In this article we will look at the Reynolds number and give examples of its values. In addition, we will talk about the importance of the discovery of a scientist for the proof of self-organization in substances, and, as a result, for the start of a new, post-non-classical period of science.

## Inconspicuous beginning

Science, like all aspects of human life, develops in stages. And the beginning of the next stage of evolution comes unnoticed. As a rule, understanding the next paradoxes (and always and everywhere abound), some ordinary scientist, by virtue of curiosity or for simplicity, gives a definition, notices the regularity and deduces a formula. For a while everyone uses this discovery, not often paying attention to its versatility or surprising utility. And then a genius comes (who most often does not consider himself as such), expresses something revolutionary, and a well-known fact emerges that confirms new theories.

So it was with Max Planck quanta: he introduced his formula for simplicity. Only Einstein realized the importance of his assumption. So it was with Mendel, who found the inheritance of the signs of hybrid plants interesting. His work was widely criticized, until Karl Correns proved it to be fair. So it was with Reynolds: studying the water flows, he introduced the criterion by which the laminar flow in front of us or turbulent was determined. Its formula was used by scientists dealing with water turbines, without assuming how important this relationship is to prove the self-organization of a substance.

## Biography of Osborne Reynolds

The mentioned scientist was born in the first half of the nineteenth century in the family of a priest, in England, in the city of Belfast. From a fairly early age, he was fond of mechanisms, moving parts of machines, worked in the workshop. He graduated from the prestigious University of Cambridge, and then taught at least titled Manchester.

All his life he devoted to scrupulous study of various phenomena from the field of mechanics, heat exchange, electricity, magnetism, astrophysics and turbulence. With special care Reynolds was preparing an experiment: being a talented mechanic, he thought over all the processes to the smallest detail, trying to eliminate parasitic phenomena. He skillfully combined the master of all trades and a thoughtful scientist. He took up any mysteries of science that could be solved by patient labor and many hours of observation.

As the reader sees, no flashes, insights, cries: “Found!” Only the daily routine of the corrosive scientist. And as a result - a lot of discoveries, including the famous Reynolds number.

## What is called the reynolds number

Water is the cradle of life on our planet. Even if the theory of panspermia and complex organic molecules came true on comets, the way people see it now, life began in the first oceans.

Water properties never cease to amaze. And if now scientists are studying the structure that this fluid forms under extreme pressure and temperature, in the nineteenth century they wondered how it flows. In particular, what distinguishes laminar flow from turbulent.

The Reynolds number is a dimensionless quantity that determines how much the viscosity of a fluid is capable of preventing the rectilinear motion of its particle at a given speed. In the flow, each particle tends to move forward, preferably in the shortest possible way. However, all the molecules of a fluid are connected and cannot exist in isolation from each other. The strength of this connection determines, among other things, the viscosity. The higher it is, the more difficult the fluid takes on new forms and gives up its molecules. Water flows easily, honey - more difficult. Bitumens have the highest viscosity. Thomas Parnell from the University of Queensland poured his variety - pitch - into the funnel. The first drop of it fell eight years later.

So, viscosity impedes the movement of fluid particles along the shortest path. Accordingly, depending on the flow velocity, the particles move more or less straight (laminar) or, overcoming the resistance of the entire volume of the liquid, they begin to randomly mix, creating mini-whirls (this is called turbulence). Reynolds number has other definitions.

## Other definitions of the Reynolds number

In mathematical terms, it means the ratio between nonlinear and dissipative terms in the Navier – Stokes equation, which describes the propagation of a finite amplitude wave in a medium.

Also, the Reynolds number determines the ratio between the kinetic energy of a liquid and the energy loss per unit length (due to internal friction). The formula for this indicator is:

Re = ρvD_{g} / η = vD_{g} / μ = QD_{g} / μA (Re is the Reynolds number, ρ is the density of a liquid, D_{g} - hydraulic diameter, v - characteristic velocity, η - dynamic viscosity of the fluid, μ - kinematic viscosity of the fluid, Q - volumetric flow rate, A - pipe section area).

In acoustics, the formula is different:

Re_{a} = ρvC_{0} / ωb (C_{0} - sound velocity in a given fluid, ω - circular frequency, b - dissipation parameter).

For the Reynolds number, depending on the substance forming the flow, roughness and shape of the pipe section, there is a certain critical indicator. It also depends on whether there is a flow without obstacles or a liquid flows around something (for example, a metal ball).

In addition, the value of the Reynolds number indicates that the flow is still laminar or already turbulent. There is also a certain intermediate state of the flow, when it can no longer be called uniform. But at the same time it still does not meet the requirements of turbulence. This Reynolds number for water in the ideal tube is 2100.

## Significance for the science of determining the Reynolds number

We have already mentioned above that new periods of science begin relatively unnoticed. Also, the reader, for sure, has already guessed that here the value associated with a very specific area — the flow of fluids — is not just described here. Let's start the story from afar.

Reynolds died in 1912. Five years later, Ilya Romanovich Prigogine was born in Moscow, and ten years later, in Germany, Hermann Haken. In the middle of the twentieth century, these two people proclaimed a new period of science, called “post-non-classical”. It continues now. Its foundation is based on a relatively new discipline - synergetics.

## Synergy

This science does not divide biology, chemistry, physics and geology. All these sections are interesting synergetics because they explore open non-equilibrium systems that are capable of self-organization. Nowhere to find a direct study of phenomena. Any science builds models, but most of them are static. Ideal vacuum, absolute zero are found in textbooks. But in reality such conditions are unattainable. Any two contiguous systems exchange mass or energy, so that any processes take place under pre-equilibrium conditions.

## Self-organization

And when some volumetric object (for example, water in a saucepan), which consists of a very large number of elements, receives energy from the outside (heats up), then at some point its behavior changes.

We were always told at school that boiling is the process chaotic, the molecules cease to form a coherent system of fluid and begin to move in different directions inconsistently. However, it is not so. The presence of a very large number of elements, if one atom "ran" up, it is likely that several tens of neighboring atoms at the same time will move up. Thus, some time there will be a cluster of water in which the particles behave the same way.

This is the essence of self-organization: in a nonequilibrium system of many components, some of them begin to behave in the same way in small portions of time and space.

## Turbulence and self-organization

The idea of self-organization, as the reader sees it, is quite simple. The proof of coherent behavior is much more complicated. The remarkable Russian scientist Klimontovich, following the paradigm of synergetics, derived: for the transition of any system from chaos to self-organization one parameter answers. When it exceeds a certain critical value, the level of difficulty increases. The same scientist for the first time in the world proved that for turbulent motion of a fluid, such a criterion is the Reynolds number. Thus, the indicator, which has been used for almost fifty years, turned out to be the key for the discovery of a new period in science.