From cc391ce3b9867d88d124e147931d33be34e756fc Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Mon, 14 Sep 2026 18:11:38 +0200
Subject: Improve knowledge base

---
 source/know/concept/prandtl-equations/index.md | 58 +++++++++++++-------------
 1 file changed, 28 insertions(+), 30 deletions(-)

(limited to 'source/know/concept/prandtl-equations/index.md')

diff --git a/source/know/concept/prandtl-equations/index.md b/source/know/concept/prandtl-equations/index.md
index d82657c..8f5c4d0 100644
--- a/source/know/concept/prandtl-equations/index.md
+++ b/source/know/concept/prandtl-equations/index.md
@@ -11,13 +11,12 @@ layout: "concept"
 
 In fluid dynamics, the **Prandtl equations** or **boundary layer equations**
 describe the movement of a [viscous](/know/concept/viscosity/) fluid
-with a large [Reynolds number](/know/concept/reynolds-number/) $$\mathrm{Re} \gg 1$$
-close to a solid surface.
+with a large [Reynolds number](/know/concept/reynolds-number/)
+$$\mathrm{Re} \gg 1$$ close to a solid surface.
 
-Fluids with a large Reynolds number
-are often approximated as having zero viscosity,
-since the simpler [Euler equations](/know/concept/euler-equations)
-can then be used instead of the [Navier-Stokes equations](/know/concept/navier-stokes-equations/).
+Fluids with a large Reynolds number are often approximated as having zero viscosity,
+since the simpler [Euler equations](/know/concept/euler-equations) can then be used
+instead of the [Navier-Stokes equations](/know/concept/navier-stokes-equations/).
 
 However, in reality, a viscous fluid obeys the *no-slip* boundary condition:
 at every solid surface the local velocity must be zero.
@@ -29,9 +28,9 @@ This is in contrast to the ideal flow far away from the surface.
 We consider a simple theoretical case in 2D:
 a large flat surface located at $$y = 0$$ for all $$x \in \mathbb{R}$$,
 with a fluid *trying* to flow parallel to it at $$U$$.
-The 2D treatment can be justified by assuming that everything is constant in the $$z$$-direction.
-We will not solve this case,
-but instead derive general equations
+The 2D treatment can be justified by assuming
+that everything is constant in the $$z$$-direction.
+We will not solve this case, but instead derive general equations
 to describe the flow close to a flat surface.
 
 At the wall, there is a very thin boundary layer of thickness $$\delta$$,
@@ -39,8 +38,7 @@ where the fluid is assumed to be completely stationary $$\va{v} = 0$$.
 We are mainly interested in the region $$\delta < y \ll L$$,
 where $$L$$ is the distance at which the fluid becomes practically ideal.
 This the so-called **slip-flow** region,
-in which the fluid is not stationary,
-but still viscosity-dominated.
+in which the fluid is not stationary, but still viscosity-dominated.
 
 In 2D, the steady Navier-Stokes equations are as follows,
 where the flow $$\va{v} = (v_x, v_y)$$:
@@ -66,13 +64,13 @@ Let $$\tilde{x}$$ and $$\tilde{y}$$ be dimenionless variables of order $$1$$:
 $$\begin{aligned}
     x
     = L \tilde{x}
-    \qquad \quad
+    \qquad \qquad
     y
     = \delta \tilde{x}
-    \qquad \quad
+    \qquad \qquad
     \pdv{}{x}
     = \frac{1}{L} \pdv{}{\tilde{x}}
-    \qquad \quad
+    \qquad \qquad
     \pdv{}{y}
     = \frac{1}{\delta} \pdv{}{\tilde{y}}
 \end{aligned}$$
@@ -85,10 +83,10 @@ by [Bernoulli's theorem](/know/concept/bernoullis-theorem/):
 $$\begin{aligned}
     v_x
     = U \tilde{v}_x
-    \qquad \quad
+    \qquad \qquad
     v_y
     = \frac{U \delta}{L} \tilde{v}_y
-    \qquad \quad
+    \qquad \qquad
     p
     = \rho U^2 \tilde{p}
 \end{aligned}$$
@@ -124,15 +122,15 @@ and that faster velocities $$U$$ give thinner layers.
 Furthermore, we expect *downstream thickening*:
 with distance $$x$$, viscous stresses slow down the slip-flow,
 leading to a gradual increase of $$\delta(x)$$.
-Some dimensional analysis thus yields the following estimate:
+Some dimensional analysis yields the following estimate:
 
 $$\begin{aligned}
     \delta
     \approx \sqrt{\frac{\nu x}{U}}
-    \sim \sqrt{\frac{\nu L}{U}}
+    \approx \sqrt{\frac{\nu L}{U}}
 \end{aligned}$$
 
-We thus insert $$\delta = \sqrt{\nu L / U}$$ into the Navier-Stokes equations, giving us:
+So we insert $$\delta = \sqrt{\nu L / U}$$ into the Navier-Stokes equations, giving us:
 
 $$\begin{aligned}
     \tilde{v}_x \pdv{\tilde{v}_x}{\tilde{x}} + \tilde{v}_y \pdv{\tilde{v}_x}{\tilde{y}}
@@ -163,15 +161,14 @@ so we can drop many terms, leaving us with these redimensionalized equations:
 $$\begin{aligned}
     v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y}
     = - \frac{1}{\rho} \pdv{p}{x} + \nu \pdvn{2}{v_x}{y}
-    \qquad \quad
+    \qquad \qquad
     \pdv{p}{y}
     = 0
 \end{aligned}$$
 
 The second one tells us that for a given $$x$$-value,
-the pressure is the same at the surface
-as in the main flow $$y > L$$, where the fluid is ideal.
-In the latter regime, we apply Bernoulli's theorem to rewrite $$p$$,
+the pressure is the same at the surface as in the ideal main flow $$y > L$$.
+In the latter regime, we can use Bernoulli's theorem to rewrite $$p$$,
 using the *Bernoulli head* $$H$$ and the mainstream velocity $$U(x)$$:
 
 $$\begin{aligned}
@@ -185,18 +182,19 @@ we arrive at the Prandtl equations:
 
 $$\begin{aligned}
     \boxed{
-        v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y}
-        = U \dv{U}{x} + \nu \pdvn{2}{v_x}{y}
-        \qquad \quad
-        \pdv{v_x}{x} + \pdv{v_y}{y}
-        = 0
+        \begin{aligned}
+            v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y}
+            &= U \dv{U}{x} + \nu \pdvn{2}{v_x}{y}
+            \\
+            \pdv{v_x}{x} + \pdv{v_y}{y}
+            &= 0
+        \end{aligned}
     }
 \end{aligned}$$
 
 A notable application of these equations is
 the [Blasius boundary layer](/know/concept/blasius-boundary-layer/),
-where the surface in question
-is a semi-infinite plane.
+where the surface in question is a semi-infinite plane.
 
 
 
-- 
cgit v1.3