Navier–Stokes equations

The Navier–Stokes equations (/nævˈjeɪ ˈstoʊks/ nav-YAY STOHKS) describe the motion of viscous fluids. This system of partial differential equations was named after Claude-Louis Navier and George Gabriel Stokes, who developed them over a few decades of progressive work, from 1822 (Navier) to 1842–1850 (Stokes).

The Navier–Stokes equations mathematically express momentum balance for Newtonian fluids and make use of the conservation of mass. They are sometimes accompanied by an equation of state relating pressure, temperature and density. They arise from applying Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing viscous flow. The Navier–Stokes equations generalize the Euler equations in that the latter model only considers inviscid flow.

The Navier–Stokes equations are of great scientific and engineering interest because they may be used to model a wide variety of scenarios. In their full or simplified forms, they can assist in the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other problems. Coupled with Maxwell's equations, they comprise the fundamentals of magnetohydrodynamics.

The Navier–Stokes equations are also of great interest in a purely mathematical sense. Despite their wide range of practical uses, the conjecture that they have smooth (meaning infinitely differentiable) or bounded solutions in three dimensions has not yet been proven. This is called the Navier–Stokes existence and smoothness problem. The Clay Mathematics Institute has called this one of the seven most important open problems in mathematics and has offered a $1 million prize for a solution or a counterexample.