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Navier-Stokes equations (Rev #14, changes)

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Content

Idea

The Navier–Stokes equations describe the motion of fluids: gases and liquids. These equations arise from applying Newton’s second law to fluid motion, together with the assumption that the fluid feels forces due to pressure, viscosity and perhaps an external force. The Navier-Stokes equations are used to model the weather, ocean currents, water flow in a pipe, air flow around a wing, and even the motion of stars inside a galaxy.

In their full generality, the Navier-Stokes equations are:

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

Here ρ\rho is the density of the fluid, vv is its velocity vector field, pp is the pressure, TT is the stress tensor, and FF is a specified external force (again a vector field). To solve these equations, we must supplement them by a formula for the pressure pp, a formula for the stress tensor TT, a formula for the force FF, and an equation expressing the conservation of mass:

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

Details

Derivation

The Navier-Stokes equations are often presented for ‘Newtonian’ fluids, where the stress tensor TT is related to the velocity vv in a simple linear way. The Navier-Stokes equations for a Newtonian fluid are:

ρ(vt+vv)=p+μ 2v+F \rho \left(\frac{\partial v}{\partial t} + v \cdot \nabla v \right) = -\nabla p + \mu \nabla^2 v + F

Here ρ\rho is the density of the fluid, vv is its velocity vector field, pp is the pressure, μ\mu is a constant called the viscosity?, and FF is a specified external force (again a vector field).

To understand where these equations come from, let us first consider the left-hand side. This describes the rate of change of momentum per unit volume of a little ‘piece’ of the fluid. The reason is that the left-hand side above equals the density times

vt+vv, \frac{\partial v}{\partial t} + v \cdot \nabla v ,

which is the acceleration of a little piece of fluid. This acceleration involves the rate of change of velocity as you just sit in one place, namely vt\frac{\partial v}{\partial t}, but also the rate of change of velocity caused by the fluid moving to a new place where the velocity is different, namely vvv \cdot \nabla v. The sum

vt+vv \frac{\partial v}{\partial t} + v \cdot \nabla v

is called the convective derivative of the velocity vector field, also known as the convective acceleration.

In short, the left-hand side of the Navier-Stokes equations is a close relative of mam a from Newton’s law. Simiarly, the right-hand side is a close relative of FF. It’s the force per unit volume on a little piece of fluid:

p+μ 2v+F -\nabla p + \mu \nabla^2 v + F

Here pp is the pressure, so p-\nabla p, minus the gradient of the pressure, is the force per unit volume due to pressure. Similarly μ\mu is the viscosity, and μ 2v \mu \nabla^2 v is the force per unit volume due to viscosity. Finally, FF is any additional force per unit volume that we might be imposing on the liquid.

The viscosity term μ 2v\mu \nabla^2 v takes a little work to understand. For example, if a little piece of fluid is moving slower in the xx direction than the average of its neighbors, the xx component of μ 2v\mu \nabla^2 v will be positive, meaning that its neighbors will be pushing on it, trying to make it go faster in that direction! We define a Newtonian fluid to be one obeying this simple linear formula for the force per unit volume due to viscosity. (Blood is a notably non-Newtonian fluid, because red blood cells push up against their neighbors in a much more complicated way.)

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

Turbulence

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. One other interesting method is the Lattice Boltzmann method, mainly due to its simplicity.

References

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

General

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)

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