Airplane Vortex

The Navier Stokes Existence is one of the seven Millennium Prize Problems , which are math problems stated by the Clay Mathematics Institute in 2000 . They offer a 1,000,000 $ prize for solving one of the 7 math problems .

For now , the only Millennium Prize problem to be solved is the Poincaré conjecture , which was solved in 2003 by Grigori Perelman who turned down the $1 million prize.

At present , 6 problems are still waiting to be solved , that's a total of $6 million , and the Navier Stokes Equations problem is one of them.

What Are The Navier Stokes Equations ?

Not only it is one of the 7 Millennium Math Prize Problems , the Navier–Stokes existence and smoothness is one of the greatest unsolved problems in physics too . 

These set of equations are named after the physicist George Gabriel Stokes and the engineer Claude-Louis Navier.

The Navier Stokes equations describes the motion of fluids , they arise from applying Newton's second law to fluid dynamics (fluid motion) , these equations form the pillars of fluid mechanics .

Cigarette Smoke

Why Would We Want To Solve The Problem ?

By solving the Navier-Stokes Problem , not only will you win a $1 million prize, but it will open many blocked doors in mathematics , physics and engineering . The Navier Stokes equations can help with the design of aircraft and cars , and to study the blood flow , we can model weather , ocean currents and water flowing in a pipe using these equations , in brief everything known as fluid (liquid and gas) can be described using these equations known as the Navier Stokes equations. 

All of these examples are directly linked to physics and engineering , but what could solving the problem benefits the field of mathematics ? By solving the problem a new level of math should emerge , it is similar to when Newton created calculus in order to explain why planets orbit is elliptical .

Turbulence 

What's The Problem ?

If we have the equations that describes fluid dynamics , what should we solve then ? 

Fluids are difficult to analyse because of the way they move, it was referred as chaos , the Navier Stokes equations were made to predict the motion of fluids, but it is not proven that a solution for these equations always exist for any given fluid . In fact we can solve the Navier Stokes equations but it is difficult and it may need supercomputers especially for the 3 dimensions problem it gets harder and harder . An example on the problem : imagine that you are adding sugar cubes in your coffee cup , after dissolving the sugar cubes can we track the motion of each molecule of sugar ? In fluids it's chaos , each molecule of sugar will behave randomly in the coffee cup, till now it is impossible to track the motion of each molecule of sugar.

The same for waves followed by a boat , and turbulent air currents that follow modern jet.  Solving these equations is very difficult except for simple problems. Mathematicians have yet to prove general solutions exist . It is considered the sixth most important unsolved problem in all of math !

The phenomenon of turbulence which is a flow regime in fluid dynamics , is considered the last unsolved problem of classical mechanics. We know more about quantum particles and supernova than we do about the swirling of creamer in a steaming cup of coffee! 

swirling of creamer in a steaming cup of coffee

Explaining The Equation !

The conservation form of the equation is :

• ρ is the density,
• V is the flow velocity,
• t is time
• ∇ is the divergence
• p is the pressure
• g represents body accelerations acting on the continuum 

The Equation for 3 dimensions from NASA :

 

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References : 

 Navier–Stokes equations

 Navier–Stokes existence and smoothness

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