From aab299218975a8e775cda26ce256ffb1fe36c863 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sun, 30 May 2021 15:54:40 +0200
Subject: Expand knowledge base

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+---
+title: "Prandtl equations"
+firstLetter: "P"
+publishDate: 2021-05-29
+categories:
+- Physics
+- Fluid mechanics
+- Fluid dynamics
+
+date: 2021-05-10T18:41:20+02:00
+draft: false
+markup: pandoc
+---
+
+# Prandtl equations
+
+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.
+
+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.
+This implies the existence of a **boundary layer**:
+a thin layer of fluid "stuck" to solid objects in the flow,
+where viscosity plays an important role.
+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
+to describe the flow close to a flat surface.
+
+At the wall, there is a very thin boundary layer of thickness $\delta$,
+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 2D, the steady Navier-Stokes equations are as follows,
+where the flow $\va{v} = (v_x, v_y)$:
+
+$$\begin{aligned}
+    v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y}
+    &= - \frac{1}{\rho} \pdv{p}{x} + \nu \Big( \pdv[2]{v_x}{x} + \pdv[2]{v_x}{y} \Big)
+    \\
+    v_x \pdv{v_y}{x} + v_y \pdv{v_y}{y}
+    &= - \frac{1}{\rho} \pdv{p}{y} + \nu \Big( \pdv[2]{v_y}{x} + \pdv[2]{v_y}{y} \Big)
+    \\
+    \pdv{v_x}{x} + \pdv{v_y}{y}
+    &= 0
+\end{aligned}$$
+
+The latter represents the fluid's incompressibility.
+We non-dimensionalize these equations,
+and assume that changes along the $y$-axis
+happen on a short scale (say, $\delta$),
+and along the $x$-axis on a longer scale (say, $L$).
+Let $\tilde{x}$ and $\tilde{y}$ be dimenionless variables of order $1$:
+
+$$\begin{aligned}
+    x
+    = L \tilde{x}
+    \qquad \quad
+    y
+    = \delta \tilde{x}
+    \qquad \quad
+    \pdv{x}
+    = \frac{1}{L} \pdv{\tilde{x}}
+    \qquad \quad
+    \pdv{y}
+    = \frac{1}{\delta} \pdv{\tilde{y}}
+\end{aligned}$$
+
+Furthermore, we choose velocity scales
+to be consistent with the incompressibility condition,
+and a pressure scale inspired
+by [Bernoulli's theorem](/know/concept/bernoullis-theorem/):
+
+$$\begin{aligned}
+    v_x
+    = U \tilde{v}_x
+    \qquad \quad
+    v_y
+    = \frac{U \delta}{L} \tilde{v}_y
+    \qquad \quad
+    p
+    = \rho U^2 \tilde{p}
+\end{aligned}$$
+
+We insert these scalings into the Navier-Stokes equations, yielding:
+
+$$\begin{aligned}
+    \frac{U^2}{L} \tilde{v}_x \pdv{\tilde{v}_x}{\tilde{x}} + \frac{U^2}{L} \tilde{v}_y \pdv{\tilde{v}_x}{\tilde{y}}
+    &= - \frac{U^2}{L} \pdv{\tilde{p}}{\tilde{x}}
+    + \nu \Big( \frac{U}{L^2} \pdv[2]{\tilde{v}_x}{\tilde{x}} + \frac{U}{\delta^2} \pdv[2]{\tilde{v}_x}{\tilde{y}} \Big)
+    \\
+    \frac{U^2 \delta}{L^2} \tilde{v}_x \pdv{\tilde{v}_y}{\tilde{x}} + \frac{U^2 \delta}{L^2} \tilde{v}_y \pdv{\tilde{v}_y}{\tilde{y}}
+    &= - \frac{U^2}{\delta} \pdv{\tilde{p}}{\tilde{y}}
+    + \nu \Big( \frac{U \delta}{L^3} \pdv[2]{\tilde{v}_y}{\tilde{x}} + \frac{U}{L \delta} \pdv[2]{\tilde{v}_y}{\tilde{y}} \Big)
+\end{aligned}$$
+
+For future convenience,
+we multiply the former equation by $L / U^2$, and the latter by $\delta / U^2$:
+
+$$\begin{aligned}
+    \tilde{v}_x \pdv{\tilde{v}_x}{\tilde{x}} + \tilde{v}_y \pdv{\tilde{v}_x}{\tilde{y}}
+    &= - \pdv{\tilde{p}}{\tilde{x}}
+    + \nu \Big( \frac{1}{U L} \pdv[2]{\tilde{v}_x}{\tilde{x}} + \frac{L}{U \delta^2} \pdv[2]{\tilde{v}_x}{\tilde{y}} \Big)
+    \\
+    \frac{\delta^2}{L^2} \tilde{v}_x \pdv{\tilde{v}_y}{\tilde{x}} + \frac{\delta^2}{L^2} \tilde{v}_y \pdv{\tilde{v}_y}{\tilde{y}}
+    &= - \pdv{\tilde{p}}{\tilde{y}}
+    + \nu \Big( \frac{\delta^2}{U L^3} \pdv[2]{\tilde{v}_y}{\tilde{x}} + \frac{1}{U L} \pdv[2]{\tilde{v}_y}{\tilde{y}} \Big)
+\end{aligned}$$
+
+We would like to estimate $\delta$.
+Intuitively, we expect that higher viscosities $\nu$ give thicker layers,
+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:
+
+$$\begin{aligned}
+    \delta
+    \approx \sqrt{\frac{\nu x}{U}}
+    \sim \sqrt{\frac{\nu L}{U}}
+\end{aligned}$$
+
+We thus insert $\delta = \sqrt{\nu L / U}$ into the Navier-Stokes equations, giving us:
+
+$$\begin{aligned}
+    \tilde{v}_x \pdv{\tilde{v}_x}{\tilde{x}} + \tilde{v}_y \pdv{\tilde{v}_x}{\tilde{y}}
+    &= - \pdv{\tilde{p}}{\tilde{x}}
+    + \nu \Big( \frac{1}{U L} \pdv[2]{\tilde{v}_x}{\tilde{x}} + \frac{1}{\nu} \pdv[2]{\tilde{v}_x}{\tilde{y}} \Big)
+    \\
+    \frac{\nu}{U L} \tilde{v}_x \pdv{\tilde{v}_y}{\tilde{x}} + \frac{\nu}{U L} \tilde{v}_y \pdv{\tilde{v}_y}{\tilde{y}}
+    &= - \pdv{\tilde{p}}{\tilde{y}}
+    + \nu \Big( \frac{\nu}{U^2 L^2} \pdv[2]{\tilde{v}_y}{\tilde{x}} + \frac{1}{U L} \pdv[2]{\tilde{v}_y}{\tilde{y}} \Big)
+\end{aligned}$$
+
+Here, we recognize the definition of the Reynolds number $\mathrm{Re} = U L / \nu$:
+
+$$\begin{aligned}
+    \tilde{v}_x \pdv{\tilde{v}_x}{\tilde{x}} + \tilde{v}_y \pdv{\tilde{v}_x}{\tilde{y}}
+    &= - \pdv{\tilde{p}}{\tilde{x}}
+    + \frac{1}{\mathrm{Re}} \pdv[2]{\tilde{v}_x}{\tilde{x}} + \pdv[2]{\tilde{v}_x}{\tilde{y}}
+    \\
+    \frac{1}{\mathrm{Re}} \tilde{v}_x \pdv{\tilde{v}_y}{\tilde{x}} + \frac{1}{\mathrm{Re}} \tilde{v}_y \pdv{\tilde{v}_y}{\tilde{y}}
+    &= - \pdv{\tilde{p}}{\tilde{y}}
+    + \frac{1}{\mathrm{Re}^2} \pdv[2]{\tilde{v}_y}{\tilde{x}} + \frac{1}{\mathrm{Re}} \pdv[2]{\tilde{v}_y}{\tilde{y}}
+\end{aligned}$$
+
+Recall that we are only considering large Reynolds numbers $\mathrm{Re} \gg 1$,
+in which case $\mathrm{Re}^{-1} \ll 1$,
+so we can drop many terms, leaving us with these redimensionalized equations:
+
+$$\begin{aligned}
+    v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y}
+    = - \frac{1}{\rho} \pdv{p}{x} + \nu \pdv[2]{v_x}{y}
+    \qquad \quad
+    \pdv{p}{y}
+    = 0
+\end{aligned}$$
+
+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 $p$,
+using the *Bernoulli head* $H$ and the mainstream velocity $U(x)$:
+
+$$\begin{aligned}
+    p
+    = \rho H - \frac{1}{2} \rho U^2
+    = p_0 - \frac{1}{2} \rho U^2
+\end{aligned}$$
+
+Inserting this into the reduced Navier-Stokes equations,
+we arrive at the Prandtl equations:
+
+$$\begin{aligned}
+    \boxed{
+        v_x \pdv{v_x}{x} + v_y \pdv{v_x}{y}
+        = U \dv{U}{x} + \nu \pdv[2]{v_x}{y}
+        \qquad \quad
+        \pdv{v_x}{x} + \pdv{v_y}{y}
+        = 0
+    }
+\end{aligned}$$
+
+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.
+
+
+
+## References
+1.  B. Lautrup,
+    *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition,
+    CRC Press.
-- 
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