# What is Navier-stokes: Definition and 84 Discussions

In physics, the Navier–Stokes equations () are a set of partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes.
The Navier–Stokes equations mathematically express conservation of momentum and conservation of mass for Newtonian fluids. They are sometimes accompanied by an equation of state relating pressure, temperature and density. They arise from applying Isaac 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 difference between them and the closely related Euler equations is that Navier–Stokes equations take viscosity into account while the Euler equations model only inviscid flow. As a result, the Navier–Stokes are a parabolic equation and therefore have better analytic properties, at the expense of having less mathematical structure (e.g. they are never completely integrable).
The 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.
48. ### Pompeiu Problem: Solved & Related to Navier-Stokes?

Hi, http://en.wikipedia.org/wiki/Pompeiu_problem Can someone rephrase the problem so I better understand its meaning? And has it been solved? Solution to Pompeiu Problem http://arxiv.org/abs/1304.2297 Thanks PS. does this problem have anything to do with Navier-Stokes?
49. ### Understanding the Navier-Stokes Equations: Conservation of Momentum and More

Hi, Stupid question: the Navier-Stokes equations, do they only consist of the equations for a statement of the conservation of momentum or do they also include the equations for conservation of mass (continuity equation), conservation of energy and an equation of state? Thanks.
50. ### Solution of Navier-Stokes eq for a single particle?

Solution of Navier-Stokes eq for a single particle? Hi! I'm reading this paper on fluid dynamics: http://jcp.aip.org/resource/1/jcpsa6/v50/i11/p4831_s1 Its equation (13) is the velocity distribution around a single bead of radius a subjecting to force fi in solution. (the subscript i is...