In
physics, the
Navier–Stokes equations , named after
Claude-Louis Navier and
George Gabriel Stokes, describe the motion of
viscous fluid substances. These balance equations 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 main difference between them and the simpler
Euler equations for
inviscid flow is that Navier–Stokes equations also in the
Froude limit (no external field) are not
conservation equations, but rather a
dissipative system, in the sense that they cannot be put into the quasilinear homogeneous form:
Navier–Stokes equations are useful because they describe the physics of many phenomena of
scientific and
engineering interest. They may be used to
model the weather,
ocean currents, water
flow in a pipe and air flow around a
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.