From 71b9e1aa3050dd492761973ad4be73c6d65e7eb1 Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sat, 10 Apr 2021 19:58:09 +0200
Subject: Expand knowledge base

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+---
+title: "Vorticity"
+firstLetter: "V"
+publishDate: 2021-04-03
+categories:
+- Physics
+- Fluid mechanics
+- Fluid dynamics
+
+date: 2021-04-03T09:24:42+02:00
+draft: false
+markup: pandoc
+---
+
+# Vorticity
+
+In fluid mechanics, the **vorticity** $\va{\omega}$
+is a measure of the local circulation in a fluid.
+It is defined as the curl of the flow velocity field $\va{v}$:
+
+$$\begin{aligned}
+    \boxed{
+        \va{\omega}
+        \equiv \nabla \cross \va{v}
+    }
+\end{aligned}$$
+
+Just as curves tangent to $\va{v}$ are called *streamlines*,
+curves tangent to $\va{\omega}$ are **vortex lines**,
+which are to be interpreted as the "axes" that $\va{v}$ is circulating around.
+
+The vorticity is a local quantity,
+and the corresponding global quantity is the **circulation** $\Gamma$,
+which is defined as the projection of $\va{v}$ onto a close curve $C$.
+Then, by Stokes' theorem:
+
+$$\begin{aligned}
+    \boxed{
+        \Gamma(C, t)
+        \equiv \oint_C \va{v} \cdot \dd{\va{l}}
+        = \int_S \va{\omega} \cdot \dd{\va{S}}
+    }
+\end{aligned}$$
+
+
+## Ideal fluids
+
+For an inviscid, incompressible fluid,
+consider the *Bernoulli field* $H$, which is defined as:
+
+$$\begin{aligned}
+    H
+    \equiv \frac{1}{2} \va{v}^2 + \Phi + \frac{p}{\rho}
+\end{aligned}$$
+
+Where $\Phi$ is the gravitational potential,
+$p$ is the pressure, and $\rho$ is the (constant) density.
+We then take the gradient of this scalar field:
+
+$$\begin{aligned}
+    \nabla H
+    &= \frac{1}{2} \nabla \va{v}^2 + \nabla \Phi + \frac{\nabla p}{\rho}
+    \\
+    &= \va{v} \cdot (\nabla \va{v}) - \Big( \!-\! \nabla \Phi - \frac{\nabla p}{\rho} \Big)
+\end{aligned}$$
+
+Since $-\nabla \Phi = \va{g}$,
+the rightmost term is the right-hand side of
+the [Euler equation](/know/concept/euler-equations/).
+We substitute the other side of said equation, yielding:
+
+$$\begin{aligned}
+    \nabla H
+    &= \va{v} \cdot (\nabla \va{v}) - \frac{\mathrm{D} \va{v}}{\mathrm{D} t}
+    = \va{v} \cdot (\nabla \va{v}) - \pdv{\va{v}}{t} - (\va{v} \cdot \nabla) \va{v}
+\end{aligned}$$
+
+We isolate this equation for $\pdv*{\va{v}}{t}$,
+and apply a vector identity to reduce it to the following:
+
+$$\begin{aligned}
+        \pdv{\va{v}}{t}
+        = \va{v} \cdot (\nabla \va{v}) - (\va{v} \cdot \nabla) \va{v} - \nabla H
+        = \va{v} \cross (\nabla \cross \va{v}) - \nabla H
+\end{aligned}$$
+
+Here, the definition of the vorticity $\va{\omega}$ is clear to see,
+leading us to an equation of motion for $\va{v}$:
+
+$$\begin{aligned}
+    \boxed{
+        \pdv{\va{v}}{t}
+        = \va{v} \cross \va{\omega} - \nabla H
+    }
+\end{aligned}$$
+
+More about this later.
+Now, we take the curl of both sides of this equation, giving us:
+
+$$\begin{aligned}
+    \nabla \cross \pdv{\va{v}}{t}
+    = \nabla \cross (\va{v} \cross \va{\omega}) - \nabla \cross (\nabla H)
+\end{aligned}$$
+
+On the left, we swap $\nabla$ with $\pdv*{t}$,
+and on the right, the curl of a gradient is always zero.
+We are thus left with the equation of motion of the vorticity $\va{\omega}$:
+
+$$\begin{aligned}
+    \boxed{
+        \pdv{\va{\omega}}{t}
+        = \nabla \cross (\va{v} \cross \va{\omega})
+    }
+\end{aligned}$$
+
+Let us now return to the equation of motion for $\va{v}$.
+For *steady* flows where $\pdv*{\va{v}}{t} = 0$, in which case
+[Bernoulli's theorem](/know/concept/bernoullis-theorem/) applies,
+it reduces to:
+
+$$\begin{aligned}
+    \nabla H
+    = \va{v} \cross \va{\omega}
+\end{aligned}$$
+
+If a fluid has $\va{\omega} = 0$ in some regions, it is known as **irrotational**.
+From this equation, we see that, in that case, $\nabla H = 0$,
+meaning that $H$ is a constant in those regions,
+a fact sometimes referred to as **Bernoulli's stronger theorem**.
+
+Furthermore, irrotationality $\va{\omega} = 0$
+implies that $\va{v}$ is the gradient of a potential $\Psi$:
+
+$$\begin{aligned}
+    \va{v}
+    = \nabla \Psi
+\end{aligned}$$
+
+This fact allows us to rewrite the Euler equations in a particularly simple way.
+Firstly, the condition of incompressibility becomes the well-known Laplace equation:
+
+$$\begin{aligned}
+    0
+    = \nabla \cdot \va{v}
+    = \nabla^2 \Psi
+\end{aligned}$$
+
+And second, the main equation of motion for $\va{v}$ states
+that the quantity $H + \pdv*{\Psi}{t}$ is spatially constant
+in the irrotational region:
+
+$$\begin{aligned}
+    \pdv{\va{v}}{t}
+    = \nabla \pdv{\Psi}{t}
+    = - \nabla H
+    \quad \implies \quad
+    \nabla \Big( H + \pdv{\Psi}{t} \Big)
+    = 0
+\end{aligned}$$
+
+
+## References
+1.  B. Lautrup,
+    *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition,
+    CRC Press.
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