navierstokes.tex

We shall state the variational formulation of the Navier--Stokes equations. This can be further discretized using the same quadrature rule as for the energy equation
Let us introduce a new space for the pressure
$$
L^2_0(\Omega)=\lbrace f \in L^2(\Omega) |\quad \int_{\Omega} f \d \Omega\ =\ 0,\ \rbrace
$$

The Navier--Stokes equations governing the flow are 

\begin{eqnarray}
\rho(\frac{\partial \vect u}{\partial t} +\vect u \cdot \nabla \vect u ) &=& -\nabla p+ \nabla \cdot \vect \tau+\vect f\\
\nabla \cdot \vect u &=& 0  \nonumber
\end{eqnarray}
where 
\begin{equation}
\tau =\mu[\nabla \vect u+\nabla \vect u^T-\frac{2}{3}\nabla \cdot \vect u \vect I]
\end{equation}
The Navier--Stokes equations are of mixed hyperbolic-parabolic type. In fact we can regard the steady NS equations as being a  non-linear advection-diffusion equation with a constraint given by the pressure meant to enforce the divergence free property of the velocity field. 

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\section{$\mathbb{P}_N-\mathbb{P}_{N-2}$ discretization}

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