• Navier-Stokes Equation

Posted: **Mon Oct 05, 2026 9:09 am**

The Navier–Stokes equations are a set of partial differential equations that describe the motion of viscous fluid substances. They express conservation of momentum for a fluid element, accounting for inertia, pressure, viscous forces, and body forces such as gravity. Together with the continuity equation (conservation of mass), they form the fundamental governing equations of fluid mechanics and are the basis for most computational fluid dynamics (CFD) analyses.

$\rho \left(\frac{\partial \mathbf{u}}{\partial t} + \mathbf{u} \cdot \nabla \mathbf{u}\right)=-\nabla p + \mu \nabla^2 \mathbf{u}+ \rho \mathbf{g}$

with the continuity equation ;

$\nabla \cdot \mathbf{u} = 0$

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$\mathbf{u}(x,t) = (u, v, w)$ velocity vector field

$p(x,t)$ pressure field

$\rho$ fluid density

$\mu$ dynamic viscosity

$\mathbf{g}$ body force per unit mass (e.g., gravity)

$\frac{\partial \mathbf{u}}{\partial t}$ local (time) acceleration

$\mathbf{u} \cdot \nabla \mathbf{u}$ convective acceleration

$-\nabla p$ pressure-gradient force

$\mu \nabla^2 \mathbf{u}$ viscous diffusion of momentum

Posted: **Mon Oct 05, 2026 9:37 am**