🌊 Navier-Stokes equations

The Navier–Stokes equations (/nævˈjeɪ stoʊks/ nav-YAY STOHKS) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842-1850 (Stokes).

—wikipedia

Incompressibility

In a connected, bounded domain \(\Omega \subset \mathbb{R}^{d}\), \(d\in\left\lbrace2,3\right\rbrace\) with a Lipschitz boundary \(\partial \Omega\), the incompressible (more strictly speaking, constant density) Navier-Stokes equations are of the generic dimensionless form,

(4)\[\begin{split}\begin{aligned} \partial_t\boldsymbol{u} - \mathcal{C}(\boldsymbol{u}) - \mathrm{Re}^{-1}\mathcal{D}(\boldsymbol{u}) +\nabla p &= \boldsymbol{f},\\ \nabla\cdot\boldsymbol{u} &= 0,\\ \end{aligned}\end{split}\]

where \(\boldsymbol{u}\) is the velocity field, \(p\) is the static pressure, \(\boldsymbol{f}\) is the body force, \(\mathrm{Re}\) is the Reynolds number, \(\mathcal{C}(\boldsymbol{u})\) and \(\mathcal{D}(\boldsymbol{u})\) represent the nonlinear convective term and the linear dissipative term, respectively.

Numerical Examples

For numerical simulations of Navier-Stokes flows with phyem, see

• Backward facing step flow
• Lid-driven cavity
• Normal dipole collision
• Shear layer rollup
• Taylor-Green vortex

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