.. _GALLERYNS:
==========================
🌊 NavierStokes equations
==========================
The **Navier–Stokes equations** (/nævˈjeɪ stoʊks/ navYAY STOHKS) are partial differential equations which
describe the motion of viscous fluid substances, named after French engineer and physicist ClaudeLouis Navier
and AngloIrish physicist and mathematician George Gabriel Stokes. They were developed over several decades
of progressively building the theories, from 1822 (Navier) to 18421850 (Stokes).
 wikipedia
Incompressibility
=================
In a connected, bounded domain :math:`\Omega \subset \mathbb{R}^{d}`, :math:`d\in\left\lbrace2,3\right\rbrace` with
a Lipschitz boundary :math:`\partial \Omega`, the incompressible (more strictly speaking, constant density)
NavierStokes equations are of the generic dimensionless form,
.. math::
:label: genericNS
\begin{equation}
\begin{aligned}
\partial_t\boldsymbol{u}  \mathcal{C}(\boldsymbol{u})  \mathrm{Re}^{1}\mathcal{D}(\boldsymbol{u})
+\nabla p &= \boldsymbol{f},\\
\nabla\cdot\boldsymbol{u} &= 0,\\
\end{aligned}
\end{equation}
where :math:`\boldsymbol{u}` is the velocity field, :math:`p` is the static pressure,
:math:`\boldsymbol{f}` is the body force,
:math:`\mathrm{Re}` is the Reynolds number, :math:`\mathcal{C}(\boldsymbol{u})` and
:math:`\mathcal{D}(\boldsymbol{u})` represent the nonlinear convective term and the linear dissipative term,
respectively.
Numerical Examples
==================
For numerical simulations of NavierStokes flows with *phyem*, see
.. toctree::
:maxdepth: 1
shear_layer_rollup/index
normal_dipole_collision/index

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