The end of a space flight is considered to be landing on a planet. To date, only three countries have learned to return spacecraft to Earth: Russia, the USA and China.
For planets with an atmosphere (Fig. 3.19), the landing problem comes down mainly to solving three problems: overcoming a high level of overload; protection against aerodynamic heating; managing the time to reach the planet and the coordinates of the landing point.
Rice. 3.19. Scheme of spacecraft descent from orbit and landing on a planet with an atmosphere:
N- turning on the brake motor; A- spacecraft deorbit; M- separation of the spacecraft from the orbital spacecraft; IN- SA entry into the dense layers of the atmosphere; WITH - start of operation of the parachute landing system; D- landing on the surface of the planet;
1 – ballistic descent; 2 – gliding descent
When landing on a planet without an atmosphere (Fig. 3.20, A, b) the problem of protection from aerodynamic heating is eliminated.
A spacecraft located in the orbit of an artificial satellite of a planet or approaching a planet with an atmosphere to land on it has a large supply of kinetic energy associated with the speed of the spacecraft and its mass, and potential energy due to the position of the spacecraft relative to the surface of the planet.
Rice. 3.20. Descent and landing of a spacecraft on a planet without an atmosphere:
A- descent to the planet with preliminary entry into a holding orbit;
b- soft landing of a spacecraft with a braking engine and landing gear;
I - hyperbolic trajectory of approach to the planet; II - orbital trajectory;
III - trajectory of descent from orbit; 1, 2, 3 - active flight sections during braking and soft landing
Upon entry into the dense layers of the atmosphere, a shock wave appears in front of the bow of the spacecraft, heating the gas to a high temperature. As the spacecraft sinks into the atmosphere, it slows down, its speed decreases, and the hot gas heats the spacecraft more and more. The kinetic energy of the device is converted into heat. 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 solar apparatus.
The strongest shock waves occur with a blunted shape of the nose, which is why blunted shapes are used for SA, rather than pointed ones, characteristic of flight at low speeds.
With increasing speeds and temperatures, most of the heat is transferred to the apparatus not due to friction with the compressed layers of the atmosphere, but due to radiation and convection from the shock wave.
The following methods are used to remove heat from the SA surface:
– heat absorption by the heat-protective layer;
– radiation cooling of the surface;
– application of blow-off coatings.
Before entering the dense layers of the atmosphere, the trajectory of the spacecraft obeys the laws of celestial mechanics. In the atmosphere, in addition to gravitational forces, the apparatus is subject to aerodynamic and centrifugal forces that change the shape of its trajectory. The gravitational force is directed towards the center of the planet, the aerodynamic drag force is in the direction opposite to the velocity vector, the centrifugal and lift forces are perpendicular to the direction of motion of the SA. The aerodynamic drag force reduces the speed of the vehicle, while the centrifugal and lift forces impart acceleration to it in a direction perpendicular to its movement.
The nature of the descent trajectory in the atmosphere is determined mainly by its aerodynamic characteristics. In the absence of lifting force in a spacecraft, the trajectory of its movement in the atmosphere is called ballistic (the descent trajectories of the spacecraft of the Vostok and Voskhod series of spacecraft), and in the presence of lifting force, it is called gliding (SA Soyuz and Apollo, as well as Space Shuttle"), or ricocheting (SA KK Soyuz and Apollo). Movement in a planetocentric orbit does not place high demands on the accuracy of guidance during reentry, since it is relatively easy to adjust the trajectory by turning on the propulsion system for braking or acceleration. When entering the atmosphere at a speed exceeding the first cosmic speed, errors in calculations are most dangerous, since a descent that is too steep can lead to the destruction of the spacecraft, and a descent that is too gentle can lead to distance from the planet.
At ballistic descent the vector of the resultant aerodynamic forces is directed directly opposite to the velocity vector of the vehicle. Descent along a ballistic trajectory does not require control. The disadvantage of this method is the large steepness of the trajectory, and, as a result, the vehicle enters the dense layers of the atmosphere at high speed, which leads to strong aerodynamic heating of the device and to overloads, sometimes exceeding 10 g - close to the maximum permissible values for humans.
At aerodynamic descent The outer body of the apparatus, as a rule, has a conical shape, and the axis of the cone makes a certain angle (angle of attack) with the velocity vector of the apparatus, due to which the resultant of the aerodynamic forces has a component perpendicular to the velocity vector of the apparatus—the lifting force. Thanks to the lifting force, the vehicle descends more slowly, the trajectory of its descent becomes flatter, while the braking section stretches both in length and in time, and the maximum overloads and the intensity of aerodynamic heating can be reduced several times, compared with ballistic braking, which is done by the glider the descent is safer and more comfortable for people.
The angle of attack during descent changes depending on the flight speed and the current air density. In the upper, rarefied layers of the atmosphere it can reach 40°, gradually decreasing with the descent of the apparatus. This requires the presence of a gliding flight control system on the SA, which complicates and weighs down the apparatus, and in cases where it is used to lower only equipment that can withstand higher overloads than a person, ballistic braking is usually used.
The Space Shuttle orbital stage, which performs the function of a descent vehicle when returning to Earth, plans throughout the entire descent phase from entry into the atmosphere until the landing gear touches the landing strip, after which the braking parachute is released.
After the speed of the vehicle has decreased to subsonic in the aerodynamic braking section, the descent of the spacecraft can then be carried out using parachutes. A parachute in a dense atmosphere reduces the speed of the vehicle to almost zero and ensures a soft landing on the surface of the planet.
In the thin atmosphere of Mars, parachutes are less effective, so during the final part of the descent the parachute is detached and the landing rocket engines are turned on.
The descent manned spacecraft of the Soyuz TMA-01M series, designed for landing on land, also have solid-fuel braking engines that turn on a few seconds before touching the ground to ensure a safer and more comfortable landing.
The descent vehicle of the Venera-13 station, after descending by parachute to an altitude of 47 km, dropped it and resumed aerodynamic braking. This descent program was dictated by the peculiarities of the atmosphere of Venus, the lower layers of which are very dense and hot (up to 500 ° C), and fabric parachutes would not have withstood such conditions.
It should be noted that in some projects of reusable spacecraft (in particular, single-stage vertical take-off and landing, for example, Delta Clipper), it is also assumed at the final stage of descent, after aerodynamic braking in the atmosphere, to also perform a parachute-free motor landing using rocket engines. Structurally, landers can differ significantly from each other depending on the nature of the payload and 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 eliminated, but for landing the speed is reduced using a braking propulsion system, which must operate in a programmable thrust mode, and the mass of the fuel can significantly exceed the mass of the spacecraft itself.
ELEMENTS OF Continuum MECHANICS
A medium is considered continuous if it is characterized by a uniform distribution of matter – i.e. medium with the same density. These are liquids and gases.
Therefore, in this section we will look at the basic laws that apply in these environments.
7.1. General properties of liquids and gases. Kinematic description of fluid motion. Vector fields. Flow and circulation of a vector field. Stationary flow of an ideal fluid. Current lines and tubes. Equations of motion and equilibrium of fluid. Continuity equation for incompressible fluid
Continuum mechanics is a branch of mechanics devoted to the study of the motion and equilibrium of gases, liquids, plasma and deformable solids. The main assumption of continuum mechanics is that matter can be considered as a continuous medium, neglecting its molecular (atomic) structure, and at the same time the distribution of all its characteristics (density, stress, particle velocities) in the medium can be considered continuous.
A liquid is a substance in a condensed state, intermediate between solid and gaseous. The region of existence of a liquid is limited on the low temperature side by a phase transition to a solid state (crystallization), and on the high temperature side by a phase transition to a gaseous state (evaporation). When studying the properties of a continuous medium, the medium itself appears to consist of particles whose sizes are much larger than the sizes of molecules. Thus, each particle includes a huge number of molecules.
To describe the motion of a fluid, you can specify the position of each fluid particle as a function of time. This method of description was developed by Lagrange. But you can follow not the particles of liquid, but individual points in space, and note the speed with which individual particles of liquid pass through each point. The second method is called Euler's method.
The state of fluid motion can be determined by indicating the velocity vector for each point in space as a function of time.
Set of vectors 
, given for all points in space, forms a velocity vector field, which can be depicted as follows. Let us draw lines in the moving fluid so that the tangent to them at each point coincides in direction with the vector 
(Fig. 7.1). These lines are called streamlines. Let us agree to draw streamlines so that their density (the ratio of the number of lines 
to the size of the area perpendicular to them 
, through which they pass) 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 current lines will be denser.
Number of streamlines passing through the site 
, perpendicular to the streamlines, is equal to 
, if the site is oriented arbitrarily towards the streamlines, the number of streamlines is equal to, where 
- angle between vector direction 
and normal to the site 
. The notation is often used 
. Number of current lines across the site 
finite dimensions is determined by the integral: 
. An integral of this type is called a vector flow 
through the platform 
.
IN 
magnitude and direction of the vector 
changes over time, therefore, the pattern of lines does not remain constant.
The flow of a vector through a certain surface and the circulation of the vector along a given contour make it possible to judge the nature of the vector field.
However, these quantities give an average characteristic of the field within the volume covered by the surface through which the flow is determined, or in the vicinity of the contour along which the circulation is taken. By reducing the dimensions of a surface or contour (contracting them to a point), one can arrive at values that will characterize the vector field at a given point. Let us consider the velocity vector field of an incompressible continuous fluid. The velocity vector flux through a certain surface is equal to the volume of fluid flowing through this surface per unit time. Let's construct a point in the neighborhood R imaginary closed surface . S (Fig. 7.2) If in volume
V 
, limited by the surface, the liquid does not appear or disappear, then the flow flowing out through the surface will be zero. A difference in flux from zero will indicate that there are sources or sinks of liquid inside the surface, i.e. points at which liquid enters the volume (sources) or is removed from the volume (sinks). The magnitude of the flow determines the total power of the sources and sinks. When sources predominate over sinks, the flow is positive; when sinks predominate, it is negative. The quotient of dividing the flow by the volume from which the flow flows out is, is the average specific power of sources contained in the volume V. The smaller the volume V, including a point 
R, V, 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 called divergence (divergence) of the vector. (Fig. 7.2) The resulting expression is valid for any vector. Integration is carried out over a closed surface 
S, limiting the volume. 
Divergence is determined by the behavior of the vector function Let us consider the velocity vector field of an incompressible continuous fluid. The velocity vector flux through a certain surface is equal to the volume of fluid flowing through this surface per unit time. near the point
R. Divergence is a scalar function of the coordinates defining n point position 
in space.
Let's find the flow through a pair of faces perpendicular to the axis X in Fig. 7.3 faces 1 and 2) .
Outer normal 
to face 2 coincides with the direction of the axis X. 
That's why 
and the flux through edge 2 is 
.Normal has a direction opposite to the axis X. 
Vector projections X per axis 
and to normal 
have opposite signs 
, and the flux through face 1 is equal to X. 
Total flow in direction 
equals
. X Difference 
represents the increment 
when displaced along the axis 
on . Due to the small size . Then we get 
Due to the small size 
.
,
Similarly, through pairs of faces perpendicular to the axes 
Y Let us consider the velocity vector field of an incompressible continuous fluid. The velocity vector flux through a certain surface is equal to the volume of fluid flowing through this surface per unit time.:
.
And 
Z R, the flows are equal 
.
Total flow through a closed surface. 
Dividing this expression by 
find the divergence of the vector 
at the point 
Knowing the divergence of the vector R at each point in space, one can calculate the flow of this vector through any surface of finite dimensions. (Fig. 7.2):
To do this, we divide the volume limited by the surface V., to an infinitely large number of infinitesimal elements
.
(Fig. 7.4). 
For any element 
,
vector flow through the surface of this element is equal to. Summing over all elements
, we get the flow through the surface 
, limiting the volume 
, integration is performed on the volume 
or 
E 
then the Ostrogradsky–Gauss theorem. Here 
- unit vector normal to the surface 
dS 
at this point. 
Let's return to the flow of incompressible fluid. Let's build a contour 
. 
, its value will remain constant. 
at this point.
-
The interaction between liquid particles will cause a redistribution of momentum between them that will equalize the velocities of all particles. In this case, the algebraic sum of the impulses is preserved, therefore
circulation speed,
- tangential component of fluid velocity in volume
,
at the time preceding the hardening of the walls. Divided by 
.
we get 
C 
circulation characterizes the field properties averaged over an area with dimensions on the order of the contour diameter Let us consider the velocity vector field of an incompressible continuous fluid. The velocity vector flux through a certain surface is equal to the volume of fluid flowing through this surface per unit time.. Let us consider the velocity vector field of an incompressible continuous fluid. The velocity vector flux through a certain surface is equal to the volume of fluid flowing through this surface per unit time. To obtain the field characteristic at a point 
, you need to reduce the size of the outline, tightening it to a point 
. Let us consider the velocity vector field of an incompressible continuous fluid. The velocity vector flux through a certain surface is equal to the volume of fluid flowing through this surface per unit time. In this case, the limit of the vector circulation ratio is taken as a characteristic of the field R:
along a flat contour Let us consider the velocity vector field of an incompressible continuous fluid. The velocity vector flux through a certain surface is equal to the volume of fluid flowing through this surface per unit time., contracting to a point 
, to the size of the contour plane 
. 
:
.
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 different values, and for opposite normal directions these values differ in sign. For a certain direction of the normal, the limit value will be maximum. Thus, the value of the limit behaves as a projection of a certain vector onto the direction of the normal to the plane of the contour along which the circulation is taken. The maximum value of the limit determines the magnitude of this vector, and the direction of the positive normal at which the maximum is reached gives the direction of the vector. This vector is called a rotor or vortex 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
X, for which the normal 
to the site coincides with one of the axes 
X,Y,Z. If, for example, you send along the axis 
Due to the small size 
, we'll find 
. 
Due to the small size 
Circuit located in this case in a plane parallel to YZ
, take a contour in the form of a rectangle with sides
.
At
values
on each of the four sides of the contour can be considered unchanged. Section 1 of the contour (Fig. 7.6) is opposite to the axis
.
Z 
, That's why
in this area coincides with . Difference 
represents the increment 
, on site 2 
, on site 3
, on site 4
.
. For circulation along this circuit we obtain the value:
,
Difference
-
contour area.
Dividing the circulation into 
, let's find the projection of the rotor onto
axis X:
.
Likewise,
,
.
Then the rotor of the vector
+
,
is determined by the expression: 
.
or 
Z R the rotor of the vector at each point of some surface 
, we can calculate the circulation of this vector along the contour R, bounding the surface
.
To do this, we divide the surface into very small elements
(Fig. 7.7). Circulation along a contour limiting 
equal to
.
, Where R- positive normal to the element
Summing these expressions over the entire surface
and substituting the expression for circulation, we get 
.
This is Stokes' theorem. R The part of the liquid bounded by streamlines is called a stream tube. Vector 
, being tangent to the stream line at each point, will be tangent to the surface of the stream tube, and the liquid particles do not cross the walls of the stream tube. R Let us consider the section of the current tube perpendicular to the direction of velocity 
(Fig. 7.8.). 
We will assume that the speed of liquid particles is the same at all points of this section. During 
through the section R 
all particles whose distance will pass R at the initial moment does not exceed the value 
. 
Due to the small size 
Therefore, during the time 
Due to the small size 
through the section
.
, and per unit time through the section 
a volume of liquid will pass through equal to
.
.. We will assume that the current tube is so thin that the speed of particles in each section 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 sections
,
(Fig. 7.9.) will remain unchanged. Then the volumes of fluid flowing per unit time through the sections, must be the same: Thus, for an incompressible fluid the quantity in any section of the same tube the current should be the same:
-
This statement is called the jet continuity theorem. The motion of an ideal fluid is described by the Navier-Stokes equation: Where
t
7.1. General properties of liquids and gases. Kinematic description of fluid motion. Vector fields. Flow and circulation of a vector field. Stationary flow of an ideal fluid. Current lines and tubes. Equations of motion and equilibrium of fluid. Continuity equation for incompressible fluid
Continuum mechanics is a branch of mechanics devoted to the study of the motion and equilibrium of gases, liquids, plasma and deformable solids. The main assumption of continuum mechanics is that matter can be considered as a continuous medium, neglecting its molecular (atomic) structure, and at the same time the distribution of all its characteristics (density, stress, particle velocities) in the medium can be considered continuous.
A liquid is a substance in a condensed state, intermediate between solid and gaseous. The region of existence of a liquid is limited on the low temperature side by a phase transition to a solid state (crystallization), and on the high temperature side by a phase transition to a gaseous state (evaporation). When studying the properties of a continuous medium, the medium itself appears to consist of particles whose sizes are much larger than the sizes of molecules. Thus, each particle includes a huge number of molecules.
To describe the motion of a fluid, you can specify the position of each fluid particle as a function of time. This method of description was developed by Lagrange. But you can follow not the particles of liquid, but individual points in space, and note the speed with which individual particles of liquid pass through each point. The second method is called Euler's method.
The state of fluid motion can be determined by indicating the velocity vector for each point in space as a function of time.
The set of vectors specified for all points in space forms a velocity vector field, which can be depicted as follows. Let us draw lines in the moving fluid so that the tangent to them at each point coincides in direction with the vector (Fig. 7.1). These lines are called streamlines. Let us agree to draw streamlines so that their density (the ratio of the number of lines to the size of the area perpendicular to them through which they pass) is proportional to the magnitude of the speed in a given place. Then, from the pattern of streamlines, it will be possible to judge not only the direction, but also the magnitude of the vector at different points in space: where the speed is greater, the streamlines will be denser.
The number of streamlines passing through the pad perpendicular to the streamlines is equal to , if the pad is oriented arbitrarily to the streamlines, the number of streamlines is equal to , where is the angle between the direction of the vector and the normal to the pad. The notation is often used. The number of streamlines through an area of finite dimensions is determined by the integral: . An integral of this type is called vector flow through the area.
The magnitude and direction of the vector changes over time, therefore, the pattern of lines does not remain constant. If at each point in space the velocity vector remains constant in magnitude and direction, then the flow is called steady or stationary. In a stationary flow, any fluid particle passes a given point in space with the same speed value. The pattern of streamlines in this case does not change, and the streamlines coincide with the trajectories of the particles.
The flow of a vector through a certain surface and the circulation of the vector along a given contour make it possible to judge the nature of the vector field. However, these quantities give an average characteristic of the field within the volume covered by the surface through which the flow is determined, or in the vicinity of the contour along which the circulation is taken. By reducing the dimensions of a surface or contour (contracting them to a point), one can arrive at values that will characterize the vector field at a given point.
Let us consider the velocity vector field of an incompressible continuous fluid. The velocity vector flux through a certain surface is equal to the volume of fluid flowing through this surface per unit time. Let's construct an imaginary closed surface S in the neighborhood of point P (Fig. 7.2). If in a volume V bounded by a surface, liquid does not appear or disappear, then the flow flowing out through the surface will be zero. A difference in flow from zero will indicate that there are sources or sinks of liquid inside the surface, i.e. points at which liquid enters the volume (sources) or is removed from the volume (sinks). The magnitude of the flow determines the total power of the sources and sinks. When sources predominate over sinks, the flow is positive; when sinks predominate, it is negative.
The quotient of the flow divided by the volume from which the flow flows out, , is the average specific power of the sources contained in volume V. The smaller the volume V that includes point P, the closer this average value is to the true specific power at this point. In the limit at , i.e. when contracting the volume to a point, we obtain the true specific power of the sources at point P, called the divergence (divergence) of the vector: . The resulting expression is valid for any vector. Integration is carried out over a closed surface S, limiting the volume V. Divergence is determined by the behavior of a vector function near point P. Divergence is a scalar function of coordinates that determine the position of point P in space.
Let us find an expression for divergence in the Cartesian coordinate system. Let us consider in the vicinity of the point P(x,y,z) a small volume in the form of a parallelepiped with edges parallel to the coordinate axes (Fig. 7.3). Due to the smallness of the volume (we will tend to zero), the values within each of the six faces of the parallelepiped can be considered unchanged. The flow through the entire closed surface is formed from the flows flowing through each of the six faces separately.
Let's find the flow through a pair of faces perpendicular to the axis X in Fig. 7.3 (faces 1 and 2). The outer normal to face 2 coincides with the direction of the X axis. Therefore, the flux through face 2 is equal to . The normal has a direction opposite to the X axis. The projections of the vector onto the X axis and onto the normal have opposite signs, , and the flux through face 1 is equal to . The total flux in the X direction is . The difference represents the increment when moving along the X axis by . Due to its smallness, this increment can be represented as . Then we get . Similarly, through pairs of faces perpendicular to the Y and Z axes, the fluxes are equal to and . Total flow through a closed surface. Dividing this expression by , we find the divergence of the vector at point P:
Knowing the divergence of a vector at each point in space, one can calculate the flow of this vector through any surface of finite dimensions. To do this, we divide the volume limited by the surface S into an infinitely large number of infinitesimal elements (Fig. 7.4).
For any element, the vector flux through the surface of this element is equal to . Summing over all elements, we obtain the flow through the surface S, limiting the volume V: , integration is carried out over the volume V, or
This is the Ostrogradsky–Gauss theorem. Here , is the unit normal vector to the surface dS at a given point.
Let's return to the flow of incompressible fluid. Let's build a contour. Let's imagine that we have somehow instantly frozen the liquid in its entire volume, with the exception of a very thin closed channel of constant cross-section, which includes a contour (Fig. 7.5). Depending on the nature of the flow, the liquid in the formed channel will be either stationary or moving (circulating) along the contour in one of the possible directions. As a measure of this movement, a value is chosen equal to the product of the fluid velocity in the channel and the length of the contour, . This quantity is called vector circulation along the contour (since the channel has a constant cross-section and the velocity module does not change). At the moment of solidification of the walls, for each liquid particle in the channel the velocity component perpendicular to the wall will be extinguished and only the component tangent to the contour will remain. Associated with this component is the impulse , the modulus of which for a liquid particle enclosed in a channel segment of length , is equal to , where is the density of the liquid and is the cross-section of the channel. The liquid is ideal - there is no friction, so the action of the walls can only change the direction, its value will remain constant. The interaction between liquid particles will cause a redistribution of momentum between them that will equalize the velocities of all particles. In this case, the algebraic sum of the impulses is preserved, therefore , where is the circulation speed, is the tangential component of the fluid velocity in the volume at the moment of time preceding the solidification of the walls. Dividing by , we get .
Circulation characterizes the field properties averaged over an area with dimensions on the order of the contour diameter. To obtain a characteristic of the field at point P, it is necessary to reduce the size of the contour, contracting it to point P. In this case, as a characteristic of the field, take the limit of the ratio of the vector circulation along a flat contour contracting to point P to the value of the plane of the contour S: . The value of this limit depends not only on the properties of the field at point P, but also on the orientation of the contour in space, which can be specified by the direction of the positive normal to the plane of the contour (the normal associated with the direction of traversal of the contour by the rule of the right screw is considered positive). By defining this limit for different directions, we will obtain different values, and for opposite normal directions these values differ in sign. For a certain direction of the normal, the limit value will be maximum. Thus, the value of the limit behaves as a projection of a certain vector onto the direction of the normal to the plane of the contour along which the circulation is taken. The maximum value of the limit determines the magnitude of this vector, and the direction of the positive normal at which the maximum is reached gives the direction of the vector. This vector is called the rotor or vortex of the vector: .
To find the projections of the rotor on the axes of the Cartesian coordinate system, it is necessary to determine the limit values for such orientations of the site S at which the normal to the site coincides with one of the axes X, Y, Z. If, for example, we direct along the X axis, we find . In this case, the contour is located in a plane parallel to YZ; let’s take the contour in the form of a rectangle with sides and . At the values of and on each of the four sides of the contour can be considered unchanged. Section 1 of the contour (Fig. 7.6) is opposite to the Z axis, therefore in this section it coincides with, in section 2, in section 3, in section 4. For circulation along this contour we obtain the value: . The difference represents the increment when moving along Y by . Due to its smallness, this increment can be represented as . Similarly, the difference . Then circulation along the considered contour,
where is the contour area. Dividing the circulation by , we find the projection of the rotor onto the X axis: . Likewise, , . Then the rotor of the vector is determined by the expression: + ,
Knowing the rotor of a vector at each point of a certain surface S, we can calculate the circulation of this vector along the contour that bounds the surface S. To do this, we divide the surface into very small elements (Fig. 7.7). The circulation along the bounding contour is equal to , where is the positive normal to the element . Summing these expressions over the entire surface S and substituting the expression for circulation, we obtain . This is Stokes' theorem.
The part of the liquid bounded by streamlines is called a stream tube. The vector, being tangent to the stream line at each point, will be tangent to the surface of the stream tube, and the liquid particles do not intersect the walls of the stream tube.
Let us consider the section of the current tube S perpendicular to the direction of velocity (Fig. 7.8.). We will assume that the speed of liquid particles is the same at all points of this section. During the time, all particles whose distance at the initial moment does not exceed the value will pass through the section S. Consequently, in a time, a volume of liquid equal to . will pass through section S, and in a unit of time, a volume of liquid will pass through section S, equal to .. We will assume that the current tube is so thin that the speed of particles in each of its sections can be considered constant. If the fluid is incompressible (that is, its density is the same everywhere and does not change), then the amount of fluid between sections and (Fig. 7.9.) will remain unchanged. Then the volumes of liquid flowing per unit time through the sections and should be the same:
Thus, for an incompressible fluid, the value in any section of the same current tube should be the same:
This statement is called the jet continuity theorem.
The motion of an ideal fluid is described by the Navier-Stokes equation:
where t is time, x,y,z are the coordinates of the liquid particle, are the projections of the body force, p is the pressure, ρ is the density of the medium. This equation allows us to determine the projection of the velocity of a particle of the medium as a function of coordinates and time. To close the system, the continuity equation is added to the Navier-Stokes equation, which is a consequence of the jet continuity theorem:
To integrate these equations, it is necessary to set the initial (if the motion is not stationary) and boundary conditions.
7.2. Pressure in a flowing liquid. Bernoulli's equation and its corollary
When considering the movement of liquids, in some cases it can be assumed that the movement of some liquids relative to others is not associated with the occurrence of friction forces. A fluid in which internal friction (viscosity) is completely absent is called ideal.
Let us select a current tube of small cross-section in a stationary flowing ideal fluid (Fig. 7.10). Let us consider the volume of liquid limited by the walls of the stream tube and sections perpendicular to the stream lines and. During the time this volume will move along the stream tube, and the cross section will move to the position having passed the path, the cross section will move to the position having passed the path. Due to the continuity of the jet, the shaded volumes will have the same size:
The energy of each fluid particle is equal to the sum of its kinetic energy and potential energy in the gravity field. Due to the stationarity of the flow, a particle located after time at any point in the unshaded part of the volume under consideration (for example, point O in Fig. 7.10) has the same speed (and the same kinetic energy) as the particle that was at the same point at the initial moment had time. Therefore, the increment in the energy of the entire volume under consideration is equal to the difference in the energies of the shaded volumes and .
In an ideal fluid there are no frictional forces, therefore the increment of energy (7.1) is equal to the work done on the selected volume by pressure forces. The pressure forces on the lateral surface are perpendicular at each point to the direction of movement of the particles and do not do any work. The work of forces applied to the sections and is equal to
Equating (7.1) and (7.2), we obtain
Since the sections were taken arbitrarily, it can be argued that the expression remains constant in any section of the current tube, i.e. in a stationary flowing ideal fluid along any streamline the following condition is satisfied:
This is Bernoulli's equation. For a horizontal streamline, equation (7.3) takes the form:
7.3. LIQUID OUTLET FROM THE HOLE
Let us apply Bernoulli's equation to the case of fluid flowing out of a small hole in a wide open vessel. Let us select a current tube in the liquid, the upper section of which lies on the surface of the liquid, and the lower section coincides with the hole (Fig. 7.11). In each of these sections, the speed and height above a certain initial level can be considered the same, the pressure in both sections is equal to atmospheric and also the same, the speed of movement of the open surface will be considered equal to zero. Then equation (7.3) takes the form:
Pulse
7.4 Viscous liquid. Internal friction forces
An ideal liquid, i.e. a fluid without friction is an abstraction. All real liquids and gases exhibit viscosity or internal friction to a greater or lesser extent.
Viscosity is manifested in the fact that the movement that has arisen in a liquid or gas gradually ceases after the cessation of the forces that caused it.
Let's consider two plates parallel to each other placed in a liquid (Fig. 7.12). The linear dimensions of the plates are much greater than the distance between them d. The lower plate is held in place, the upper one is driven relative to the lower one with some
speed It has been experimentally proven that in order to move the upper plate at a constant speed, it is necessary to act on it with a very specific constant force. The plate does not receive acceleration, therefore, the action of this force is balanced by a force equal to it in magnitude, which is the friction force acting on the plate as it moves in the liquid. Let's denote it, and the part of the fluid lying under the plane acts on the part of the fluid lying above the plane with a force. In this case, and are determined by formula (7.4). Thus, this formula expresses the force between contacting layers of liquid.
It has been experimentally proven that the speed of liquid particles changes in the z direction perpendicular to the plates (Fig. 7.6) according to a linear law
Liquid particles in direct contact with the plates seem to stick to them and have the same speed as the plates themselves. From formula (7.5) we obtain
The modulus sign in this formula is placed for the following reason. When the direction of motion changes, the derivative of the speed will change sign, while the ratio is always positive. Taking into account the above, expression (7.4) takes the form
The SI unit of viscosity is the viscosity at which the velocity gradient with modulus , leads to the appearance of an internal friction force of 1 N on 1 m of the contact surface of the layers. This unit is called the Pascal second (Pa s).
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Plan
1. The concept of continuum. General properties of 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. Energy of an elastically deformed body.
Theses
1. The volume of gas is determined by the volume of the container that the gas occupies. In liquids, unlike gases, the average distance between molecules remains almost constant, so the liquid has an almost constant 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 a liquid depends little on pressure. The density of gases depends significantly on pressure. It is known from experience that the compressibility of liquid and gas in many problems can be neglected and the single 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, i.e., an imaginary liquid in which there are no forces of internal friction. An ideal fluid is an imaginary fluid in which there are no internal friction forces. A viscous liquid contradicts it. A physical quantity determined by the normal force acting on the part of a liquid per unit area is called pressure The motion of an ideal fluid is described by the Navier-Stokes equation: liquids The unit of pressure is pascal (Pa): 1 Pa is equal to the pressure created by a force of 1 N, uniformly distributed over a surface normal to it with an area of 1 m 2 (1 Pa = 1 N/m 2). Pressure in the equilibrium of liquids (gases) obeys Pascal's law: the pressure in any place of a liquid at rest is the same in all directions, and the pressure is equally transmitted throughout the entire volume occupied by the liquid at rest.
Pressure varies linearly with altitude. Pressure P= rgh called hydrostatic. The pressure force on the lower layers of the liquid is greater than on the upper ones, therefore, a body immersed in a liquid is acted upon by a buoyant force determined by Archimedes’ law: a body immersed in a liquid (gas) is acted upon by an upward buoyant force equal to its weight from the side of this liquid liquid (gas) displaced by the body, where r is the density of the liquid, (Fig. 7.2)- the volume of a body immersed in a liquid.
The movement of liquids is called flow, and the collection of particles of a moving liquid 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 fluid velocity vector at the corresponding points in space (Fig. 45). From the pattern of streamlines one can judge the direction and magnitude of velocity at different points in space, i.e., one can determine the state of fluid motion. The part of the liquid bounded by stream lines is called a stream tube. The flow of a fluid is called steady (or stationary) if the shape and location of the streamlines, as well as the velocity values at each point do not change over time.
Let's consider some current tube. Let us choose two of its sections R 1 and R 2 , perpendicular to the direction of speed (Fig. 46). If the fluid is incompressible (r=const), then through the section R 2 will pass in 1 s the same volume of liquid as through the section R 1, i.e. The product of the flow velocity of an incompressible fluid and the cross-section of a current tube is a constant value for a given current tube. The relationship is called the continuity equation for an incompressible fluid. - Bernoulli equation - expression of the law of conservation of energy in relation to the steady flow of an ideal fluid ( here p - static pressure (fluid pressure on the surface of a body flowing around it), value - dynamic pressure, - hydrostatic pressure). For a horizontal current tube, Bernoulli's equation is written in the form where left side called total pressure. - Torricelli formula
Viscosity is the property of real liquids to resist the movement of one part of the liquid relative to another. When some layers of a real liquid move relative to others, internal friction forces arise, directed tangentially to the surface of the layers. The internal friction force F is greater, the larger the surface area of the layer S under consideration, and depends on how quickly the fluid flow velocity changes when moving from layer to layer. The Dv/Dx value shows how quickly the speed changes when moving from layer to layer in the direction X, 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 coefficient of proportionality h , depending on the nature of the liquid, 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 that arise in it. Viscosity depends on temperature, and the nature of this dependence is different for liquids and gases (for liquids it decreases with increasing temperature, for gases, on the contrary, it increases), which indicates a difference in the mechanisms of internal friction in them. The viscosity of oils depends especially strongly on temperature. Methods for determining viscosity:
1) Stokes formula; 2) Poiseuille formula
2. Deformation is called elastic if, after the cessation of external forces, the body returns to its original size and shape. Deformations that remain in the body after the cessation of external forces are called plastic. The force acting per unit 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. Relative change in rod length (longitudinal deformation), relative transverse tension (compression), where d -- rod diameter. Deformations e and e " always have different signs, where m is a positive coefficient depending on the properties of the material, called Poisson's ratio.
Robert Hooke experimentally established that for small deformations the relative elongation e and 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 that causes an elongation equal to unity. Then Hooke's law can be written like this, where k- elasticity coefficient:elongation of a rod during elastic deformation is proportional to the force acting on core strength. Potential energy of an elastically stretched (compressed) rod Deformations of solid bodies obey Hooke's law only for elastic deformations. The relationship between strain and stress is represented in the form of a stress diagram (Fig. 35). The figure shows that the linear dependence s (e), established by Hooke, is satisfied only within very narrow limits up to the so-called limit of proportionality (s p). With a further increase in stress, the deformation is still elastic (although the dependence s (e) is no longer linear) and up to the elastic limit (s y) residual deformations do not occur. Beyond the elastic limit, residual deformations occur in the body and the graph describing the return of the body to its original state after the cessation of the force will not be drawn as a curve VO, ah parallel to it - CF. The stress at which noticeable residual deformation appears (~=0.2%) is called the yield strength (s t) - point WITH on the curve. In area CD the deformation increases without increasing the stress, i.e. the body seems to “flow”. This region is called the yield region (or plastic deformation region). Materials for which the yield region is significant are called viscous, while for which it is practically absent - brittle. With further stretching (beyond the point D) the body is destroyed. The maximum stress that occurs in a body before failure is called the ultimate strength (s p).
