🌊 Navier-Stokes equations

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🌊 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 $Ω \subset R^{d}$ , $d \in {2, 3}$ with a Lipschitz boundary $\partial Ω$ , the incompressible (more strictly speaking, constant density) Navier-Stokes equations are of the generic dimensionless form,

(1)#

\begin{array}{r} \begin{aligned} \partial_{t} u - C (u) - {Re}^{- 1} D (u) + \nabla p & = f, \\ \nabla \cdot u & = 0, \end{aligned} \end{array}

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

Numerical Examples#

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

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