Equations are more important to me, because politics is for the present, but an equation is something for eternity.

-Albert Einstein, when he rejected the offer to become the president of Israel.

What the hell is Navier-Stokes Eqn?
To be frank, I have no idea wht is it! Ok, from my understanding (correct me if ur a PhD in Fluid Dynamics. I always have interest in knowing this kind of info regardless how complex they were), it analysizes a fluid flow in all dimiention using either polar coordinate or cartesion coorditnate. Thus, we’ll get an very detail aspect in calculating a flow. This includes from (for cartesiuon coordinate) X-direction, Y-direction and Z-direction.

Though from wht I read from the text book and the internet, the general equation is juz a bundle of differential equation, but wht makes is more complicate when u include and considers these conditions in the equation itself:

1) conservation of momentum
2) inviscid flow
3) incompressible flow
4) fully-develop flow
5) Newtonian fluid
6) Rotational and irrotational flow

And other factor which Dr Tio didn’t teach. Maybe it’s not in the syllabus. I found it online:

1) Bingham fluid
2) Power Law fluid

If we considers these factor, then the Navier-Stokes eqn can be used to calculate many kind of things in our daily life.

Application

Mainly from this equation, we manage to get:

1) model weather forecasting
2) ocean currents
3) airfoil wing of a plane
4) designing from an automobile on the land to the plane tht flies
5) water flow in a pipe (this is the most popular example when u learn this eqn for the 1st time)
6) motion of stars inside a galaxy (this one i believe contaiins in Astrophysics)
7) blood flow in our body
8) design of powerstation (for steam flow and so on)
9) to analysis the effects of pollution (El Nino and La Nina)
10) Magnetohydrodynamics (if combined with Maxwell’s Equations)*

*Please refer to my article I wrote on Dec 18th 2005 about Magnetohydrodynamics [MHD} Propulsion

Conclusion

For more complex situations, such as global weather systems like El Niño or lift in a wing, solutions of the Navier-Stokes equations must be found with the help of computers. This is a field of sciences by its own called computational fluid dynamics. Even though turbulence is an everyday experience it is extremely difficult to find solutions for this class of problems. A $1,000,000 prize was offered in May 2000 by the Clay Mathematics Institute to whoever makes substantial progress toward a mathematical theory which will help in the understanding of this phenomenon.

As we can see, from the list of the application itself, it effects of our life from this earth until accross the universe. As a Muslim, I always take this as the wonders of Allah. It teaches me not only about the use of the equation itself, but also it helps me to understand better how everythings around u works as it is. Even Albert Einstein would rather give up the President post and stated that equation would last for eternity. It’s shows how fascinated he was with deriving these equations (as such E=mc^2). Perhaps this wht happens to those who are “too genuis”.Maybe another 1000 years from now, our great-great-grandchildren will use this same equation for something that we never see or knew it’s gonna invented. Only God knows.

A simplified form of the Navier-Stokes equations

George Stokes and Claude Navier

The Formula 1 car is one to the apllication tht scratched the Navier-Stokes Equation to it’s extent. The F1 car aerodynamic design along with the aerofoil front to back, theoritically says tht at 300 kmph, the car will stick on the ceiling if it was driven upside down.

Further reading:
Article 1:Navier-Stokes Equation
Article 2:Claude-Louis Navier, equation founder
Article 3:Sir George Gabriel Stokes, also another eqn founder
Article 4:About Fluid Dynamics