Navier–Stokes Equations (Overview)

Key idea

Equations of fluid flow.

In depth

These govern viscous fluid flow: ρ(v_t + v·∇v) = −∇p + μΔv + f with ∇·v = 0 for incompressible flow. They express momentum conservation. Whether smooth solutions in 3D always exist is a Millennium Prize problem.

Worked example

Simplified to Stokes flow, they model slow, viscous flow such as honey.

Real-world uses

• Modelling airflow over wings, water in pipes and weather systems

Common mistake

Assuming the equations are fully understood mathematically; they are not.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related expert topics in Differential Equations, Abstract Algebra & Applied Math

• Exact Equations – M dx + N dy = 0 with Mᵧ = Nₓ.
• Bernoulli Equations – Substitute v = y¹⁻ⁿ to linearise.
• Runge–Kutta Methods – Higher-accuracy numerical solvers.
• Second-Order Homogeneous Equations – Use the characteristic equation.
• Undetermined Coefficients – Guess the particular solution form.
• Variation of Parameters – General method for particular solutions.