What is Perfect fluid: Definition and 25 Discussions
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.
Hello. Could anyone help me with some insight in an extra term appearing in the motion equations of a relativistic fluid? I say extra term, because it's not present on the motion for a test particle, as it follows:
Let's propose Minkowski space-time, the motion equations for a fluid with zero...
On page 353 of Schutz's textbook he writes the following:
So it seems that the ether is replaced by a "homogeneous perfect fluid".
It seems the medium which fills the universe is not ether but "homogeneous perfect fluid".
But in that case what characteristics did the proposed ether have that a...
Stress tensor for the fluid is ##T_{ab} = \rho u_a u_b + P(\eta_{ab} + u_a u_b)##, whilst the equation of motion (assuming the system is isolated) is given by ##\partial^a T_{ab} = 0##. So I tried$$\begin{align*}
\partial^a T_{ab} &= \partial^a \rho u_a u_b + \partial^a P(\eta_{ab} + u_a u_b)...
The energy momentum tensor of a perfect relativistic fluid is given by
$$T^{\mu\nu} = (\rho + p)u^\mu u^\nu + p g^{\mu\nu}$$
I don't understand why this is a tensor, i.e. why it transforms properly under coordinate changes.
##u^\mu u^\nu## and ##g^{\mu\nu}## are tensors, so for ##T^{\mu\nu}##...
Starting from the following definition of stress-energy tensor for a perfect fluid in special relativity :
$${\displaystyle T^{\mu \nu }=\left(\rho+{\frac {p}{c^{2}}}\right)\,v^{\mu }v^{\nu }-p\,\eta ^{\mu \nu }\,}\quad(1)$$
with ##v^{\nu}=\dfrac{\text{d}x^{\nu}}{\text{d}\tau}## and...
During CFD modeling of a gas-solid flow, flow of solid was modeled as a perfect fluid using OpenFOAM.
The density of the perfect fluid is calculated using the following equation as given in the documentation.
ρ = P/RT + ρo , where ρo is the density at T = 0 kelvin, ρ is the density of the...
Hello guys,
I've came up with three statements in a discussion with a friend where we were trying to check if we had a clear vision of what isotropy and group invariance would imply in an arbitrary theory of gravity at the level of its matter lagrangian. We got stuck at some point so I came here...
Homework Statement
This should be pretty simple and I guess I am doing something stupid?
##T_{bv}=(p+\rho)U_bU_v-\rho g_{bv}##
compute ##T^u_v##:
##T^0_0=\rho, T^i_i=-p##Homework Equations
##U^u=\delta^t_u##
##g_{uv}## is the FRW metric,in particular ##g_{tt}=1##
##g^{bu}T_{bv}=T^u_v##
##...
Homework Statement
Derive the relativistic Euler equation by contracting the conservation law $$\partial _\mu {T^{\mu \nu}} =0$$ with the projection tensor $${P^{\sigma}}_\nu = {\delta^{\sigma}}_\nu + U^{\sigma} U_{\nu}$$ for a perfect fluid.
Homework Equations
$$\partial _\mu {T^{\mu \nu}} =...
I believe this thread is sufficiently different from one that was recently closed to not violate any guidelines, though there are unfortunately some similarities as the closed thread sparked the questions in my mind.
If we look at the stress energy tensor of a perfect fluid in geometric units...
In Hawking-Ellis Book(1973) "The large scale structure of space-time" p69-p70, they derive the energy-momentum tensor for perfect fluid by lagrangian formulation. They imply if ##D## is a sufficiently small compact region, one can represent a congruence by a diffeomorphism ##\gamma: [a,b]\times...
The stress-energy tensor of a perfect fluid in its rest frame is:
(1) Tij= diag [ρc2, P, P, P]
where ρc2 is the energy density and P the pressure of the fluid.
If Tij is as stated in eq.(1), the metric tensor gij of the system composed by an indefinitely extended perfect fluid in...
Hi everyone,
I want to derive the Friedmann equations from Einstein Field Equations. However, I have a problem that stems from the energy-momentum tensor. I am also trying to keep track of ## c^2 ## terms.
FRW Metric:
$$ ds^2= -c^2dt^2 + a^2(t) \left( {\frac{dr^2}{1-kr^2} + r^2 d\theta^2 + r^2...
Homework Statement
For a system of discrete point particles the energy momentum takes the form
T_{\mu \nu} = \sum_a \frac{p_\mu^{(a)}p_\nu^{(a)}}{p^{0(a)}} \delta^{(3)}(\vec{x}-\vec{x}^{(a)}),
where the index a labels the different particles. Show that, for a dense collection of particles...
Homework Statement
[
For a system of discrete point particles the energy momentum takes the form
T_{\mu \nu} = \sum_a \frac{p_\mu^{(a)}p_\nu^{(a)}}{p^{0(a)}} \delta^{(3)}(\vec{x}-\vec{x}^{(a)}),
where the index a labels the different particles. Show that, for a dense collection of particles...
we use perfect fluid which is characterized by a energy density and isotropic pressure for general forms of matter.
When guessing the values of energy momentum tensor indices we can use the physical insight that they are the flux of four momentum in a constant surface of spacetime.
The...
So the textbook I am reading says that a perfect fluid cannot sustain shear stress, or that p_{yx}=2\mu \dot \epsilon_{yx} =0, where \mu is the viscosity, and \dot \epsilon_{yx} is the rate of angular deformation. Then it says when \dot \epsilon_{yx} =0, this means that "two adjacent...
The stress-energy tensor is usually defined in standard GR treatments as
T_{\mu\nu} = -\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{g}L_m)}{\delta g^{\mu\nu}})
with the Lm the matter Lagrangian.
I'm curious what Lm is for a perfect fluid with density ρ and pressure P that would lead to the...
Hi everyone,
If anyone could point me in the right direction with this problem I'd really appreciate it.
"Show that the cosmological constant can be interpreted as a perfect fluid having an equation of state w=-1."
I have a rough idea of how to do the second part of the proof: if the...
Hi everyone,
If anyone could point me in the right direction with this problem I'd really appreciate it.
"Show that the cosmological constant can be interpreted as a perfect fluid having an equation of state w=-1."
I have a rough idea of how to do the second part of the proof: if the...
If I had a perfect fluid I could write the stress energy tensor in the rest frame of the fluid as a nice diagonal tensor with [rho,p,p,p] and use this to solve the field equations.
Pressure is just random internal motions, so what happens if all the particles in my perfect fluid decide to...
The energy momentum tensor of perfect fluid is
T^{\alpha \beta} = \left( \rho + p \right) \, U^\alpha U^\beta - p \, g^{\alpha \beta}
It must be derived by varying the metric in the action of matter fields but I've never seen that action. Anyone knows it?
The stress energy tensor of a perfect fluid is composed of two terms of which only one term contains the metric tensor gab. (product of metric tensor and pressure). For curved spacetime, one replaces the flat spacetime metric tensor by the metric tensor of curved space. What I find bizar however...