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Navier-Stokes equations (Rev #11, changes)

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from Wikipedia

In physics the Navier–Stokes equations, describe the motion of fluid substances. These equations arise from applying Newton’s second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous term (proportional to the gradient of velocity), plus a pressure term.

The equations are useful because they describe the physics of many things of academic and economic interest. They may be used to model the weather, ocean currents, water flow in a pipe, air flow around a wing, and motion of stars inside a galaxy. 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 general form of Navier-Stokes (1) is

ρ(vt+v.v)=p+T+f\rho \left( \frac {\partial v}{\partial t} + v.\nabla v\right) = -\nabla p + \nabla T +f

and v is the flow velocity vector, ρ\rho is the fluids density, p is pressure and T is a stress tensor and f are the body forces.

and conservation of mass:

vt+(ρv)=0\frac {\partial v}{\partial t} + \nabla \left(\rho v\right) = 0

It is possible to state stochastic differential equations such that the expectation value of the solution process is a solution to the Navier-Stokes equations. This seems to be a fairly recent result of Peter Constantin and Gautam Iyer:

  • Peter Constantin, Gautam Iyer: A stochastic Lagrangian representation of the 3-dimensional incompressible Navier-Stokes equations (arXiv), A stochastic-Lagrangian approach to the Navier-Stokes equations in domains with boundary (arXiv).

Recently there has been published the claim that there is a close connection of the incompressible Navier-Stokes equations and the Einstein field equations for the vacuum of general relativity, see

  • Irene Bredberg, Cynthia Keeler, Vyacheslav Lysov, Andrew Strominger: From Navier-Stokes To Einstein arXiv


One of the most mysterious and striking features of solutions of the Navier-Stokes equations is turbulence.

Numerical Simulation

Two common approaches for the numerical simulation of the Navier-Stokes equations used in meteorology and for climate models are finite element methods and spectral methods.


There are also a lot of references on the page turbulence.


Exact Solutions

The following book provides a list of exact solutions:

  • Philip G. Drazin, Norman Riley: The Navier-Stokes equations. A classification of flows and exact solutions. (ZMATH)