Shulou(Shulou.com)11/24 Report--

When Eiffel built the famous Eiffel Tower in 1889, he selected 72 famous 19th century French scientists and engraved their names on the tower as a sign of reverence. The most striking ones are Lagrange, Laplace and Legendre. You will also find the name of Navid, a famous engineer who studied for some time with the great mathematician Fourier. After 1820, Navid began to think about fluid-related mathematics. Between 1821 and 1822 he discovered the famous Navid-Stokes equations.

In the first half of the 18th century, the Swiss mathematician Daniel Bernoulli used calculus to describe the equation of motion of a fluid under multiple forces. On the basis of Bernoulli, Euler constructed a set of equations that can accurately describe the motion of inviscid fluids.

In 1822, Navid improved Euler's equation so that it could be applied to fluids with a certain degree of viscosity. Navid's mathematical derivation is flawed. But the equation he got in the end was correct. A few years later, the Irish mathematician Stokes made a correct derivation. From the beginning, Stokes focused on using calculus to explain the motion of the fluid. He discovered the formula Navid introduced 20 years ago (but his derivation was correct).

Based on the work of Navid Stokes, mathematicians were one step closer to developing a complete theory of fluid motion by the end of the 19th century. There is only one problem to be solved. No one can prove whether the Navid-Stokes equations have a solution. The mathematics of fluid motion seems extremely difficult.

From discrete to continuous when mathematicians in the 16th and early 17th centuries tried to write formulas to describe planetary motion, they encountered a basic problem. Mathematical tools are static in nature. Numbers, points, lines, etc., are excellent for calculation and measurement, but they alone cannot describe motion. In order to study objects in continuous motion, mathematicians must find a way to apply these static tools to dynamic motion. In the middle of the 17th century, Newton and Leibniz of Germany independently "invented" calculus, which made mathematics a big step forward.

Newton and Leibniz thought that continuous motion was made up of a series of stationary forms. Each static form can be analyzed with existing mathematical skills, and the difficulty lies in how to combine all the still forms. To form continuous motion mathematically, Newton and Leibniz must "show" these static forms at an infinite speed, and each form can only last for an infinitely short time. Calculus is a set of techniques developed by Newton and Leibniz to carry out the work of arranging infinite forms in order.

The basic operation of differential calculus is a process called differentiation. The purpose of differentiation is to obtain the rate of change of some amount of change. In order to do this, the value, location, or path of the change must be given by an appropriate formula. Then differentiate this formula and produce another formula that gives the rate of change. Therefore, differentiation is the process of transforming one formula into another.

In the eighteenth century, calculus was used to study the continuous motion of solid objects such as planets, or the continuously changing slopes of continuous geometry. Bernoulli tried to apply this method to the continuous motion of a fluid (liquid or gas).

For Newton and Leibniz, the continuous motion analyzed is the continuous motion of isolated, discrete objects (planets or particles, or points of a figure or surface). However, in the case of fluid, not only the motion, but also the matter itself is continuous.

Bernoulli sees continuous fluids as infinitely small discrete regions (or "droplets") close together, each of which can be treated by Newton and Leibniz. Another way is to write an equation describing the path of any particular point in the fluid (an infinitesimal point). This requires the grasp of two kinds of infinitesimal.

The motion of each infinitesimal particle is regarded as a series of "fixed frames", which is the standard calculus method used to study the continuous motion of a single object. Motion is regarded as a sequence formed by arranging a series of static states in time.

There is an infinitesimal geometric change between the path taken by one "point" and the path followed by another "point" infinitely close to it.

The thorny problem is to grasp these two kinds of infinitesimal at the same time-time infinitesimal and geometric infinitesimal. This took most of Bernoulli's adult life. In 1738, he published his results in his book Hydrodynamics. The key idea is to take the solution as the so-called vector field. To put it simply, a vector field is a function with three independent variables x, y, z, which tells you the speed and direction of fluid flow at any point in it.

There is an equation in hydrodynamics which shows that when a fluid flows through a surface, the pressure acting on the surface decreases with the increase of the flow velocity. Why is this conclusion worth mentioning? Because the Bernoulli equation lays the foundation of modern aviation theory and explains why airplanes can fly in the air.

On the basis of Bernoulli's work, Euler established a set of equations describing the motion of frictionless fluid under known forces, but he failed to solve these equations. Navid Stokes later improved Euler's equations to make them applicable to viscous fluids. The equation they got is called the Navid-Stokes equation.

Although these equations can be solved in the hypothetical two-dimensional case of infinitely thin plane film fluid, people do not know whether there is a solution in the three-dimensional case. Please note that the crux of the problem is not what the solution of the equation is, but whether the equation has a solution.

Let's start with Euler's system of equations about fluid motion. This set of equations describes the flow of a frictionless fluid that extends infinitely in all directions.

We assume that every point in the fluid P = (x _ 1 _ 4 _ y _ z) is subjected to a force that varies with time. Suppose the force acting on point P at t moment is

Let p be the pressure of the fluid at point P at the moment t.

The motion of the time t fluid at point P can be described by giving its velocity in the direction of the three axes. The velocity of the fluid at the P point along the x axis, the velocity of the fluid at the P point along the y axis, and the velocity of the fluid at the P point along the y axis, and the velocity along the z axis.

We assume that the fluid is incompressible, that is, when a force is applied to it, it can flow in a certain direction, but it cannot be compressed or expanded. This property is expressed by the following equation

Suppose we know the state of motion at t = 0. Moreover, these initial functions are assumed to be well-behaved functions.

"good state" is a technical term in mathematics, but it does not affect the understanding of equations. However, the precise expression of "good state" has something to do with the statement of the Navid-Stokes problem as a millennium puzzle. Therefore, people who want to solve this problem still need to know its accurate statement.

Apply Newton's law to every point P in the fluid

Force = mass × acceleration

Euler obtained the following equations and combined them with the above incompressibility equation to describe the motion ∶ of the fluid.

This is the Euler equation of fluid motion. In order to apply to viscous fluids, Navid-Stokes introduced a viscosity constant v, which is a measure of the internal friction of the fluid, and added an additional force to the right of the equation-viscous force.

In the x direction, the term added to the right of the equation is

Y and z direction are the same.

Here, the symbol

Represents the second-order partial derivative, which is obtained by first finding the differential with respect to x for Utrex, and then finding the differential with respect to x from the obtained results, that is,

In the case of y and z, the definition is similar.

The Euler equation looks very scary. Mathematicians also feel powerless. After careful observation, it can be found that there is little difference between the Euler equations in the z direction, and the addition of three additional viscosity terms is also based on the same form of change.

In the 19th century, mathematicians invented a symbol and a method to deal with directional motion in a simple way. The idea is to introduce a new class of quantities, called vectors. The vector has both size and direction. Using vectors, mathematicians can write Navid-Stokes equations in a more compact ∶.

Here, f and u are vector functions, symbols.

Represents the operation of vector calculus.

Progress in solving the Navid-Stokes equation was so small that the Clay Promotion Association decided to set up a $1 million bonus to solicit answers to any change in the question. One of the simplest forms (though not necessarily the easiest to solve) is that if you make the force functions fcorrecxgravity and fyogz both zero, can you find the functions p (xpenydhozpeng t), upright x (xmemydjinzpent), upright (xpeny yjorzjint) and uplift (xmemydjorzpart). Make them meet the improved version of the Euler equation (that is, including the viscosity term) and be "well-behaved" enough to make them look consistent with physical reality?

I would like to mention that the similar problem of zero viscosity (that is, the Euler equation) has not been solved.

If the Navid-Stokes problem is reduced to a two-dimensional case (so that all z terms are equal to zero), the equation can be solved. But it does not help to solve the three-dimensional situation.

A complete three-dimensional problem can also be solved in a highly restricted way. Given all kinds of initial conditions, we can always find a positive number T, which makes the equation solvable for all times of 0 ≤ t ≤ T. Generally speaking, counting T is too small, so this solution is not particularly useful in reality. Counting T is called the "blowup" time of this particular system.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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