Physics is full of equations, which dominate the operation of everything. In all equations, a set of equations have been proposed for the first time (1822) 200 years ago (1822), and it has become an irreplaceable mountain, that is

Navavi Stokes equation


NS equation




Navavi Stocks equation. (Photo / Principle)

The difficulty of understanding the NS equation in mathematics has even surpassed

Einstein field equation

This is because it is still the most difficult part of the physical world behind it, but it is also a phenomenon that can be seen everywhere in our daily life.


What is turbulent?

What is turbulent? The answer to this question may be unexpected. In fact, in the scientific community, there is no definition of turbulence that is generally recognized, but this is another thing that “you will know when you see it”.

For example, if you pour a little milk in a cup of black coffee, you can see the white liquid blooming like the clouds, spread in the liquid, complex vortex, and eventually integrate into a perfect cup of coffee. Or, a gentle river flows slowly, but when the river water encounters a pier, a large -scale turbulence event may occur.

The turbulence simulation in the tea cup. (Figure /

In fact, even if you just wave your hand in the air, there will be turbulence that cannot be seen, and

Their complexity is beyond imagination

Essence In turbulence, a large number of vortex will appear in the fluid, that is, many tiny vortex and the reverse flow they produce. These vortex continue to change the size, speed, and direction. Bring huge challenges.

An example of non -turbulence is a stable river, and each part of this river moves in the same direction at the same rate. The turbulence is the break of this river. It allows different parts of the river to move in different directions at different speeds. (Figure / Joseasorrentino, Wiki)

However, although turbulence is everywhere, it has a headache in mathematics, even if it is a scientific master

Wilner Haysenburg

(Werner Heisenberg)

It is also trapped. According to rumors, Haysenburg said in his later years that if he could ask God, he would ask two questions, “Why is there the theory of relativity? Why is there a turbulence?” Then he said, “I believe God will have the first question of the first question s answer”.

The credibility of this story may not be so high, but it is enough to see the “psychological shadow” left by turbulence to scientists.

NS equation and millennial prize problems

In the first half of the 19th century, French engineer and physicist

Claude Louis Navi

(Claude-Louis Navier)

And Irish mathematician and physicist

George Stols

(George Gabriel STOKES)


The NS equation describing fluid movements is gradually developed.


Navavi Stokes equation and various meanings. (Photo / Principle)

This is a set of non -linear partial differential equations, which describes sticky, non -compressed fluids, and how to exercise under the given viscosity, collective speed and external pressure. It can be understood so simple, it is like the fluid version of Newton’s second law. Newton’s second law connects acceleration and force, and

The NS equation connects the change rate of fluid speed with the force on the fluid.

In the past 200 years, the NS equation has helped countless physicists and engineers solve a large number of fluid problems. This equation can almost always predict the movement of fluids


The consistency of this prediction and experiment may be enough for experimental physics and engineering.

However, mathematicians also need to solve more basic problems, that is, the so -called NS equation

Exisivity and smoothness

question. This problem is extremely difficult, so that it

Crey Mathematics Research Institute listed as seven

Millennium award problem


This problem can be divided into two parts: the first is about the existence of equation solution; the second is about whether these solutions have the border

(It’s limited value)

The first part is that for a mathematical model, no matter how complicated it is, if you want to represent this physical world, then it must first have solution. At first glance, you might think that if we can’t determine whether these equations are solved, why are they still using them? In fact, in practice, these equations provide many good predictions for the motion of the fluid, but these solutions are the approximation value of the complete solution of the NS equation. Although we are very confident that these approximate solutions are correct, it lacks a mathematical certificate that can formally indicate that these equations do have.

The second part needs to explore whether the solution of these equations will appear strange

(Or infinity)

Essence The history of fluid mechanics is full of simplified version of the NS equation, and these equations produce strange solutions. In this case, the strange solution implies some physical phenomena that have not been considered in the simplified model. The identification of this new physical phenomenon has prompted researchers to further improve their mathematical models, thereby improving the consistency between the model and the reality. If NS does have a strange solution, maybe the next Millennium Award will be awarded to the people who discover what new physics is needed to eliminate the strange points.

Turbulent mystery

When you look at a slowly flowing river and start analyzing the turbulence, it may destroy the poetic meaning at that moment. However, if this slowly flowing river becomes a destructive torrent, the behavior of understanding complex turbulence will become essential.

However, due to the complexity of turbulence, for scholars who study turbulence, even if they use the fastest supercomputers at present, they may only simulate a few centimeters around the wings of civil aircraft. Therefore, many scientists are trying to try to develop various algorithms and models, and use existing technology to “squeeze” as much information as possible.

Turbine also affects various aspects such as energy consumption, and a more comprehensive understanding of turbulence can bring far -reaching economic benefits. It may also lead great progress in engineering, medical equipment design, vehicles, weather forecasting and climate change research.

Written article: Gaviota

Capture / Design: Wenwen

Reference source: of-Turbulence/


Cover chart/first picture source: gene wilburn/flowr