In physics, the Navier–Stokes equations, named after Claude-Louis Navier and Sir George Stokes, 1st Baronet|George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to Fluid dynamics|fluid motion, together with the assumption that the fluid stress (physics)|stress is the sum of a diffusion|diffusing viscosity|viscous term (proportional to the gradient of velocity) and a pressure term - hence describing viscous flow.
The equations are useful because they describe the physics of many things of academic and economic interest. They may be used to model (abstract)|model the weather, ocean currents, water flow conditioning|flow in a pipe and air flow around a airfoil|wing. The Navier–Stokes equations in their full and simplified forms help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell's equations they can be used to model and study magnetohydrodynamics.
The Navier–Stokes equations are also of great interest in a purely mathematical sense. Somewhat surprisingly, given their wide range of practical uses, mathematicians have not yet proven that in three dimensions solutions always exist (Existence theorem|existence), or that if they do exist, then they do not contain any Mathematical singularity|singularity (smoothness). These are called the Navier–Stokes existence and smoothness problems. The Clay Mathematics Institute has called this one of the Millennium Prize Problems|seven most important open problems in mathematics and has offered a US$1,000,000 prize for a solution or a counter-example.

In physics, the Navier–Stokes equations, named after Claude-Louis Navier and Sir George Stokes, 1st Baronet|George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying Newton's second law to Fluid dynamics|fluid motion, together with the assumption that the fluid stress (physics)|stress is the sum of a diffusion|diffusing viscosity|viscous term (proportional to the gradient of velocity) and a pressure term - hence describing viscous flow.
The equations are useful because they describe the physics of many things of academic and economic interest. They may be used to model (abstract)|model the weather, ocean currents, water flow conditioning|flow in a pipe and air flow around a airfoil|wing. The Navier–Stokes equations in their full and simplified forms help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell's equations they can be used ...