Elements of mechanics of solid media. Elements of continuous media laminar and turbulent

The conclusion of the space flight is considered to land on the planet. To date, only three countries have learned to return spacecraft to Earth: Russia, USA and China.

For planets with an atmosphere (Fig. 3.19) The planting problem is mainly reduced to solving three tasks: overcoming a high level of overload; protection against aerodynamic heating; Manage the time to achieve the planet and the coordinates of the landing point.

Fig. 3.19. The descent scheme with orbits and landing on the planet with the atmosphere:

N.- turning on the brake engine; BUT- gathering with orbits; M.- separation of Ca from orbital ka; IN- Entrance system in dense layers of the atmosphere; FROM -getting started by a parachute planting system; D.- landing on the surface of the planet;

1 - ballistic descent; 2 - Planning Descent

When landing on the planet without an atmosphere (Fig. 3.20, but, b.) The problem of protection against aerodynamic heating is removed.

Ka, located in the orbit of the planet's artificial satellite or an approaching planet with an atmosphere to put on it, has a large margin of kinetic energy associated with the speed of ka and its mass, and potential energy caused by the position of the spacecraft relative to the surface of the planet.

Fig. 3.20. Descent and landing on the planet without the atmosphere:

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

b.- soft landing with brake engine and landing device;

I - the hyperbolic trajectory of a flow to the planet; II - orbital trajectory;

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

At the entrance to the dense layers of the atmosphere in front of the nasal part, a shock wave occurs, heating gas to a high temperature. As it is immersed in the atmosphere of sa, the speed is reduced, and the hot gas is increasingly heating sa. The kinetic energy of the apparatus turns into heat. At the same time, most of the energy is discharged into the surrounding space in two ways: most of the heat is discharged into the surrounding atmosphere due to the action of strong shock waves and due to heat emission with the heated surface of the C.

The strongest shock waves occur during the blunt form of the nasal part, which is why the battered forms are used for Ca, and not pointed, characteristic of flight at low speeds.

With increasing speeds and temperatures, most of the heat is transmitted to the apparatus not by friction on compressed atmospheric layers, but by radiation and convection from the shock wave.

The following methods are applied to heat heat from SA surface:

- heat absorption with heat shielding layer;

- radiation cooling of the surface;

- Applications of worn coatings.

Before the entrance to the dense layers of the atmosphere, the trajectory is subject to the laws of heavenly mechanics. In the atmosphere on the device, in addition to the gravitational forces, there are aerodynamic and centrifugal forces, changing the form of the trajectory of his movement. The attraction force is directed towards the center of the planet, the strength of the aerodynamic resistance in the direction opposite to the velocity vector, centrifugal and lifting force - perpendicular to the direction of motion sa. The power of aerodynamic resistance reduces the speed of the device, while the centrifugal and lifting force informs it acceleration in the direction perpendicular to its movement.

The character of the trajectory of the descent in the atmosphere is determined mainly by its aerodynamic characteristics. In the absence of lifting force, the trajectory of its movement in the atmosphere is called the ballistic (the path of the spacecraft of the Space ships of the East and Sunrise series), and in the presence of a lifting force - either planning (SA KK Union and Apollo, as well as Space Shuttle), or ricastant (CA KK Union and Apollo). Movement on a planet center orbit does not impose high requirements for the accuracy of guidance when entering the atmosphere, because by turning on the motor installation for braking or acceleration, relatively easy to adjust the trajectory. When entering the atmosphere at a speed exceeding the first cosmic, errors in the calculations are most dangerous, since too steep descent can lead to the destruction of Ca, but too gently - to removal from the planet.

For ballistic descent The vector of the automatic aerodynamic forces is directed directly oppositely vector vehicle velocity of the device. The descent on the ballistic trajectory does not require management. The disadvantage of this method is the large steepness of the trajectory, and, as a result, the entry of the apparatus into the dense layers of the atmosphere at high speed, which leads to a strong aerodynamic heating of the apparatus and to overloads, sometimes exceeding 10g - close to the maximum permissible values \u200b\u200bfor humans.

For aerodynamic descent The external body of the apparatus has, as a rule, a conical shape, and the axis of the cone is some angle (an angle of attack) with a velocity vector of the device, due to the equality of the aerodynamic forces, it has a component perpendicular to the velocity vector of the apparatus - lifting force. Due to the lifting force, the device is reduced slower, the trajectory of its descent becomes more common, while the braking section is stretched and in length and in time, and the maximum overload and the intensity of aerodynamic heating can be reduced several times compared with ballistic braking, which makes the planning The descent for people is safer and comfortable.

The angle of attack during the descent varies depending on the flight speed and the current air density. In the upper, sparse layers of the atmosphere, it can reach 40 °, gradually decreasing with a decrease in the device. This requires the availability of a planning flight control system that complicates and weights the device, and in cases where it serves to descend only equipment that is able to withstand higher overloads than a person is used, as a rule, ballistic braking.

The orbital step "Space Shuttle", when returning to the Earth, performing the function of the descendable apparatus, plans on the entire sections of the descent from the entrance to the atmosphere before the landing gear chassis is touched, after which the brake paracter is produced.

After on the aerodynamic braking section, the speed of the device decreases to the dialing further, the SA can be carried out with parachutes. Parachute in a dense atmosphere quench the speed of the device almost to zero and provides a soft planting it to the surface of the planet.

In a rarefied atmosphere of Mars, parachutes are less efficient, therefore, at the final section of the descent, the parachute is unfolded and the landing rocket engines are included.

Despicable manned vessels of the Space Ships of the TMA-01M Union of TMA-01M, intended for landing to land, also have solid fuel brake engines, which are included in a few seconds before the land touch to provide a safer and comfortable landing.

The descended apparatus of the Venus station-13 after the descent on the parachute to the height 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 the parachutes from the tissue would not withstand such conditions.

It should be noted that in some projects of cosmic vehicles of reusable (in particular, single-stage vertical takeoff and landing, for example, Delta Clipper) are assumed at the final stage of descent, after aerodynamic braking in the atmosphere, also produce a non-parasite motor landing on rocket engines. Constructively descended devices 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 produced.

When landing on the planet without the atmosphere, the problem of aerodynamic heating is removed, but for the installation of the velocity, it is carried out using a braking motor installation, which should operate in programmable thrust mode, and the mass of fuel can significantly exceed the mass of the CA itself.

Elements of solid media

The medium for which the uniform distribution of the substance is characterized by the uniform distribution - i.e. Wednesday with the same density. Such are liquids and gases.

Therefore, in this section, we consider the basic laws that are performed in these environments.

Plan

1. The concept of a solid medium. General properties of liquids and gases. Perfect and viscous liquid. Bernoulli equation. Laminar and turbulent fluids. Stokes formula. Formula Poiseil.

2. Elastic stresses. Energy of elastically deformed body.

Abstracts

1. The volume of gas is determined by the volume of the vessel that gas takes. In liquids, in contrast to the gases, the average distance between molecules remains almost constant, so the fluid has almost unchanged volume. In mechanics with a large degree of accuracy of fluid and gases are considered solid, continuously distributed in the part of the space. The density of the fluid depends on the pressure. The density of gases on pressure depends substantially. From experience, it is known that the compressibility of fluid and gas in many tasks can be neglected and use the uniform concept of incompressible fluid, the density of which is the same everywhere and does not change over time. Perfect liquid - physical abstractioni.e. imaginary fluid, in which there are no forces of internal friction. The perfect liquid is an imaginary liquid in which there are no internal friction forces. It contradicts a viscous liquid. The physical value determined by the normal force acting from the liquid per unit area is called pressure rliquids. Pressure unit - Pascal (PA): 1 Pa is equal to the pressure generated by force 1 H, evenly distributed over a normal surface to it with an area of \u200b\u200b1 m 2 (1 Pa \u003d 1 N / m 2). Pressure in equilibrium liquids (gases) is subject to Pascal's law: the pressure in any place of a resting liquid is equally in all directions, and the pressure is equally transmitted throughout the volume occupied by a resting liquid.

Pressure varies linearly with a height. Pressure P \u003d. rGHcalled hydrostatic. The power of the pressure on the lower layers of the liquid is greater than on the top, therefore the body, immersed in the liquid, acts the ejecting force determined by the Archimedes law: on the body immersed in liquid (gas), acts on the side of this liquid directed upwards, equal to weight displaced fluid (gas), where R is the density of the liquid, V.- The volume of the body immersed in fluid.

The movement of fluids is called the flow, and the combination of particles of the moving fluid - flow. Graphically movement of liquids is depicted using current lines that are conducted in such a way that the tangents are coincided in the direction of the fluid velocity vector at the respective points of space (Fig. 45). On the picture of the current line, you can judge the direction and module of the speed at different points of space, i.e., you can determine the state of the fluid movement. Part of the fluid bounded by current lines is called the current tube. The fluid flow is called the installed (or stationary), if the form and location of the current lines, as well as the speeds of the speeds at each point do not change over time.

Consider any current tube. Choose two sections S. 1 I. S. 2 , perpendicular to the direction of speed (Fig. 46). If the liquid is incompressible (r \u003d const), then through the section S. 2 will be held for 1 with the same fluid, as through the section S. 1, i.e., the product of the flow rate of an incompressible fluid on the transverse section of the current tube there is a permanent value for this current tube. The ratio is called the equation of continuity for incompressible fluid. - Bernoulli equation - the expression of the law of conservation of energy in relation to the established flow of perfect fluid ( here p -static pressure (fluid pressure on the surface of the body streamped by it), the value is dynamic pressure, - hydrostatic pressure). For the horizontal tube current, the Bernoulli equation is written in the form where left part called full pressure. - Formula Torricelli

Viscosity is the property of real liquids to resist the movement of one part of the fluid relative to the other. When moving alone layers of real liquid relative to others, there are internal friction forces, aimed at the surface of the layers arise. The internal friction force F is the greater the greater the greater the surface area of \u200b\u200bthe layer S, and depends on how quickly the flow rate of the fluid changes during the transition from the layer to the layer. The amount of DV / DX shows how the speed changes quickly when moving from a layer to a layer in the direction x,perpendicular to the direction of movement of the layers, and is called a speed gradient. Thus, the module of the internal friction force is where the proportionality coefficient H , the nature-dependent fluid is called a dynamic viscosity (or simply viscosity). Viscosity unit - Pascal second (Pa C) (1 Pa C \u003d 1 N C / m 2). The greater the viscosity, the stronger the liquid differs from the ideal, the greater the inner friction forces in it arise. The viscosity depends on the temperature, and the nature of this dependence for liquids and gases is poured (for liquids with increasing temperature decreases, in gases, on the contrary, increases), which indicates the difference in internal friction mechanisms. The viscosity of the oil depends on the temperature of the oil. Viscosity definition methods:

1) Formula of Stokes; 2) Formula Poazeil

2. The deformation is called elastic, if after stopping the action of external forces, the body takes the initial dimensions and shape. The deformations that are stored in the body after the cessation of external forces are called plastic. The force acting on the unit of cross-sectional area is called voltage and is measured in Pascal. A quantitative measure characterizing the degree of deformation tested by the body is its relative deformation. Relative change in the length of the rod (longitudinal deformation), relative transverse stretching (compression), where d -rod diameter. Deformation e and e " always have different signs where M is a positive coefficient depending on the properties of a material called the Poisson coefficient.

Robert Gum experimentally found that for small deformations, the relative elongation E and the voltage S is directly proportional to each other: where the proportionality coefficient E.called the Jung module.

The Jung module is determined by the voltage causing a relative elongation equal to one. Then law Guka. can be written so where k.- coefficient of elasticity:the elongation of the rod with elastic deformation is proportional to the acting onrod strength. The potential energy of the elastic stretched (compressed) rod of deformation of solids is obeyed by the law of the thickness only for elastic deformations. The relationship between deformation and voltage is represented as a voltage diagram (Fig. 35). It can be seen from the figure that the linear dependence S (E) mounted in a bitter is performed only in very narrow limits to the so-called proportionality limit (S P). With a further increase in voltage, the deformation is still elastic (although the dependence S (E) is no longer linear) and to the limit of elasticity (S y), residual deformations do not arise. For the limit of elasticity in the body there are residual deformations and a schedule describing the return of the body to the original state after the termination of the force is shown not the curve. WITH, A.parallel to her - CF.The voltage at which a noticeable residual deformation appears (~ \u003d 0.2%), is called the yield limit (s T) - point FROMon the curve. In area CDthe deformation increases without increasing the voltage, i.e. the body is "flowing". This area is called the turnover area (or the area of \u200b\u200bplastic deformations). Materials for which the turning area is significant, are called viscous, for which it is practically absent - fragile. With further stretching (per point D)body destruction occurs. The maximum voltage arising in the body before destruction is called the strength limit (s P).

7.1. General properties of liquids and gases. Kinematic description of fluid movement. Vector fields. Flow and circulation of vector field. Stationary flow of perfect fluid. Lines and current tubes. Equations of motion and equilibrium fluid. EXTENSION EXTENSION FOR EXCELATED LIQUID

The mechanics of solid media are a section of mechanics dedicated to the study of the movement and equilibrium of gases, liquids, plasma and deformable solids. The main assumption of solid media is that the substance can be considered as a continuous solid medium, neglecting it with a molecular (atomic) structure, and at the same time consider continuous distribution in the medium of all its characteristics (density, voltages, particle rates).

The liquid is a substance in a condensed state, intermediate between solid and gaseous. The field of fluid existence is limited from low temperatures by a phase transition to a solid state (crystallization), and from high temperatures - in gaseous (evaporation). When studying the properties of a continuous medium itself, the medium itself is consisting of particles, the dimensions of which are many more than the dimensions of molecules. Thus, each particle includes a huge amount of molecules.

To describe the fluid movement, you can set the position of each particle of fluid as a function of time. This method of description was developed by Lagrange. But it is possible to monitor not behind the particles of the liquid, but for certain points of space, and note the speed with which the individual particles of the liquid pass through each point. The second method is called the Euler method.

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

The combination of vectors specified for all points of space forms a speed vector field that can be depicted as follows. We carry out the line in the moving fluid so that the tangent to them at each point coincided in the direction with the vector (Fig. 7.1). These lines are called current lines. We treat the current lines so that their thickness (the ratio of the number of lines to the value perpendicular to them the site through which they pass) was proportional to the speed in this place. Then, on the picture of the current lines, it will be possible to judge not only about the direction, but also about the magnitude of the vector at different points of space: where the speed is greater, the current line will be thicker.

The number of current lines passing through the pad perpendicular to the current lines is equal if the site is oriented randomly to the current lines, the number of current lines is equal to the angle between the direction of the vector and the normal to the site. Often use the designation. The number of current lines through the end-dimensional area is determined by the integral :. The integral of this species is called the vector stream through the platform.

The magnitude and direction of the vector varies with the times, therefore, and the picture of the lines does not remain constant. If at each point of space, the speed vector remains constant in magnitude and direction, the current is called the installed or stationary. With inpatient flow, any particle of fluid undergoes this point of space with the same speed value. The pattern of current lines in this case does not change, and the current lines coincide with the trajectories of the particles.

Vector stream through some surface and circulation of vector on a given circuit make it possible to judge the nature of the vector field. However, these values \u200b\u200bgive the 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, according to which circulation is taken. Reducing the size of the surface or contour (tightening them to the point), you can come to the values \u200b\u200bthat will characterize the vector field at this point.

Consider the field of the speed vector of incompressible inseparable fluid. The flow of velocity vector through a certain surface is equal to the volume of the fluid flowing through this surface per unit of time. We construct an imaginary closed surface S (Fig.7.2) in the neighborhood of the point p (Fig. 7.2). If in the volume V, a limited surface, the liquid does not occur and does not disappear, then the stream flowing out through the surface will be zero. The difference between the stream from zero will indicate that there are sources or drainage of liquid inside the surface, i.e. reader, in which the liquid enters the volume (sources) or is removed from the volume (drainage). The flow of the flow determines the total power of sources and wastewater. With the predominance of sources above the drains, the flow is positive, with the predominance of effluents - negative.

A private from dividing the flow by the amount of volume from which the flow follows, there is an average specific power of sources enclosed in volume V. The smaller volume V, which includes the point p, the closer this is the average value to the true specific power at this point. In the limit, i.e. When tightening the volume to the point, we obtain the true specific power of the sources at the point P, called the divergence (discrepancy) of the vector :. The resulting expression is valid for any vector. Integration is conducted along a closed surface S, limiting the volume V. Divergence is determined by the behavior of the vector function near the point R. Divergence is a scalar function of the coordinates that determine the position of the point P in space.

We find an expression for divergence in the Cartesian coordinate system. Consider in the neighborhood of the point P (x, y, z) a small volume in the form of a parallelepiped with ribs parallel to the axes of coordinates (Fig. 7.3). In view of the smell of volume (we will strive for zero) the values \u200b\u200bwithin each of the six faces of the parallelepiped can be considered unchanged. The flow across the entire closed surface is formed from streams current through each of the six faces separately.

We find a stream after a couple of faces, perpendicular to the OST X in Fig. 7.3 facets 1 and 2). The outer normal to the face 2 coincides with the direction of the x axis. Therefore, the flow through the face 2 is equal to. Normal has a direction opposite to the axis of X. The design of the vector on the x axis and the normal signs have opposite signs, and the flow through the face 1 is equal. The total flow towards x is equal. The difference is an increment when shifting along the x axis. In view of the smallness, this increment can be represented as. Then we get. Similarly, through pairs of faces perpendicular to the axes y and z, the flows are equal and. Full flow through a closed surface. Sharing this expression on, we find the vector divergence at the point P:

Knowing the vector divergence at each point of space, you can calculate the flow of this vector through any surface of the final sizes. To do this, we break the volume bounded by the surface S, on the infinitely a large number of infinitely small elements (Fig. 7.4).

For any element, the stream of vector through the surface of this element is equal. Having arise through all the elements, we obtain the flow through the surface S, limiting the volume V:, the integration is performed by volume V, or

This is the Ostrogradsky Theorem - Gauss. Here, a single vector of normal to the DS surface at this point.

Let's return to the flow of incompressible fluid. Build contour. Imagine that we somehow froze instantly fluid throughout the volume except for a very thin closed channel of a constant cross section, which includes the contour (Fig. 7.5). Depending on the nature of the flow, the fluid in the resulting channel will be either a fixed or moving (circulating) along the contour in one of the possible directions. As a measure of this movement, the value is selected equal to the product of the fluid velocity in the channel and the length of the contour ,. This value is called the circulation of the vector along the contour (as the channel has a constant section and the speed module does not change). At the time of hardening the walls, each particle of fluid in the channel will quench the velocity component, perpendicular to the wall and will only remain the component, tangent to the contour. The impulse is connected with this component, the module of which for a particle of the liquid concluded in the length of the channel length is equal to where the density of the liquid is the cross-section of the canal. The perfect fluid - the friction is not, so the action of the walls can only change the direction, its value will remain constant. The interaction between fluid particles will cause such a redistribution of the pulse between them, which lines the speed of all particles. In this case, the algebraic sum of pulses persists, therefore, where - the rate of circulation is the tangent component of the fluid rate in the amount at the time of the time preceding the solidification of the walls. Sharing on, we get.

Circulation characterizes the properties of a field averaged over the size of the contour diameter order. To obtain the characteristic of the field at the point P point, it is necessary to reduce the contour dimensions, tightening it to the point R. At the same time, the limit of the vector circulation ratio of the flat circuit is taken as the field characteristic of the field, which is tightened to the Point P, to the size of the contour of the S :. The magnitude of this limit depends not only on the properties of the field at the point P, but also on the orientation of the contour in the space, which can be specified by the direction of positive normal to the circuit plane (the normal is considered to be a positive contour direction of the contour of the right screw). Determining this limit for different directions, we will get different values, and for the opposite direction, these values \u200b\u200bare familiar with the sign. For some direction, the normal limit value will be maximum. Thus, the limit value behaves as a projection of some vector to the direction of normal to the circuit plane, according to which circulation is taken. The maximum limit value determines the module of this vector, and the direction of positive normal, in which the maximum is achieved, gives the direction of the vector. This vector is called rotor or vortex vector :.

To find the projection of the rotor on the axis of the Cartesian coordinate system, it is necessary to determine the limit values \u200b\u200bfor such orientation of the s platform S, at which normal to the site coincides with one of the x, y, z axes. If, for example, to send along the x axis, we will find. The contour is located in this case in the plane parallel to YZ, take the contour in the form of a rectangle with the sides and. At the value and on each of the four sides, the contour can be considered unchanged. The contour site 1 is opposite to the z axis, so this section coincides with, in section 2, in section 3, on the plot 4. For circulation on this contour we obtain the value :. The difference is an increment when the shift along y is on. In view of the smallness, this increment can be represented as analogously, the difference. Then circulation on the contour under consideration,

where is the area of \u200b\u200bthe contour. Sharing the circulation on, we find the projection of the rotor on the X axis :. Similarly,. Then the vector rotor is determined by the expression: +,

Knowing the vector rotor at each point of some surface S, it is possible to calculate the circulation of this vector along the contour limiting the surface S. To do this, we break the surface to very small elements (Fig. 7.7). Circulation by contour limiting is equal to, where - positive normal to the element. Having arouses these expressions along the entire surface S and substituting the expression for circulation, we get. This is the Stokes theorem.

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

Consider perpendicular to the direction of the velocity section of the current tube S (Fig.7.8.). We assume that the velocity of the particles of the fluid is the same in all points of this section. During the time, all particles will be held through the cross section, the distance of which at the initial moment does not exceed the value. Consequently, during the cross section S, the volume of the liquid will pass, and the volume of the liquid is passed per unit of time through the cross section S, it will take that the current tube is so thin that the particle speed in each of its cross section can be considered constant. If the liquid is incompressible (i.e. its density is the same everywhere and does not change), then the amount of fluid between sections and (Fig.7.9.) It will remain unchanged. Then the volume of fluid flowing per unit of time through the sections and should be the same:

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

This statement is called the theorem on the continuity of the jet.

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

where T is the time, x, y, z coordinates of the liquid particle, - the projection of the bulk force, p - pressure, ρ is the density of the medium. This equation allows you to determine the projection of the velocity of the medium as the coordinate and time functions. To close the system, the equation of continuity is added to the Navier - Stokes equation, which is a consequence of the continuity theorem of the jet:

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

7.2. Pressure in the current fluid. Bernoulli equation and a consequence of it

Considering the movement of fluids, in some cases we can assume that the movement of some liquids relative to others is not associated with the occurrence of friction forces. The liquid that inner friction (viscosity) is completely absent, is called ideal.

We highlight in the stationary current perfect fluid a small cross section tube (Fig. 7.10). Consider the volume of fluid, limited by the walls of the current tube and perpendicular to the current lines by sections and Same value:

The energy of each particle of the liquid is equal to the sum of its kinetic energy and the potential in the field of gravity. Due to the stationarity of the particle flow, which occurs after a time in any of the points of the unlocked part of the volume under consideration (for example, point O in Fig. 7.10), it has the same speed (and the same kinetic energy), which particle had at the same point at the initial moment time. Therefore, the increment of the energy of the entire volume under consideration is equal to the difference in the energy of shaded volumes and.

In the ideal fluid, the friction force is absent, therefore the increment of energy (7.1) is equal to the work performed on the highlighted pressure for pressure. Pressure forces on the side surface are perpendicular to each point to the direction of movement of particles and work are not performed. The work of the forces attached to the sections is equal

Equating (7.1) and (7.2), we get

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 the stationary current ideal fluid along any current line, the condition is performed

This is Bernoulli equation. For the horizontal current line, equation (7.3) takes the form:

7.3. Estation of the liquid from the hole

Apply Bernoulli equation to the case of the expiration of the liquid from the small hole in a wide open vessel. We highlight the current tube in the liquid, the upper cross section of which lies on the surface of the liquid, and the bottom coincides with the hole (Fig. 7.11). In each of these sections, the speed and height above some initial level can be considered the same, the pressure in both sections are equal to the 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. Combine liquid. Internal friction forces

Perfect liquid, i.e. Fluid without friction is an abstraction. All real liquids and gases are more or less inherent viscosity or internal friction.

Viscosity is manifested in the fact that the movement arising in liquid or gas after the termination of the forces that caused it, gradually stops.

Consider two parallel plates placed in a liquid (Fig. 7.12). Linear dimensions of plates a lot more distances between them d.. The lower plate is held in place, the upper is driven relative to the bottom with some

speed. It is experimentally proven that to move the upper plate at a constant speed, it is necessary to affect it a completely defined permanent force. The plate does not receive acceleration, therefore, the effect of this force is balanced equal to it by force, which is the force of friction acting on the plate during its movement in the liquid. Denote it, and a part of the fluid lying under the plane acts onto a piece of liquid lying above the plane, with force. At the same time, and are determined by the formula (7.4). Thus, this formula expresses the force between the contacting layers of the liquid.

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

Fluid particles directly contact with plates, as if stick to them and have the same speed as the plates themselves. From formula (7.5) we get

The module sign in this formula is supplied for the following reason. When the direction of movement changes, the speed derivative will change the sign, while the ratio is always positive. Taking into account the expression said (7.4) takes

The unit of viscosity with si is such a viscosity in which the gradient of the velocity with the module leads to the emergence of the inner friction force in 1 H per 1 m surface of the layers. This unit is called Pascal - Second (Pa · s).

1 | | | |

Lecture №5 Elements of mechanics of solid media
Physical model: A solid medium is a model of substance, in
which is neglecting the inner structure of the substance,
believing that the substance is continuously distributed
throughout
The volume occupied by them and fully fills this volume.
Uniformly called medium having the same one at each point
Properties.
Isotropic is called the medium whose properties are the same for all
directions.
Aggregate states of matter
Solid body - the state of the substance characterized by
Fixed volume and immutability of the form.
Liquid
–
state
substances
Characterized
Fixed volume, but having a certain form.
Gas - the state of the substance at which the substance fills the whole
Granted to him volume.

Mechanics of the deformable body
Deformation is a change in the shape and size of the body.
Elasticity - the property of the tel to resist the change in their volume and
Forms under the influence of loads.
Deformation is called elastic if it disappears after removal
load and - plastic, if it is not after removing the load
disappears.
In the theory of elasticity, it is proved that all types of deformations
(stretching - compression, shift, bending, tapping) can be reduced to
At the same time tensile deformations - compression and
shift.

Stretching deformation - compression
Stretching - Compression - Increase (or
reduction) cylindrical body length or
prismatic shape caused by force
directed along the longitudinal axis.
Absolute deformation - value equal
Change
body sizes caused by
External influence:
L L L0.
,
(5.1)
where L0 and L is the initial and final body length.
Law Dungal (I) (Robert Guk, 1660): force
Elasticity
Proportional
magnitude
absolute deformation and sent to
side of its reduction:
F K L,
where k is the coefficient of elasticity of the body.
(5.2)

Relative deformation:
L L0.
.
(5.3)
Mechanical voltage - value
Condition
deformed body \u003d PA:
F S.
,
(5.4)
where F is the force caused by deformation,
S - body cross section.
Truck Law (II): Mechanical Voltage,
arising in the body proportionally
The magnitude of its relative deformation:
E.
,
(5.5)
where E is the Jung module - the value
characterizing
Elastic
Properties
material, numerically equal to voltage,
arising in the body with a single
Relative deformation, [E] \u003d PA.

Deformations of solid bodies obey the law of the throat to
famous limit. Communication between deformation and voltage
It seems in the form of a voltage chart, a qualitative course
which is considered for the metal bar.

Energy elastic deformation
When tensile - compression energy of elastic deformation
L.
k l 2 1 2
(5.8)
KXDX
E V,
2
2
0
where V is the volume of the deformable body.
Bulk density
Tensile - compression
W.
Energy
1 2
E.
V 2.
Bulk density
Shift deformations
Elastic
.
Energy
1
W G 2.
2
for
(5.9)
Elastic
.
deformations
deformations
(5.10)
for

Elements of mechanics of liquids and gases
(hydro and aeromechanics)
Being in a solid aggregate state, the body at the same time
possesses both elasticity of form and elasticity of volume (or that
The same, when deforming in a solid, arise as
Normal and tangential mechanical stresses).
Liquids
and gases have only elasticity volume, but not
have elasticity of form (they take the form of a vessel, in
which
Liquids
There are).
and
Gas
Consequence
is an
This
General
Equity
in
Features
High quality
respect most mechanical properties of liquids and gases, and
their differences are
only
Quantitative characteristics
(for example, as a rule, liquid density is more density
Gas). Therefore, within the framework of solid media, used
A single approach to the study of liquids and gases.

Source characteristics
The density of the substance is scalar physical quantity,
characterizing the mass distribution by volume of substance and
determined by the mass ratio of the substance concluded in
Some volume, to the magnitude of this volume \u003d m / kg3.
In the case of a homogeneous medium, the density of the substance is calculated by
Formula
M v.
(5.11)
In the general case of an inhomogeneous medium mass and density of matter
Related by relationship
V.
(5.12)
M DV.
0
Pressure
- scalar value characterizing the condition
liquid or gas and equal strength that acts on a single
The surface towards normal to it [p] \u003d Pa:
P Fn S.
.
(5.13)

Elements of hydrostatics
Features of forces acting inside resting liquid
(gas)
1) if there is a small volume inside the resting liquid, then
Liquid on this volume has the same pressure in all
directions.
2) the resting liquid acts on in contact with it
The surface of the solid with force directed by normal to this
Surfaces.

EXTRACTION EQUATION
Tube current - part of a liquid bounded by current lines.
Stationary (or installed) is called such a flow
fluids in which the form and location of current lines, as well as
velocity values \u200b\u200bat each point of moving fluid with
Time does not change.
Mass fluid flow - a mass of liquid passing through
Cross section of current tube per unit time \u003d kg / s:
Qm M T SV,
(5.15)
where and V is the density and speed of fluid flow in section S.

The equation
Inseparable
–
Mathematical
ratio,
in
in accordance with which in the stationary flow of its fluid
Mass flow in each cross section of the current tube is the same:
1S1V 1 2S2V 2 or SV Const
,
(5.16)

Incompressible is the liquid, the density of which does not depend on
Temperatures and pressure.
Volumetric fluid flow - the volume of fluid passing through
Cross section of the current tube per unit time \u003d m3 / s:
QV V T SV,
(5.17)
Equation of continuity of incompressible homogeneous liquid -
Mathematical ratio, in accordance with which
stationary flow of an incompressible homogeneous fluid
Volumetric flow in each cross section of the current tube is the same:
S1V 1 S2V 2 or SV Const
,
(5.18)

Viscosity - the property of gases and liquids to provide resistance
The movement of one part of them relative to the other.
Physical Model: Perfect Liquid - Imaginary
incompressible liquid in which there is no viscosity and
thermal conductivity.
Bernoulli equation (Daniel Bernoulli 1738) - equation,
Being
consequence
law
Conservation
Mechanical
Energy for the stationary stream of perfect incompressible fluid
and recorded for an arbitrary cross section of the current tube located in
Field of gravity:
V 12.
V 22.
V 2.
GH1 P1.
GH2 P2 or
GH P Const. (5.19)
2
2
2

In the Bernoulli equation (5.19):
P - static pressure (fluid pressure on the surface
streamlined her body;
V 2.
- dynamic pressure;
2
GH - hydrostatic pressure.

Internal friction (viscosity). Newton law
Newton Law (Isaac Newton, 1686): Internal friction force,
per unit area of \u200b\u200bmoving liquid layers or
Gas, directly proportional to the gradient of the movement speed of the layers:
F.
S.
DV
DY.
,
(5.20)
where is the internal friction coefficient (dynamic viscosity),
\u003d m2 / s.

Types of flow of viscous fluid
Laminar flow - the shape of the course in which the liquid or
Gas moves with layers without stirring and ripples (that is,
indiscriminate rapid changes in speed and pressure).
Turbulent flow - form of fluid or gas flow, with
which
them
Elements
Made
disordered
unsteady movements on complex trajectories, which leads to
Intensive stirring between layers of moving fluid
or gas.

The number of Reynolds
Criteria for the transition of the laminar mode of fluid flow in
Turbulent mode is based on the use of Reynolds
(Osborne Réinolds, 1876-1883).
In case of fluid movement on the pipe number Reynolds
defined as
v D.
Re.
,
(5.21)
where V is the medium in cross section of the pipe fluid; D - diameter
pipes; and - density and coefficient of internal friction
liquids.
With the values \u200b\u200bof RE<2000 реализуется ламинарный режим течения
Pipe fluids, and when re\u003e 4000 - turbulent mode. For
values \u200b\u200b2000. There is a mixture of laminar and turbulent flows).

Consider the course of a viscous fluid by contacting directly
to experience. With the help of the rubber hose, connect to the plumbing
Crane thin horizontal glass tube with soldered into it
vertical manometric tubes (see Figure).
With a small flow rate, a decrease in the level is clearly visible.
Water in manometer tubes in the direction of the flow (H1\u003e H2\u003e H3). it
Indicates the presence of a pressure gradient along the axis of the tube -
Static pressure in the fluid decreases downstream.

Laminar flow of viscous fluid in a horizontal pipe
With a uniform rectilinear flow of pressure force fluid
Balanced by viscosity.

Distribution
Section
Flood
Speed
viscous
in
transverse
liquids
can
observe when it leaks from the vertical
tubes through a narrow hole (see Figure).
If, for example, with a closed crane to pour
at first
uncommon glycerin and then
From above, carefully add tinted, then in
a state of equilibrium
horizontal.
If the crane is to open, the border will take
The form similar to a paraboloid of rotation. it
Indicates
on the
Existence
Distributions
Speeds in the section Tube with a viscous course
Glycerin.

Formula Poiseil
The distribution of speeds in the cross section of the horizontal pipe with
Laminar flow of viscous fluid is determined by the formula
P 2 2.
V R.
R R.
4 L.
,
(5.23)
where R and L radius and the length of the pipe, respectively, p is the difference
Pressure at the ends of the pipe, R is the distance from the axis of the pipe.
Volumetric flow rate is determined by the formula of Poiseil
(Jean Poazeil, 1840):
R 4 P.
.
(5.24)
QV.
8 L.

Movement of bodies in a viscous environment
When moving tel in liquid or gas on the body
There is an internal friction force depending on
Body speed. At low speeds
observed
Laminar
Thought
Body
Liquid or gas and internal friction force
It turns out
proportional
Speed
Body movement and determined by the Stokes formula
(George Stokes, 1851):
F b L V
,
(5.25)
where b is a constant depending on the shape of the body and
its orientation relative to the stream, L -
Characteristic body size.
For a ball (B \u003d 6, L \u003d R) the power of internal friction:
F 6 RV
Where R is a ball radius.
,

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

Set of vectors specified for all points of space forms the speed vector field that can be depicted as follows. We carry out the line in the moving fluid so that the tangent to them on each point coincided in the direction with the vector (Fig. 7.1). These lines are called current lines. We agree to carry out the current lines so that their delicate (the ratio of the number of lines
to the magnitude of perpendicular platform
Through which they pass) was proportional to the speed of speed in this place. Then, in the picture of the current lines, it will be possible to judge not only about the direction, but also the magnitude of the vector at different points of space: where the speed is larger, the current line will be thicker.

The number of current lines passing through the platform
perpendicular to current lines, equal
if the site is oriented randomly to current lines, the number of current lines is equal to where
- angle between the direction of the vector and normal to the site . Often use the designation
. The number of current lines via the site the final sizes is determined by the integral:
. The integral of this type is called the vector stream through the playground .

IN vinchin and direction vector changing over time, therefore, the line of lines does not remain constant. If at each point of space, the speed vector remains constant in magnitude and direction, the current is called the installed or stationary. With inpatient flow, any particle of fluid undergoes this point of space with the same speed value. The pattern of current lines in this case does not change, and the current lines coincide with the trajectories of the particles.

Consider the field of the speed vector of incompressible inseparable fluid. The flow of velocity vector through a certain surface is equal to the volume of the fluid flowing through this surface per unit of time. Build in the neighborhood of the point R Imaginary closed surface S.(Fig. 7.2) . If in volume V., limited surface, the liquid does not occur and does not disappear, then the stream flowing out through the surface will be zero. The difference between the stream from zero will indicate that there are sources or drainage of liquid inside the surface, i.e. reader, in which the liquid enters the volume (sources) or is removed from the volume (drainage). The flow of the flow determines the total power of sources and wastewater. With the predominance of sources above the drains, the flow is positive, with the predominance of effluents - negative.

Private from the flow of flow by the amount of volume from which the flow flows,
, there is a medium specific power of sources enclosed in volume V. The smaller volume V,including point R,the closer is the average value to the true specific power at this point. In the limit of
. When tightening the volume to the point, we will get the true specific power of the sources at the point R, called divergence (discrepancy) vector :
. The resulting expression is valid for any vector. Integration is conducted on a closed surface S,limiting volume V.. Divergence is determined by the behavior of vector function near the point R. Divergence is a scalar function of coordinates defining point Movement R in space.

We find an expression for divergence in the Cartesian coordinate system. Consider in the neighborhood of the point P (x, y, z) Small volume in the form of parallelepiped with ribs parallel to axes of coordinates (Fig. 7.3). In mind the smell of volume (we will strive for zero)
within each of the six faces of parallelepiped can be considered unchanged. The flow across the entire closed surface is formed from streams current through each of the six faces separately.

We will find a stream after a couple of faces perpendicular to H.figure 7.3 facets 1 and 2) . External Normal to face 2 coincides with the direction of the axis H.. therefore
and the stream through the face 2 is equal
.Normal has the direction opposite to the axis H.Projections vector on the axis H. And on Normal have opposite signs
, and the stream through the face 1 is equal
. Total flow towards H. Raven
. Difference
represents increment when shifting along the axis H. on the
. In view of the smallness

. Then get
. Similarly, through pairs of faces perpendicular to the axes Y.and Z. , streams are equal
and
. Full flow through a closed surface. Sharing this expression on
, we find the divergence of the vector at point R:

Knowing divergence vector at each point of space, you can calculate the flow of this vector through any surface of the final sizes. To do this, we break the volume limited to the surface S., infinitely a large number of infinitely small elements
(Fig. 7.4).

For any element
stream vector through the surface of this element is equal
. Having aroused over all elements
, we get a flow through the surface S.Limiting volume V.:
integration is made volume V,or

E. that theorem of Ostrogradsky - Gauss. Here
,- single vector normal to the surface ds. At this point.

Let's return to the flow of incompressible fluid. Build contour . Imagine that we somehow frozen the instantly fluid throughout the volume except for a very thin closed channel of a constant cross section, which includes the contour (Fig. 7.5). Depending on the nature of the flow, the fluid in the resulting channel will be either a fixed or moving (circulating) along the contour in one of the possible directions. As a measure of this movement, the value is selected equal to the product of the fluid velocity in the channel and the length of the contour,
. This value is called vector circulation by contour (Since the channel has a constant section and the speed module does not change). At the time of hardening the walls, each particle of fluid in the channel will quench the velocity component, perpendicular to the wall and will only remain the component, tangent to the contour. This component is connected by impetus
, whose module for a particle of a liquid concluded in the length of the channel length
Raven
where - liquid density, - Channel cross section. The fluid perfect - friction is not, so the wall action can change only the direction
His value will remain constant. The interaction between fluid particles will cause such a redistribution of the pulse between them, which lines the speed of all particles. In this case, the algebraic sum of pulses persists, so
where - circulation rate - the tangent component of fluid velocity in the amount
at the time of time preceding the solidification of the walls. Sharing on
, receive
.

C. ircoulation characterizes the properties of the field averaged over the size of the contour diameter . To get the field characteristic at the point R, you need to reduce the size of the contour, tightening it to the point R. At the same time, as a field characteristic, vector circulation ratios take flat contour tie R, to the magnitude of the contour plane S.:
. The magnitude 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 that can be given by the direction of positive normal to the contour plane (the normal is considered to be a positive, associated with the direction of circuit by the rule of the right screw). Determining this limit for different directions , we get different meanings, and for the opposite directions, these values \u200b\u200bdiffer in the sign. For some direction, the normal limit value will be maximum. Thus, the limit value behaves as a projection of some vector to the direction of normal to the circuit plane, according to which circulation is taken. The maximum limit value determines the module of this vector, and the direction of positive normal, in which the maximum is achieved, gives the direction of the vector. This vector is called rotor or vector swivel. :
.

To find the projection of the rotor on the axis of the Cartesian coordinate system, you need to determine the limits for such site orientations S. under which normal to the site coincides with one of the axes X, y, z.If, for example, send along the axis H.We find
. Circuit located in this case in the plane parallel Yz., take the contour in the form of a rectangle with the parties
and
. For
values and on each of the four sides, the contour can be considered unchanged. Plot 1 contour (Fig. 7.6) is the opposite axis Z., so on this site coincides with
on site 2
on site 3
on site 4
. For circulation on this contour we get a value: . Difference
represents increment when offset along Y. on the
. In view of the smallness
this increment can be represented as
.Alogically, difference
. Then circulation according to the contour
,

where
- contour area. Sharing the circulation by
we will find the projection of the rotor on axis H.:
. Similarly,
,
. Then rotor vector determined by the expression:

+
,

or
.

Z. naya vector rotor at every point of some surface S., it is possible to calculate the circulation of this vector by contour Limiting the surface S.. To do this, we break the surface on very small items.
(Fig.7.7). Circulation by contour limiting
equal
where - Positive Normal to Element
. Having arise these expressions over the entire surface S.and substituting the expression for circulation, we get
. This is the Stokes theorem.

Part of the fluid bounded by current lines is called a current tube. Vector While at every point tangent to the current line, it will be tangent to the surface of the current tube, and the particles of the liquid do not intersect the walls of the current tube.

Consider perpendicular to the direction of the velocity section of the current tube S.(Fig. 7.8.). We assume that the velocity of the particles of the fluid is the same in all points of this section. During
through section S.all particles will be held, whose distance at the initial moment does not exceed the value
. Therefore during
through section S.
, and per unit of time through the section S. Will passes the volume of liquid equal
.. We assume that the current tube is so thin that the particle speed in each of its cross section can be considered constant. If the liquid is incompressible (i.e. its density is the same everywhere and does not change), then the amount of fluid between sections and (Fig. 7.9.) It will remain unchanged. Then the volume of fluid flowing per unit of time through the sections and , Must be the same:

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

.This statement is called the theorem on the continuity of the jet.

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

where t. - time, x, Y, Z - coordinates of the liquid particle,

- surround projection r - Pressure, ρ is the density of the medium. This equation allows you to determine the projection of the velocity of the medium as the coordinate and time functions. To close the system, the equation of continuity is added to the Navier - Stokes equation, which is a consequence of the continuity theorem of the jet:

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