Elements of continuum mechanics. Magnetic field in vacuum

The end of a space flight is considered to be landing on a planet. To date, only three countries have learned to return to Earth spacecraft: Russia, USA and China.

For planets with an atmosphere (Fig. 3.19), the landing problem comes down mainly to solving three problems: overcoming a high level of overload; protection against aerodynamic heating; managing the time to reach the planet and the coordinates of the landing point.

Rice. 3.19. Scheme of spacecraft descent from orbit and landing on a planet with an atmosphere:

N- turning on the brake motor; A- spacecraft deorbit; M- separation of the spacecraft from the orbital spacecraft; IN- SA entry into the dense layers of the atmosphere; WITH - start of operation of the parachute landing system; D- landing on the surface of the planet;

1 – ballistic descent; 2 – gliding descent

When landing on a planet without an atmosphere (Fig. 3.20, A, b) the problem of protection from aerodynamic heating is eliminated.

A spacecraft located in the orbit of an artificial satellite of a planet or approaching a planet with an atmosphere to land on it has a large supply of kinetic energy associated with the speed of the spacecraft and its mass, and potential energy due to the position of the spacecraft relative to the surface of the planet.

Rice. 3.20. Descent and landing of a spacecraft on a planet without an atmosphere:

A- descent to the planet with preliminary entry into a holding orbit;

b- soft landing of a spacecraft with a braking engine and landing gear;

I - hyperbolic trajectory of approach to the planet; II - orbital trajectory;

III - trajectory of descent from orbit; 1, 2, 3 - active flight sections during braking and soft landing

Upon entry into the dense layers of the atmosphere, a shock wave appears in front of the bow of the spacecraft, heating the gas to a high temperature. As the spacecraft sinks into the atmosphere, it slows down, its speed decreases, and the hot gas heats the spacecraft more and more. The kinetic energy of the device is converted into heat. Wherein most of energy is removed into the surrounding space in two ways: most of the heat is removed into the surrounding atmosphere due to the action of strong shock waves and due to heat radiation from the heated surface of the SA.

The strongest shock waves occur with a blunted shape of the nose, which is why blunted shapes are used for SA, rather than pointed ones, characteristic of flight at low speeds.

With increasing speeds and temperatures, most of the heat is transferred to the apparatus not due to friction with the compressed layers of the atmosphere, but due to radiation and convection from the shock wave.

The following methods are used to remove heat from the SA surface:

– heat absorption by the heat-protective layer;

– radiation cooling of the surface;

– application of blow-off coatings.

Before entering the dense layers of the atmosphere, the trajectory of the spacecraft obeys the laws of celestial mechanics. In the atmosphere, in addition to gravitational forces, the apparatus is subject to aerodynamic and centrifugal forces that change the shape of its trajectory. The gravitational force is directed towards the center of the planet, the aerodynamic drag force is in the direction opposite to the velocity vector, the centrifugal and lift forces are perpendicular to the direction of motion of the SA. The aerodynamic drag force reduces the speed of the vehicle, while the centrifugal and lift forces impart acceleration to it in a direction perpendicular to its movement.

The nature of the descent trajectory in the atmosphere is determined mainly by its aerodynamic characteristics. In the absence of lifting force in a spacecraft, the trajectory of its movement in the atmosphere is called ballistic (the descent trajectories of the spacecraft of the Vostok and Voskhod series of spacecraft), and in the presence of lifting force, it is called gliding (SA Soyuz and Apollo, as well as Space Shuttle"), or ricocheting (SA KK Soyuz and Apollo). Movement in a planetocentric orbit does not place high demands on the accuracy of guidance during reentry, since it is relatively easy to adjust the trajectory by turning on the propulsion system for braking or acceleration. When entering the atmosphere at a speed exceeding the first cosmic speed, errors in calculations are most dangerous, since a descent that is too steep can lead to destruction of the spacecraft, and a descent that is too gentle can lead to distance from the planet.

At ballistic descent the vector of the resultant aerodynamic forces is directed directly opposite to the velocity vector of the vehicle. Descent along a ballistic trajectory does not require control. The disadvantage of this method is the large steepness of the trajectory, and, as a consequence, the vehicle enters the dense layers of the atmosphere at high speed, which leads to strong aerodynamic heating of the device and to overloads, sometimes exceeding 10g - close to the maximum permissible values ​​for humans.

At aerodynamic descent The outer body of the apparatus, as a rule, has a conical shape, and the axis of the cone makes a certain angle (angle of attack) with the velocity vector of the apparatus, due to which the resultant of the aerodynamic forces has a component perpendicular to the velocity vector of the apparatus—the lifting force. Thanks to the lifting force, the vehicle descends more slowly, the trajectory of its descent becomes flatter, while the braking section stretches both in length and in time, and the maximum overloads and the intensity of aerodynamic heating can be reduced several times, compared with ballistic braking, which is done by the glider the descent is safer and more comfortable for people.

The angle of attack during descent changes depending on the flight speed and the current air density. In the upper, rarefied layers of the atmosphere it can reach 40°, gradually decreasing with the descent of the apparatus. This requires the presence of a gliding flight control system on the SA, which complicates and weighs down the apparatus, and in cases where it is used to lower only equipment that can withstand higher overloads than a person, ballistic braking is usually used.

The Space Shuttle orbital stage, which performs the function of a descent vehicle when returning to Earth, plans throughout the entire descent phase from entry into the atmosphere until the landing gear touches the landing strip, after which the braking parachute is released.

After the speed of the vehicle has decreased to subsonic in the aerodynamic braking section, the descent of the spacecraft can then be carried out using parachutes. A parachute in a dense atmosphere reduces the speed of the vehicle to almost zero and ensures a soft landing on the surface of the planet.

In the thin atmosphere of Mars, parachutes are less effective, so during the final part of the descent the parachute is detached and the landing rocket engines are turned on.

The descent manned spacecraft of the Soyuz TMA-01M series, designed for landing on land, also have solid-fuel braking engines that turn on a few seconds before touching the ground to ensure a safer and more comfortable landing.

The descent vehicle of the Venera-13 station, after descending by parachute to an altitude of 47 km, dropped it and resumed aerodynamic braking. This descent program was dictated by the peculiarities of the atmosphere of Venus, the lower layers of which are very dense and hot (up to 500 ° C), and fabric parachutes would not have withstood such conditions.

It should be noted that in some projects of reusable spacecraft (in particular, single-stage vertical take-off and landing, for example, Delta Clipper), it is also assumed at the final stage of descent, after aerodynamic braking in the atmosphere, to also perform a parachute-free motor landing using rocket engines. Structurally, descent vehicles can differ significantly from each other depending on the nature of the payload and on physical conditions on the surface of the planet being landed on.

When landing on a planet without an atmosphere, the problem of aerodynamic heating is eliminated, but for landing the speed is reduced using a braking propulsion system, which must operate in a programmable thrust mode, and the mass of the fuel can significantly exceed the mass of the spacecraft itself.

ELEMENTS OF Continuum MECHANICS

A medium is considered continuous if it is characterized by a uniform distribution of matter – i.e. medium with the same density. These are liquids and gases.

Therefore, in this section we will look at the basic laws that apply in these environments.

Lecture 4. Mechanical elements continuum

Let us consider the motion of an ideal fluid - a continuous medium, the compressibility and viscosity of which can be neglected. Let us select a certain volume in it, at several points of which the velocity vectors of movement of liquid particles at a moment in time are determined. If the pattern of the vector field remains unchanged over time, then such fluid motion is called steady. In this case, the particle trajectories are continuous and non-intersecting lines. They are called current lines , and the volume of liquid limited by streamlines is current tube (Fig. 4.1).

Since liquid particles do not intersect the surface of such a tube, it can be considered as a real tube with walls immovable for the liquid. Let us select arbitrary sections in the current tube and those perpendicular to the direction of particle velocity in sections and, respectively (Fig. 4.1).

In a short period of time, volumes of liquid flow through these sections

. (4.1)

So the liquid is incompressible and... And then for any section of the current tube the equality holds

. (4.2)

Fig.4.1

It is called the jet continuity equation. In accordance with (4.2), where the cross section is smaller, the fluid flow velocity is greater and vice versa.

Bernoulli's equation.Let the cross sections of the ideal liquid flow tube under consideration be small, so that the values ​​of velocity and pressure in them can be considered constant, i.e. and, in section and, in (Fig. 4.2).

When a fluid moves over a short period of time, the section will move to the position having passed the path, and the section will move to the position having passed. The volume of liquid contained between the sections and due to the continuity equation will be

equal to the volume of liquid contained in the gap

Rice. 4.2 between and. The tube has some slope

and the centers of its sections and are at heights and above a given

horizontal level. Taking into account that and, the change in the total energy of the released mass of liquid, located at the initial moment between sections and, can be represented in the form

. (4.3)

This change, according to the law of conservation of energy, is caused by the work of external forces. In this case, these are pressure forces and, acting, respectively, on sections and, where and are the corresponding pressures. For any current tube section

, (4.4)

where fluid density Equality (4.4) expresses the fundamental law of hydrodynamics, which is also called the Bernoulli equation after the scientist who first obtained it.

Pressure in a fluid flow.It should be noted that in expression (4.4) all terms have the dimension of pressure and are respectively called: dynamic, hydrostatic or weight, static pressure, and their sum is total pressure. Taking this into account, relation (4.4) can be expressed in words: in a stationary flow of an ideal fluid, the total pressure in any section of the stream tube (in the limit of the stream line) is a constant value, and the flow velocity

. (4.5)

Fluid leaking from the hole.Let the hole located near the bottom of the vessel filled with liquid be open (Fig. 4.3). Let us select a current tube with sections - at the level of the open surface of the liquid in the vessel; - at the level of the hole -. For them, the Bernoulli equation has the form

. (4.6)

Here, where is atmospheric pressure. Therefore, from (4.6) we have

(4.7)

If, then you can be a member

Rice. 4.3 neglect. Then from (4.7) we obtain

Therefore, the fluid flow rate will be equal to:

, (4.8)

Where. Formula (4.8) was first obtained by Torricelli and bears his name. In a short period of time, a volume of liquid flows out of the vessel. The corresponding mass, where is the density of the liquid. She has momentum. Consequently, the vessel imparts this impulse to the outflowing mass, i.e. acts by force

According to Newton's third law, a force will act on the vessel, i.e.

. (4.9)

Here is the reaction force of the flowing fluid. If the vessel is on a trolley, then under the influence of force it will move, which is called reactive motion.

Laminar and turbulent flows. Viscosity.The flow of a liquid in which each layer slides relative to other similar layers and there is no mixing is calledlaminar or layered. If the formation of vortices and intense mixing of layers occurs inside the liquid, then such a flow is called turbulent.

Steady (stationary) flow of an ideal fluid is laminar at any speed. In real liquids, internal friction forces arise between the layers, i.e. real liquids have viscosity. Therefore, each layer slows down the movement of the neighboring layer. The magnitude of the internal friction force is proportional to the contact area of ​​the layers and the velocity gradient, i.e.

, (4.10)

where is the proportionality coefficient, called the viscosity coefficient. Its unit is (Pascal second). Viscosity depends on the type of liquid and temperature. As temperature increases, viscosity decreases.

If the internal friction force is small and the flow speed is low, then the movement is practically laminar. When internal friction forces are high, the layered nature of the flow is disrupted and intense mixing begins, i.e. there is a transition to turbulence. The conditions for this transition when liquid flows through pipes are determined by the quantity kr, called Reynolds number

, (4.11)

where is the density of the liquid, is the average flow velocity over the cross section of the pipe, and is the diameter of the pipe. Experiments show that when the flow is laminar, when it becomes turbulent. For round pipes of radius Reynolds number. The influence of viscosity leads to the fact that the flow velocity through a round pipe is different for different layers. Its average value is determinedPoiseuille's formula

, (4.12)

where is the radius of the pipe, () is the pressure difference at the ends of the pipe, is its length.

The influence of viscosity is also detected during the interaction of a flow with a stationary body. Usually, in accordance with the mechanical principle of relativity, the inverse problem is considered, For example, Stokes It has been established that when a friction force acts on a ball moving in a liquid

, (4.13)

where r - radius of the ball, - speed of its movement. Stokes formula (4.13) is used in laboratory practice to determine the viscosity coefficient of liquids.

Oscillations and waves

Oscillatory motion, or simply oscillation, is a motion characterized by varying degrees of repeatability in time of the values ​​of the physical quantities that determine this motion. We encounter oscillations when studying a wide variety of physical phenomena: sound, light, alternating currents, radio waves, pendulum swings, etc. Despite the wide variety of oscillatory processes, they all occur according to some common patterns. The simplest of them is harmonic oscillatory motion. Oscillatory motion is called harmonic if the change in physical quantity X (displacement) occurs according to the law of cosine (or sine)

, (4.14)

where the value A equal to maximum displacement X system from the equilibrium position, is called the amplitude of oscillation, (, determines the magnitude of the displacement x at a given time and is called the phase of oscillation. At the moment the time begins (the phase of oscillation is equal. Therefore, the value is called the initial phase. The phase is measured in radians or degrees, - cyclic frequency , equal to the number of complete oscillations occurring during time s.

A period is the time of one complete oscillation. It is related to the cyclic frequency by the following relation

. (4.15)

Obviously, the linear frequency (the number of oscillations per unit time) is related to the period T in the following way

(4.16)

The unit of frequency is the frequency of such an oscillation, the period of which is 1 s. This unit is called the hertz (Hz). Frequency in 10 3 Hz is called kilohertz (kHz), in 10 6 Hz, megahertz (MHz).

Oscillatory motion is characterized not only by displacement X, but also speed and acceleration A. Their values ​​can be determined from expression (4.14).

Differentiating (4.14) with respect to time, we obtain the speed formula

. (4.17)

As can be seen from (4.17), the speed also changes according to a harmonic law, and the amplitude of the speed is equal. From a comparison of (4.14) and (4.17) it follows that the speed is ahead of the phase displacement by.

Differentiating (4.14) again with respect to time, we find an expression for acceleration

. (4.18)

As follows from (4.14) and (4.18), acceleration and displacement are in antiphase. This means that at the moment when the displacement reaches its largest positive value, the acceleration reaches its largest negative value, and vice versa.

Plane traveling wave equation

Wave equationis an expression that describes a job And The magnitude of the displacement of an oscillating particle from coordinates and time:

. (4.20)

Let points located in the plane oscillate according to the law. Vibrations of particles of the medium at a point (Fig. 4.4) located at a distance I changes from the source of oscillations will occur according to the same A kon, but will lag in time from fluctuations in the source And ka on (where is the speed of wave propagation). The vibration equation of these particles has the form: (4.20)

Fig.4.4

Since the point was chosen arbitrarily, equation (5.7) allows us to determine the displacement of any point in the medium involved in the oscillatory process at any time, therefore it is calledequation of a plane running in l us. In general, it looks like: (4.21) where is the amplitude of the wave; ¶ plane wave phase; – cyclic wave frequency; – initial phase of oscillations and niy.

Substituting expressions for speed () and cyclic frequency () into equation (4.21), p about ray:

(4.22)

If we introduce the wave number, then the plane wave equation can be written as:

. (4.23)

The speed in these equations is sk O the growth of the phase movement of the wave, and it is calledphase velocity. Indeed, let the phase in the wave process be constant. To find the speed of its movement, divide the expression for the phase by and differentiate with respect to time neither. We get:

Where.

Standing wave. If several waves simultaneously propagate in a medium, thenprinciple of superposition): to a each wave behaves as if there were no other waves, and the result is Yu The total displacement of particles of the medium at any moment of time is equal to geometric sum displacements that often receive And cy, participating in each of the constituent wave processes from owls

Of great practical interest is the superposition of two plane waves

And, (4.24)

with identical frequencies and amplitudes, propagating towards each other along the axis. Adding these equations, p O we obtain the equation of the resulting wave, called standing wave (4.25)

Table 4.1

In a running wave	In a standing wave
Oscillation amplitude
All points of the medium oscillate with the same y ampl and there ami	All points of the medium oscillate with different a m slabs
Oscillation phase
The phase of oscillations depends on the coordinate and the chosen point	All points between two nodes oscillate in the same phase . When passing through a node, phase count e baniya changes to.
Energy transfer
The energy of vibrational motion is transferred in the direction of distribution O wandering waves.	There is no transfer of energy, only mutual transformations of energy occur within.

At points in the environment where ampl. And there the waves go to zero (). These points are called nodes () standing wave. Node coordinates.

The distance between two adjacent nodes (or between two O middle antinodes), calledstanding wave length,equal to half the length of the running she waves . Thus, when two traveling waves are added together, a standing wave is formed, the nodes and antinodes of which are always in the same places.

The characteristics of traveling and standing waves are given in Table 5.1.

Basic 1 , 5 . 6

Add. 18, 22 [25-44]

Control questions:

Basic 18 .

Control questions:

1. Can the pressure be the same at two points lying at different levels in an installed obliquely tapering tube through which an ideal liquid flows?

2. Why does the stream of liquid flowing from the hole become more and more compressed as it moves away from the hole?

3. How do the phases of acceleration and displacement oscillations relate to harmonic oscillations?

7.1. General properties of liquids and gases. Kinematic description of fluid motion. Vector fields. Flow and circulation of a vector field. Stationary flow of an ideal fluid. Current lines and tubes. Equations of motion and equilibrium of fluid. Continuity equation for incompressible fluid

Continuum mechanics is a branch of mechanics devoted to the study of the motion and equilibrium of gases, liquids, plasma and deformable solids. The main assumption of continuum mechanics is that matter can be considered as a continuous medium, neglecting its molecular (atomic) structure, and at the same time the distribution of all its characteristics (density, stress, particle velocities) in the medium can be considered continuous.

A liquid is a substance in a condensed state, intermediate between solid and gaseous. The region of existence of the liquid is limited from the side low temperatures phase transition to a solid state (crystallization), and from high temperatures to a gaseous state (evaporation). When studying the properties of a continuous medium, the medium itself appears to consist of particles whose sizes are much larger than the sizes of molecules. Thus, each particle includes a huge number of molecules.

To describe the motion of a fluid, you can specify the position of each fluid particle as a function of time. This method of description was developed by Lagrange. But you can follow not the particles of liquid, but individual points in space, and note the speed with which individual particles of liquid pass through each point. The second method is called Euler's method.

The state of fluid motion can be determined by specifying the velocity vector for each point in space as a function of time.

The set of vectors specified for all points in space forms a velocity vector field, which can be depicted as follows. Let us draw lines in the moving fluid so that the tangent to them at each point coincides in direction with the vector (Fig. 7.1). These lines are called streamlines. Let us agree to draw streamlines so that their density (the ratio of the number of lines to the size of the area perpendicular to them through which they pass) is proportional to the magnitude of the speed in a given place. Then, from the pattern of streamlines, it will be possible to judge not only the direction, but also the magnitude of the vector at different points in space: where the speed is greater, the streamlines will be denser.

The number of streamlines passing through the pad perpendicular to the streamlines is equal to , if the pad is oriented arbitrarily to the streamlines, the number of streamlines is equal to , where is the angle between the direction of the vector and the normal to the pad. The notation is often used. The number of streamlines through an area of ​​finite dimensions is determined by the integral: . An integral of this type is called vector flow through the area.

The magnitude and direction of the vector changes over time, therefore, the pattern of lines does not remain constant. If at each point in space the velocity vector remains constant in magnitude and direction, then the flow is called steady or stationary. In a stationary flow, any fluid particle passes a given point in space with the same speed value. The pattern of streamlines in this case does not change, and the streamlines coincide with the trajectories of the particles.

The flow of a vector through a certain surface and the circulation of the vector along a given contour make it possible to judge the nature of the vector field. However, these quantities give an average characteristic of the field within the volume covered by the surface through which the flow is determined, or in the vicinity of the contour along which the circulation is taken. By reducing the dimensions of a surface or contour (contracting them to a point), one can arrive at values ​​that will characterize the vector field at a given point.

Let us consider the velocity vector field of an incompressible continuous fluid. The velocity vector flux through a certain surface is equal to the volume of fluid flowing through this surface per unit time. Let us construct an imaginary closed surface S in the neighborhood of point P (Fig. 7.2). If in a volume V bounded by a surface, liquid does not appear or disappear, then the flow flowing out through the surface will be zero. A difference in flux from zero will indicate that there are sources or sinks of liquid inside the surface, i.e. points at which liquid enters the volume (sources) or is removed from the volume (sinks). The magnitude of the flow determines the total power of the sources and sinks. When sources predominate over sinks, the flow is positive; when sinks predominate, it is negative.

The quotient of the flow divided by the volume from which the flow flows out, , is the average specific power of the sources contained in volume V. The smaller the volume V that includes point P, the closer this average value is to the true specific power at this point. In the limit at , i.e. when contracting the volume to a point, we obtain the true specific power of the sources at point P, called the divergence (divergence) of the vector: . The resulting expression is valid for any vector. Integration is carried out over a closed surface S, limiting the volume V. Divergence is determined by the behavior of a vector function near point P. Divergence is a scalar function of coordinates that determine the position of point P in space.

Let us find an expression for divergence in the Cartesian coordinate system. Let us consider in the vicinity of the point P(x,y,z) a small volume in the form of a parallelepiped with edges parallel to the coordinate axes (Fig. 7.3). Due to the smallness of the volume (we will tend to zero), the values ​​within each of the six faces of the parallelepiped can be considered unchanged. The flow through the entire closed surface is formed from the flows flowing through each of the six faces separately.

Let's find the flow through a pair of faces perpendicular to the axis X in Fig. 7.3 (faces 1 and 2). The outer normal to face 2 coincides with the direction of the X axis. Therefore, the flux through face 2 is equal to . The normal has a direction opposite to the X axis. The projections of the vector onto the X axis and onto the normal have opposite signs, , and the flux through face 1 is equal to . The total flux in the X direction is . The difference represents the increment when moving along the X axis by . Due to its smallness, this increment can be represented as . Then we get . Similarly, through pairs of faces perpendicular to the Y and Z axes, the fluxes are equal to and . Total flow through a closed surface. Dividing this expression by , we find the divergence of the vector at point P:

Knowing the divergence of a vector at each point in space, one can calculate the flow of this vector through any surface of finite dimensions. To do this, we divide the volume limited by the surface S into an infinitely large number of infinitesimal elements (Fig. 7.4).

For any element, the vector flux through the surface of this element is equal to . Summing over all elements, we obtain the flow through the surface S, limiting the volume V: , integration is performed over the volume V, or

This is the Ostrogradsky–Gauss theorem. Here , is the unit normal vector to the surface dS at a given point.

Let's return to the flow of incompressible fluid. Let's build a contour. Let's imagine that we have somehow instantly frozen the liquid in its entire volume, with the exception of a very thin closed channel of constant cross-section, which includes a contour (Fig. 7.5). Depending on the nature of the flow, the liquid in the formed channel will be either stationary or moving (circulating) along the contour in one of the possible directions. As a measure of this movement, a value is chosen equal to the product of the fluid velocity in the channel and the length of the contour, . This quantity is called vector circulation along the contour (since the channel has a constant cross-section and the velocity module does not change). At the moment of solidification of the walls, for each liquid particle in the channel the velocity component perpendicular to the wall will be extinguished and only the component tangent to the contour will remain. Associated with this component is the impulse , the modulus of which for a liquid particle enclosed in a channel segment of length , is equal to , where is the density of the liquid and is the cross-section of the channel. The liquid is ideal - there is no friction, so the action of the walls can only change the direction, its magnitude will remain constant. The interaction between liquid particles will cause a redistribution of momentum between them that will equalize the velocities of all particles. In this case, the algebraic sum of the impulses is preserved, therefore , where is the circulation speed, is the tangential component of the fluid velocity in the volume at the time preceding the solidification of the walls. Dividing by , we get .

Circulation characterizes the field properties averaged over an area with dimensions on the order of the contour diameter. To obtain a characteristic of the field at point P, it is necessary to reduce the size of the contour, contracting it to point P. In this case, as a characteristic of the field, take the limit of the ratio of the vector circulation along a flat contour contracting to point P to the value of the plane of the contour S: . The value of this limit depends not only on the properties of the field at point P, but also on the orientation of the contour in space, which can be specified by the direction of the positive normal to the plane of the contour (the normal associated with the direction of traversal of the contour by the rule of the right screw is considered positive). By defining this limit for different directions, we will obtain different values, and for opposite normal directions these values ​​differ in sign. For a certain direction of the normal, the limit value will be maximum. Thus, the value of the limit behaves as a projection of a certain vector onto the direction of the normal to the plane of the contour along which the circulation is taken. The maximum value of the limit determines the magnitude of this vector, and the direction of the positive normal at which the maximum is reached gives the direction of the vector. This vector is called the rotor or vortex of the vector: .

To find the projections of the rotor on the axis cartesian system coordinates, it is necessary to determine the limit values ​​for such orientations of the site S for which the normal to the site coincides with one of axes X,Y,Z. If, for example, we direct along the X axis, we find . In this case, the contour is located in a plane parallel to YZ; let’s take the contour in the form of a rectangle with sides and . At the values ​​of and on each of the four sides of the contour can be considered unchanged. Section 1 of the contour (Fig. 7.6) is opposite to the Z axis, therefore in this section it coincides with, in section 2, in section 3, in section 4. For circulation along this contour we obtain the value: . The difference represents the increment when moving along Y by . Due to its smallness, this increment can be represented as . Similarly, the difference . Then circulation along the considered contour,

where is the contour area. Dividing the circulation by , we find the projection of the rotor onto the X axis: . Likewise, , . Then the rotor of the vector is determined by the expression: + ,

Knowing the rotor of a vector at each point of a certain surface S, we can calculate the circulation of this vector along the contour that bounds the surface S. To do this, we divide the surface into very small elements (Fig. 7.7). The circulation along the bounding contour is equal to , where is the positive normal to the element . Summing these expressions over the entire surface S and substituting the expression for circulation, we obtain . This is Stokes' theorem.

The part of the liquid bounded by streamlines is called a stream tube. The vector, being tangent to the stream line at each point, will be tangent to the surface of the stream tube, and the liquid particles do not intersect the walls of the stream tube.

Let us consider the section of the current tube S perpendicular to the direction of velocity (Fig. 7.8.). We will assume that the speed of liquid particles is the same at all points of this section. During the time, all particles whose distance at the initial moment does not exceed the value will pass through the section S. Consequently, in a time, a volume of liquid equal to . will pass through section S, and in a unit of time, a volume of liquid will pass through section S, equal to .. We will assume that the current tube is so thin that the speed of particles in each of its sections can be considered constant. If the fluid is incompressible (that is, its density is the same everywhere and does not change), then the amount of fluid between sections and (Fig. 7.9.) will remain unchanged. Then the volumes of liquid flowing per unit time through the sections and should be the same:

Thus, for an incompressible fluid, the value in any section of the same current tube should be the same:

This statement is called the jet continuity theorem.

The motion of an ideal fluid is described by the Navier-Stokes equation:

where t is time, x,y,z are the coordinates of the liquid particle, are the projections of the body force, p is the pressure, ρ is the density of the medium. This equation allows us to determine the projection of the velocity of a particle of the medium as a function of coordinates and time. To close the system, the continuity equation is added to the Navier-Stokes equation, which is a consequence of the jet continuity theorem:

To integrate these equations, it is necessary to set the initial (if the motion is not stationary) and boundary conditions.

7.2. Pressure in a flowing liquid. Bernoulli's equation and its corollary

When considering the movement of liquids, in some cases it can be assumed that the movement of some liquids relative to others is not associated with the occurrence of friction forces. A fluid in which internal friction (viscosity) is completely absent is called ideal.

Let us select a current tube of small cross-section in a stationary flowing ideal fluid (Fig. 7.10). Let us consider the volume of liquid limited by the walls of the stream tube and sections perpendicular to the stream lines and . During time, this volume will move along the stream tube, and the cross section will move to the position , having passed the path , the cross section will move to the position , having passed the path . Due to the continuity of the jet, the shaded volumes will have the same size:

The energy of each fluid particle is equal to the sum of its kinetic energy and potential energy in the gravity field. Due to the stationarity of the flow, a particle located after time at any point in the unshaded part of the volume under consideration (for example, point O in Fig. 7.10) has the same speed (and the same kinetic energy) as the particle that was at the same point at the initial moment had time. Therefore, the increment in the energy of the entire volume under consideration is equal to the difference in the energies of the shaded volumes and .

In an ideal fluid there are no frictional forces, therefore the increment of energy (7.1) is equal to the work done on the selected volume by pressure forces. The pressure forces on the lateral surface are perpendicular at each point to the direction of movement of the particles and do not do any work. The work of forces applied to the sections and is equal to

Equating (7.1) and (7.2), we obtain

Since the sections were taken arbitrarily, it can be argued that the expression remains constant in any section of the current tube, i.e. in a stationary flowing ideal fluid along any streamline the following condition is satisfied:

This is Bernoulli's equation. For a horizontal streamline, equation (7.3) takes the form:

7.3. LIQUID OUTLET FROM THE HOLE

Let us apply Bernoulli's equation to the case of fluid flowing out of a small hole in a wide open vessel. Let us select a current tube in the liquid, the upper section of which lies on the surface of the liquid, and the lower section coincides with the hole (Fig. 7.11). In each of these sections, the speed and height above a certain initial level can be considered the same, the pressure in both sections is equal to atmospheric and also the same, the speed of movement of the open surface will be considered equal to zero. Then equation (7.3) takes the form:

Pulse

7.4 Viscous liquid. Internal friction forces

An ideal liquid, i.e. a fluid without friction is an abstraction. All real liquids and gases exhibit viscosity or internal friction to a greater or lesser extent.

Viscosity is manifested in the fact that the movement that has arisen in a liquid or gas gradually ceases after the cessation of the forces that caused it.

Let's consider two plates parallel to each other placed in a liquid (Fig. 7.12). The linear dimensions of the plates are much greater than the distance between them d. The lower plate is held in place, the upper one is driven relative to the lower one with some

speed It has been experimentally proven that in order to move the upper plate at a constant speed, it is necessary to act on it with a very specific constant force. The plate does not receive acceleration, therefore, the action of this force is balanced by a force equal to it in magnitude, which is the friction force acting on the plate as it moves in the liquid. Let's denote it, and the part of the fluid lying under the plane acts on the part of the fluid lying above the plane with a force. In this case, and are determined by formula (7.4). Thus, this formula expresses the force between contacting layers of liquid.

It has been experimentally proven that the speed of liquid particles changes in the z direction perpendicular to the plates (Fig. 7.6) according to a linear law

Liquid particles in direct contact with the plates seem to stick to them and have the same speed as the plates themselves. From formula (7.5) we obtain

The modulus sign in this formula is placed for the following reason. When the direction of motion changes, the derivative of the speed will change sign, while the ratio is always positive. Taking into account the above, expression (7.4) takes the form

The SI unit of viscosity is the viscosity at which the velocity gradient with modulus , leads to the appearance of an internal friction force of 1 N on 1 m of the contact surface of the layers. This unit is called the Pascal second (Pa s).

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Liquids and gases are largely similar in their properties. They are fluid and take the shape of the vessel in which they are located. They obey the laws of Pascal and Archimedes.

When considering the movement of liquids, we can neglect the frictional forces between the layers and consider them absolutely incompressible. Such an absolutely inviscid and absolutely incompressible fluid is called ideal..

The movement of a fluid can be described by showing the trajectories of movement of its particles in such a way that the tangent at any point of the trajectory coincides with the velocity vector. These lines are called current lines. It is customary to draw streamlines so that their density is greater where the fluid flow rate is greater (Fig. 2.11).

The magnitude and direction of the velocity vector V in a liquid can change over time, and the pattern of streamlines can continuously change. If the velocity vectors at each point in space do not change, then the fluid flow is called stationary.

The part of the liquid bounded by streamlines is called current tube. Liquid particles moving inside the current tube do not cross its walls.

Consider one current tube and denote the areas by S 1 and S 2 cross section in it (Fig. 2.12). Then, per unit time, equal volumes of liquid flow through S 1 and S 2:

S 1 V 1 =S 2 V 2 (2.47)

this applies to any cross-section of the current tube. Consequently, for an ideal liquid the value SV=const in any section of the current tube. This ratio is called continuity of the jet. It follows from it:

those. the speed V of a stationary liquid flow is inversely proportional to the cross-sectional area S of the current tube, and this may be due to the pressure gradient in the liquid along the current tube. The jet continuity theorem (2.47) is also applicable to real liquids (gases) when they flow in pipes of different sections, if the friction forces are small.

Bernoulli's equation. Let us select a current tube of variable cross-section in an ideal liquid (Fig. 2.12). Due to the continuity of the jet, equal volumes of liquid ΔV flow through S 1 and S 2 at the same time.

The energy of each fluid particle is the sum of its kinetic energy and potential energy. Then, when moving from one section of the tube to another, the increment in the energy of the liquid will be:

In an ideal fluid the increment ΔW should be equal to the work of pressure forces on the change in volume ΔV, i.e. A=(P 1 -P 2) ΔV.

Equating ΔW=A and reducing by ΔV and taking into account that ( ρ -density of the liquid), we obtain:

because The cross-section of the stream tube is taken arbitrarily, then for an ideal liquid along any stream line the following holds:

. (2.48)

Where R-static pressure in a certain section S of the current tube;

Dynamic pressure for this section; V is the speed of fluid flow through this section;

ρgh-hydrostatic pressure.

Equation (2.48) is called Bernoulli's equation.

Viscous liquid. In a real liquid, when its layers move relative to each other, internal friction forces(viscosity). Let two layers of liquid be separated from each other by a distance Δх and move with speeds V 1 and V 2 (Fig. 2.13).

Then internal friction force between layers(Newton's law):

, (2.49)

Where η - coefficient of dynamic viscosity of the liquid:

Arithmetic mean speed of molecules;

Average free path of molecules;

Layer velocity gradient; ΔS– area of ​​contacting layers.

Layered fluid flow is called laminar. As the speed increases, the layered nature of the flow is disrupted and mixing of the liquid occurs. This flow is called turbulent.

In laminar flow, fluid flow Q in a pipe of radius R is proportional to the pressure drop per unit length of the pipe ΔР/ℓ:

Poiseuille's formula. (2.51)

In real liquids and gases, moving bodies experience resistance forces. For example, the drag force acting on a ball moving uniformly in a viscous medium is proportional to its speed V:

Stokes formula, (2.52)

Where r- radius of the ball.

As the speed of movement increases, the flow around the body is disrupted, vortices are formed behind the body, which additionally wastes energy. This leads to an increase in drag.