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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/] |
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| ==Irrotational barotropic flow==
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| Take the simple example of a [[barotropic]], [[inviscid]] [[vorticity-free]] fluid.
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| Then, the conjugate fields are the [[mass density]] field ''ρ'' and the [[velocity potential]] ''φ''. The [[Poisson bracket]] is given by
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| :<math>\{\varphi(\vec{x}),\rho(\vec{y})\}=\delta^d(\vec{x}-\vec{y})</math>
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| and the Hamiltonian by:
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| :<math>\mathcal{H}=\int \mathrm{d}^d x \left[ \frac{1}{2}\rho(\vec{\nabla} \varphi)^2 +e(\rho) \right],</math>
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| where ''e'' is the [[internal energy]] density, as a function of ''ρ''.
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| For this barotropic flow, the internal energy is related to the pressure ''p'' by:
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| :<math>e'' = \frac{1}{\rho}p',</math>
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| where an apostrophe ('), denotes differentiation with respect to ''ρ''.
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| This Hamiltonian structure gives rise to the following two [[equations of motion]]:
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| :<math> | |
| \begin{align}
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| \frac{\partial \rho}{\partial t}&=+\frac{\delta\mathcal{H}}{\delta\varphi}= -\vec{\nabla}\cdot(\rho\vec{v}),
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| \\
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| \frac{\partial \varphi}{\partial t}&=-\frac{\delta\mathcal{H}}{\delta\rho}=-\frac{1}{2}\vec{v}\cdot\vec{v}-e',
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| \end{align}
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| </math>
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| 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]]:
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| :<math>\frac{\partial \vec{v}}{\partial t} + (\vec{v}\cdot\nabla) \vec{v} = -e''\nabla\rho = -\frac{1}{\rho}\nabla{p}</math>
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| after exploiting the fact that the [[vorticity]] is zero:
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| :<math>\vec{\nabla}\times\vec{v}=\vec{0}.</math> | |
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| ==See also==
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| *[[Luke's variational principle]]
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| ==References==
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| *{{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 }}
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| *{{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 }}
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| [[Category:Fluid dynamics]]
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| [[Category:Hamiltonian mechanics]]
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| [[Category:Dynamical systems]]
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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.
My web blog: http://nxnn.info/