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In a beautiful summer, everyone likes to stand by the water and watch the waves lapping on the shore. But how many people have been curious about the extreme complexity of water in the process of movement? Its motion looks smooth and regular, but when it smashes on the beach, it splits into hundreds of currents and bubbles and becomes completely unpredictable. It is the Navi-Stokes equations (Navier-Stokes equations) that control this incredible complexity.

Most people are familiar with Newton's second law: the force acting on a body is equal to the product of its mass and acceleration.

The formula of Newton's second law applies to all macroscopic objects in the world. But if you want to know the state of the liquid, you need to know something else-the Navid-Stokes equations.

Around the world, engineers and physicists use them in fields ranging from aircraft design to blood circulation. These equations are very difficult to solve, which is why they are one of the seven millennium prize puzzles (the prize for solving one of them is $1 million).

The Navid-Stokes equations, like any advanced formula, may seem daunting, but the concepts they represent are not complex. We will explore their meaning one by one to understand why they are so important.

We need to make some assumptions before the introduction begins.

First of all, we study Newtonian fluid, which is the simplest mathematical model to explain the viscosity of fluid. There is no real Newtonian fluid in reality, but in most cases, air and water can be regarded as Newtonian fluids. Another very important assumption is that the fluid is incompressible. This means that its density ρ is a constant.

Mass conservation

The equation of mass conservation formula tells us that the mass of the fluid we are studying is conserved. It can change its shape, but its mass remains the same from beginning to end.

The divergence of the velocity vector now let's talk about mathematics. The letter u denotes the velocity vector of the fluid, which has three components, which we can call ugravity vrew, which represents the component of velocity in the three directions of xrecoery yrew. The Greek letter nabla ∇ plus a dot multiplication symbol represents the divergence operator, which means to differentiate the components of the vector in all directions.

The first derivative shows how the x component of velocity changes with the change of space x, and the other two derivatives represent the same meaning. Because this formula is equal to zero, the mass is conserved.

Conservation of momentum

The second equation of momentum conservation formula is actually a system of equations composed of three differential equations, which can be regarded as Newton's second law of fluid. If we expand the expression, we can get a complex set of equations:

In order to understand the extended momentum conservation formula, we will ignore this extended form and focus on momentum conservation.

When we study fluids, we can think of mass and density as the same thing (as long as they are the same volume). If we consider two kinds of fluids, we can say that denser fluids are "heavier" fluids (for example, mercury and mercury in water are heavier). The Greek letter ρ (rho) is used to represent the density of the fluid.

Now that we have mass, if we want to use Newton's second law, we also need to get the acceleration, which is the time derivative of the velocity vector.

Acceleration is the time derivative of velocity. Now, only the terms to the right of the equal sign are unknown, and they represent all the forces exerted on the fluid.

The first term ∇ p is the pressure gradient, which represents the pressure difference in the space where the fluid is located. If there is a lower pressure area and another higher pressure area, the fluid will flow from the high pressure area to the low pressure area. The gradient of p characterizes such a relationship.

The second term describes the viscosity of the fluid. Consider two different fluids, such as water and honey. When you pour out a glass of water, the water can easily fly out of the cup and fall to the ground. When you do the same thing with honey, because honey is sticky, it will fall very slowly. This is what this item means.

The last term F is the simplest term, which represents all the external forces acting on the fluid. Usually, we think of this force as gravity.

To sum up, the relationship expressed by all these strange symbols and letters is only "force = mass × acceleration".

The application of Navier-Stokes equations because the solution of these equations is extremely complex, we need to make a lot of approximations in order to use them. Two examples are Poiseuille flow and Kuyt flow (Poiseuille and Couette flow). Through a large number of assumptions, the two scientists were able to find a solution to the Navid-Stokes equation for a very specific application. However, if we want to use them in more complex situations, such as weather forecasts, we need something to add.

The most common method to use these equations is to transform them with Reynolds averages, and the Reynolds equations are obtained by this method. They are often called RANS (Reynolds averaged Navier-Stokes) equations.

RANS equations (the angle m represents the average) these equations can be used when the fluid is in a state of turbulent flow. Except for the last term, they look almost identical to the Navid-Stokes equation. The last term is called the Reynolds stress tensor, which can explain the turbulence in the fluid.

In the RANS equation, the quantity we use is the amount after averaging a certain time interval. This time interval must be small enough to observe the phenomenon we are studying. At the same time, it must be large enough to minimize the effect of turbulence.

These equations are valid under correct assumptions. We know how to use them to make F1 cars faster, to get spacecraft into the International Space Station, or to make weather forecasts.

You may also want to know how the proof of these equations is worth $1 million.

The million dollar prize from a physical point of view, these formulas are only Newton's second laws applied to fluids. When we make some reasonable assumptions and some reasonable simplification, we can use these equations to do some amazing things.

The problem is that this system of equations is very complex without the introduction of approximation. It is so difficult to solve them that it is not possible to prove the existence of analytical solutions. This is the origin of the Millennium Awards.

The official statement on this issue is:

Prove the following proposition or give its counterexample: given an initial velocity field in three-dimensional space plus one-dimensional time, a smooth and globally defined vector velocity field and a scalar pressure field can be found as the solution of the Navid-Stokes equation.

This means that if you want to get a $1 million bonus, you have to do three things:

It is proved that the solution of Navid-Stokes equation exists.

The solution exists at any point in space.

These solutions must be smooth. This means that a small change in the initial conditions will only produce a small change in the result.

For engineers, it is only necessary to know that even if the basis is only a certain degree of hypothesis, these equations are still valid; however, it is very important for mathematicians to know whether these solutions exist and what they mean.

You may now think that this formula is useful and that it is a waste of time to spend time and energy looking for proof. Well, like many technological advances in human history, this result doesn't seem to matter. What is important is the road to there, which can bring new knowledge and improvement to our lives.

For example, in the space program, if man had never thought of going for a walk on the moon, we would lose a lot of equipment that could improve our living conditions. Magnetic resonance imagers and pacemakers come from technologies developed for space exploration. Today, doctors around the world use them every day to save lives.

The same principle is also applicable to the study of Navier-Stokes equation. The process of exploring the solution of Navier-Stokes equation will help to improve our understanding of fluids or other things. It can lead us to new discoveries and may need to explore new mathematical methods. This can be used to solve many other problems, invent new technologies to improve our lives and make us better.

Original text link:

The Navier-Stokes Equations. A simple introduction to a million... | | by Alessandro Bazzi | Cantor's Paradise (medium.com) |

The translated content only represents the author's point of view, not the position of the Institute of Physics of the Chinese Academy of Sciences.

This article comes from the official account of Wechat: Institute of Physics, Chinese Academy of Sciences (ID:cas-iop), author: AlessandroBazzi, translator: Nothing, revision: zhenni

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