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Perfect fluid

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Fluid fully characterized by its density and isotropic pressure

Inphysics, aperfect fluid orideal fluid is afluid that can be completely characterized by its rest framemass density $\rho _{m}$ andisotropicpressure⁠ $p {\displaystyle p}$ ⁠.^[1] Real fluids are viscous ("sticky") and contain (and conduct) heat. Perfect fluids are idealized models in which these possibilities are ignored. Specifically, perfect fluids have noshear stresses,viscosity, orheat conduction.^[1] Aquark–gluon plasma^[2]andgraphene are examples of nearly perfect fluids that could be studied in a laboratory.^[3]

Examples

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Perfect fluids are used ingeneral relativity to model idealized distributions ofmatter, such as the interior of a star or an isotropic universe. In the latter case, the symmetry of the cosmological principle and theequation of state of the perfect fluid lead toFriedmann equation for theexpansion of the universe.^[4]

A flock of birds in the medium of air is an example of a perfect fluid without relativistic symmetry; an electron gas in a solid with possible electron-phonon coupling is also modeled as a perfect fluid.^[1]

D'Alembert paradox

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Inclassical mechanics, ideal fluids are described byEuler equations. Ideal fluids produce nodrag according tod'Alembert's paradox. If a fluid produced drag, then work would be needed to move an object through the fluid and that work would produce heat or fluid motion. However, a perfect fluid can not dissipate energy and it can't transmit energy infinitely far from the object.^[5]^: 34

Relativistic formulation

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Thestress–energy tensor of a perfect fluid contains only the diagonal components.

In space-positivemetric signature tensor notation, thestress–energy tensor of a perfect fluid can be written in the form

T^{\mu \nu }=\left(\rho _{m}+{\frac {p}{c^{2}}}\right)\,U^{\mu }U^{\nu }+p\,\eta ^{\mu \nu },

whereU is the4-velocity vector field of the fluid and where $\eta _{\mu \nu }=\operatorname {diag} (-1,1,1,1)$ is the metric tensor ofMinkowski spacetime.

In time-positivemetric signature tensor notation, thestress–energy tensor of a perfect fluid can be written in the form

T^{\mu \nu }=\left(\rho _{\text{m}}+{\frac {p}{c^{2}}}\right)\,U^{\mu }U^{\nu }-p\,\eta ^{\mu \nu },

where $U {\displaystyle U}$ is the 4-velocity of the fluid and where $\eta _{\mu \nu }=\operatorname {diag} (1,-1,-1,-1)$ is the metric tensor ofMinkowski spacetime.

This takes on a particularly simple form in the rest frame

\left[{\begin{matrix}\rho _{e}&0&0&0\\0&p&0&0\\0&0&p&0\\0&0&0&p\end{matrix}}\right]

where $\rho _{\text{e}}=\rho _{\text{m}}c^{2}$ is theenergy density and $p {\displaystyle p}$ is thepressure of the fluid.

Perfect fluids admit aLagrangian formulation, which allows the techniques used infield theory, in particular,quantization, to be applied to fluids.

References

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^^a ^b ^cde Boer, Jan; Hartong, Jelle; Obers, Niels; Sybesma, Waste; Vandoren, Stefan (2018-07-17)."Perfect fluids".SciPost Physics.5 (1): 003.arXiv:1710.04708.Bibcode:2018ScPP....5....3D.doi:10.21468/SciPostPhys.5.1.003.ISSN 2542-4653.
^WA Zajc (2008). "The fluid nature of quark–gluon plasma".Nuclear Physics A.805 (1–4):283c –294c.arXiv:0802.3552.Bibcode:2008NuPhA.805..283Z.doi:10.1016/j.nuclphysa.2008.02.285.S2CID 119273920.
^Müller, Markus (2009)."Graphene: A Nearly Perfect Fluid".Physical Review Letters.103 (2): 025301.arXiv:0903.4178.Bibcode:2009PhRvL.103b5301M.doi:10.1103/PhysRevLett.103.025301.
^Navas, S.; et al. (Particle Data Group) (2024). "Review of Particle Physics".Physical Review D.110 (3):1–708.doi:10.1103/PhysRevD.110.030001.hdl:20.500.11850/695340. 22.1.3 The Friedmann equations of motion
^Landau, Lev Davidovich; Lifšic, Evgenij M. (1959).Fluid mechanics. Their course of theoretical physics. London: Pergamon Press.ISBN 978-1-4831-4050-6.

v t e Non-Newtonian fluids
Effects	Barus Self-siphoning Kaye Toms Weissenberg
Properties	Viscoelastic materials Kelvin–Voigt Maxwell Time-dependent viscosity Rheopecty Thixotropy Non-Newtonian viscoelasticity Shear thinning Shear thickening
Generalized Newtonian fluids	Bingham Carreau Cross Herschel–Bulkley Newtonian Ideal Power-law

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Perfect fluid

Examples

D'Alembert paradox

Relativistic formulation

See also

References

Further reading