.. _navierstokes:

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The compressible Navier-Stokes equations
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In two dimensions the compressible Navier-Stokes equations can be written as

.. math::
   :nowrap:

   \begin{equation}
      \vec{Q}_t + \frac{\partial}{\partial x}(\vec{f}(\vec{Q})+\vec{F}(\vec{Q})) +
      \frac{\partial}{\partial y} (\vec{g}(\vec{Q})+\vec{G}(\vec{Q})) = 0.
   \end{equation}

Where the conservative state variables are :math:`\vec{Q} = [\rho,
\rho u, \rho v, E]^T` and the hyperbolic fluxes are

.. math::
   :nowrap:

   \begin{equation}
     \vec{f}(\vec{Q}) = \left( \begin{array}{c}
      \rho u\\
      \rho u^2 + p \\
      \rho u v \\
      u (E+p)      
     \end{array} \right), \ \ \  \vec{g}(\vec{Q}) = \left( \begin{array}{c}
      \rho v\\
      \rho u v  \\
      \rho v^2 + p \\
      v (E+p)      
     \end{array} \right)
   \end{equation}

Here we assume an equation of state  :math:`p = (\gamma-1) \rho e` and note
that we can relate :math:`E` and :math:`p` through :math:`\rho e = E - \rho (u^2+v^2)/2`. 

The viscous fluxes are: 

.. math::
   :nowrap:

   \begin{equation}
     \vec{F}(\vec{Q}) = - \left( \begin{array}{c}
      0\\
     \frac{4 \mu}{3} u_x+(\frac{\mu}{3}+\mu_b) v_y \\
      \mu (u_y+v_x) \\
      u F_2 + v F_3 + \frac{\mu \gamma}{Pr} T_x     
     \end{array} \right), \ \ \  \vec{G}(\vec{Q}) = - \left( \begin{array}{c}
      0\\
      \mu (u_y+v_x) \\
      \frac{4 \mu}{3} v_y + (\frac{mu}{3} + \mu_b) u_x\\
      u G_2 + v G_3 + \frac{\mu \gamma}{Pr} T_y     
     \end{array} \right).
   \end{equation}

Here :math:`\rho` is the density, :math:`(u,v)` is the velocity
vector, :math:`E` is the total energy, :math:`e` is the specific
internal energy, :math:`\mu` is the dynamic shear viscosity,
:math:`\mu_b` is the bulk viscosity, :math:`\gamma` is the ratio of
specific heats, :math:`Pr` is the Prandtl number, :math:`T =
(\gamma-1)e/R` is the temperature and :math:`R` is the gas constant.

Isentropic vortex
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Couette flow
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Lid driven cavity
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A pair of vortices 
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