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diff --git a/source/know/concept/vorticity/index.md b/source/know/concept/vorticity/index.md index da1205b..f7b5c6e 100644 --- a/source/know/concept/vorticity/index.md +++ b/source/know/concept/vorticity/index.md @@ -9,9 +9,9 @@ categories: layout: "concept" --- -In fluid mechanics, the **vorticity** $\va{\omega}$ +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 $\va{v}$: +It is defined as the curl of the flow velocity field $$\va{v}$$: $$\begin{aligned} \boxed{ @@ -20,13 +20,13 @@ $$\begin{aligned} } \end{aligned}$$ -Just as curves tangent to $\va{v}$ are called *streamlines*, -curves tangent to $\va{\omega}$ are **vortex lines**, -which are to be interpreted as the "axes" that $\va{v}$ is circulating around. +Just as curves tangent to $$\va{v}$$ are called *streamlines*, +curves tangent to $$\va{\omega}$$ are **vortex lines**, +which are to be interpreted as the "axes" that $$\va{v}$$ is circulating around. The vorticity is a local quantity, -and the corresponding global quantity is the **circulation** $\Gamma$, -which is defined as the projection of $\va{v}$ onto a close curve $C$. +and the corresponding global quantity is the **circulation** $$\Gamma$$, +which is defined as the projection of $$\va{v}$$ onto a close curve $$C$$. Then, by Stokes' theorem: $$\begin{aligned} @@ -41,15 +41,15 @@ $$\begin{aligned} ## Ideal fluids For an inviscid, incompressible fluid, -consider the *Bernoulli field* $H$, which is defined as: +consider the *Bernoulli field* $$H$$, which is defined as: $$\begin{aligned} H \equiv \frac{1}{2} \va{v}^2 + \Phi + \frac{p}{\rho} \end{aligned}$$ -Where $\Phi$ is the gravitational potential, -$p$ is the pressure, and $\rho$ is the (constant) density. +Where $$\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} @@ -59,7 +59,7 @@ $$\begin{aligned} &= \va{v} \cdot (\nabla \va{v}) - \Big( \!-\! \nabla \Phi - \frac{\nabla p}{\rho} \Big) \end{aligned}$$ -Since $-\nabla \Phi = \va{g}$, +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: @@ -70,7 +70,7 @@ $$\begin{aligned} = \va{v} \cdot (\nabla \va{v}) - \pdv{\va{v}}{t} - (\va{v} \cdot \nabla) \va{v} \end{aligned}$$ -We isolate this equation for $\ipdv{\va{v}}{t}$, +We isolate this equation for $$\ipdv{\va{v}}{t}$$, and apply a vector identity to reduce it to the following: $$\begin{aligned} @@ -79,8 +79,8 @@ $$\begin{aligned} = \va{v} \cross (\nabla \cross \va{v}) - \nabla H \end{aligned}$$ -Here, the definition of the vorticity $\va{\omega}$ is clear to see, -leading us to an equation of motion for $\va{v}$: +Here, the definition of the vorticity $$\va{\omega}$$ is clear to see, +leading us to an equation of motion for $$\va{v}$$: $$\begin{aligned} \boxed{ @@ -97,9 +97,9 @@ $$\begin{aligned} = \nabla \cross (\va{v} \cross \va{\omega}) - \nabla \cross (\nabla H) \end{aligned}$$ -On the left, we swap $\nabla$ with $\ipdv{}{t}$, +On the left, we swap $$\nabla$$ with $$\ipdv{}{t}$$, and on the right, the curl of a gradient is always zero. -We are thus left with the equation of motion of the vorticity $\va{\omega}$: +We are thus left with the equation of motion of the vorticity $$\va{\omega}$$: $$\begin{aligned} \boxed{ @@ -108,8 +108,8 @@ $$\begin{aligned} } \end{aligned}$$ -Let us now return to the equation of motion for $\va{v}$. -For *steady* flows where $\ipdv{\va{v}}{t} = 0$, in which case +Let us now return to the equation of motion for $$\va{v}$$. +For *steady* flows where $$\ipdv{\va{v}}{t} = 0$$, in which case [Bernoulli's theorem](/know/concept/bernoullis-theorem/) applies, it reduces to: @@ -118,13 +118,13 @@ $$\begin{aligned} = \va{v} \cross \va{\omega} \end{aligned}$$ -If a fluid has $\va{\omega} = 0$ in some regions, it is known as **irrotational**. -From this equation, we see that, in that case, $\nabla H = 0$, -meaning that $H$ is a constant in those regions, +If a fluid has $$\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$$ is a constant in those regions, a fact sometimes referred to as **Bernoulli's stronger theorem**. -Furthermore, irrotationality $\va{\omega} = 0$ -implies that $\va{v}$ is the gradient of a potential $\Psi$: +Furthermore, irrotationality $$\va{\omega} = 0$$ +implies that $$\va{v}$$ is the gradient of a potential $$\Psi$$: $$\begin{aligned} \va{v} @@ -140,8 +140,8 @@ $$\begin{aligned} = \nabla^2 \Psi \end{aligned}$$ -And second, the main equation of motion for $\va{v}$ states -that the quantity $H + \ipdv{\Psi}{t}$ is spatially constant +And second, the main equation of motion for $$\va{v}$$ states +that the quantity $$H + \ipdv{\Psi}{t}$$ is spatially constant in the irrotational region: $$\begin{aligned} |