Navier-Stokes Equations

CORE DEFINITION

The Navier-Stokes equations are a set of partial differential equations that describe the motion of fluids such as liquids and air. Named after French physicist Claude-Louis Navier and Irish physicist George Stokes, these equations describe the conservation of momentum and mass that Newtonian fluids must satisfy during motion. In some applications, the Navier-Stokes equations are listed together with the equation of state to illustrate the relationship among fluid pressure, temperature, and density. The Navier-Stokes equations result from applying Newton's second law to fluid dynamics. Unlike the Euler equations, the Navier-Stokes equations account for fluid viscosity. They assume that during fluid motion, the stress is the sum of a diffusive viscous force proportional to the velocity gradient and pressure. Therefore, the Navier-Stokes equations can describe the motion of viscous fluids, while the Euler equations can only describe inviscid flow. When fluid viscosity is zero, the Navier-Stokes equations reduce to the Euler equations. The Navier-Stokes equations have a wide range of applications.