Development of Turbulence: I. Stability and Smoothness of Navier-Stokes equations

Fluids are everywhere, from the air we breathe to the water in our rivers, and they follow a set of rules that have governed physics for centuries. These rules, known as the Navier-Stokes equations, describe how liquids and gases move, how they swirl, and how they resist flow. For more than a hundred years, these equations have been the bedrock of engineering and meteorology, allowing us to design airplanes and predict the weather. Yet, despite their success, a deep mystery has lingered at the heart of the theory: do these equations always produce a smooth, predictable result, or can they suddenly break down? In the world of mathematics, a "breakdown" means a singularity, a point where the speed of the fluid would theoretically shoot up to infinity in a finite amount of time, creating a physical impossibility. If such a point exists, it would mean the equations fail to describe reality under certain extreme conditions, leaving a gap in our understanding of how the universe works.

A recent study by Boris Shekunov tackles this question directly, not by looking at the abstract math alone, but by simulating the behavior of fluids on a computer with extreme precision. The researcher set out to see if these equations could ever allow a fluid to reach infinite speed, a scenario that would signal a collapse of the theory. To do this, he created a digital model of a fluid jet, a focused stream of air, and subjected it to a wide range of conditions. He tested the fluid at different sizes, from the scale of a human hair down to the microscopic level, and at different speeds, represented by a number called the Reynolds number. This number essentially tells us whether a flow is smooth and orderly or chaotic and turbulent. The simulations covered a vast range, from very slow, sticky flows to faster, more energetic ones, allowing the researcher to watch how the fluid reacted when pushed to its limits.

The investigation focused on a specific type of disturbance: a sudden, random jolt in the speed of the fluid at a single point. In the digital model, this was like giving a tiny, sharp push to a specific spot in the stream. The computer then tracked how this push spread through the fluid, measuring the pressure and speed at every step. The goal was to see if this disturbance would grow uncontrollably, leading to the dreaded singularity where speed becomes infinite. The results were clear and consistent. In every simulation where the flow was slow enough, specifically below a critical threshold where the Reynolds number was less than 1.5, the fluid remained stable. The random jolts did not cause the speed to explode; instead, the fluid's natural resistance, or friction, smoothed them out. The pressure and speed adjusted in a way that kept the system balanced, preventing any runaway growth.

When the researcher looked at the specific case of a "singular jet"—a theoretical stream designed to shrink in size while speeding up, mimicking the conditions that might create a singularity—the outcome was equally definitive. The simulations showed that as the jet got smaller and the flow became more viscous, the energy required to keep it speeding up would become impossible to supply. The pressure needed to squeeze the fluid into a smaller space would rise to infinity, and the friction would become so strong that it would stop the acceleration before it could ever reach an infinite speed. The study found that the equations themselves contain a built-in safety mechanism. As the scale of the flow decreases, the friction terms in the equations become dominant, acting as a brake that prevents the velocity from blowing up. This means that the idea of a fluid reaching infinite speed in a finite time is incompatible with the laws of physics as described by these equations.

The study also explored what happens when energy is constantly added to the system, such as a pump pushing fluid through a pipe. Even in these scenarios, the simulations showed that the fluid settles into a steady, smooth state rather than spiraling into chaos. The computer models revealed that the energy balance of the fluid naturally limits how fast it can go. If the flow tries to speed up too much, the energy dissipation increases, effectively draining the extra energy and keeping the speed in check. The researcher concluded that for any fluid flow that is small enough and slow enough, a smooth, steady solution always exists. The chaotic, turbulent behavior we see in large-scale flows, like a storm or a rushing river, does not imply that the underlying equations break down at the smallest scales. Instead, the equations remain valid and smooth all the way down to the tiniest possible unit of measurement.

This work suggests that the fear of a mathematical singularity in fluid dynamics may be unfounded. The simulations indicate that the Navier-Stokes equations are robust and stable, capable of describing fluid motion without ever producing a physical impossibility. The study does not claim to have solved every problem in turbulence, but it provides strong evidence that the equations do not fail, even when pushed to their theoretical limits. By showing that the fluid naturally resists the formation of infinite speeds, the research reinforces the idea that the laws of motion are consistent and reliable. The findings offer a quiet reassurance to scientists and engineers: the mathematical framework that describes the flow of water and air is sound, and the chaotic beauty of turbulence is a feature of the system, not a sign of its collapse.