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(app_Biot_Savart)=
# Biot-Savart law

```{admonition} Referred from 
{ref}`auton_lift`
```

```{admonition} References
{cite:t}`Saffman1995-ic`
```

```{figure} ../fig/Auton-BiotSavart.png
:name: Auton_Auton-BiotSavart
Biot-Savart law
```

Let us consider the following conditions for a rotational flow: 
\begin{enumerate}
	\item The velocity field, $\mathbf{v}$, satisfies $\nabla \cdot \mathbf{v} = 0$.
	\item The fluid region is singly connected.
	\item The normal component, $\mathbf{n} \cdot \mathbf{v}$, of the velocity is given at all bounding surfaces $S$.
	\item The velocity $\mathbf{v}$ vanishes at infinity when the fluid is unbounded. 
	\item The normal component, $\mathbf{n} \cdot \boldsymbol{\omega}$, of vorticity vanishes on $S$. 
	\item The vorticity field $\boldsymbol{\omega}$ is compact when the fluid is unbounded. 
\end{enumerate}
The velocity field under these conditions can be expressed as the sum of a solenoidal velocity potential component, $\mathbf{v}_{v}$, for which $\nabla \cdot \mathbf{v}_{v} = 0$, and an irrotational scalar component, $\nabla \phi$, i.e., 
```{math}
:label: eq:Auton_eq_velocity-reconstructed
	\mathbf{v} (\mathbf{r}, t) = \mathbf{v}_{v} (\mathbf{r}, t) + \nabla \phi
```
where the irrotational component is determined by the Poisson equation:
```{math}
:label: eq:app_BiotSavart_nonref_0
	\nabla^{2} \phi = 0
```
with the boundary condition 
```{math}
:label: eq:app_BiotSavart_nonref_1
	\mathbf{n} \cdot \nabla \phi = \mathbf{n} \cdot \mathbf{v} - \mathbf{n} \cdot \mathbf{v}_{v}
```
or
```{math}
:label: eq:app_BiotSavart_nonref_2
	\phi \rightarrow 0~~~~\text{as}~r \rightarrow \infty
```
when the fluid is unbounded. Taking $rot$ of Eq. {eq}`eq:Auton_eq_velocity-reconstructed` yields
```{math}
:label: eq:app_BiotSavart_nonref_3
	\nabla \times \mathbf{v}_{v} = \boldsymbol{\omega}
```
since $\nabla \times \nabla \phi = 0$. The solenoidal component is constructed by the vorticity distribution (\ref{Auton_Auton-BiotSavart})
```{math}
:label: eq:Auton_eq_Biot-Savart-1
	\mathbf{v}_{v} (\mathbf{r}, t) = \frac{1}{4 \pi} \iiint_{V'} \frac{\boldsymbol{\omega} (\mathbf{r}', t) \times (\mathbf{r} - \mathbf{r}') }{| \mathbf{r} - \mathbf{r}' |^{3}} dV'
```
where $\mathbf{r}' = (x', y', z')$ and $dV' = dx' dy' dz'$. This is analogous to the relation between a electric current density and a magnetic flux field. Eq. {eq}`eq:Auton_eq_Biot-Savart-1` is therefore called the Biot-Savart law for the fluid velocity induced by the vorticity distribution. Since 
```{math}
:label: eq:app_BiotSavart_nonref_4
	\frac{\partial }{\partial x_{k}} \frac{1}{| \mathbf{r}' - \mathbf{r} |}
	= \frac{x'_{k} - x_{k}}{| \mathbf{r}' - \mathbf{r} |^{3}}
```
we can rewrite Eq. {eq}`eq:Auton_eq_Biot-Savart-1` in the following form 
```{math}
:label: eq:Auton_eq_Biot-Savart-2
	\mathbf{v}_{v} (\mathbf{r}, t) = - \frac{1}{4 \pi} \iiint_{V'} \boldsymbol{\omega} (\mathbf{r}', t) \times \nabla \frac{1}{| \mathbf{r}' - \mathbf{r} |} dV'
```
Let us see the Biot-Savart velocity field satisfies $\nabla \cdot \mathbf{v}_{v} = 0$ and $\nabla \times \mathbf{v}_{v} = \boldsymbol{\omega}$ in the following. For simplicity, we write $\boldsymbol{\omega}' = \boldsymbol{\omega} (\mathbf{r}', t)$, $\nabla' = \partial / \partial \mathbf{r}'$ and $r = | \mathbf{r} - \mathbf{r}' |$. Taking $div$ of Eq. {eq}`eq:Auton_eq_Biot-Savart-2` we have
```{math}
:label: eq:app_BiotSavart_nonref_5
	\nabla \cdot \mathbf{v}_{v} 
	= - \frac{1}{4 \pi} \nabla \cdot \iiint_{V'} \boldsymbol{\omega}' \times \nabla \frac{1}{r} dV'	
	= - \frac{1}{4 \pi} \iiint_{V'} \nabla \cdot \left( \boldsymbol{\omega}' \times \nabla \frac{1}{r} \right) dV'	
```
We can take $\boldsymbol{\omega}'$ out from the divergence since it is a function of $\mathbf{r}'$, not $\mathbf{r}$. Therefore,
```{toggle}
$- \nabla \cdot \left( \boldsymbol{\omega}' \times \nabla \frac{1}{r} \right)
	\rightarrow - \frac{\partial }{\partial x_{k}} \left( \epsilon_{kij} \omega'_{i} \frac{\partial }{\partial x_{j}} \frac{1}{r} \right)
	= - \omega'_{i} \epsilon_{kij} \frac{\partial }{\partial x_{k}} \left( \frac{\partial }{\partial x_{j}} \frac{1}{r} \right)$
$= \omega'_{i} \epsilon_{ikj} \frac{\partial }{\partial x_{k}} \left( \frac{\partial }{\partial x_{j}} \frac{1}{r} \right)
	\rightarrow \boldsymbol{\omega}' \cdot \left( \nabla \times \nabla \frac{1}{r} \right)$
```

```{math}
:label: eq:app_BiotSavart_nonref_7
	\nabla \cdot \mathbf{v}_{v} 
	= \frac{1}{4 \pi} \iiint_{V'} \boldsymbol{\omega}' \cdot \left( \nabla \times \nabla \frac{1}{r} \right) dV'	
```
However, because of the identity $\nabla \times \nabla~\text{(any scalar)} = 0$, 
```{math}
:label: eq:app_BiotSavart_nonref_8
	\nabla \cdot \mathbf{v}_{v} = 0
```
The Biot-Savart velocity field is thus confirmed to be solenoidal. Then, we take $rot$ of Eq. {eq}`eq:Auton_eq_Biot-Savart-2`:\footnote{
```{math}
:label: eq:app_BiotSavart_nonref_9
\begin{split}	
	- \nabla \times \iiint_{V'} \boldsymbol{\omega}' \times \nabla \frac{1}{r}
	&\rightarrow - \epsilon_{ijk} \frac{\partial}{\partial x_{j}} \epsilon_{kmn} \omega'_{m} \frac{\partial}{\partial x_{n}} \frac{1}{r}
	= - \epsilon_{ijk} \epsilon_{kmn} \omega'_{m} \frac{\partial}{\partial x_{j}} \frac{\partial}{\partial x_{n}} \frac{1}{r} \\
	&= - (\delta_{im} \delta_{jn} - \delta_{in} \delta_{jm}) \omega'_{m} \frac{\partial}{\partial x_{j}} \frac{\partial}{\partial x_{n}} \frac{1}{r}
	= - \omega'_{i} \frac{\partial}{\partial x_{j}} \frac{\partial}{\partial x_{j}} \frac{1}{r} + \omega'_{j} \frac{\partial}{\partial x_{j}} \frac{\partial}{\partial x_{i}} \frac{1}{r} \\
	&\rightarrow - \boldsymbol{\omega}' \nabla^{2} (1/r) + \boldsymbol{\omega} \cdot \nabla \nabla (1/r)
\end{split}
```
}
```{math}
:label: eq:app_BiotSavart_nonref_10
	\nabla \times \mathbf{v}_{v} 
	= - \frac{1}{4 \pi} \nabla \times \iiint_{V'} \boldsymbol{\omega}' \times \nabla \frac{1}{r} dV'	
	= \frac{1}{4 \pi} \iiint_{V'} \left\{ - \boldsymbol{\omega}' \nabla^{2} \frac{1}{r} + \boldsymbol{\omega}' \cdot \nabla \left( \nabla \frac{1}{r} \right) \right\} dV'	
```
$\nabla^{2} (1/r)$ behaves like Dirac's delta, i.e., $\nabla^{2} (1/r) = - 4 \pi \delta (\mathbf{r} - \mathbf{r}')$ (see Appendix {ref}`app_Dirac_delta`). Therefore the integration of the first term gives $\boldsymbol{\omega} (\mathbf{r})$: 
```{math}
:label: eq:app_BiotSavart_nonref_11
	- \frac{1}{4 \pi} \iiint_{V'} \boldsymbol{\omega}' \nabla^{2} \frac{1}{r} dV'	
	= - \frac{1}{4 \pi} \iiint_{V'} \boldsymbol{\omega}' ( - 4 \pi \delta (\mathbf{r} - \mathbf{r}') ) dV'	
	= \boldsymbol{\omega} (\mathbf{r})
```
For the second term, using $\nabla (1/r) = - \nabla' (1/r)$ gives 
```{math}
:label: eq:app_BiotSavart_nonref_12
	\iiint_{V'} \boldsymbol{\omega}' \cdot \nabla \left( \nabla \frac{1}{r} \right) dV'
	= - \iiint_{V'} \boldsymbol{\omega}' \cdot \nabla' \left( \nabla \frac{1}{r} \right) dV'
	= - \iiint_{V'} \nabla' \cdot \left( \boldsymbol{\omega}' \nabla \frac{1}{r} \right) dV'
```
where $\nabla' \cdot \boldsymbol{\omega}' = 0$ was used. By the divergence theorem, we have 
```{math}
:label: eq:app_BiotSavart_nonref_13
	\iiint_{V'} \nabla' \cdot \left( \boldsymbol{\omega}' \nabla \frac{1}{r} \right) dV'
	= \iint_{S'} \mathbf{n}' \cdot \left( \boldsymbol{\omega}' \nabla \frac{1}{r} \right) dS'
```
However, this integral vanishes because of the condition $\mathbf{n} \cdot \boldsymbol{\omega} = 0$ on all bounding surfaces or the compactness of the vorticity field. Thus, 
```{math}
:label: eq:app_BiotSavart_nonref_14
	\nabla \times \mathbf{v}_{v} = \boldsymbol{\omega}
```