of an external body force $\va{f}$ (e.g. gravity)
-and the [Cauchy stress tensor](/know/concept/cauchy-stress-tensor/) $\hat{\sigma}$:
+$$\mathrm{D}/\mathrm{D}t$$ is the [material derivative](/know/concept/material-derivative/),
+$$\rho$$ is the density, and $$\va{f^*}$$ is the effective force density,
+expressed in terms of an external body force $$\va{f}$$ (e.g. gravity)
+and the [Cauchy stress tensor](/know/concept/cauchy-stress-tensor/) $$\hat{\sigma}$$:
 
 $$\begin{aligned}
     \va{f^*}
@@ -51,8 +51,8 @@ $$\begin{aligned}
     = - p \delta_{ij} + \eta (\nabla_i v_j + \nabla_j v_i)
 \end{aligned}$$
 
-Where $\eta$ is the dynamic viscosity.
-Inserting this, we calculate $\nabla \cdot \hat{\sigma}^\top$ in index notation:
+Where $$\eta$$ is the dynamic viscosity.
+Inserting this, we calculate $$\nabla \cdot \hat{\sigma}^\top$$ in index notation:
 
 $$\begin{aligned}
     \big( \nabla \cdot \hat{\sigma}^\top \big)_i
@@ -62,7 +62,7 @@ $$\begin{aligned}
     &= - \nabla_i p + \eta \nabla_i \sum_{j} \nabla_j v_j + \eta \sum_{j} \nabla_j^2 v_i
 \end{aligned}$$
 
-Thanks to incompressibility $\nabla \cdot \va{v} = 0$,
+Thanks to incompressibility $$\nabla \cdot \va{v} = 0$$,
 the middle term vanishes, leaving us with:
 
 $$\begin{aligned}
@@ -70,7 +70,7 @@ $$\begin{aligned}
     = \va{f} - \nabla p + \eta \nabla^2 \va{v}
 \end{aligned}$$
 
-We assume that the only body force is gravity $\va{f} = \rho \va{g}$.
+We assume that the only body force is gravity $$\va{f} = \rho \va{g}$$.
 Newton's second law then becomes:
 
 $$\begin{aligned}
@@ -78,8 +78,8 @@ $$\begin{aligned}
     = \rho \va{g} - \nabla p + \eta \nabla^2 \va{v}
 \end{aligned}$$
 
-Dividing by $\rho$, and replacing $\eta$
-with the kinematic viscosity $\nu = \eta/\rho$,
+Dividing by $$\rho$$, and replacing $$\eta$$
+with the kinematic viscosity $$\nu = \eta/\rho$$,
 yields the main equation:
 
 $$\begin{aligned}
@@ -90,7 +90,7 @@ $$\begin{aligned}
 \end{aligned}$$
 
 Finally, we can optionally allow incompressible fluids
-with an inhomogeneous "lumpy" density $\rho$,
+with an inhomogeneous "lumpy" density $$\rho$$,
 by demanding conservation of mass,
 just like for the Euler equations:
 
@@ -115,11 +115,11 @@ $$\begin{aligned}
     }
 \end{aligned}$$
 
-Due to the definition of viscosity $\nu$ as the molecular "stickiness",
-we have boundary conditions for the velocity field $\va{v}$:
-at any interface, $\va{v}$ must be continuous.
+Due to the definition of viscosity $$\nu$$ as the molecular "stickiness",
+we have boundary conditions for the velocity field $$\va{v}$$:
+at any interface, $$\va{v}$$ must be continuous.
 Likewise, Newton's third law demands that the normal component
-of stress $\hat{\sigma} \cdot \vu{n}$ is continuous there.
+of stress $$\hat{\sigma} \cdot \vu{n}$$ is continuous there.
 
 
 
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