Plan

1. The concept of a continuous medium. General properties liquids and gases. An ideal and viscous liquid. Bernoulli's equation. Laminar and turbulent flow of liquids. Stokes formula. Poiseuille's formula.

2. Elastic stresses. The energy of an elastically deformed body.

Abstracts

1. The volume of the gas is determined by the volume of the vessel that the gas occupies. In liquids, unlike gases, the average distance between molecules remains practically constant, so the liquid has an almost unchanged volume. In mechanics, with a high degree of accuracy, liquids and gases are considered as continuous, continuously distributed in the part of space they occupy. The density of the liquid depends little on pressure. The density of gases depends significantly on pressure. It is known from experience that the compressibility of a liquid and a gas in many problems can be neglected and the unified concept of an incompressible liquid can be used, the density of which is the same everywhere and does not change with time. Ideal liquid - physical abstraction, that is, an imaginary fluid in which there are no internal friction forces. An ideal fluid is an imaginary fluid in which there are no internal friction forces. It is contradicted by a viscous liquid. The physical quantity determined by the normal force acting from the side of the liquid per unit area is called pressure R liquids. The unit of pressure is pascal (Pa): 1 Pa is equal to the pressure created by a force of 1 N, evenly distributed over a surface normal to it with an area of ​​1 m 2 (1 Pa = 1 N / m 2). The pressure at equilibrium of liquids (gases) obeys Pascal's law: the pressure in any place of the liquid at rest is the same in all directions, and the pressure is equally transmitted throughout the volume occupied by the liquid at rest.

The pressure changes linearly with height. Pressure P = rgh called hydrostatic. The force of pressure on the lower layers of the liquid is greater than on the upper ones, therefore, a buoyant force acts on a body immersed in a liquid, determined by Archimedes' law: a buoyant force directed upward acts on a body immersed in a liquid (gas), equal to weight the liquid (gas) displaced by the body, where r is the density of the liquid, V- the volume of the body immersed in the liquid.

The movement of fluids is called a flow, and the collection of particles of a moving fluid is called a flow. Graphically, the movement of fluids is depicted using streamlines, which are drawn so that the tangents to them coincide in direction with the velocity vector of the fluid at the corresponding points in space (Fig. 45). From the pattern of streamlines, one can judge the direction and modulus of velocity at different points in space, i.e., the state of fluid motion can be determined. The part of the fluid bounded by the streamlines is called a stream tube. A fluid flow is called steady (or stationary) if the shape and location of the streamlines, as well as the values ​​of the velocities at each of its points, do not change over time.

Consider a current tube. Let's choose two of its sections S 1 and S 2 , perpendicular to the direction of speed (Fig. 46). If the fluid is incompressible (r = const), then through the section S 2 will pass in 1 s the same volume of liquid as through the section S 1, that is, the product of the flow velocity of an incompressible fluid and the cross section of the current tube is a constant value for a given current tube. The ratio is called the continuity equation for an incompressible fluid. - Bernoulli's equation - an expression of the law of conservation of energy as applied to the steady flow of an ideal fluid ( here p - static pressure (pressure of the fluid on the surface of the body flown around it), value - dynamic pressure, - hydrostatic pressure). For a horizontal stream tube, the Bernoulli equation is written in the form, where left side called total pressure. - Torricelli formula

Viscosity is the property of real fluids to resist the movement of one part of the fluid relative to another. When moving some layers of a real liquid relative to others, internal friction forces arise, directed tangentially to the surface of the layers. The force of internal friction F is the greater, the larger the considered surface area of ​​the layer S, and depends on how quickly the fluid flow rate changes when passing from layer to layer. The Dv / Dx value shows how quickly the velocity changes when passing from layer to layer in the direction NS, perpendicular to the direction of movement of the layers, and is called the velocity gradient. Thus, the modulus of the internal friction force is equal to, where the proportionality coefficient h , depending on the nature of the fluid is called dynamic viscosity (or simply viscosity). The unit of viscosity is pascal second (Pa s) (1 Pa s = 1 N s / m 2). The higher the viscosity, the more the liquid differs from the ideal, the greater the forces of internal friction arise in it. Viscosity depends on temperature, and the nature of this dependence for liquids and gases is different (for liquids it decreases with increasing temperature, for gases, on the contrary, it increases), which indicates a difference in the mechanisms of internal friction. The viscosity of oils depends especially strongly on temperature. Methods for determining viscosity:

1) Stokes formula; 2) Poiseuille's formula

2. Deformation is called elastic if, after the cessation of the action of external forces, the body assumes its original size and shape. Deformations that persist in the body after the cessation of external forces are called plastic. The force per unit of cross-sectional area is called stress and is measured in pascals. A quantitative measure characterizing the degree of deformation experienced by a body is its relative deformation. The relative change in the length of the bar (longitudinal deformation), the relative transverse tension (compression), where d - rod diameter. Deformations e and e " always have different signs, where m is a positive factor depending on material properties, called Poisson's ratio.

Robert Hooke experimentally found that for small deformations, the elongation e and the stress s are directly proportional to each other:, where the proportionality coefficient E called Young's modulus.

Young's modulus is determined by the stress causing elongation, equal to one... Then Hooke's law can be written like this, where k- coefficient of elasticity:the elongation of the rod under elastic deformation is proportional to the action on pivot strength. Potential energy of an elastically stretched (compressed) bar Deformation solids obey Hooke's law only for elastic deformations. The relationship between strain and stress is presented in the form of a stress diagram (Fig. 35). The figure shows that the linear dependence s (e), established by Hooke, is fulfilled only within very narrow limits up to the so-called proportionality limit (s p). With a further increase in stress, the deformation is still elastic (although the dependence s (e) is no longer linear) and no residual deformations arise up to the elastic limit (s y). Residual deformations occur in the body beyond the elastic limit, and the graph describing the return of the body to its original state after the cessation of the action of the force is not displayed as a curve IN, and parallel to it - CF. The stress at which a noticeable permanent deformation appears (~ = 0.2%) is called the yield point (s t) - point WITH on the curve. In the area of CD deformation increases without increasing stress, that is, the body "flows", as it were. This area is called the yield area (or plastic deformation area). Materials for which the yield area is significant are called viscous, for which it is practically absent - brittle. With further stretching (per point D) the body is destroyed. The maximum stress that occurs in a body before failure is called the ultimate strength (s p).

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. Lines and tubes of current. Equations of motion and equilibrium of a liquid. 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. Basic assumption of mechanics continuous media consists in the fact that a substance can be considered as a continuous continuous medium, neglecting its molecular (atomic) structure, and at the same time, the distribution in the medium of all its characteristics (density, stresses, particle velocities) can be considered continuous.

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

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

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

Collection of vectors , given for all points in space, forms the field of the velocity vector, which can be depicted as follows. Let us draw lines in a moving fluid so that the tangent to them at each point coincides in direction with the vector (Figure 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) was proportional to the magnitude of the speed at a given location. 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 site
perpendicular to the streamlines is
, if the site is oriented arbitrarily to the streamlines, the number of streamlines is, where
- the angle between the direction of the vector and normal to the site ... The notation is often used
... The number of streamlines through the site finite size is determined by the integral:
... An integral of this kind is called the flow of the vector across the site .

IN The magnitude and direction of the vector changes over time, therefore, the line pattern 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 liquid particle passes this point space with the same speed value. The streamline pattern 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 values ​​give an average characteristic of the field within the volume enclosed 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 size of the surface or contour (by pulling them to a point), you can come up with values ​​that will characterize the vector field at a given point.

Consider the field of the velocity vector of an incompressible continuous fluid. The flow of the velocity vector through a certain surface is equal to the volume of liquid flowing through this surface per unit time. We construct in the vicinity of the point R imaginary closed surface S(Figure 7.2) . If in volume V bounded by the surface, the liquid does not arise and does not disappear, then the flow outward through the surface will be equal to zero. A flow difference 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 flow rate determines the total power of sources and sinks. With a predominance of sources over effluents, the flow is positive, with a predominance of effluents - negative.

The quotient of dividing the flow by the amount of volume from which the flow flows,
, is the average specific power of the sources enclosed in the volume V. The smaller the volume V, including dot R, the closer this average is to the true power density at that point. In the limit at
, i.e. when contracting the volume to a point, we get the true specific power of the sources at the point R, 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 scope V... Divergence is determined by the behavior of the vector function near point R. Divergence is a scalar function of coordinates defining n point position R in space.

Let's find an expression for the divergence in the Cartesian coordinate system. 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 (Figure 7.3). In view of 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 across the entire enclosed surface is formed from flows flowing through each of the six faces separately.

Find the flow through a pair of faces perpendicular to the stop NS in Figure 7.3 faces 1 and 2) . Outer normal to face 2 coincides with the direction of the axis NS... therefore
and the flow through face 2 is
.Normal has a direction opposite to the axis NS. Vector projections per axis NS and to normal have opposite signs,
, and the flow through face 1 is
... The total flow in the direction NS is equal to
... Difference
is an increment when displaced along the axis NS on the
... Due to the smallness

... Then we get
... Similarly, through pairs of faces perpendicular to the axes Y and Z, the flows are equal
and
... Full flow through an enclosed surface. Dividing this expression into
, find the divergence of the vector at the point R:

.

Knowing the vector divergence at each point in space, you can calculate the flux of this vector through any surface of finite dimensions. To do this, we divide the volume bounded by the surface S, on endlessly big number infinitesimal elements
(Figure 7.4).

For any element
stream vector through the surface of this element is
... Summing over all elements
, we get the flow through the surface S limiting the volume V:
, integration is performed on the volume V, or

.

NS then the Ostrogradskii - Gauss theorem. Here
,is the unit normal vector to the surface dS at this point.

Let's return to the flow of an incompressible fluid. Let's build a contour ... Imagine that we somehow instantly frozen the liquid in the entire volume, with the exception of a very thin closed channel of constant cross section, which includes a contour (Figure 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 velocity of the fluid in the channel and the length of the circuit,
... This quantity is called the circulation of the vector 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. An impulse is associated with this component
, the modulus of which for a liquid particle enclosed in a channel section with length
, equals
, where - the density of the liquid, - channel section. The fluid is ideal - there is no friction, so the action of the walls can only change the direction
, its value will remain constant. The interaction between the particles of the liquid will cause such a redistribution of momentum between them, which will equalize the velocities of all particles. In this case, the algebraic sum of impulses is conserved, therefore
, where - circulation speed, - the tangential component of the fluid velocity in the volume
at the moment of time preceding the solidification of the walls. Dividing into
, get
.

C The circulation characterizes the properties of the field, averaged over a region with dimensions of the order of the contour diameter ... To obtain the characteristic of the field at the point R, you need to reduce the size of the contour, pulling it to a point R... In this case, the limit of the circulation ratio of the vector is taken as a characteristic of the field on a flat contour contracting to the point R, to the size of the contour plane S:
... The value of this limit depends not only on the properties of the field at the point R, 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 traversing the contour by the rule of the right screw is considered positive). Determining this limit for different directions , we will get its different values, and for the opposite normal directions these values ​​differ in sign. For some direction of the normal, the value of the limit will be maximum. Thus, the value of the limit behaves like 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 modulus 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 projection of the rotor on the axis of the Cartesian coordinate system, you need to determine the limit values ​​for such site orientations S for which the normal to the site coincides with one of the axes X, Y, Z. If, for example, you send along the axis NS, find
... Circuit is located in this case in a plane parallel to YZ, take a contour in the form of a rectangle with sides
and
... At
meaning and on each of the four sides of the contour can be considered unchanged. Section 1 of the contour (Figure 7.6) is opposite to the axis Z, so on this site coincides with
, in section 2
, in section 3
, on site 4
... For circulation along this circuit, we get the value: . Difference
is an increment when displaced along Y on the
... Due to the smallness
this increment can be represented as
.Likewise, difference
. Then the circulation along the considered contour
,

where
- contour area. Dividing the circulation into
, we find the projection of the rotor on axis NS:
. Similarly,
,
... Then the rotor of the vector defined by the expression:

+
,

or
.

Z the rotor of the vector at each point of some surface S, we can calculate the circulation of this vector along the contour bounding the surface S... To do this, we divide the surface into very small elements
(Figure 7.7). Circulation along the boundary
is equal to
, where - positive normal to the element
. Summing up these expressions over the entire surface S and substituting the expression for the circulation, we get
... This is Stokes' theorem.

The part of the fluid bounded by the streamlines is called the stream tube. Vector , being at each point tangent to the streamline, will be tangent to the surface of the stream tube, and the fluid particles do not cross the walls of the stream tube.

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

.

Thus, for an incompressible fluid, the quantity
in any section of the same flow tube must 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- time, x, y, z- coordinates of a liquid particle,

- body force projections, R- pressure, ρ - density of the medium. This equation allows you to determine the projection of the velocity of the 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 required to set the initial (if the motion is not stationary) and boundary conditions.

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

When considering the motion of fluids, one can neglect the forces of friction between the layers and consider them absolutely incompressible. Such an absolutely inviscid and absolutely incompressible fluid is called ideal..

The motion of a fluid can be described by showing the trajectories of motion 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 streamlines... It is customary for streamlines to be drawn so that their density is greater where more speed fluid flow (Figure 2.11).

The magnitude and direction of the velocity vector V in a liquid can change with time, then the pattern of streamlines can change continuously. 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... Particles of liquid, moving inside the flow tube, do not cross its walls.

Consider one stream tube and denote by S 1 and S 2 the cross-sectional areas in it (Figure 2.12). Then, per unit time, the same volumes of liquid flow through S 1 and S 2:

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

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

those. the velocity V of the steady flow of the liquid is inversely proportional to the cross-sectional area S of the flow tube, and this may be due to the pressure gradient in the liquid along the flow 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 equation... Let's select a current tube of variable cross section in an ideal fluid (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 liquid particle is the sum of its kinetic energy and potential energy. Then, when passing from one section of the tube, the currents to another, the increase 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 canceling by ΔV and taking into account that ( ρ is the density of the liquid), we get:

since the cross section of the stream tube is taken arbitrarily, then for an ideal fluid along any stream line the following is fulfilled:

. (2.48)

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

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

ρgh-hydrostatic pressure.

Equation (2.48) is called Bernoulli 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 spaced apart from each other at a distance Δх and move with velocities V 1 and V 2 (Figure 2.13).

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

, (2.49)

where η -coefficient dynamic viscosity liquids:

Average arithmetic speed of molecules;

Average free path of molecules;

Layer speed gradient; ΔS- the area of ​​contacting layers.

Layered fluid flow is called laminar... As the velocity increases, the layered nature of the flow is violated, and the liquid is mixed. 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 the action of a resistance force. For example, the resistance force acting on a ball moving uniformly in a viscous medium is proportional to its velocity V:

Stokes formula, (2.52)

where r is the radius of the ball.

With an increase in the speed of movement, the flow around the body is disturbed, vortices are formed behind the body, for which additional energy is spent. This leads to an increase in drag.

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

For planets with an atmosphere (Fig. 3.19), the landing problem is reduced mainly to solving three problems: overcoming high level overloads; protection against aerodynamic heating; control of the time of reaching the planet and the coordinates of the landing point.

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

N- turning on the brake motor; BUT- spacecraft descent from orbit; M- separation of the spacecraft from the orbiting spacecraft; IN- the entry of the SA into the dense layers of the atmosphere; WITH - the beginning of the 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 atmosphere (Fig. 3.20, but, b) the problem of protection against aerodynamic heating is removed.

Spacecraft in orbit artificial satellite planets or those approaching a planet with an atmosphere to land on it has a large amount of kinetic energy associated with the spacecraft speed and mass, and potential energy due to the spacecraft position relative to the planet's surface.

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

but- descent to the planet with a preliminary exit to the waiting orbit;

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

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

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

When entering the dense layers of the atmosphere, a shock wave arises in front of the bow of the spacecraft, heating the gas to a high temperature. As it sinks into the atmosphere, the SA is decelerated, its speed decreases, and the hot gas heats up the SA more and more. The kinetic energy of the apparatus is converted into heat. In this case, most of the 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 blunt nose, which is why blunt forms are used for the SA, rather than the sharp ones characteristic of flight at low speeds.

With an increase in speeds and temperatures, most of the heat is transferred to the vehicle not due to friction against 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 CA surface:

- heat absorption by the heat-shielding layer;

- radiation cooling of the surface;

- the use of carry-over coatings.

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

The nature of the descent trajectory in the atmosphere is mainly determined by its aerodynamic characteristics. In the absence of lift from the SA, the trajectory of its motion in the atmosphere is called ballistic (the trajectory of the descent of the SA spaceships series "Vostok" and "Voskhod"), and in the presence of lift - either gliding (CA SSC Soyuz and Apollo, as well as Space Shuttle), or ricocheting (CA SSC Soyuz and Apollo). Movement in a planetary orbit does not impose high requirements on guidance accuracy when entering the atmosphere, since it is relatively easy to correct the trajectory by turning on the propulsion system for braking or accelerating. When entering the atmosphere at a speed exceeding the first cosmic velocity, errors in calculations are most dangerous, since a too steep descent can lead to the destruction of the spacecraft, and too shallow - to a distance from the planet.

At ballistic descent the vector of the resultant of the aerodynamic forces is directed directly opposite to the vector of the velocity 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 entry of the apparatus into the dense layers of the atmosphere on high speed, which leads to strong aerodynamic heating of the vehicle and to overloads, sometimes exceeding 10g - close to the maximum permissible values ​​for a person.

At aerodynamic descent The outer casing of the vehicle has, as a rule, a conical shape, and the axis of the cone makes a certain angle (angle of attack) with the velocity vector of the vehicle, due to which the resultant of aerodynamic forces has a component perpendicular to the velocity vector of the vehicle - the lifting force. Due to the lifting force, the vehicle descends more slowly, the trajectory of its descent becomes flatter, while the braking section is stretched 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 makes the gliding the descent for people is safer and more comfortable.

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 makes the apparatus heavier, and in cases where it serves to launch only equipment that is capable of withstanding higher overloads than a person, ballistic braking is usually used.

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

After the vehicle speed decreases to subsonic in the section of aerodynamic braking, then the descent of the SA can be carried out with the help of parachutes. Parachute in dense atmosphere extinguishes the speed of the device to almost zero and ensures its soft landing on the surface of the planet.

In the rarefied atmosphere of Mars, parachutes are less effective, therefore, at the final stage of descent, the parachute is unhooked 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 brake motors 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. Such a 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 parachutes made of cloth would not withstand such conditions.

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

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

ELEMENTS OF CONTINUOUS MEDIA MECHANICS

A medium is considered to be continuous, for which a uniform distribution of matter is characteristic - 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. Elements of continuum mechanics

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

Since the liquid particles do not cross the surface of such a tube, it can be considered as a real tube with walls that are immovable for the liquid. Let us select in the stream tube arbitrary sections and perpendicular to the direction of particle velocity in sections and, respectively (Figure 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 stream tube, the equality

. (4.2)

Figure 4.1

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

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

When the 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 fluid enclosed between the sections and due to the equation of continuity will be

is equal to the volume of liquid contained in the interval

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 selected mass of liquid located at the initial moment between the sections and can be represented in the form

. (4.3)

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

, (4.4)

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

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

. (4.5)

Liquid outflow from the hole.Let the hole located near the bottom of the vessel filled with liquid be open (Fig. 4.3). Let's select a stream tube with cross-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 - Atmosphere pressure... Therefore, from (4.6) we have

(4.7)

If, then a member can be

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

Consequently, the rate of fluid flow 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 action of force it will set in motion, which is called reactive motion.

Laminar and turbulent flows. Viscosity.The flow of a liquid, in which each of its layers slides relative to other similar layers, and there is no mixing, is calledlaminar or layered... If vortices are formed inside the liquid and the layers are intensively mixed, then such a flow is called turbulent.

The steady (stationary) flow of an ideal fluid is laminar at any speed. In real fluids, internal friction forces arise between the layers, i.e. real fluids are viscous. Therefore, each of the layers slows down the movement of the adjacent 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 coefficient of proportionality, called the coefficient of viscosity. Its unit is (Pascal-second). The viscosity depends on the type of liquid and on the temperature. As the temperature rises, the viscosity decreases.

If the force of internal friction is small and the flow rate is small, then the motion is practically laminar. At high forces of internal friction, the layered nature of the flow is violated, intensive mixing begins, i.e. there is a transition to turbulence. The conditions for this transition in the flow of liquid through pipes is determined by the value cr called Reynolds number

, (4.11)

where is the density of the liquid, is the flow velocity averaged over the pipe section, and is the pipe diameter. Experiments show that when the flow is laminar, when it becomes turbulent. For pipes with a circular cross-section of radius, the Reynolds number. The effect of viscosity leads to the fact that at the rate of flow through a pipe of circular cross-section, different layers are different. Its average value is determinedby the Poiseuille formula

, (4.12)

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

The effect of viscosity is also detected when the flow interacts with a stationary body. Usually, in accordance with the mechanical principle of relativity, the inverse problem is considered, For example, Stokes it was found that at, the friction force acts on a ball moving in a liquid

, (4.13)

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

Oscillations and waves

Oscillatory movement, or simply oscillation, is a movement characterized by one or another degree of repetition in time of values physical quantities determining this movement. We encounter hesitation in the study of the most varied physical phenomena: sound, light, alternating currents, radio waves, pendulum swing, etc. Despite the wide variety of oscillatory processes, they all occur according to some common laws for them. The simplest of them is harmonic oscillatory motion. Oscillatory motion is called harmonic if the change in physical quantity NS (displacement) occurs according to the cosine (or sine) law

, (4.14)

where the value A - equal to maximum displacement NS system from the equilibrium position, called the amplitude of the oscillation, (, determines the magnitude of the displacement x at a given moment in time and is called the phase of the oscillation. , equal to the number full fluctuations occurring during s.

A period is the time of one full swing. It is related to the cyclic frequency by the following relationship

. (4.15)

Obviously, line frequency(the number of oscillations per unit of time) is associated with the period T in the following way

(4.16)

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

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

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

. (4.17)

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

Differentiating (4.14) again in time, we find an expression for the acceleration

. (4.18)

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

Plane Traveling Wave Equation

Wave equationis called an expression describing the head and Simplicity of the displacement of an oscillating particle from coordinates and time:

. (4.20)

Let the points located in the plane oscillate according to the law. Oscillations of the particles of the medium at a point (Figure 4.4) located at a distance I from the source of oscillations will occur according to the same but at stake, but will lag behind the fluctuations of the source and ka on (where is the speed of wave propagation). The equation of oscillation of these particles has the form: (4.20)

Figure 4.4

Since the point was chosen arbitrarily, then equation (5.7) allows you to determine the displacement of any point of the medium involved in the oscillatory process at any time, therefore it is calledthe equation of the plane traveling in l we. In the general case, it has the form: (4.21) where is the amplitude of the wave; - plane wave phase; – cyclical wave frequency; – initial phase of oscillation a niy.

Substituting into equation (4.21) the expressions for the velocity () and cyclic frequency (), n we will receive:

(4.22)

If we introduce a wave number, then the equation of a plane wave can be written in the form:

. (4.23)

The velocity in these equations is ck O the rate of movement of the phase of the wave, and it is calledphase velocity... Indeed, let the phase be constant in the wave process... To find the speed of its movement, we divide the expression for the phase by and differentiate by time e ni. We get:

Where.

Standing wave. If several waves propagate simultaneously in the medium, thensuperposition principle (superposition): to a waiting wave behaves as if other waves are absent, and the result Yu the displacement of the particles of the medium at any moment of time is geometric sum displacements that receive frequent and participating in each of the components of the wave processes with owls.

Of great practical interest is the superposition of two plane waves

And, (4.24)

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

Table 4.1

In a traveling wave	In a standing wave
Amplitude of vibration
All points of the environment fluctuate with the same yi amp and tud ami	All points of the environment fluctuate with different a m with slabs
Oscillation phase
The oscillation phase depends on the coordinate. and measured point	All points between two nodes oscillate in the same phase e ... When passing through the node, the phase count e bania 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 within the limits there are mutual transformations of energy.

At the points of the medium, where the amplitude and there the waves become zero (). These points are called knots () standing wave. The coordinates of the nodes.

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

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

Main one , 5 . 6

Add. 18, 22 [25-44]

Test questions:

Main 18 .

Test questions:

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

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

3.How are the phases of acceleration and displacement oscillations related to harmonic oscillations?