Navier–Stokes equations

The Navier–Stokes equations are nonlinear partial differential equations governing fluid motion through conservation of momentum and mass, with viscous stresses represented by the fluid’s viscosity.

Continuity equation: A conservation equation stating that mass cannot appear or disappear within a flow. It supplies the mass-conservation condition paired with momentum conservation in fluid motion.
Conservation of mass: The physical principle that mass is neither created nor destroyed in an isolated system. It produces the continuity equation that constrains admissible fluid motion.
Computational fluid dynamics: The numerical simulation of fluid flow using computational methods. Most practical solutions of complex Navier–Stokes flows are computed rather than derived exactly.
Euler equations: Fluid-motion equations that neglect viscosity and describe an idealized inviscid fluid. They arise when viscous stresses are omitted from the Navier–Stokes momentum balance.
Navier–Stokes existence and smoothness: The open mathematical problem of whether three-dimensional incompressible Navier–Stokes solutions remain smooth for all time. A proof or counterexample would settle a foundational question about the equations’ solutions.
Cauchy momentum equation: A continuum equation expressing momentum conservation for a material, before specifying a constitutive stress law. The Navier–Stokes momentum equation follows by specifying how a fluid’s stress depends on its motion.
Conservation of momentum: The physical principle that total momentum changes only through external forces. It is the foundation of the equations’ balance of fluid acceleration and forces.
Laminar flow: Fluid motion in orderly layers with limited mixing between neighboring paths. It provides regimes where solutions and approximations can often be analyzed cleanly.
Stokes flow: A viscous flow regime in which inertial effects are negligible compared with viscous effects. Neglecting inertia in Navier–Stokes yields the linear Stokes equations.
Clay Millennium Prize Problems: Seven major mathematical problems announced by the Clay Mathematics Institute in 2000, each with a million-dollar prize. Navier–Stokes existence and smoothness is one of the seven problems.

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