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Physical Sciences
Fluid Mechanics
1845
Research

Navier-Stokes Equation

ρ(∂v∂t+v⋅∇v)=−∇P+μ∇2v+f\rho\left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v}\cdot\nabla\mathbf{v}\right) = -\nabla P + \mu\nabla^2\mathbf{v} + \mathbf{f}

The master equation of fluid motion—balancing inertia, pressure, viscosity, and external forces.

By Claude-Louis Navier, George Gabriel Stokes

Physical Sciences
Navier-Stokes Equation
1845 · Claude-Louis Navier
Source Verified
89%

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Five doors into the universe behind this equation. Choose your path.

Story PortalWho discovered this?Discover how Navier and Stokes and others shaped this equation.Visual PortalWhat does it look like?See the equation come alive in the Visual Studio.Machine PortalWhere is it used?Explore machines powered by this equation — aircraft wings and wind turbines.Math PortalWhat does it derive from?Trace the mathematical lineage from conservation of momentum in fluids.Future PortalWhere is it going?Millennium Prize open problem on smooth solutions
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Why it matters: One of seven Millennium Prize problems—governs weather, blood flow, and turbulence.

Discoverers: Claude-Louis Navier, George Gabriel Stokes (1845)

What does it mean?

The master equation of fluid motion—balancing inertia, pressure, viscosity, and external forces.

Why should I care?

One of seven Millennium Prize problems—governs weather, blood flow, and turbulence.

Equation Compass

North — Prerequisites

West — History

East — Applications

South — Derivations

Variables & Units

SymbolNameUnitMeaning
ρρDensity—Fluid density
vvVelocity—Flow velocity field
μμViscosity—Dynamic viscosity
PPPressure—Pressure field

Worked Example

Pipe flow profiles solved numerically for given Reynolds number.

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Continue your trail

Aircraft WingWind TurbineAerospaceClimate Science

Sources & further reading

Search WikipediaSearch Wolfram MathWorldOpen in Wolfram|AlphaWatch explainers on YouTube

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Equation Universe

Navier-Stokes Equation

ρ(∂v∂t+v⋅∇v)=−∇P+μ∇2v+f\rho\left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v}\cdot\nabla\mathbf{v}\right) = -\nabla P + \mu\nabla^2\mathbf{v} + \mathbf{f}

Real-world impact

Planetary stewardship

Dynamics equations inform sustainability.

Photo: Unsplash — forest ecosystem

The master equation of fluid motion—balancing inertia, pressure, viscosity, and external forces.

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