Navier Stokes Equation

Statement

The Navier-Stokes equations describe the motion of viscous fluid substances. They are derived by applying Newton’s second law to fluid motion, assuming the fluid stress is the sum of a diffusing viscous term and a pressure term.

Mathematical Form

For an incompressible Newtonian fluid: $\rho\left(\frac{\partial\mathbf{v}}{\partial t} + (\mathbf{v}\cdot\nabla)\mathbf{v}\right) = -\nabla p + \mu\nabla^2\mathbf{v} + \rho\mathbf{g}$

where:

• $\rho$: Density
• $\mathbf{v}$: Velocity field
• $p$: Pressure
• $\mu$: Dynamic viscosity
• $\mathbf{g}$: Body force (gravity)

Relationships

• Related: Continuity Equation, Reynolds Number
• General Form: Cauchy Momentum Equation