Perfect fluid explained
and
isotropic pressure .
[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 no
shear stresses,
viscosity, or
heat conduction.
[1] A
quark–gluon plasma[2] and
graphene are examples of nearly perfect fluids that could be studied in a laboratory.
[3] Non-relativistic fluid mechanics
See main article: Inviscid flow. 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.[4]
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.[5] [6] 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
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.[7]
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(\rhom+
\right)U\muU\nu+pη\mu\nu,
where
U is the 4-velocity
vector field of the fluid and where
η\mu=\operatorname{diag}(-1,1,1,1)
is the metric tensor of
Minkowski spacetime.
The case where p=0 describes a dust solution. When
, 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(\rhom+
\right)U\muU\nu-pη\mu\nu,
where
is the 4-velocity of the fluid and where
η\mu=\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}\rhoe&0&0&0\\0&p&0&0\\0&0&p&0\\0&0&0&p\end{matrix}\right]
where
is the
energy density and
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
in the non relativistic limit, these equations reduce to the usual Euler equations.
[8] See also
Further reading
- , (pbk.)
- Jackiw . R . Nair . V P . Pi . S-Y . Polychronakos . A P . 2004-10-22 . Perfect fluid theory and its extensions . Journal of Physics A: Mathematical and General . 37 . 42 . R327–R432 . 10.1088/0305-4470/37/42/R01 . 0305-4470. hep-ph/0407101 . Topical review.
Notes and References
- de Boer . Jan . Hartong . Jelle . Obers . Niels . Sybesma . Waste . Vandoren . Stefan . 2018-07-17 . Perfect fluids . SciPost Physics . en . 5 . 1 . 003 . 10.21468/SciPostPhys.5.1.003 . free . 1710.04708 . 2018ScPP....5....3D . 2542-4653.
- WA Zajc. 2008. The fluid nature of quark–gluon plasma. Nuclear Physics A. 805. 1–4. 283c–294c. 0802.3552. 2008NuPhA.805..283Z. 10.1016/j.nuclphysa.2008.02.285. 119273920.
- Müller . Markus . 2009 . Graphene: A Nearly Perfect Fluid . Physical Review Letters . 103 . 2 . 025301 . 10.1103/PhysRevLett.103.025301. 0903.4178 . 2009PhRvL.103b5301M .
- Book: Landau . Lev Davidovich . Fluid mechanics . Lifšic . Evgenij M. . 1959 . Pergamon Press . 978-1-4831-4050-6 . Their course of theoretical physics . London.
- Book: Annett, James F. . Superconductivity, Superfluids and Condensates . 2004-03-25 . OUP Oxford . 978-0-19-850756-7 . en.
- Book: Grioli, G. . Macroscopic Theories of Superfluids . 1991-09-26 . CUP Archive . 978-0-521-37572-6 . en.
- Navas, S. . 2024 . Review of Particle Physics . . 110 . 3 . 1–708 . 10.1103/PhysRevD.110.030001 . free . . 20.500.11850/695340. 22.1.3 The Friedmann equations of motion
- Book: Luscombe, James . Core Principles of Special and General Relativity . 2018-12-07 . CRC Press . 978-0-429-65953-9 . en.
