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A constant stream of discoveries related to fluid equations

An amazing experimental discovery related to the behavior of fluids triggered a wave of mathematical evidence



The complex flows of fluid in a cup of tea inspired scientists to some important evidence.

Scientific progress does not always move in a straight line. Researchers begin to deal with some issues, and then drop them. Results no longer inspire. It could take decades to form a theory.

But sometimes the accumulation of scientific knowledge goes a direct way, and one discovery gives rise to another, as if falling dominoes.

A similar thing has recently happened in the field of mathematics studying fluid mechanics. The amazing experimental discovery of 2013 launched a series of mathematical proofs that destroyed centuries-old ideas.

“It was a very dynamic and amazing story,” said Alexander Kiselev, a mathematician at Duke University, co-author of one of the evidence.

The discoveries are based on the Euler equations formulated by Leonard Euler in 1757. Mathematicians and physicists used them to simulate the behavior of fluids over time. If you throw a stone in the pond, how will the fluid move in five seconds? Euler equations help answer this question.

But not literally. Euler equations describe an idealized world in which a fluid has several properties that are not found in reality. For example, the equations assume the absence of viscosity in liquids (internal flows do not create friction), as well as incompressibility (you cannot shove a liquid into a volume smaller than it already takes).

In this idealized world, these equations use Newton's laws of motion to predict the future state of a fluid. Mathematicians studying Euler equations ultimately need to understand whether they always work. Are there any scenarios that cause the equations to break down and prevent them from describing the behavior of the fluid in the future?

In 2013, a couple of mathematicians decided to find such a scenario. Thomas Howe of the California Institute of Technology and Guo Luo of Hong Kong City University practiced digital simulations on a computer. They gave a numerical description of the initial state of the fluid and instructed the computer to apply the Euler equations to determine the motion of this fluid in the future.

Howe and Luo concentrated on a specific scenario, which can be repeated almost exactly even at home. Before discussing the surprisingly complex ways that liquids can move, let's do an experiment that everyone can repeat.

Imagine a cylindrical mug with a flat bottom into which tea is poured. Some tea leaves rest at the bottom. Stir the tea clockwise. First, all the liquid will begin to rotate as a whole, and will take tea leaves with it.

However, some time after the start of the movement, the centrifugal force of the rotating fluid will begin to interact with the walls of the cup, creating, as physicists say, a “secondary flow” - a more complex movement that occurs in response to the initial stirring. These secondary streams, going down along the walls of the cylinder and up in the center, demonstrate the tea leaves well: they gather in the center at the bottom of the cup and remain practically motionless, despite the fact that tea continues to rotate around them.

This phenomenon, which has been observed for many centuries, is called the " tea leaf paradox ." In 1926, Albert Einstein gave the first mathematical explanation of this behavior.

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- Albert Einstein (from a report at a meeting of the Prussian Academy of Sciences on January 7, 1926)




The script reviewed by Howe and Luo is a bit more complicated. Imagine the fluid in the cylinder again. But now the liquid at the top of the cylinder rotates clockwise, as in a cup of tea, and at the bottom - counterclockwise. This movement creates several secondary flows. Whirlpools appear, moving up and down along the cylinder walls.

“From above, the liquid spirals downward, and from below it swirls in the opposite direction,” Howe said.

Starting a numerical simulation, Howe and Luo saw something unexpected happen in the middle of the cup, where conflicting streams meet. The Euler equations indicate that the vorticity of the liquid in this place increases extremely. Their simulation showed that, according to Euler's equations, the vorticity in this place grows so fast that it becomes infinite in a finite time.



Such infinite values ​​are called singularities. If Euler’s equations give out a singularity, they break - or, as mathematicians say, “explode” - and can no longer describe the future movement of the fluid. Equations cannot be calculated with infinite values.

The opening of Hou and Luo caused a sensation. For more than 200 years, mathematicians have hunted for scenarios that break Euler's equations. Many carried out numerical simulations, which, in their opinion, should have led to singularities, but none of them passed the test on fast computers. Hou and Luo seem to have finally found such a scenario.

“Many researchers consider this scenario to obtain a singularity the most convincing of all,” said Vladimir Sverak of the University of Minnesota.

However, computer simulations are just evidence. This is not proof.

“Computers are limited in the sense that they cannot operate with infinitely small quantities,” Kiselev said. “The results may look convincing, but we cannot be sure of them.” Perhaps if you take a better supercomputer, you can see how everything collapses. "

Therefore, mathematicians hurried to check whether it is possible to prove that the result of Hou and Luo is correct from a mathematical point of view.


Thomas Howe, a mathematician at the California Institute of Technology

Kiselev and Sverak learned about this simulation in 2013 during the presentation of Howe at the summer conferenceat Stanford. This prompted them to start working on one of the important unsolved problems concerning the growth rate of vorticity in two-dimensional fluids. They managed to prove a long-standing hypothesis regarding the properties of growth rate by considering the scenario that Howe and Luo used in their simulation.

“It’s as if we got a goal that we could strive for,” said Kiselev. - It's one thing when you hunt and do not see the target. And it's completely different when you know where she is. ”

Subsequent evidence expanded the mathematical understanding of the formation of singularities in Euler equations. In 2019, Tarek Elgindi of the University of California, San Diego published two evidencedescribing the conditions under which the Euler equations give out singularities. And the earlier work of Kiselev and Sverak was one of his starting points.

Elgindi's proofs use special conditions, and they do not give a complete understanding of the formation of singularities in the Euler equations that mathematicians need. Nevertheless, these are some of the strongest results achieved in this area.
As whirlpools in a stream change its properties, so the work of Elgindi prompted scientists to a new wave of mathematical discoveries. In October 2019, Hou and Jiajie Chen adapted some of Elgindi's methods to create rigorous mathematical proof.a scenario closely related to what was used in the 2013 experiment. They proved that in a slightly modified version of the scenario, the singularity in the Euler equations does appear.
“They took Elgindi’s ideas and applied them to the 2013 scenario,” said Sverak. The circle is closed.

Of course, there is still a lot of work. Certain technical features of Howe’s new proof do not allow him to determine the existence of a singularity exactly in the situation that he modeled in 2013. But after an outstanding six-year period of work and with new support, Howe believes that he will soon be able to overcome these difficulties. “It seems to me that we are already very close,” he said.

Source: https://habr.com/ru/post/undefined/

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