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Mon Sep 28 14:47:26 CEST 2009

Navier-Stokes

Time to look at the Navier-Stokes equation[1] and fluid dynamics.
There is a description in the Princeton Companion to Mathematics[2]
section III.23 p196.  Another starting point is Feynman's Lectures on
Physics, part II chapters 40-41.

The Navier-Stokes (and Euler) equations are non-linear partial
differential equations in terms of a velocity vector field u(x) and a
pressure distribution p(x).  The Euler equation is N-S with viscosity
v=0.  Apparently, even for small v these behave quite different.

Additionally one can pose a constraint that expresses the fluid is not
compressable.

E and N-S are Newton's law applied to an infinitesimal portion of the
fluid.

The nonlinearity (which causes turbulence) is due to convective
acceleration, which is an acceleration associated with the change in
velocity over position.

In practice, the interaction between large and fine scale behaviour
requires such a fine mesh that the problem is often approximated,
i.e. using the Reynolds-averaged Navier-Stokes equiation[3].


Following Feynman in chapter II.40 (The Flow of Dry Water).  A liquid
moves under shear stress.  Hydrostatics, fluid in rest, amounts to the
absence of shear stress.  The important result here is that the force
per unit volume is -grad(p).  The important conclusion is that for a
force defined by a potential energy, there is no general equilibrium
solution if the density is not constant. (An exception arises when
density depends only on pressure).

For dynamic fluids, the clue seems to be to translate a velocity
vector field description to a particle path description, and formulate
Newton's law of conservation of momentum for that path.  I.e. with dx
denoting infinitesimals and a 2D equation.  This yields an equation
nonlinear in the velocity components.

The force component of the N-S equation is where most of the variation
lies.





[1] http://en.wikipedia.org/wiki/Navier-Stokes_equations
[2] isbn://0691118809
[3] http://en.wikipedia.org/wiki/Reynolds-averaged_Navier-Stokes_equations



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