Perfect fluid

{{Short description|Fluid fully characterized by its density and isotropic pressure}}

In physics, a perfect fluid or ideal fluid is a fluid that can be completely characterized by its rest frame mass density $\rho_m$ and isotropic pressure {{tmath|1= p }}.{{Cite journal |last1=de Boer |first1=Jan |last2=Hartong |first2=Jelle |last3=Obers |first3=Niels |last4=Sybesma |first4=Waste |last5=Vandoren |first5=Stefan |date=2018-07-17 |title=Perfect fluids |journal=SciPost Physics |language=en |volume=5 |issue=1 |pages=003 |doi=10.21468/SciPostPhys.5.1.003 |doi-access=free |arxiv=1710.04708 |bibcode=2018ScPP....5....3D |issn=2542-4653}} 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 no shear stresses, viscosity, or heat conduction.

A quark–gluon plasma{{cite journal|author=WA Zajc|year=2008|title=The fluid nature of quark–gluon plasma|journal=Nuclear Physics A|volume=805|issue=1–4|pages=283c–294c|arxiv=0802.3552|bibcode=2008NuPhA.805..283Z|doi=10.1016/j.nuclphysa.2008.02.285|s2cid=119273920}}

and graphene are examples of nearly perfect fluids that could be studied in a laboratory.{{Cite journal |last=Müller |first=Markus |date=2009 |title=Graphene: A Nearly Perfect Fluid |url=https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.103.025301 |journal=Physical Review Letters |volume=103 |issue=2 |page=025301 |doi=10.1103/PhysRevLett.103.025301|arxiv=0903.4178 |bibcode=2009PhRvL.103b5301M }}

Non-relativistic fluid mechanics

In classical mechanics, ideal fluids are described by Euler equations. Ideal fluids produce no drag according to d'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.{{Cite book |last1=Landau |first1=Lev Davidovich |title=Fluid mechanics |last2=Lifšic |first2=Evgenij M. |date=1959 |publisher=Pergamon Press |isbn=978-1-4831-4050-6 |series=Their course of theoretical physics |location=London}}{{rp|34}}

A flock of birds in the medium of air is an example of a perfect fluid; an electron gas is also modeled as a perfect fluid.

= Superfluidity =

Superfluids are fluids with zero viscosity, however in practice superfluids cannot be accurately described as a perfect fluid.{{Cite book |last=Annett |first=James F. |url=https://www.google.fr/books/edition/Superconductivity_Superfluids_and_Conden/WZcXmBrZIc8C?hl=en&gbpv=1&dq=superfluids+are+not+perfect+fluids+landau&pg=PA40&printsec=frontcover |title=Superconductivity, Superfluids and Condensates |date=2004-03-25 |publisher=OUP Oxford |isbn=978-0-19-850756-7 |language=en}}{{Cite book |last=Grioli |first=G. |url=https://www.google.fr/books/edition/Macroscopic_Theories_of_Superfluids/0fc8AAAAIAAJ?hl=en&gbpv=1&dq=superfluids+are+not+perfect+fluids&pg=PA151&printsec=frontcover |title=Macroscopic Theories of Superfluids |date=1991-09-26 |publisher=CUP Archive |isbn=978-0-521-37572-6 |language=en}} In the two-fluid model, superfluids are macroscopically considered as having two-coexisting phases, a mixture between a normal fluid and a perfect fluid.

Cosmology and astrophysics

File:StressEnergyTensor contravariant.svg of a perfect fluid contains only the diagonal components.]]

Perfect fluids are a fluid solution used in general relativity to model idealized distributions of matter, such as the interior of a star or an isotropic universe. In the latter case, the symmetry of the cosmological principle and the equation of state of the perfect fluid lead to Friedmann equation for the expansion of the universe.{{Cite journal |last=Navas, S. |year=2024 |title=Review of Particle Physics |journal=Physical Review D |volume=110 |issue=3 |pages=1–708 |doi=10.1103/PhysRevD.110.030001 |hdl-access=free |collaboration=Particle Data Group |hdl=20.500.11850/695340}} 22.1.3 The Friedmann equations of motion

= Formulation =

In space-positive metric signature tensor notation, the stress–energy tensor of a perfect fluid can be written in the form

: $T^{\mu\nu} = \left( \rho_m + \frac{p}{c^2} \right) \, U^\mu U^\nu + p \, \eta^{\mu\nu} ,$

where U is the 4-velocity vector field of the fluid and where $\eta_{\mu \nu} = \operatorname{diag}(-1,1,1,1)$ is the metric tensor of Minkowski spacetime.

The case where p=0 describes a dust solution. When $p=\rho_m c^2/3$ , it describes a photon gas (radiation).

In time-positive metric signature tensor notation, the stress–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 $U$ is the 4-velocity of the fluid and where $\eta_{\mu \nu} = \operatorname{diag}(1,-1,-1,-1)$ is the metric tensor of Minkowski 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 the energy density and $p$ is the pressure of the fluid.

Perfect fluids admit a Lagrangian formulation, which allows the techniques used in field theory, in particular, quantization, to be applied to fluids.

Relativistic Euler equations read

: $\partial_\nu T^{\mu\nu}=0$

in the non relativistic limit, these equations reduce to the usual Euler equations.{{Cite book |last=Luscombe |first=James |url=https://www.google.fr/books/edition/Core_Principles_of_Special_and_General_R/ToqADwAAQBAJ?hl=en&gbpv=1&dq=relativistic+euler+equation+nonrelativistic+limit&pg=PA181&printsec=frontcover |title=Core Principles of Special and General Relativity |date=2018-12-07 |publisher=CRC Press |isbn=978-0-429-65953-9 |language=en}}

Perfect fluid

Non-relativistic fluid mechanics

= Superfluidity =

Cosmology and astrophysics

= Formulation =

See also

References

Further reading