Navier-Stokes Equations

The Navier-Stokes equations describe the motion of viscous fluids. They represent conservation of momentum for a fluid element, accounting for pressure forces, viscous forces, and body forces.

Equations

$\frac{\partial \rho u_k}{\partial x}+\nabla \bullet (\rho u{u}_k-\mu \nabla u_k)=\frac{-\partial p}{\partial x_s}+F_k$

Where:

• $\rho$ = fluid density
• $u_k$ = quantity of interest
• $v$ = velocity vector
• $p$ = pressure
• $\mu$ = dynamic viscosity
• $F_k$ = sources

Vector Form

$\rho \frac{D\vec{v}}{Dt}=-\nabla p+\mu {\nabla }^2\vec{v}+\rho \vec{g}$

Component Form (x-direction)

$\rho \left(\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{\partial z}\right)=-\frac{\partial p}{\partial x}+\mu \left(\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}+\frac{{\partial }^{2}u}{\partial z^2}\right)+\rho g_x$

Notes

Left Side (Acceleration):

• ∂v/∂t: Local acceleration (unsteady flow)
• v·∇v: Convective acceleration (velocity changes with position)

Right Side (Forces per unit volume):

• -∇p: Pressure force
• μ∇²v: Viscous force
• ρg: Body force (gravity)

Navier-Stokes reduces to Bernoulli’s Equation when viscous terms are negligible.

Navier-Stokes must be solved simultaneously with the Continuity Equation. Together, they form a complete system for velocity and pressure.

For Steady State flows, ∂v/∂t = 0.

Low Reynolds Number (Stokes Flow):

• Neglect convective terms: v·∇v ≈ 0
• Linear equations, easier to solve
• Valid for very viscous or very slow flows

High Reynolds Number:

• Viscous effects confined to boundary layers
• Can use inviscid flow theory in core regio

Related

• Reynolds Number
• Bernoulli’s Equation
• Continuity Equation