- #1
Ondrej Certik
- 12
- 0
Hi,
I would like to start from the stress energy tensor for the perfect fluid:
[tex]
T^{\mu\nu}=\begin{pmatrix} \rho c^2 & 0 & 0 & 0\cr 0 & p & 0 & 0\cr 0 & 0 & p & 0\cr 0 & 0 & 0 & p\cr\end{pmatrix}
[/tex]
where [tex]\rho[/tex] is the mass density and [tex]p[/tex] is the pressure, and I would like to derive the nonrelativistic Euler equations: continuity equation, momentum equation and the energy equation:
[tex]
\partial_t\rho + \partial_i(\rho v^i) = 0
[/tex]
[tex]
\partial_t (\rho v^i) + \partial_j (\rho v^i v^j + p\delta^{ij}) = 0
[/tex]
[tex]
\partial_t E + \partial_j\left(v^j\left(E + p \right)\right) = 0
[/tex]
in the above, [tex]v^i[/tex] is the nonrelativistic velocity, [tex]E={1\over2}\rho v^2 + \rho e[/tex] the kinetic plus internal energy density, [tex]\partial_t\equiv {\partial\over\partial t}[/tex] is the time derivative and [tex]\partial_i\equiv {\partial\over \partial x^i}[/tex] are the spatial derivatives and we sum over [tex]i[/tex].
Those are the equations found for example on wikipedia:
http://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)
Here are my attempts to derive it:
http://certik.github.com/theoretical-physics/book/src/fluid-dynamics/general.html#perfect-fluids
http://certik.github.com/theoretical-physics/book/src/fluid-dynamics/general.html#relativistic-derivation-of-the-energy-equation
essentially, I start from
[tex]\partial_\nu T^{\mu\nu} = 0 [/tex]
and for [tex]\mu=i[/tex] we get the momentum equation in the limit [tex]c\to\infty[/tex], and for
[tex]\mu=0[/tex] we get:
[tex]
\partial_t \left(\left(\rho c^2 + p{v^2\over c^2}\right)\gamma^2\right)
+
\partial_i\left(\left(\rho c^2 + p\right)v^i \gamma^2\right)
=0
[/tex]
where [tex]\gamma=(1-{v^2\over c^2})^{-1/2}[/tex]. If we let [tex]E=\rho c^2[/tex], and neglect the term [tex]p {v^2\over c^2}[/tex], we almost get the right energy equation, the only problem is that this energy [tex]\rho c^2[/tex] also contains the rest mass energy, and thus the pressure [tex]p[/tex] is negligible, and the nonrelativistic limit of this equation just gives the equation of the continuity.
How do I derive the energy equation? Somehow, I guess I need to separate the rest mass, use the equation of continuity for the rest mass (that follows from the conservation of the baryon number), and then I should be just left with the kinetic+internal energy and the right equation for nonrelativistic energy. Does anyone know how to do that in all details?
I tried to find this question already answered, but didn't find exactly what I want (I am new to the forums, so I might have missed that --- please point me to the right direction).
Thanks,
Ondrej Certik
I would like to start from the stress energy tensor for the perfect fluid:
[tex]
T^{\mu\nu}=\begin{pmatrix} \rho c^2 & 0 & 0 & 0\cr 0 & p & 0 & 0\cr 0 & 0 & p & 0\cr 0 & 0 & 0 & p\cr\end{pmatrix}
[/tex]
where [tex]\rho[/tex] is the mass density and [tex]p[/tex] is the pressure, and I would like to derive the nonrelativistic Euler equations: continuity equation, momentum equation and the energy equation:
[tex]
\partial_t\rho + \partial_i(\rho v^i) = 0
[/tex]
[tex]
\partial_t (\rho v^i) + \partial_j (\rho v^i v^j + p\delta^{ij}) = 0
[/tex]
[tex]
\partial_t E + \partial_j\left(v^j\left(E + p \right)\right) = 0
[/tex]
in the above, [tex]v^i[/tex] is the nonrelativistic velocity, [tex]E={1\over2}\rho v^2 + \rho e[/tex] the kinetic plus internal energy density, [tex]\partial_t\equiv {\partial\over\partial t}[/tex] is the time derivative and [tex]\partial_i\equiv {\partial\over \partial x^i}[/tex] are the spatial derivatives and we sum over [tex]i[/tex].
Those are the equations found for example on wikipedia:
http://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)
Here are my attempts to derive it:
http://certik.github.com/theoretical-physics/book/src/fluid-dynamics/general.html#perfect-fluids
http://certik.github.com/theoretical-physics/book/src/fluid-dynamics/general.html#relativistic-derivation-of-the-energy-equation
essentially, I start from
[tex]\partial_\nu T^{\mu\nu} = 0 [/tex]
and for [tex]\mu=i[/tex] we get the momentum equation in the limit [tex]c\to\infty[/tex], and for
[tex]\mu=0[/tex] we get:
[tex]
\partial_t \left(\left(\rho c^2 + p{v^2\over c^2}\right)\gamma^2\right)
+
\partial_i\left(\left(\rho c^2 + p\right)v^i \gamma^2\right)
=0
[/tex]
where [tex]\gamma=(1-{v^2\over c^2})^{-1/2}[/tex]. If we let [tex]E=\rho c^2[/tex], and neglect the term [tex]p {v^2\over c^2}[/tex], we almost get the right energy equation, the only problem is that this energy [tex]\rho c^2[/tex] also contains the rest mass energy, and thus the pressure [tex]p[/tex] is negligible, and the nonrelativistic limit of this equation just gives the equation of the continuity.
How do I derive the energy equation? Somehow, I guess I need to separate the rest mass, use the equation of continuity for the rest mass (that follows from the conservation of the baryon number), and then I should be just left with the kinetic+internal energy and the right equation for nonrelativistic energy. Does anyone know how to do that in all details?
I tried to find this question already answered, but didn't find exactly what I want (I am new to the forums, so I might have missed that --- please point me to the right direction).
Thanks,
Ondrej Certik
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