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Navier-Stokes equations (Rev #4, changes)

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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.

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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.

Details

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, 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

References

category: methodology