<commit_before>.. index:: fluid dynamics

Fluid Dynamics
==============

.. index::
    pair: stress-energy; tensor

Stress-Energy Tensor
--------------------


In general, the stress energy tensor is the flux of momentum $p^\mu$ over the
surface $x^\nu$. The Navier-Stokes equations can be derived from the
conservation law:

.. math::

    \partial_\nu T^{\mu\nu} + f^\mu = 0

Perfect Fluids
~~~~~~~~~~~~~~

Perfect fluids have no heat conduction ($T^{i0} = T^{0i} = 0$) and no
viscosity ($T^{ij} = p\one$), so in the MCRF:

.. math::

    T^{\alpha\beta} = \diag(\rho, p, p, p) =
    (\rho+p)u^\alpha u^\beta + p g^{\alpha\beta}

where in MCRF we have $g^{\mu\nu} = \diag(-1, 1, 1, 1)$, $u^0=1$ and $u^i=0$,
but $\partial_\alpha U^i\neq0$. The last equation is a tensor equation so it
holds in any frame. The nonrelativistic limit of this stress energy tensor is:

.. math::

    T^{00} = (\rho+p)u^0u^0 - p \approx \rho

    T^{0j} = T^{j0} = (\rho+p)u^0u^j \approx \rho u^j

    T^{jk} = (\rho+p)u^ju^k + p \delta^{jk} \approx \rho u^j u^k + p
        \delta^{jk}


Substituting this into the conversation law $\partial_\beta
T^{\alpha\beta}=0$ we get for $\alpha=0$:

.. math::

    \partial_\beta T^{0\beta} = 0

    \partial_0 T^{00} + \partial_i T^{0i} = 0

    \partial_0 \rho + \partial_i (\rho u^i) = 0

    {\partial \rho\over\partial t} + \nabla \cdot (\rho {\bf u}) = 0

This is the continuity equation. For $\alpha=i$ we get:

.. math::

    \partial_\beta T^{i\beta} = 0

    \partial_0 T^{i0} + \partial_j T^{ij} = 0

    \partial_0 (\rho u^i) + \partial_j (\rho u^i u^j + p\delta^{ij}) = 0

    {\partial (\rho{\bf u})\over\partial t} + \nabla \cdot
        (\rho {\bf u}{\bf u}^T) + \nabla p = 0

Which are the Euler equations.


Navier-Stokes Equations
-----------------------

When we write the relativistic conservation law in a nonrelativistic limit (for
a general fluid), we get the Cauchy momentum equation:

.. math::

    \rho\left({\partial {\bf v}\over\partial t} +{\bf v}\cdot\nabla{\bf v} \right) = \nabla \cdot \mathds{\sigma} + {\bf f}
<commit_msg>"c" constants added<commit_after>.. index:: fluid dynamics

Fluid Dynamics
==============

.. index::
    pair: stress-energy; tensor

Stress-Energy Tensor
--------------------


In general, the stress energy tensor is the flux of momentum $p^\mu$ over the
surface $x^\nu$. The Navier-Stokes equations can be derived from the
conservation law:

.. math::

    \partial_\nu T^{\mu\nu} + f^\mu = 0

Perfect Fluids
~~~~~~~~~~~~~~

Perfect fluids have no heat conduction ($T^{i0} = T^{0i} = 0$) and no
viscosity ($T^{ij} = p\one$), so in the MCRF:

.. math::

    T^{\alpha\beta} = \diag(\rho, p, p, p) =
    (\rho+p)u^\alpha u^\beta + p g^{\alpha\beta}

where in MCRF we have $g^{\mu\nu} = \diag(-1, 1, 1, 1)$, $u^0=c$ and $u^i=0$,
but $\partial_\alpha U^i\neq0$. The last equation is a tensor equation so it
holds in any frame. The nonrelativistic limit of this stress energy tensor is:

.. math::

    T^{00} = (\rho+p)u^0u^0 - p \approx \rho c

    T^{0j} = T^{j0} = (\rho+p)u^0u^j \approx \rho u^j

    T^{jk} = (\rho+p)u^ju^k + p \delta^{jk} \approx \rho u^j u^k + p
        \delta^{jk}


Substituting this into the conversation law $\partial_\beta
T^{\alpha\beta}=0$ we get for $\alpha=0$:

.. math::

    \partial_\beta T^{0\beta} = 0

    \partial_0 T^{00} + \partial_i T^{0i} = 0

    \partial_0 (\rho c) + \partial_i (\rho u^i) = 0

    {\partial \rho\over\partial t} + \nabla \cdot (\rho {\bf u}) = 0

This is the continuity equation. For $\alpha=i$ we get:

.. math::

    \partial_\beta T^{i\beta} = 0

    \partial_0 T^{i0} + \partial_j T^{ij} = 0

    \partial_0 (\rho u^i) + \partial_j (\rho u^i u^j + p\delta^{ij}) = 0

    {\partial (\rho{\bf u})\over\partial t} + \nabla \cdot
        (\rho {\bf u}{\bf u}^T) + \nabla p = 0

Which are the Euler equations.


Navier-Stokes Equations
-----------------------

When we write the relativistic conservation law in a nonrelativistic limit (for
a general fluid), we get the Cauchy momentum equation:

.. math::

    \rho\left({\partial {\bf v}\over\partial t} +{\bf v}\cdot\nabla{\bf v} \right) = \nabla \cdot \mathds{\sigma} + {\bf f}

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