the steady-state Navier-Stokes equations for incompressible fluids#
The steady-state Navier Stokes problem for an incompressible fluid, with homogeneous Dirichlet boundary conditions (``no slip’’ condition), is defined as
\[\begin{split}
\begin{equation}
\left\{
\begin{aligned}
\left( \mathbf{u} \cdot \nabla \right) \mathbf{u} + \nabla p - 2 \nabla \cdot \left( \frac{1}{R_e} D\left( \mathbf{u} \right) \right) &= \mathbf{f} && \text{in $\Omega$},
\\
\nabla \cdot \mathbf{u} &= 0 && \text{in $\Omega$},
\\
\mathbf{u} &= 0 && \text{on $\partial\Omega$},
\end{aligned}
\right.
\end{equation}
\end{split}\]
where \(R_e\) is the Reynolds number and \(D\left( \mathbf{u} \right) := \frac{1}{2}\left( \nabla \mathbf{u} + \nabla \mathbf{u}^T \right)\) is the strain rate tensor. The weak formulation reads
\[
\begin{align}
\text{find $\mathbf{u} \in W$ such that} \quad
a(\mathbf{u},\mathbf{v}) + b(\mathbf{u},\mathbf{v};\mathbf{u}) = l(\mathbf{v}) \quad
\forall \mathbf{v} \in W
\end{align}
\]
where \(W \subset \{ \mathbf{v} \in \mathbf{H}_0^1(\Omega): \nabla \cdot \mathbf{v} = 0 \}\) and
\[
\begin{align*}
a(\mathbf{u}, \mathbf{v}) := \frac{2}{R_e} \int_{\Omega} D\left( \mathbf{u} \right) : D\left(\mathbf{v} \right) ~d\Omega,
\quad
b(\mathbf{u}, \mathbf{v}; \mathbf{w}) := \int_{\Omega} \left( \left( \mathbf{w} \cdot \nabla \right) \mathbf{u} \right) \cdot \mathbf{v} ~d\Omega,
\quad
l(\mathbf{v}) := \int_{\Omega} \mathbf{f} \cdot \mathbf{v} ~d\Omega.
\end{align*}
\]