Spatial frequency - Revision history

en>BG19bot: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (10242)

2014-06-16T08:30:15Z

WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB (10242)

← Older revision		Revision as of 10:30, 16 June 2014
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	~~'''Hamiltonian fluid mechanics'''~~ is ~~the application of [[Hamiltonian mechanics\|Hamiltonian]] methods to [[fluid mechanics]]~~. ~~This formalism can only apply~~ to ~~non[[dissipative]] fluids.~~		Hello and welcome. My name is Irwin and I totally dig that title. California is exactly where her home is but she needs to transfer because of her family members. Doing ceramics is what my family members and I enjoy. Managing people has been his working day occupation for a while.<br><br>My web blog: [http://nxnn.info/user/G8498 http://nxnn.info/]

	~~==Irrotational barotropic flow==~~
	~~Take the simple example~~ of ~~a [[barotropic]], [[inviscid]] [[vorticity-free]] fluid~~.

	~~Then, the conjugate fields are the [[mass density]] field ''ρ''~~ and ~~the [[velocity potential]] ''φ''~~. ~~The [[Poisson bracket]] is given by~~

	~~:<math>\{\varphi(\vec{x}),\rho(\vec{y})\}=\delta^d(\vec{x}-\vec{y})</math>~~

	~~and the Hamiltonian by:~~

	~~:<math>\mathcal{H}=\int \mathrm{d}^d x \left[ \frac{1}{2}\rho(\vec{\nabla} \varphi)^2 +e(\rho) \right],</math>~~

	~~where ''e'' is the [[internal energy]] density, as~~ a ~~function of ''ρ''~~.
	~~For this barotropic flow, the internal energy is related to the pressure ''p'' by:~~

	:<~~math~~>~~e'' = \frac{1}{\rho}p',~~<~~/math~~>

	~~where an apostrophe ('), denotes differentiation with respect to ''ρ''.~~

	~~This Hamiltonian structure gives rise to the following two [~~[~~equations of motion]]:~~

	:~~<math>~~
	~~\begin{align}~~
	~~\frac{\partial \rho}{\partial t}&=+\frac{\delta\mathcal{H}}{\delta\varphi}= -\vec{\nabla}\cdot(\rho\vec{v}),~~
	\\
	~~\frac{\partial \varphi}{\partial t}&=-\frac{\delta\mathcal{H}}{\delta\rho}=-\frac{1}{2}\vec{v}\cdot\vec{v}-e',~~
	~~\end{align}~~
	</~~math>~~

	~~where <math>\vec{v}\ \stackrel{\mathrm{def}}{=}\ \nabla \varphi<~~/~~math> is the velocity and is [[vorticity-free]]~~. ~~The second equation leads to the [[Euler equations]]:~~

	~~:<math>\frac{\partial \vec{v}}{\partial t} + (\vec{v}\cdot\nabla) \vec{v} = -e''\nabla\rho = -\frac{1}{\rho}\nabla{p}<~~/~~math>~~

	~~after exploiting the fact that the [[vorticity]] is zero:~~

	:~~<math>\vec{\nabla}\times\vec{v}=\vec{0}.<~~/~~math>~~

	~~==See also==~~
	*[[Luke's variational principle]]

	~~==References==~~
	*{{cite journal \| journal=Annual Review of Fluid Mechanics \| volume=20 \| pages=225–256 \| year=1988 \| doi=10.1146/~~annurev.fl.20.010188.001301 \| title=Hamiltonian Fluid Mechanics \| author=R. Salmon\|bibcode = 1988AnRFM..20..225S }}~~
	*{{cite journal \| doi=10.~~1016~~/S0065-2687(08)60429-X \| title=Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics \| author=T. G. Shepherd \| year=1990 \| journal=Advances in Geophysics \| volume=32 \| pages=287–338 \|bibcode = 1990AdGeo..32..287S }}


	~~[[Category:Fluid dynamics]]~~
	~~[[Category:Hamiltonian mechanics]]~~
	~~[[Category:Dynamical systems]~~]

en>Monkbot: /* Spatial-frequency theory */Fix CS1 deprecated date parameter errors

2014-01-30T07:00:30Z

Spatial-frequency theory: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 09:00, 30 January 2014
Line 1:		Line 1:
	~~They contact me Emilia~~. ~~North Dakota is exactly where me~~ and ~~my spouse live~~. ~~Doing ceramics~~ is ~~what my family members~~ and ~~I appreciate~~. For ~~years I~~'~~ve been working as a payroll clerk~~.<br><br>~~my web~~-~~site at home std testing~~ ([~~http~~://~~www~~.~~pinaydiaries~~.~~com~~/~~user/LConsiden click for more info~~])		'''Hamiltonian fluid mechanics''' is the application of [[Hamiltonian mechanics\|Hamiltonian]] methods to [[fluid mechanics]]. This formalism can only apply to non[[dissipative]] fluids.

			==Irrotational barotropic flow==
			Take the simple example of a [[barotropic]], [[inviscid]] [[vorticity-free]] fluid.

			Then, the conjugate fields are the [[mass density]] field ''ρ'' and the [[velocity potential]] ''φ''. The [[Poisson bracket]] is given by

			:<math>\{\varphi(\vec{x}),\rho(\vec{y})\}=\delta^d(\vec{x}-\vec{y})</math>

			and the Hamiltonian by:

			:<math>\mathcal{H}=\int \mathrm{d}^d x \left[ \frac{1}{2}\rho(\vec{\nabla} \varphi)^2 +e(\rho) \right],</math>

			where ''e'' is the [[internal energy]] density, as a function of ''ρ''.
			For this barotropic flow, the internal energy is related to the pressure ''p'' by:

			:<math>e'' = \frac{1}{\rho}p',</math>

			where an apostrophe ('), denotes differentiation with respect to ''ρ''.

			This Hamiltonian structure gives rise to the following two [[equations of motion]]:

			:<math>
			\begin{align}
			\frac{\partial \rho}{\partial t}&=+\frac{\delta\mathcal{H}}{\delta\varphi}= -\vec{\nabla}\cdot(\rho\vec{v}),
			\\
			\frac{\partial \varphi}{\partial t}&=-\frac{\delta\mathcal{H}}{\delta\rho}=-\frac{1}{2}\vec{v}\cdot\vec{v}-e',
			\end{align}
			</math>

			where <math>\vec{v}\ \stackrel{\mathrm{def}}{=}\ \nabla \varphi</math> is the velocity and is [[vorticity-free]]. The second equation leads to the [[Euler equations]]:

			:<math>\frac{\partial \vec{v}}{\partial t} + (\vec{v}\cdot\nabla) \vec{v} = -e''\nabla\rho = -\frac{1}{\rho}\nabla{p}</math>

			after exploiting the fact that the [[vorticity]] is zero:

			:<math>\vec{\nabla}\times\vec{v}=\vec{0}.</math>

			==See also==
			*[[Luke's variational principle]]

			==References==
			*{{cite journal \| journal=Annual Review of Fluid Mechanics \| volume=20 \| pages=225–256 \| year=1988 \| doi=10.1146/annurev.fl.20.010188.001301 \| title=Hamiltonian Fluid Mechanics \| author=R. Salmon\|bibcode = 1988AnRFM..20..225S }}
			*{{cite journal \| doi=10.1016/S0065-2687(08)60429-X \| title=Symmetries, conservation laws, and Hamiltonian structure in geophysical fluid dynamics \| author=T. G. Shepherd \| year=1990 \| journal=Advances in Geophysics \| volume=32 \| pages=287–338 \|bibcode = 1990AdGeo..32..287S }}


			[[Category:Fluid dynamics]]
			[[Category:Hamiltonian mechanics]]
			[[Category:Dynamical systems]]

en>Thermochap: units clarify the various conventions used in the literature

2012-04-10T14:52:59Z

units clarify the various conventions used in the literature

New page

They contact me Emilia. North Dakota is exactly where me and my spouse live. Doing ceramics is what my family members and I appreciate. For years I've been working as a payroll clerk.<br><br>my web-site at home std testing ([http://www.pinaydiaries.com/user/LConsiden click for more info])