Differential operators in cylindrical coordinates - Revision history

Apwillis: original form seems most common in fluids

2020-11-23T20:11:08Z

original form seems most common in fluids

← Older revision		Revision as of 13:11, 23 November 2020
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	=== Velocity gradient tensor ===		=== Velocity gradient tensor ===

	$~~\mathrm{L} = (~~\bnabla\vec{u}~~)^T~~ ~=~ \left[		$\bnabla\vec{u} ~=~ \left[
	\begin{array}{ccc}		\begin{array}{ccc}
	\partial_r u_r & \partial_r u_\theta & \partial_r u_z \\		\partial_r u_r & \partial_r u_\theta & \partial_r u_z \\

Apwillis: /* Velocity gradient tensor */

2020-11-23T19:48:56Z

Velocity gradient tensor

← Older revision		Revision as of 12:48, 23 November 2020
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	=== Velocity gradient tensor ===		=== Velocity gradient tensor ===

	$(\bnabla\vec{u})^T ~=~ \left[		$\mathrm{L} = (\bnabla\vec{u})^T ~=~ \left[
	\begin{array}{ccc}		\begin{array}{ccc}
	\partial_r u_r & \partial_r u_\theta & \partial_r u_z \\		\partial_r u_r & \partial_r u_\theta & \partial_r u_z \\

Apwillis: /* Velocity gradient tensor */

2020-11-23T17:11:59Z

Velocity gradient tensor

← Older revision		Revision as of 10:11, 23 November 2020
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	=== Velocity gradient tensor ===		=== Velocity gradient tensor ===

	$\bnabla\vec{u} ~=~ \left[		$(\bnabla\vec{u})^T ~=~ \left[
	\begin{array}{ccc}		\begin{array}{ccc}
	\partial_r u_r & \partial_r u_\theta & \partial_r u_z \\		\partial_r u_r & \partial_r u_\theta & \partial_r u_z \\

Apwillis at 11:00, 28 January 2020

2020-01-28T11:00:11Z

← Older revision		Revision as of 04:00, 28 January 2020
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	$(\bnabla f)_r = \partial_{r}f,		$(\bnabla f)_r = \partial_{r}f,
	\quad		\quad
	(\bnabla f)_\theta = \frac1{r}\, \partial_{\theta} f,		(\bnabla f)_\theta = \frac1{r}\, \partial_{\theta} f,
	\quad		\quad
	(\bnabla f)_z = \partial_{z} f .$		(\bnabla f)_z = \partial_{z} f .$


	$\bnabla \cdot \vec{A} = (\frac1{r} + \partial_{r}) A_r		$\bnabla \cdot \vec{A} = (\frac1{r} + \partial_{r}) A_r
	+ \frac1{r}\,\partial_{\theta}A_\theta + \partial_{z} A_z .$		+ \frac1{r}\,\partial_{\theta}A_\theta + \partial_{z} A_z .$


	$(\bnabla \wedge \vec{A})_r =		$(\bnabla \wedge \vec{A})_r =
	\frac1{r}\,\partial_{\theta} A_z - \partial_{z} A_\theta ,$		\frac1{r}\,\partial_{\theta} A_z - \partial_{z} A_\theta ,$

	$(\bnabla \wedge \vec{A})_\theta =		$(\bnabla \wedge \vec{A})_\theta =
	\partial_{z} A_r - \partial_{r} A_z ,$		\partial_{z} A_r - \partial_{r} A_z ,$

	$(\bnabla \wedge \vec{A})_z =		$(\bnabla \wedge \vec{A})_z =
	(\frac1{r}+\partial_{r}) A_\theta - \frac1{r}\,\partial_{\theta} A_r .$		(\frac1{r}+\partial_{r}) A_\theta - \frac1{r}\,\partial_{\theta} A_r .$

	=== Laplacian ===		=== Laplacian ===

	$\nabla^2 f = (\frac1{r}\partial_{r} + \partial_{rr} ) f		$\nabla^2 f = (\frac1{r}\partial_{r} + \partial_{rr} ) f
	+\frac1{r^2}\,\partial_{\theta\theta} f + \partial_{zz} f .$		+\frac1{r^2}\,\partial_{\theta\theta} f + \partial_{zz} f .$

	$(\nabla^2 \vec{A})_r =		$(\nabla^2 \vec{A})_r =
	\nabla^2 A_r - \frac{2}{r^2}\,\partial_{\theta}A_\theta		\nabla^2 A_r - \frac{2}{r^2}\,\partial_{\theta}A_\theta
	- \frac{A_r}{r^2} ,$		- \frac{A_r}{r^2} ,$

	$(\nabla^2 \vec{A})_\theta =		$(\nabla^2 \vec{A})_\theta =
	\nabla^2 A_\theta + \frac{2}{r^2}\,\partial_{\theta}A_r		\nabla^2 A_\theta + \frac{2}{r^2}\,\partial_{\theta}A_r
	- \frac{A_\theta}{r^2} ,$		- \frac{A_\theta}{r^2} ,$

	$(\nabla^2 \vec{A})_z =		$(\nabla^2 \vec{A})_z =
	\nabla^2 A_z .$		\nabla^2 A_z .$

	=== Advective operator ===		=== Advective operator ===

	$(\vec{A}\cdot\bnabla\,\vec{B})_r =		$(\vec{A}\cdot\bnabla\,\vec{B})_r =
	A_r\,\partial_{r}B_r + \frac{A_\theta}{r}\,\partial_{\theta}B_r		A_r\,\partial_{r}B_r + \frac{A_\theta}{r}\,\partial_{\theta}B_r
	+ A_z\,\partial_{z}B_r - \frac{A_\theta B_\theta}{r} ,$		+ A_z\,\partial_{z}B_r - \frac{A_\theta B_\theta}{r} ,$

	$(\vec{A}\cdot\bnabla\,\vec{B})_\theta =		$(\vec{A}\cdot\bnabla\,\vec{B})_\theta =
	A_r\,\partial_{r}B_\theta + \frac{A_\theta}{r}\,\partial_{\theta}B_\theta		A_r\,\partial_{r}B_\theta + \frac{A_\theta}{r}\,\partial_{\theta}B_\theta
	+ A_z\,\partial_{z}B_\theta + \frac{A_\theta B_r}{r} ,$		+ A_z\,\partial_{z}B_\theta + \frac{A_\theta B_r}{r} ,$

	$(\vec{A}\cdot\bnabla\,\vec{B})_z =		$(\vec{A}\cdot\bnabla\,\vec{B})_z =
	A_r\,\partial_{r}B_z + \frac{A_\theta}{r}\,\partial_{\theta}B_z		A_r\,\partial_{r}B_z + \frac{A_\theta}{r}\,\partial_{\theta}B_z
	+ A_z\,\partial_{z}B_z .$		+ A_z\,\partial_{z}B_z .$

	=== Velocity gradient tensor ===		=== Velocity gradient tensor ===
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	$\begin{eqnarray*}		$\begin{eqnarray*}
	&		&
	\vec{A} = \bnabla \wedge(\vechat{z}\psi) + \bnabla \wedge\bnabla \wedge(\vechat{z}\phi) ,		\vec{A} = \bnabla \wedge(\vechat{z}\psi) + \bnabla \wedge\bnabla \wedge(\vechat{z}\phi) ,
	& \\ &		& \\ &
	\psi=\psi(r,\theta,z), \quad		\psi=\psi(r,\theta,z), \quad
	\phi=\phi(r,\theta,z).		\phi=\phi(r,\theta,z).
	\nonumber &		\nonumber &
	\end{eqnarray*}$		\end{eqnarray*}$

	i.e.		i.e.

	$\vec{A}_r =		$\vec{A}_r =
	\frac1{r}\,\partial_{\theta} \psi + \partial_{rz} \phi ,$		\frac1{r}\,\partial_{\theta} \psi + \partial_{rz} \phi ,$

	$\vec{A}_\theta =		$\vec{A}_\theta =
	-\partial_{r} \psi + \frac1{r}\,\partial_{\theta z} \phi ,$		-\partial_{r} \psi + \frac1{r}\,\partial_{\theta z} \phi ,$

	$\vec{A}_z =		$\vec{A}_z =
	- \nabla^2_h \phi \, ,$		- \nabla^2_h \phi \, ,$

	where		where

	$\nabla^2_h f =		$\nabla^2_h f =
	(\frac1{r}\,\partial_{r}+\partial_{rr}) f + \frac1{r^2}\,\partial_{\theta\theta} f .$		(\frac1{r}\,\partial_{r}+\partial_{rr}) f + \frac1{r^2}\,\partial_{\theta\theta} f .$


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	$\begin{eqnarray*}		$\begin{eqnarray*}
	&		&
	\vec{A} = \psi_0\,\vechat{\theta} + \phi_0\,\vechat{z}		\vec{A} = \psi_0\,\vechat{\theta} + \phi_0\,\vechat{z}
	+ \bnabla \wedge(\vec{r}\psi) + \bnabla \wedge\bnabla \wedge(\vec{r}\phi) ,		+ \bnabla \wedge(\vec{r}\psi) + \bnabla \wedge\bnabla \wedge(\vec{r}\phi) ,
	& \\ &		& \\ &
	\psi_0=\psi_0(r), \quad		\psi_0=\psi_0(r), \quad
	\phi_0=\phi_0(r), \quad		\phi_0=\phi_0(r), \quad
	\psi=\psi(r,\theta,z), \quad		\psi=\psi(r,\theta,z), \quad
	\phi=\phi(r,\theta,z).		\phi=\phi(r,\theta,z).
	\nonumber &		\nonumber &
	\end{eqnarray*}$		\end{eqnarray*}$

	i.e.		i.e.

	$\vec{A}_r =		$\vec{A}_r =
	-r \nabla^2_c \phi ,$		-r \nabla^2_c \phi ,$

	$\vec{A}_\theta =		$\vec{A}_\theta =
	\psi_0 + r \partial_{z} \psi + \partial_{r\theta} \phi ,$		\psi_0 + r \partial_{z} \psi + \partial_{r\theta} \phi ,$

	$\vec{A}_z =		$\vec{A}_z =
	\phi_0 - \partial_{\theta} \psi + (2+r\partial_{r})\partial_{z} \phi \, ,$		\phi_0 - \partial_{\theta} \psi + (2+r\partial_{r})\partial_{z} \phi \, ,$

	where		where

	$\nabla^2_c f = \frac1{r^2}\,\partial_{\theta\theta} f + \partial_{zz} f .$		$\nabla^2_c f = \frac1{r^2}\,\partial_{\theta\theta} f + \partial_{zz} f .$

Apwillis: /* Toroidal-poloidal decompositions */

2017-10-12T08:41:04Z

Toroidal-poloidal decompositions

← Older revision		Revision as of 01:41, 12 October 2017
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	\end{eqnarray*}$		\end{eqnarray*}$

	~~where~~		i.e.

	$\vec{A}_r =		$\vec{A}_r =
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	$\vec{A}_z =		$\vec{A}_z =
	- \nabla^2_h \phi .$		- \nabla^2_h \phi \, ,$

			where

	$\nabla^2_h f =		$\nabla^2_h f =
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	\end{eqnarray*}$		\end{eqnarray*}$

	~~where~~		i.e.

	$\vec{A}_r =		$\vec{A}_r =
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	$\vec{A}_z =		$\vec{A}_z =
	\phi_0 - \partial_{\theta} \psi + (2+r\partial_{r})\partial_{z} \phi .$		\phi_0 - \partial_{\theta} \psi + (2+r\partial_{r})\partial_{z} \phi \, ,$

			where

	$\nabla^2_c f = \frac1{r^2}\,\partial_{\theta\theta} f + \partial_{zz} f .$		$\nabla^2_c f = \frac1{r^2}\,\partial_{\theta\theta} f + \partial_{zz} f .$

Apwillis at 11:31, 7 March 2017

2017-03-07T11:31:24Z

← Older revision		Revision as of 04:31, 7 March 2017
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	=== Velocity gradient tensor ===		=== Velocity gradient tensor ===

	$\left[		$\bnabla\vec{u} ~=~ \left[
	\begin{array}{ccc}		\begin{array}{ccc}
	\partial_r u_r & \partial_r u_\theta & \partial_r u_z \\		\partial_r u_r & \partial_r u_\theta & \partial_r u_z \\

Apwillis: /* grad, div, curl */

2016-12-13T09:25:31Z

grad, div, curl

← Older revision		Revision as of 02:25, 13 December 2016
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	{{latexPreamble}}		{{latexPreamble}}
	=== ~~grad~~, ~~div~~, curl ===		=== Gradient, divergence, curl ===

	$(\bnabla f)_r = \partial_{r}f,		$(\bnabla f)_r = \partial_{r}f,

Apwillis at 11:55, 6 December 2016

2016-12-06T11:55:26Z

@@ Line 51: / Line 51: @@
        A_r\,\partial_{r}B_z + \frac{A_\theta}{r}\,\partial_{\theta}B_z
        + A_z\,\partial_{z}B_z .$
 === Toroidal-poloidal decompositions ===

Apwillis at 10:28, 15 August 2014

2014-08-15T10:28:19Z

← Older revision		Revision as of 03:28, 15 August 2014
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	+ A_z\,\partial_{z}B_z .$		+ A_z\,\partial_{z}B_z .$

	=== Toroidal-poloidal ~~decomposition, axial~~ ===		=== Toroidal-poloidal decompositions ===
			'''Axial form:'''

	$\begin{eqnarray*}		$\begin{eqnarray*}
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	\end{eqnarray*}$		\end{eqnarray*}$

	~~$\nabla^2_h f =~~		where
	~~(\frac1{r}\,\partial_{r}+\partial_{rr}) f + \frac1{r^2}\,\partial_{\theta\theta} f .$~~

	$\vec{A}_r =		$\vec{A}_r =
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	- \nabla^2_h \phi .$		- \nabla^2_h \phi .$

	=~~== Toroidal-poloidal decomposition~~, ~~radial ===~~		$\nabla^2_h f =
			(\frac1{r}\,\partial_{r}+\partial_{rr}) f + \frac1{r^2}\,\partial_{\theta\theta} f .$


			'''Radial form:'''

	$\begin{eqnarray*}		$\begin{eqnarray*}
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	\end{eqnarray*}$		\end{eqnarray*}$

	~~$\nabla^2_c f = \frac1{r^2}\,\partial_{\theta\theta} f + \partial_{zz} f .$~~		where

	$\vec{A}_r =		$\vec{A}_r =
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	$\vec{A}_z =		$\vec{A}_z =
	\phi_0 - \partial_{\theta} \psi + (2+r\partial_{r})\partial_{z} \phi .$		\phi_0 - \partial_{\theta} \psi + (2+r\partial_{r})\partial_{z} \phi .$

			$\nabla^2_c f = \frac1{r^2}\,\partial_{\theta\theta} f + \partial_{zz} f .$

Apwillis: Apwillis moved page Differential operators in cylindrical geometry to Differential operators in cylindrical coordinates

2014-08-15T09:07:52Z

Apwillis moved page Differential operators in cylindrical geometry to Differential operators in cylindrical coordinates

← Older revision	Revision as of 02:07, 15 August 2014
(No difference)