As we mentioned, in pipes that are not very long and wide enough, the friction is so small that it can be neglected. Under these conditions, the pressure drop is so small that in a constant-section tube, the liquid in the gauge tubes is almost at the same height. However, if the pipe has different places unequal cross section, then even in those cases where friction can be neglected, experience reveals that the static pressure is different in different places.

Let's take a pipe of unequal cross-section (Fig. 311) and we will pass a constant stream of water through it. From the levels in the manometric tubes, we will see that the static pressure is less in the narrowed places of the pipe than in the wide ones. This means that when moving from a wide part of the pipe to a narrower one, the degree of compression of the liquid decreases (pressure decreases), and when moving from a narrower part to a wide one, it increases (pressure increases).

Rice. 311. In narrow parts of a pipe, the static pressure of the flowing liquid is less than in wide parts.

This is explained by the fact that in the wide parts of the pipe, the liquid must flow more slowly than in the narrow parts, since the amount of liquid flowing over the same time intervals is the same for all sections of the pipe. Therefore, when passing from the narrow part of the pipe to the wide part of the pipe, the velocity of the liquid decreases: the liquid slows down, as if flowing onto an obstacle, and the degree of its compression (as well as its pressure) increases. On the contrary, when passing from the wide part of the pipe to the narrow part, the velocity of the liquid increases and its compression decreases: the liquid, accelerating, behaves like a straightening spring.

So we see that the pressure of the fluid flowing through the pipe is greater where the fluid velocity is less, and vice versa: the pressure is less where the fluid velocity is greater. This The relationship between the speed of a fluid and its pressure is called Bernoulli's law named after the Swiss physicist and mathematician Daniel Bernoulli (1700-1782).

Bernoulli's law holds for both liquids and gases. It remains valid for the movement of fluid not limited by the walls of the pipe - in a free flow of fluid. In this case, Bernoulli's law must be applied as follows.

Assume that the movement of a liquid or gas does not change over time (steady flow). Then we can imagine lines inside the flow along which the fluid moves. These lines are called current lines; they break the liquid into separate streams that flow side by side without mixing. Streamlines can be made visible by introducing into a stream of water liquid paint through thin tubes. Jets of paint are located along the streamlines. In the air to receive visible lines current, you can use wisps of smoke. It can be shown that Bernoulli's law applies to each jet separately: the pressure is greater in those parts of the jet where the speed in it is less and, therefore, where the cross section of the jet is greater, and vice versa. From fig. 311 shows that the cross section of the jet is large in those places where the streamlines diverge; where the jet cross section is smaller, the streamlines approach each other. That's why Bernoulli's law can also be formulated as follows: in those places of the flow where the streamlines are thicker, the pressure is less, and in those places where the streamlines are rarer, the pressure is greater.

Let's take a pipe that has a narrowing, and we will pass water through it at high speed. According to Bernoulli's law, the pressure will be reduced in the narrowed part. You can choose the shape of the pipe and the flow rate in such a way that in the narrowed part the water pressure will be less than atmospheric. If we now attach a drain pipe to the narrow part of the pipe (Fig. 312), then the outside air will be sucked into a place with less pressure: getting into the stream, the air will be carried away by the water. Using this phenomenon, one can construct dilution pump - the so-called water jet pump. In the one shown in Fig. 313 of the water jet pump model, air is sucked in through the annular gap 1, near which water moves at high speed. Branch 2 is attached to the evacuated vessel. Water jet pumps do not have moving solid parts (like a piston in conventional pumps), which is one of their advantages.

A lot of the world around us obeys the laws of physics. This should not be surprising, because the term "physics" comes from the Greek word, which means "nature" in translation. And one of these laws, constantly working around us, is Bernoulli's law.

The law itself acts as a consequence of the principle of conservation of energy. This interpretation of it allows us to give a new understanding to many previously well-known phenomena. To understand the essence of the law, it is simply enough to recall a flowing stream. Here it flows, runs between stones, branches and roots. In some places it is made wider, somewhere narrower. You can see that where the stream is wider, the water flows more slowly, where it is narrower, the water flows faster. This is the Bernoulli principle, which establishes the relationship between the pressure in a fluid flow and the speed of such a flow.

True, physics textbooks formulate it somewhat differently, and it has to do with hydrodynamics, and not with a flowing stream. In a fairly popular Bernoulli, it can be stated in this way - the pressure of a fluid flowing in a pipe is higher where its speed is less, and vice versa: where the speed is greater, the pressure is less.

To confirm, it is enough to the simplest experience. You need to take a sheet of paper and blow along it. The paper will rise up in the direction along which the air flow passes.

Everything is very simple. As Bernoulli's law says, where the speed is higher, the pressure is lower. This means that along the surface of the sheet, where the flow passes less, and at the bottom of the sheet, where there is no air flow, the pressure is greater. Here the sheet rises in the direction where the pressure is less, i.e. where the air flow is.

The described effect is widely used in everyday life and in technology. As an example, consider a spray gun or airbrush. They use two tubes, one with a larger section, the other with a smaller one. The one with a larger diameter is connected to a container with paint, while the one with a smaller section passes air at high speed. Due to the resulting pressure difference, the paint enters the air stream and is transferred by this stream to the surface to be painted.

The same principle applies to the pump. In fact, what is described above is the pump.

No less interesting is Bernoulli's law applied to the drainage of swamps. As always, everything is very simple. The wetlands are connected by ditches to the river. There is a current in the river, but not in the swamp. Again there is a pressure difference, and the river begins to suck water out of the wetlands. Happens in pure form demonstration of the law of physics.

This effect can also be destructive. For example, if two ships pass close to each other, then the speed of water movement between them will be higher than on the other side. As a result, an additional force will arise that will attract the ships to each other, and a catastrophe will be inevitable.

All of the above can be stated in the form of formulas, but it is not at all necessary to write the Bernoulli equations to understand the physical essence of this phenomenon.

For a better understanding, we give another example of the use of the described law. Everyone imagines a rocket. In a special chamber, fuel is burned, and a jet stream is formed. To accelerate it, a specially narrowed section is used - a nozzle. Here there is an acceleration of the jet of gases and, as a result, an increase

There are many more various options the use of Bernoulli's law in technology, but it is simply impossible to consider all of them within the framework of this article.

So, Bernoulli's law is formulated, an explanation of the physical essence of the ongoing processes is given, examples from nature and technology show possible options application of this law.

In this section, we will apply the law of conservation of energy to the movement of liquid or gas through pipes. The movement of fluid through pipes is often found in technology and everyday life. Water pipes supply water in the city to houses, to places of its consumption. In machines, pipes supply oil for lubrication, fuel to engines, etc. The movement of fluid through pipes is often found in nature. Suffice it to say that the circulation of animals and humans is the flow of blood through tubes - blood vessels. To some extent, the flow of water in rivers is also a kind of flow of fluid through pipes. The river bed is a kind of pipe for flowing water.

As you know, a stationary liquid in a vessel, according to Pascal's law, transfers external pressure in all directions and to all points of the volume without change. However, when a fluid flows without friction through a pipe whose cross-sectional area is different areas different, the pressure is not the same along the pipe. Let us find out why the pressure in a moving fluid depends on the cross-sectional area of ​​the pipe. But first, let's take a look at one important feature any fluid flow.

Let us assume that the liquid flows through a horizontally located pipe, the section of which is different in different places, for example, through a pipe, part of which is shown in Figure 207.

If we mentally drew several sections along the pipe, the areas of which are respectively equal, and measured the amount of liquid flowing through each of them over a certain period of time, we would find that the same amount of liquid flowed through each section. This means that all the liquid that passes through the first section in the same time passes through the third section in the same time, although it is much smaller in area than the first. If this were not the case and, for example, less liquid passed through the cross section with an area over time than through the cross section with an area, then the excess liquid would have to accumulate somewhere. But the liquid fills the entire pipe, and there is nowhere for it to accumulate.

How can a liquid that has flowed through a wide section have time to "squeeze" through a narrow one in the same time? Obviously, for this, when passing through narrow parts of the pipe, the speed of movement must be greater, and just as many times as the cross-sectional area is smaller.

Indeed, let us consider a certain section of a moving liquid column, coinciding at the initial moment of time with one of the sections of the pipe (Fig. 208). During the time, this area will move a distance that is equal to where is the speed of the fluid flow. The volume V of the liquid flowing through the pipe section is equal to the product of the area of ​​\u200b\u200bthis section and the length

In a unit of time, the volume of liquid flows -

The volume of fluid flowing per unit time through the pipe section is equal to the product of the cross-sectional area of ​​the pipe and the flow velocity.

As we have just seen, this volume must be the same in different sections of the pipe. Therefore, the smaller the cross section of the pipe, the greater the speed of movement.

How much liquid passes through one section of the pipe in a certain time, the same amount must pass for such

the same time through any other section.

Moreover, we assume that a given mass of liquid always has the same volume, that it cannot compress and reduce its volume (a liquid is said to be incompressible). It is well known, for example, that in the narrow places of the river the speed of the flow of water is greater than in the wide ones. If we designate the fluid flow velocity in sections by areas through then we can write:

From this it can be seen that when a liquid passes from a pipe section with a larger cross-sectional area to a section with a smaller cross-sectional area, the flow velocity increases, i.e., the liquid moves with acceleration. And this, according to Newton's second law, means that a force acts on the liquid. What is this power?

This force can only be the difference between the pressure forces in the wide and narrow sections of the pipe. Thus, in a wide section of the pipe, the fluid pressure must be greater than in a narrow section of the pipe.

The same follows from the law of conservation of energy. Indeed, if the velocity of the liquid increases in the narrow places of the pipe, then its kinetic energy also increases. And since we have assumed that the fluid flows without friction, this increase in kinetic energy must be compensated by a decrease in potential energy, because the total energy must remain constant. What is the potential energy here? If the pipe is horizontal, then the potential energy of interaction with the Earth in all parts of the pipe is the same and cannot change. This means that only the potential energy of elastic interaction remains. The pressure force that causes the liquid to flow through the pipe is the elastic force of compressing the liquid. When we say that a liquid is incompressible, we only mean that it cannot be compressed so much that its volume changes noticeably, but a very small compression, causing the appearance of elastic forces, inevitably occurs. These forces create fluid pressure. This is the compression of the liquid and decreases in the narrow parts of the pipe, compensating for the increase in speed. In narrow places of pipes, the fluid pressure must therefore be less than in wide ones.

This is the law discovered by the Petersburg academician Daniil Bernoulli:

The pressure of the flowing fluid is greater in those sections of the flow in which the speed of its movement is less, and,

on the contrary, in those sections in which the speed is greater, the pressure is less.

Strange as it may seem, but when the liquid "squeezes" through narrow sections pipe, its compression does not increase, but decreases. And experience confirms this well.

If the pipe through which the liquid flows is provided with open tubes soldered into it - pressure gauges (Fig. 209), then it will be possible to observe the distribution of pressure along the pipe. In narrow places of the pipe, the height of the liquid column in the manometric tube is less than in wide ones. This means that there is less pressure in these places. The smaller the cross section of the pipe, the greater the flow rate in it and the lower the pressure. It is possible, obviously, to choose such a section in which the pressure is equal to the external atmospheric pressure (the height of the liquid level in the manometer will then be equal to zero). And if we take an even smaller cross section, then the pressure of the liquid in it will be less than atmospheric.

This fluid flow can be used to pump air. The so-called water jet pump operates on this principle. Figure 210 shows a diagram of such a pump. A jet of water is passed through tube A with a narrow hole at the end. The water pressure at the pipe opening is less than atmospheric pressure. That's why

gas from the evacuated volume through tube B is drawn to the end of tube A and is removed together with water.

Everything said about the movement of liquid through pipes applies to the movement of gas. If the gas flow rate is not too high and the gas is not compressed enough to change its volume, and if, in addition, friction is neglected, then Bernoulli's law is also true for gas flows. In the narrow parts of the pipes, where the gas moves faster, its pressure is less than in the wide parts, and may become less than atmospheric pressure. In some cases, this does not even require pipes.

You can do a simple experiment. If you blow on a sheet of paper along its surface, as shown in Figure 211, you can see that the paper will rise up. This is due to the decrease in pressure in the air stream above the paper.

The same phenomenon takes place during the flight of an aircraft. The oncoming air flow runs into the convex upper surface of the wing of a flying aircraft, and due to this, a decrease in pressure occurs. The pressure above the wing is less than the pressure below the wing. That is why the lifting force of the wing arises.

Exercise 62

1. Permissible speed of oil flow through pipes is 2 m/sec. What volume of oil passes through a pipe with a diameter of 1 m in 1 hour?

2. Measure the amount of water flowing out of faucet for a certain time Determine the speed of water flow by measuring the diameter of the pipe in front of the faucet.

3. What should be the diameter of the pipeline through which water must flow per hour? Permissible water flow rate 2.5 m/sec.

Documentary educational films. Series "Physics".

Daniel Bernoulli (January 29 (February 8), 1700 - March 17, 1782) was a Swiss universal physicist, mechanic and mathematician, one of the creators of the kinetic theory of gases, hydrodynamics and mathematical physics. Academician and foreign honorary member (1733) Petersburg Academy Sciences, member of the Academies: Bologna (1724), Berlin (1747), Paris (1748), Royal Society of London (1750). Son of Johann Bernoulli.

Bernoulli's law (equation) is (in the simplest cases) a consequence of the law of conservation of energy for a stationary flow of an ideal (that is, without internal friction) incompressible fluid:

Here

is the density of the liquid, - flow rate, is the height at which the fluid element under consideration is located, - pressure at the point in space where the center of mass of the fluid element under consideration is located, - acceleration of gravity.

The Bernoulli equation can also be derived as a consequence of the Euler equation expressing the momentum balance for a moving fluid.

In scientific literature, Bernoulli's law is usually called Bernoulli equation(not to be confused with differential equation Bernoulli), Bernoulli's theorem or Bernoulli integral.

The constant on the right hand side is often called full pressure and depends, in the general case, on the streamline.

The dimension of all terms is a unit of energy per unit volume of liquid. The first and second terms in the Bernoulli integral have the meaning of the kinetic and potential energy per unit volume of the liquid. It should be noted that the third term in its origin is the work of pressure forces and does not represent a reserve of any special kind energy (“pressure energy”).

A ratio close to the one given above was obtained in 1738 by Daniil Bernoulli, whose name is usually associated Bernoulli integral. IN modern form the integral was obtained by Johann Bernoulli around 1740.

For horizontal pipe the height is constant and the Bernoulli equation takes the form: .

This form of the Bernoulli equation can be obtained by integrating the Euler equation for a stationary one-dimensional fluid flow, with constant density : .

According to Bernoulli's law, the total pressure in a steady flow of fluid remains constant along this flow.

Full pressure consists of weight, static and dynamic pressures.

It follows from Bernoulli's law that as the flow cross section decreases, due to an increase in velocity, that is, dynamic pressure, the static pressure decreases. This is the main reason for the Magnus effect. Bernoulli's law is also valid for laminar gas flows. The phenomenon of a decrease in pressure with an increase in the flow rate underlies the operation of various types of flow meters (for example, a Venturi tube), water and steam jet pumps. And the consistent application of Bernoulli's law led to the emergence of a technical hydromechanical discipline - hydraulics.

Bernoulli's law is valid in its pure form only for liquids whose viscosity is zero. For an approximate description of the flows of real fluids in technical hydromechanics (hydraulics), the Bernoulli integral is used with the addition of terms that take into account losses due to local and distributed resistances.

Generalizations of the Bernoulli integral are known for certain classes of viscous fluid flows (for example, for plane-parallel flows), in magnetohydrodynamics, and ferrohydrodynamics.

Take a pipe through which a liquid flows. Our pipe is not the same along the entire length, but has a different cross-sectional diameter. Bernoulli's law is expressed in the fact that, despite the different diameter, the same volume of liquid flows through any section in this pipe at the same time.

Those. how much liquid passes through one section of the pipe in a certain time, the same amount must pass through any other section in the same time. And since the volume of the liquid does not change, and the liquid itself practically does not compress, something else changes.

In the narrower part of the pipe, the fluid velocity is higher and the pressure is lower. Conversely, in wide parts of the pipe, the velocity is lower and the pressure is higher.

Fluid pressure and velocity change. If the pipe through which the liquid flows is provided with open manometer tubes soldered into it (Fig. 209), then it will be possible to observe the distribution of pressure along the pipe.

Everything said about the movement of liquid through pipes applies to the movement of gas. If the gas flow rate is not too high and the gas is not compressed enough to change its volume, and if, in addition, friction is neglected, then Bernoulli's law is also true for gas flows. In the narrow parts of the pipes, where the gas moves faster, its pressure is less than in the wide parts.

As applied to aerodynamics, Bernoulli's law is expressed in the fact that the air flow on the wing has a different speed and pressure under the wing and above the wing, as a result of which the lifting force of the wing arises.

Let's do a simple experiment. Take a small piece of paper and place it directly in front of us in this way:

And then we blow over its surface, then a piece of paper, contrary to expectations, instead of bending even more towards the Earth, on the contrary, straightens up. The thing is that by blowing air above the surface of the sheet, we reduce its pressure, while the air pressure under the sheet remains the same. It turns out that there is an area of ​​low pressure above the leaf, and an area of ​​increased pressure under the leaf. Air masses are trying to "get over" from the area high pressure to the low area, and this causes the sheet to straighten out.

You can also do another experiment. Taking 2 sheets of paper and placing them in front of you as follows:

And then blowing into the area between them, the pieces of paper, contrary to our expectations, instead of moving away from each other, on the contrary, will approach. Here we observe the same effect. Air masses with outside parties have a leaflet more pressure than the air accelerated by us between the sheets. This leads to the fact that the pieces of paper are attracted to each other.

The same principle is used to carry out their flights paragliders, hang gliders, airplanes, gliders, helicopters, etc. aircrafts. This is what allows a multi-ton passenger aircraft to take off.