Navier–Stokes equation

Push water and it starts moving. Stop pushing and wall friction and viscosity gradually slow it down. A fluid's velocity changes because forces act on it.

The Navier–Stokes equation says: fluid acceleration equals the effects of pressure, viscosity and body forces such as gravity. It is Newton's second law written for a fluid.

Imagine two neighbouring fluid layers. If the upper layer moves faster, viscosity holds it back while pulling the slower lower layer forward. Their velocity difference therefore becomes smaller.

Some momentum from the faster layer is transferred to the slower layer. This is momentum diffusion. Honey smooths a velocity difference more strongly than water because its dynamic viscosity is larger.

For a Newtonian fluid, shear stress $\boldsymbol{\tau}$ is proportional to the rate $\mathrm d u/\mathrm d y$ at which velocity changes with position. The coefficient $\mu$ is dynamic viscosity.

Take one small parcel of fluid. Its velocity is changed mainly by pressure differences, viscosity from neighbouring fluid and body forces such as gravity. Their resultant produces the parcel's acceleration.

Equation (3) collects this idea in a form that works in every direction. The left side is fluid-particle acceleration; the three terms on the right represent pressure, viscosity and body force.

Now restrict the problem to an incompressible Newtonian fluid with constant density and viscosity. Under these conditions the viscous force can be written as $\mu\nabla^2\boldsymbol{u}$ [1,2].

Replacing the viscous term in Equation (3) gives Equation (4). Although longer, its structure is unchanged: acceleration on the left and pressure, viscosity and body force on the right.

Equation (5) says that incompressible fluid is not created or destroyed at a point. In a calculation, Equations (4) and (5) are solved together for velocity and pressure.

Each term in Equation (4) has units of force per volume, $\mathrm{N/m^3}$. Dynamic viscosity $\mu$ has SI unit $\mathrm{Pa{\cdot}s}$ and kinematic viscosity $\nu=\mu/\rho$ has SI unit $\mathrm{m^2/s}$ [4].

The equation balances momentum change of a particle against the forces that cause it.

Local acceleration

$\partial\boldsymbol{u}/\partial t$: change with time at a fixed point.

Advective acceleration

$(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}$: change as a particle enters a region with a different velocity.

Pressure

$-\nabla p$: pushes from high toward low pressure.

Viscosity

$\mu\nabla^2\boldsymbol{u}$: diffuses momentum and smooths velocity differences.

Body force

$\rho\boldsymbol{b}$: gravity and other forces acting throughout the volume.

Equation (4) assumes incompressibility, Newtonian behavior and constant viscosity. High-speed gases, strong property variation and non-Newtonian fluids require the general balances and suitable constitutive laws.

Incompressible

Constant-density approximation, often suitable for low-speed liquids.

Newtonian

Viscous stress is linear in rate of deformation.

Boundary conditions

A solid wall normally uses no slip: fluid velocity equals wall velocity.

Turbulence

The instantaneous equation remains valid, but resolving all scales or using a turbulence model is required.

Consider fully developed pressure-driven flow between parallel plates, with $x$ along the flow, $y$ normal to the walls, $u=u(y)$, steady conditions and no $x$-directed gravity.

The $x$ equation reduces to Equation (6). Integrating twice for constant pressure gradient gives the parabolic profile, with constants fixed by no slip.

Record which assumptions remove each term instead of memorizing only the final equation.

Incompressible means constant pressure

It means constant density; pressure may vary.

Steady removes the whole left side

Local acceleration vanishes but advection may remain.

Viscosity is always μ∇²u

That simplification needs constant-viscosity incompressible Newtonian flow.

The equation alone determines the solution

Initial and boundary conditions plus continuity are required.

Applications include pipe flow, boundary layers, external aerodynamics, heat transfer, lubrication, weather and ocean flow, blood flow and CFD. Most practical geometries require numerical solution.

The Reynolds number next compares inertia and viscosity, helping determine dominant physics and appropriate approximations.