Hamiltonian fluid mechanics

Hamiltonian fluid mechanics is the application of Hamiltonian methods to fluid mechanics. This formalism can only apply to nondissipative 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

:{varphi(vec{x}), ho(vec{y})}=delta^d(vec{x}-vec{y})

and the Hamiltonian by:

:mathcal{H}=int mathrm{d}^d x left [ frac{1}{2} ho(vec{ abla} varphi)^2 +e( ho) ight] ,

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:

:e" = frac{1}{ ho}p',

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

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

:egin{align} frac{partial ho}{partial t}&=+frac{deltamathcal{H{deltavarphi}= -vec{ abla}cdot( hovec{v}), \ frac{partial varphi}{partial t}&=-frac{deltamathcal{H{delta ho}=-frac{1}{2}vec{v}cdotvec{v}-e',end{align}

where vec{v} stackrel{mathrm{def{=} abla varphi is the velocity and is vorticity-free. The second equation leads to the Euler equations:

:frac{partial vec{v{partial t} + (vec{v}cdot abla) vec{v} = -e" abla ho = -frac{1}{ ho} abla{p}

after exploiting the fact that the vorticity is zero:

:vec{ abla} imesvec{v}=vec{0}.

ee 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
*cite journal | 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


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Hamiltonian — may refer toIn mathematics : * Hamiltonian system * Hamiltonian path, in graph theory * Hamiltonian group, in group theory * Hamiltonian (control theory) * Hamiltonian matrix * Hamiltonian flow * Hamiltonian vector field * Hamiltonian numbers (or …   Wikipedia

  • Mechanics — This article is about an area of scientific study. For other uses, see Mechanic (disambiguation). Mechanics (Greek Μηχανική) is the branch of physics concerned with the behavior of physical bodies when subjected to forces or displacements, and… …   Wikipedia

  • mechanics — /meuh kan iks/, n. 1. (used with a sing. v.) the branch of physics that deals with the action of forces on bodies and with motion, comprised of kinetics, statics, and kinematics. 2. (used with a sing. v.) the theoretical and practical application …   Universalium

  • Classical mechanics — This article is about the physics sub field. For the book written by Herbert Goldstein and others, see Classical Mechanics (book). Classical mechanics …   Wikipedia

  • Continuum mechanics — Continuum mechanics …   Wikipedia

  • Quantum mechanics — For a generally accessible and less technical introduction to the topic, see Introduction to quantum mechanics. Quantum mechanics …   Wikipedia

  • Dynamic fluid film equations — An example of dynamic fluid films. Fluid films, such as soap films, are commonly encountered in everyday experience. A soap film can be formed by dipping a closed contour wire into a soapy solution as in the figure on the right. Alternatively, a… …   Wikipedia

  • Liouville's theorem (Hamiltonian) — In physics, Liouville s theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that the phase space distribution function is constant along the trajectories… …   Wikipedia

  • Partition function (statistical mechanics) — For other uses, see Partition function (disambiguation). Partition function describe the statistical properties of a system in thermodynamic equilibrium. It is a function of temperature and other parameters, such as the volume enclosing a gas.… …   Wikipedia

  • celestial mechanics — the branch of astronomy that deals with the application of the laws of dynamics and Newton s law of gravitation to the motions of heavenly bodies. [1815 25] * * * Branch of astronomy that deals with the mathematical theory of the motions of… …   Universalium


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.