“I am an old man now, and when I die and go to Heaven there are two matters on which I hope for enlightenment. One is quantum electrodynamics and the other is the turbulent motion of fluids. And about the former I am rather more optimistic”- Sir Horace Lamb
Navier-Stokes equation, which governs the motion of nearly most of the fluids, has been obtained by three scientists in three different methods. Namely by George Gabriel stokes, Claude Louis Navier, and Simeon Denis Poisson. It is still remaining as one of the greatest unsolved problems in physics; its solutions include turbulence, which is of massive importance to engineers and scientists.
Even though it looks like an ordinary differential equation, the non linear quadratic term leads it to describe the most complex phenomena of fluid dynamics.
Derivation
It’s derived by using basic principles
1. Gauss law of pressure
2. Newton’s second law - momentum conservation
3. Newton’s law of viscosity
4. Mass conservation
The assumptions are that the flow is incompressible as well as Newtonian
The rest I would upload as an image since it's hard to convert equations to html format
The pdf of the document
https://docs.google.com/open?id=0B2fTwviJVzRYY2ZhMWRjM2UtNTY1Mi00MjA5LTg3NDUtMTBiNmZmM2VhNmM2
references
Turbulence: an introduction for scientists and engineers By Peter Alan Davidson
http://itunes.apple.com/us/itunes-u/fluid-mechanics-2010-eng-me303/id452560560
Introduction to turbulence By Paul A. Libby
http://www.hitech-projects.com/euprojects/artic/index/Low%20Reynolds%20number%20flows.pdf
