Perfect fluid

In physics, a perfect fluid is a fluid that can be completely characterized by its rest frame mass density




ρ

m




{\displaystyle \rho _{m}}
and isotropic pressure p.
Real fluids are "sticky" and contain (and conduct) heat. Perfect fluids are idealized models in which these possibilities are neglected. Specifically, perfect fluids have no shear stresses, viscosity, or heat conduction.
In space-positive metric signature tensor notation, the stress–energy tensor of a perfect fluid can be written in the form





T

μ
ν


=

(


ρ

m


+


p

c

2





)



U

μ



U

ν


+
p


η

μ
ν





{\displaystyle 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




η

μ
ν


=
diag
⁡
(
−
1
,
1
,
1
,
1
)


{\displaystyle \eta _{\mu \nu }=\operatorname {diag} (-1,1,1,1)}
is the metric tensor of Minkowski spacetime.
In time-positive metric signature tensor notation, the stress–energy tensor of a perfect fluid can be written in the form





T

μ
ν


=

(


ρ

m


+


p

c

2





)



U

μ



U

ν


−
p


η

μ
ν





{\displaystyle 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




η

μ
ν


=
diag
⁡
(
1
,
−
1
,
−
1
,
−
1
)


{\displaystyle \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





[





ρ

e




0


0


0




0


p


0


0




0


0


p


0




0


0


0


p




]



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




ρ

e


=

ρ

m



c

2




{\displaystyle \rho _{\text{e}}=\rho _{\text{m}}c^{2}}
is the energy density and



p


{\displaystyle 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. This formulation can be generalized, but unfortunately, heat conduction and anisotropic stresses cannot be treated in these generalized formulations.Perfect fluids are 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 equation of state of the perfect fluid may be used in Friedmann–Lemaître–Robertson–Walker equations to describe the evolution of the universe.
In general relativity, the expression for the stress–energy tensor of a perfect fluid is written as





T

μ
ν


=

(


ρ

m


+


p

c

2





)



U

μ



U

ν


+
p


g

μ
ν





{\displaystyle T^{\mu \nu }=\left(\rho _{m}+{\frac {p}{c^{2}}}\right)\,U^{\mu }U^{\nu }+p\,g^{\mu \nu }\,}
where U is the 4-velocity vector field of the fluid and where




g

μ
ν




{\displaystyle g_{\mu \nu }}
is the inverse metric,
written with a space-positive signature.
An example of an ideal fluid is superfluid helium-4.

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