A Few Thoughts on the Navier-Stokes Equation

In short, the Navier-Stokes equations are used to explain Fluid Motion.

From the movement of airplanes and ships to the flow of blood in veins and arteries, Fluid Motion is everywhere.

After the deaths of scientists Navier and Stokes, research on the Navier-Stokes equations continued and significant progress was made.

Now only one problem remains, and it is the most fundamental one: no one can find a formula for solving the Navier-Stokes equations.

To be precise, no one can prove in principle that the Navier-Stokes equations have solutions.

Scientists' persistence lies in their pursuit of truth. Over the past one or two hundred years, humanity has developed numerous theories, created various tools, and applied them to various engineering scenarios using an equation whose solution is unknown.

Scientists felt a little uneasy, especially those like Shen Qi and Witten who considered issues more deeply.

A conference room in the center.

Zhou Yu'an spoke eloquently, "Fluid Motion satisfies the conservation of mass, linear momentum, and angular momentum. As a result, it is necessary to introduce six additional motion characterization quantities, which are the Reynolds Stress with three independent components, the Dynamic Viscosity with two independent components, and the Dynamic Pressure with one independent component."

"However, I think there's a problem here. While satisfying these three conservation laws, it negates the Navier-Stokes equations. Moreover, the choice of Reynolds Stress determines the Dynamic Viscosity and Dynamic Pressure, which means there are six free quantities to choose from! This leads to the Velocity Field being a function of these six degrees of freedom." Zhou Yu'an had his thoughts, and he stated them.

Xiao Junlong continued, "Old Zhou, this problem is a fatal flaw in Fluid Mechanics. The original derivation of the Navier-Stokes equations only requires the conservation of linear momentum and angular momentum, but the fundamental reason for adding the conservation of mass is that it cannot characterize the conservation of angular momentum. You and I must have learned about the two mechanical viewpoints of the Navier-Stokes equations when we were doing our doctoral studies: first, conservation of mass + conservation of linear momentum; second, conservation of linear momentum + conservation of angular momentum."

"The first point of view introduces the concepts of Material Velocity and Euler Velocity, thus maintaining the basic framework of solving three Velocity Field components using three equations, which is the currently popular Fluid Mechanics theory. However, I regret to find that, based on the first mainstream theoretical system, the Navier-Stokes equations have no solution." Zhou Yu'an shook his head, his expression one of utter bewilderment.

"You're absolutely right, Lao Zhou! Great minds think alike!" The clash of ideas fostered a steady and progressive friendship. Xiao Junlong was also perplexed. "The second viewpoint also has problems. It introduces the Vorticity Vector, but it still uses three equations to calculate three Velocity Field components, which is contradictory. At this point, the derivative of velocity with respect to spatial coordinates is not commutative! Although some have introduced Lie Algebra forms, they have little engineering value because the theoretical foundation is conceptually confused."

"So, the Navier-Stokes equations have no solution?" Zhou Yu'an and Xiao Junlong discussed it and came to this conclusion.

Shen Qi and Witten, who had been listening attentively, smiled but remained silent.

In order to make sure Witten understood, Zhou Yu'an and Xiao Junlong's conversation was entirely in English.

Through intense and rigorous deduction, exchange, and discussion, both Deputy Leader, Zhou and Xiao, concluded that the Navier-Stokes equations have no solution.

This is actually the current state of academia.

Engineers can always introduce additional conditions to the equations for specific problems, thereby making the Navier-Stokes equations solvable.

However, while the results were correct for their specific problems, they lacked general applicability.

This is why the vast majority of scholars believe that the Navier-Stokes equations do not have a general solution.

A general solution is a golden key with universal laws. Finding this golden key, or providing convincing evidence to completely prove that this golden key does not exist at all, are all scientifically valuable operations.

Zhou Yu'an and Xiao Junlong exchanged a glance, shook their heads, and smiled.

That's too naive. Handing over a ten million dollar project like this is bound to get us attacked.

Zhou Yu'an and Xiao Junlong looked at Shen Qi and Witten, the two big shots.

According to conventional script design, after the blushing supporting character finishes their questionable lines, the calm and composed protagonist should step in, give a definitive and correct answer, and lead the team onto a smooth path.

"Edward, you go first," Shen Qi said to Witten.

"This isn't a system problem, it's an algorithm problem," Witten said calmly.

Shen Qi, Zhou Yu'an, and Xiao Junlong: "Hmm."

Witten walked to the water dispenser, grabbed a paper cup, placed it under the spout, and pressed the water dispensing switch.

The water jet flows down at a constant rate and into the paper cup.

When the paper cup is filled with purified water, it overflows, splashing everywhere.

"This is the first experiment." Witten handed Zhou Yu'an a paper cup filled with water and said, "Zhou, drink it."

Glug glug.

Zhou Yu'an drank the water in one gulp. The water had neither sweet nor salty taste; it was just the original taste of ordinary bottled purified water.

Witten took the empty cup from Zhou Yu'an and went back to the water dispenser.

This time, Witten crushed the paper cup, but not completely; it was only partially crushed.

Splash!

Under Witten's operation, the water dispenser spouted a small stream of water, which flowed into the half-collapsed paper cup.

Because Witten applied external force to the paper cup, the cup did not expand to its original size despite being continuously filled with water.

Witten maintains the external force applied to the paper cup, preserving its irregular shape when filled with water.

"This is the second experiment." Witten walked up to Xiao Junlong, pretending to feed him water, and said, "Xiao, drink it."

"But... Zhou just used this paper cup!" Although Xiao Junlong wasn't handsome, he was a clean and hygienic man.

"Don't worry about the details," Witten said.

"Okay, Edward, I'll drink." Xiao Junlong glanced at Zhou Yu'an and asked in Chinese, "Old Zhou, you don't have hepatitis, do you?"

"Look at you, what are you doing, researching?" Zhou Yu'an forced himself not to laugh.

Xiao Junlong closed his eyes, steeled his resolve, and finished the water in his glass with Witten's help.

Witten stated, "Through two experiments, we discovered that, within the same water dispenser system, the same paper cup held less water the second time due to different additional conditions."

Zhou Yu'an and Xiao Junlong: "Hmm."

Shen Qi smiled, without making a sound.

Witten likes to use readily available, simple props to demonstrate and explain profound scientific principles, "You mentioned before that engineers can always introduce additional conditions into equations for specific problems, thereby making the Navier-Stokes equations solvable. However, these additional conditions are deliberate, and even if a solution to the Navier-Stokes equations is found, it can only satisfy the specific conditions they originally set, and it does not have universality."

"The same paper cup was filled with water twice, resulting in significant differences in the amount of water due to different additional conditions. However, it is undeniable that both achieved an engineering effect, solving Zhou and Xiao's thirst problem. Therefore, I believe that the fundamental problem with the Navier-Stokes equations is not a system problem, but an algorithm problem."

Witten walked to the water dispenser and put the paper cup back in its original position. "Who can design a third experiment, also based on this water dispenser system, to arrive at a universal solution?"