General properties of liquids and gases. Equilibrium equation and fluid motion. Hydrostatics of an incompressible fluid. Stationary motion of an ideal fluid. Bernoulli equation. Ideally elastic body. Elastic stresses and deformations. Hooke's law. Young's modulus.
Relativistic mechanics.
The principle of relativity and transformations of Galileo. Experimental substantiations of the special theory of relativity (SRT). Einstein's postulates of the special theory of relativity. Lorentz transformations. The concept of simultaneity. Relativity of lengths and intervals of time. Relativistic law of addition of velocities. relativistic momentum. Equation of motion of a relativistic particle. Relativistic expression for kinetic energy. Interrelation of mass and energy. Relationship between total energy and momentum of a particle. Limits of applicability of classical (Newtonian) mechanics.
Basics molecular physics and thermodynamics
Thermodynamic systems. Ideal gas.
Dynamic and statistical laws in physics. Statistical and thermodynamic methods for studying macroscopic phenomena.
Thermal motion of molecules. Interaction between molecules. Ideal gas. State of the system. Thermodynamic parameters of the state. Equilibrium states and processes, their representation on thermodynamic diagrams. The equation of state for an ideal gas.
Fundamentals of molecular-kinetic theory.
Basic equation of molecular kinetic theory ideal gases and its comparison with the Clapeyron-Mendeleev equation. Average kinetic energy of molecules. Molecular-kinetic interpretation of thermodynamic temperature. The number of degrees of freedom of a molecule. The law of uniform distribution of energy over the degrees of freedom of molecules. Internal energy and heat capacity of an ideal gas.
Maxwell's law for the distribution of molecules over velocities and energies thermal motion. Ideal gas in a force field. Boltzmann distribution of molecules in a force field. barometric formula.
Effective molecular diameter. The number of collisions and the mean free path of molecules. transfer phenomena.
Fundamentals of thermodynamics.
The work done by a gas when its volume changes. Quantity of heat. First law of thermodynamics. Application of the first law of thermodynamics to isoprocesses and the adiabatic process of an ideal gas. The dependence of the heat capacity of an ideal gas on the type of process. The second law of thermodynamics. Thermal engine. circular processes. Carnot cycle, efficiency of the Carnot cycle.
3 .Electrostatics
Electric field in vacuum.
The law of conservation of electric charge. Electric field. Main characteristics electric field: tension and potential. Tension as a potential gradient. Calculation of electrostatic fields by the superposition method. Tension vector flow. The Ostrogradsky-Gauss theorem for an electrostatic field in vacuum. Application of the Ostrogradsky-Gauss theorem to the calculation of the field.
Electric field in dielectrics.
Free and bound charges. Types of dielectrics. Electronic and orientational polarizations. Polarization. Dielectric susceptibility of matter. electrical displacement. Dielectric permittivity of the medium. Calculation of the field strength in a homogeneous dielectric.
conductors in an electric field.
The field inside the conductor and at its surface. Distribution of charges in a conductor. Electric capacity of a solitary conductor. Mutual capacitance of two conductors. Capacitors. Energy of charged conductor, capacitor and system of conductors. The energy of the electrostatic field. Volumetric energy density.
DC electric current
Current strength. current density. Conditions for the existence of a current. Third party forces. Electromotive force of the current source. Ohm's law for an inhomogeneous section of an electrical circuit. Kirchhoff's rules. Work and power of electric current. Joule-Lenz law. Classical theory of electrical conductivity of metals. Difficulties of classical theory.
Electromagnetism
Magnetic field in vacuum.
Magnetic interaction of direct currents. A magnetic field. Magnetic induction vector. Ampere's law. The magnetic field of the current. Biot-Savart-Laplace law and its application to calculation magnetic field straight conductor with current. Magnetic field of circular current. The total current law (circulation of the magnetic induction vector) for a magnetic field in vacuum and its application to the calculation of the magnetic field of a toroid and a long solenoid. magnetic flux. The Ostrogradsky-Gauss theorem for a magnetic field. Vortex nature of the magnetic field The action of the magnetic field on a moving charge. Lorentz force. Movement of charged particles in a magnetic field. Rotation of a circuit with current in a magnetic field. The work of moving a conductor and a circuit with current in a magnetic field.
Electromagnetic induction.
The phenomenon of electromagnetic induction (experiments of Faraday). Lenz's rule. The law of electromagnetic induction and its derivation from the law of conservation of energy. The phenomenon of self-induction. Inductance. Currents during the closing and opening of an electrical circuit containing inductance. Energy coil with current. Volumetric energy density of the magnetic field.
Magnetic field in matter.
Magnetic moment of atoms. types of magnets. Magnetization. Micro and macro currents. Elementary theory of dia- and paramagnetism. The total current law for a magnetic field in matter. Magnetic field strength. Magnetic permeability of the medium. Ferromagnets. Magnetic hysteresis. Curie point. Spin nature of ferromagnetism.
Maxwell's equations.
Faraday and Maxwell's interpretations of the phenomenon of electromagnetic induction. bias current. Maxwell's system of equations in integral form.
oscillatory motion
The concept of oscillatory processes. Unified approach to vibrations of different physical nature.
Amplitude, frequency, phase of harmonic oscillations. Addition of harmonic vibrations. Vector diagrams.
Pendulum, weight on a spring, oscillating circuit. Free damped vibrations. Differential equation damped oscillations Damping coefficient, logarithmic decrement, quality factor.
Forced vibrations under sinusoidal action. Amplitude and phase during forced oscillations. resonance curves. Forced oscillations in electrical circuits.
Waves
The mechanism of wave formation in an elastic medium. Longitudinal and transverse waves. Plane sine wave. Running and standing waves. Phase velocity, wavelength, wave number. One-dimensional wave equation. Group velocity and wave dispersion. Energy ratios. Umov vector. Plane electromagnetic waves. Wave polarization. Energy ratios. Pointing vector. dipole radiation. radiation pattern
8 . wave optics
Light interference.
Coherence and monochromaticity of light waves. Calculation of the interference pattern from two coherent sources. Young's experience. Interference of light in thin films. Interferometers.
Diffraction of light.
Huygens-Fresnel principle. Fresnel zone method. Rectilinear propagation of light. Fresnel diffraction by a circular hole. Fraunhofer diffraction at a single slit. Diffraction grating as a spectral device. The concept of the holographic method of obtaining and restoring the image.
polarization of light.
Natural and polarized light. Polarization upon reflection. Brewster's law. Analysis of linearly polarized light. Malus' law. Double refraction. Artificial optical anisotropy. Electro-optical and magneto-optical effects.
dispersion of light.
Regions of normal and anomalous dispersion. Electronic theory of light dispersion.
The quantum nature of radiation
Thermal radiation.
Characteristics of thermal radiation. absorption capacity. Black body. Kirchhoff's law for thermal radiation. Stefan-Boltzmann law. Distribution of energy in the spectrum of a completely black body. Wien's displacement law. Quantum hypothesis and Planck's formula.
The quantum nature of light.
External photoelectric effect and its laws. Einstein's equation for the external photoelectric effect. Photons. Mass and momentum of a photon. Light pressure. Lebedev's experiments. Quantum and wave explanation of light pressure. Corpuscular-wave dualism of light.
Landing on a planet is considered the completion of a space flight. To date, only three countries have learned how to return to Earth spacecraft: Russia, USA and China.
For planets with an atmosphere (Fig. 3.19), the problem of landing comes down mainly to solving three problems: overcoming high level overloads; protection against aerodynamic heating; control of 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; BUT- spacecraft deorbit; M- separation of the SA from the orbital spacecraft; IN- SA entry into the dense layers of the atmosphere; FROM - start of operation of the parachute landing system; D- landing on the surface of the planet;
1 - ballistic descent; 2 - planning descent
When landing on a planet without an atmosphere (Fig. 3.20, but, b) the problem of protection against aerodynamic heating is removed.
spacecraft in orbit artificial satellite planet or approaching a planet with an atmosphere to land on it has a large reserve of kinetic energy associated with the speed of the spacecraft and its mass, and potential energy, determined by 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:
but- descent to the planet with a preliminary entry into the waiting orbit;
b- soft landing of a spacecraft with a braking engine and a landing gear;
I - hyperbolic trajectory of approach to the planet; II - orbital trajectory;
III - trajectory of descent from orbit; 1, 2, 3 - active flight segments during braking and soft landing
When entering the dense layers of the atmosphere, a shock wave arises in front of the nose of the SA, heating the gas to a high temperature. As the SA descends into the atmosphere, it slows down, its speed decreases, and the hot gas heats the SA more and more. The kinetic energy of the apparatus is converted into heat. Wherein most of energy is removed to the surrounding space in two ways: most of the heat is removed in surrounding atmosphere due to the action of strong shock waves and due to heat radiation from the heated surface of the SA.
The strongest shock waves occur with a blunt nose shape, which is why CA uses blunt shapes, rather than the pointed ones that are characteristic of low-speed flight.
With increasing speeds and temperatures, most of the heat is transferred to the apparatus not due to friction against 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-shielding layer;
– radiation cooling of the surface;
– application of blown coatings.
Before entering the dense layers of the atmosphere, the spacecraft trajectory obeys the laws of celestial mechanics. In the atmosphere, in addition to gravitational forces, the apparatus is affected by aerodynamic and centrifugal forces that change the shape of the trajectory of its movement. The force of attraction is directed towards the center of the planet, the force of aerodynamic resistance is in the direction opposite to the velocity vector, the centrifugal and lifting forces are perpendicular to the direction of the SA motion. The force of aerodynamic resistance reduces the speed of the vehicle, while the centrifugal and lift forces give it accelerations in the 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 a lifting force from the SA, the trajectory of its movement in the atmosphere is called ballistic (the trajectory of the descent of the SA spaceships series "Vostok" and "Voskhod"), and in the presence of a lifting force - either gliding (CA CC Soyuz and Apollo, as well as the Space Shuttle) or ricocheting (CA CC Soyuz and Apollo). Movement in a planetocentric orbit does not impose high requirements on the accuracy of pointing during reentry, since it is relatively easy to correct the trajectory by turning on the propulsion system for braking or accelerating. When entering the atmosphere at a speed exceeding the first cosmic one, errors in calculations are most dangerous, since a too steep descent can lead to the destruction of the SA, and a too gentle descent can lead to a 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 high 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 strong aerodynamic heating of the apparatus and to overloads, sometimes exceeding 10g - close to the maximum permissible values for a person.
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 aerodynamic forces has a component perpendicular to the velocity vector of the apparatus - lifting force. Due to the lifting force, the vehicle descends more slowly, the trajectory of its descent becomes more gentle, while the braking section is stretched both in length and in time, and the maximum overloads and the intensity of aerodynamic heating can be reduced several times compared to ballistic braking, which makes the glider descent for people more safe and comfortable.
The angle of attack during descent changes depending on the airspeed and the current air density. In the upper, rarefied layers of the atmosphere, it can reach 40°, gradually decreasing as the apparatus descends. This requires the presence of a gliding flight control system on the SA, which complicates and makes the device heavier, and in cases where it serves to launch only equipment that can withstand higher overloads than a person, ballistic braking is usually used.
The Space Shuttle orbital stage, which, when returning to Earth, performs the function of a descent vehicle, glides throughout the entire descent section from entry into the atmosphere to touching the landing gear, after which a braking parachute is released.
After the speed of the apparatus decreases to subsonic in the aerodynamic deceleration section, further descent of the SA can be carried out using parachutes. Parachute in dense atmosphere dampens the vehicle's speed to near zero and ensures a soft landing on the surface of the planet.
In the rarefied atmosphere of Mars, parachutes are less effective, so at the final stage of the descent, the parachute is unhooked and the landing rocket engines are turned on.
The Soyuz TMA-01M series manned descent vehicles designed for landing on land also have solid-fuel deceleration engines that are activated a few seconds before touchdown to ensure a safer and more comfortable landing.
The descent vehicle of the Venera-13 station, after descending on a parachute to an altitude of 47 km, dropped it and resumed aerodynamic braking. Such a descent program was dictated by the peculiarities of the atmosphere of Venus, the lower layers of which are very dense and hot (up to 500 ° C), and cloth parachutes would not have survived 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 assumed that at the final stage of descent, after aerodynamic braking in the atmosphere, a non-parachute powered landing on rocket engines is also expected. Descent vehicles can differ significantly in design depending on the nature of the payload and on the physical conditions on the surface of the planet on which they are landing.
When landing on a planet without an atmosphere, the problem of aerodynamic heating is removed, but for landing, speed reduction is carried out using a braking propulsion system, which must operate in the programmable thrust mode, and the mass of fuel in this case can significantly exceed the mass of the SA itself.
ELEMENTS OF CONTINUOUS MEDIA MECHANICS
A continuous medium is considered to be 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 consider the main laws that hold in these environments.
Under the action of applied forces, the bodies change their shape and volume, that is, they are deformed.
For solids There are deformations: elastic and plastic.
Elastic deformations are called deformations that disappear after the termination of the action of forces, and the bodies restore their shape and volume.
Plastic deformations are called deformations that persist after the termination of the action of forces, and the bodies do not restore their original shape and volume.
Plastic deformation occurs during cold working of metals: stamping, forging, etc.
The deformation will be elastic or plastic depends not only on the properties of the material of the body, but also on the magnitude of the applied forces.
Bodies that experience only elastic deformations under the action of any forces are called perfectly elastic.
For such bodies, there is an unambiguous relationship between the acting forces and the elastic deformations caused by them.
We restrict ourselves to elastic deformations, which obey the law Hooke.
All solids can be divided into isotropic and anisotropic.
Isotropic bodies are bodies whose physical properties are the same in all directions.
Anisotropic bodies are bodies whose physical properties are different in different directions.
The above definitions are relative, since real bodies can behave as isotropic with respect to some properties and as anisotropic with respect to others.
For example, crystals of the cubic system behave as isotropic if light propagates through them, but they are anisotropic if their elastic properties are considered.
In what follows, we confine ourselves to the study of isotropic bodies.
The most widespread in nature are metals with a polycrystalline structure.
Such metals consist of many tiny randomly oriented crystals.
As a result of plastic deformation, the randomness in the orientation of crystals can be broken.
After the termination of the action of forces, the substance will be anisotropic, which is observed, for example, when the wire is pulled and twisted.
The force per unit area of the surface on which they act is called mechanical stress. n .
If the stress does not exceed the elastic limit, then the deformation will be elastic.
The limiting stresses applied to the body, after the action of which it still retains its elastic properties, is called the elastic limit.
There are stresses of compression, tension, bending, torsion, etc.
If under the action of forces applied to the body (rod), it is stretched, then the resulting stresses are called tension
If the rod is compressed, then the resulting stresses are called pressure:
.
(7.2)
Consequently,
T = - R. (7.3)
If 
- the length of the undeformed rod, then after the application of force, it receives an elongation 
.
Then the length of the rod
. (7.4)
Attitude 
to 
, is called relative elongation, i.e.
. (7.5)
Based on experiments, Hooke established the law: within the limits of elasticity, the stress (pressure) is proportional to the relative elongation (compression), i.e.
(7.6)
,
(7.7)
where E is Young's modulus.
Relations (7.6) and (7.7) are valid for any rigid body, but up to a certain limit.
Up to point A (elastic limit), after the termination of the force, the length of the rod returns to its original (elastic deformation region).
Beyond the limits of elasticity, the deformation becomes partially or completely irreversible (plastic deformation). For most solids, linearity is maintained almost to the elastic limit. If the body continues to stretch, it will collapse.
The maximum force that can be applied to a body without breaking it is called tensile strength(P. B, Fig. 7.1).
Consider an arbitrary continuous medium. Let it be divided into parts 1 and 2 along the surface A-a-B-b (Fig. 7.2).
If the body is deformed, then its parts interact with each other along the interface along which they border.
To determine the resulting stresses, in addition to the forces acting in the section A-a-B-b, you need to know how these forces are distributed over the section.
,
(7.8)
where 
is the unit vector of the normal to the area dS.
.
(7.9)
To determine the mechanical stress in the medium, on an oppositely oriented site, at any point, it is enough to set the stresses on three mutually perpendicular sites: S x, S y, S–, passing through this point, for example, point 0 (Fig. 7.3 ).
This position is valid for a medium at rest or moving with arbitrary acceleration.
In this case
,
(7.10)
where 
(8.11)
S is the area of the ABC face; n is the outer normal to it.
Consequently, the stress at each point of an elastically deformed body can be characterized by three vectors 
or nine of their projections on the X, Y, Z coordinate axes:
(7.12)
who are called elastic stress tensor.
LECTURE №5 Elements of mechanics continuous media
Physical model: a continuous medium is a model of matter, in
neglected internal structure substances
assuming that matter is continuously distributed
throughout
the volume it occupies and completely fills this volume.
A medium is called homogeneous if it has the same values at every point.
properties.
An isotropic medium is one whose properties are the same in all
directions.
Aggregate states of matter
A solid is a state of matter characterized by
fixed volume and invariable form.
Liquid
–
condition
substances
characterized
fixed volume, but not having a definite shape.
A gas is a state of matter in which a substance fills the entire
the amount given to him.
Deformation is a change in the shape and size of the body.
Elasticity - the property of bodies to resist changes in their volume and
shape under load.
A deformation is said to be elastic if it disappears after removal.
load and - plastic, if after removal of the load it does not
disappears.
In the theory of elasticity, it is proved that all types of deformations
(tension - compression, shear, bending, torsion) can be reduced to
simultaneously occurring tensile-compressive and
shift. Tensile-compressive deformation
Tension - compression - increase (or
reduction) of the body length of a cylindrical or
prismatic shape, caused by force,
directed along its longitudinal axis.
Absolute deformation is a value equal to
change
body size caused
external influence:
l l l0
,
(5.1)
where l0 and l are the initial and final body lengths.
Hooke's Law (I) (Robert Hooke, 1660): strength
elasticity
proportional
size
absolute deformation and is directed to
direction of decrease:
F kl ,
where k is the coefficient of elasticity of the body.
(5.2)Relative deformation:
l l0
.
(5.3)
Mechanical stress - value,
characterizing the state
deformed body =Pa:
F S
,
(5.4)
where F is the force causing the deformation,
S is the sectional area of the body.
Hooke's Law (II): Mechanical stress,
arising in the body, in proportion
the value of its relative deformation:
E
,
(5.5)
where E - Young's modulus - value,
characterizing
elastic
properties
material, numerically equal to the stress,
arising in the body with a single
relative deformation, [E]=Pa. Deformations of solids obey Hooke's law up to
known limit. Relationship between strain and stress
represented in the form of a voltage diagram, the qualitative course
which is considered for a metal bar. Elastic strain energy
In tension - compression, the 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
tension - compression
w
energy
1 2
E
V 2
Bulk density
shear deformations
elastic
.
energy
1
w G 2
2
at
(5.9)
elastic
.
deformations
deformations
(5.10)
at Elements of mechanics of liquids and gases
(hydro- and aeromechanics)
Being in solid state of aggregation, body at the same time
has both elasticity of form and elasticity of volume (or, what
the same, under deformations in a solid, they arise as
normal and tangential stresses).
Liquids
and gases have only volume elasticity, but not
have elasticity of form (they take the form of a vessel, in
which
liquids
are located).
And
gases
Consequence
is an
this
general
sameness
in
peculiarities
quality
most of the mechanical properties of liquids and gases, and
their difference is
only
quantitative characteristics
(for example, as a rule, the density of a liquid is greater than the density
gas). Therefore, within the framework of continuum mechanics, one uses
unified approach to the study of liquids and gases. Initial characteristics
The density of matter is a scalar physical quantity,
characterizing the distribution of mass over the volume of a substance and
determined by the ratio of the mass of a substance enclosed in
some volume, to the value of this volume =m/kg3.
In the case of a homogeneous medium, the density of a substance is calculated from
formula
m V .
(5.11)
In the general case of an inhomogeneous medium, the mass and density of matter
related by the ratio
V
(5.12)
m dV .
0
Pressure
is a scalar value characterizing the state
liquid or gas and equal to the force that acts on a unit
surface in the direction of the normal to it [p]=Pa:
p Fn S
.
(5.13)Elements of hydrostatics
Peculiarities of forces acting inside a fluid at rest
(gas)
1) If a small volume is allocated inside a fluid at rest, then
liquid exerts the same pressure on this volume in all
directions.
2) A fluid at rest acts on the fluid that comes into contact with it
the surface of a rigid body with a force directed along the normal to this
surfaces. Continuity equation
A stream tube is a part of a fluid bounded by stream lines.
Such a flow is called stationary (or steady)
liquid, in which the shape and location of the streamlines, as well as
velocity values at each point of the moving fluid with
do not change over time.
Mass flow rate is the mass of fluid passing through
cross section of the current tube per unit time = kg / s:
Qm m t Sv ,
(5.15)
where and v are the density and velocity of the fluid flow in the section S. The equation
continuity
–
mathematical
ratio,
in
according to which, in a stationary flow of a liquid, its
the mass flow in each section of the current tube is the same:
1S1v 1 2S2v 2 or Sv const
,
(5.16)An incompressible liquid is a liquid whose density does not depend on
temperature and pressure.
The volumetric flow rate of a liquid is the volume of liquid passing through
cross section of the current tube per unit time = m3/s:
QV V t Sv ,
(5.17)
The continuity equation for an incompressible homogeneous fluid is
mathematical relation according to which
stationary flow of an incompressible homogeneous fluid, its
the volume flow in each 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 resist
movement of one part relative to another.
Physical model: ideal fluid - imaginary
an incompressible fluid that has no viscosity and
thermal conductivity.
Bernoulli's equation (Daniel Bernoulli 1738) is an equation that
being
consequence
law
conservation
mechanical
energy for a stationary flow of an ideal incompressible fluid
and written for an arbitrary section of the current tube located in
gravity field:
v 12
v 22
v2
gh1 p1
gh2 p2 or
gh p const . (5.19)
2
2
2In the Bernoulli equation (5.19):
p - static pressure (liquid pressure on the surface
body streamlined by it;
v2
- dynamic pressure;
2
gh - hydrostatic pressure. Internal friction (viscosity). Newton's law
Newton's law (Isaac Newton, 1686): the force of internal friction,
per unit area of moving layers of liquid or
gas, is directly proportional to the velocity gradient of the layers:
F
S
dv
dy
,
(5.20)
where is the coefficient of internal friction (dynamic viscosity),
= m2 / s. Types of viscous fluid flow
Laminar flow is a form of flow in which a liquid or
gas moves in layers without mixing and pulsations (i.e.
erratic rapid changes in velocity and pressure).
Turbulent flow - a form of flow of a liquid or gas, with
which
them
elements
commit
disordered,
unsteady movements along complex trajectories, which leads to
intensive mixing between layers of moving liquid
or gas. Reynolds number
The criterion for the transition of the laminar regime of fluid flow into
turbulent regime is based on the use of the Reynolds number
(Osborne Reynolds, 1876-1883).
In the case of fluid movement through a pipe, the Reynolds number
defined as
v d
Re
,
(5.21)
where v is the fluid velocity averaged over the pipe section; d - diameter
pipes; and - density and coefficient of internal friction
liquids.
For Re<2000 реализуется ламинарный режим течения
fluid through the pipe, and at Re>4000 - turbulent mode. At
values 2000
Consider the flow of a viscous fluid by referring directly to
to experience. Using a rubber hose, connect to the water supply
tap a thin horizontal glass tube with soldered into it
vertical manometric tubes (see figure).
At a low flow rate, a decrease in the level is clearly visible.
water in manometric tubes in the direction of flow (h1>h2>h3). This
indicates the presence of a pressure gradient along the axis of the tube -
the static pressure in the fluid decreases with the flow. Laminar flow of a viscous fluid in a horizontal pipe
With a uniform rectilinear flow of a liquid, the pressure forces
balanced by viscous forces. Distribution
section
flow
speeds
viscous
in
transverse
liquids
can
observe when it flows out of the vertical
tubes through a narrow hole (see figure).
If, for example, when the tap K is closed, pour
at first
uncolored glycerin, and then
carefully add tinted on top, then in
state of equilibrium, the interface D will be
horizontal.
If tap K is opened, then the border will take
a shape similar to a paraboloid of revolution. This
indicates
on the
Existence
distribution
velocities in the cross section of the tube in viscous flow
glycerin. Poiseuille formula
The distribution of velocities in the cross section of a horizontal pipe at
laminar flow of a viscous fluid is determined by the formula
p 2 2
vr
R r
4l
,
(5.23)
where R and l are the radius and length of the pipe, respectively, p is the difference
pressure at the ends of the pipe, r is the distance from the pipe axis.
The volumetric flow rate of the liquid is determined by the Poiseuille formula
(Jean Poiseuille, 1840):
R4p
.
(5.24)
Qv
8l Motion of bodies in a viscous medium
When moving bodies in a liquid or gas on a body
there is an internal friction force that depends on
body movement speed. At low speeds
observed
laminar
flow around
body
liquid or gas and internal friction force
turns out
proportional
speed
body movements and is 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 flow, l -
typical body size.
For a ball (b=6 , l=R) internal friction force:
F6Rv
where R is the radius of the ball.
,
7.1. General properties of liquids and gases. Kinematic description of fluid motion. Vector fields. Flow and circulation of a vector field. Stationary flow of an ideal fluid. Lines and tubes of current. Equations of motion and equilibrium of a fluid. Continuity equation for an incompressible fluid
Continuum mechanics is a branch of mechanics devoted to the study of the motion and equilibrium of gases, liquids, plasmas, and deformable solids. The main assumption of continuum mechanics is that matter can be considered as a continuous continuum, neglecting its molecular (atomic) structure, and at the same time, the distribution of all its characteristics (density, stresses, 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 from the side of low temperatures by a phase transition in solid state(crystallization), and from the side high temperatures- into gaseous (evaporation). When studying the properties of a continuous medium, the medium itself is represented as consisting of particles whose sizes are many more sizes molecules. Thus, each particle includes a huge number of molecules.
To describe the motion of a fluid, one 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 the liquid, but individual points in space, and note the speed with which individual particles of the liquid pass through each point. The second method is called the Euler method.
The state of fluid motion can be determined by specifying for each point in space the velocity vector as a function of time.
The set of vectors given for all points in space forms the field of the velocity vector, which can be represented as follows. Let's draw lines in a moving liquid so that the tangent to them at each point coincides in direction with the vector (Fig. 7.1). These lines are called streamlines. We 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 speed at a given location. Then, according to 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 thicker.
The number of streamlines passing through the area perpendicular to the streamlines is , if the area is arbitrarily oriented to the streamlines, the number of streamlines is , where is the angle between the direction of the vector and the normal to the area. The notation is often used. The number of streamlines through a platform of finite dimensions is determined by the integral: . An integral of this kind is called the vector flow through the area.
The magnitude and direction of the vector changes with 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 velocity. The streamline pattern in this case does not change, and the streamlines coincide with the particle trajectories.
The flow of a vector through a certain surface and the circulation of a vector along a given contour make it possible to judge the nature of the vector field. However, these values give average characteristic fields within the volume enclosed by the surface through which the flow is determined, or in the vicinity of the contour along which the circulation is taken. Reducing the dimensions of the surface or contour (contracting them to a point), one can arrive at values that will characterize the vector field at a given point.
Consider the field of the velocity vector of an incompressible inseparable fluid. The flow of the velocity vector through a certain surface is equal to the volume of fluid flowing through this surface per unit time. We construct an imaginary closed surface S in the vicinity of the point P (Fig. 7.2). If in the volume V bounded by the surface, the liquid does not appear and does not disappear, then the flow flowing outward through the surface will be equal to zero. If the flow differs from zero, it 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 sources and sinks. With the predominance of sources over sinks, the flow is positive, with the predominance of sinks, it is negative.
The quotient of dividing the flow by the value of the volume from which the flow flows, , is the average specific power of the sources contained in the volume V. The smaller the volume V, which includes the point P, the closer this average value is to the true specific power at this point. In the limit at , i.e. when the volume is contracted to a point, we will get the true specific power of the sources at the 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, bounding the volume V. Divergence is determined by the behavior of the vector function near the point P. Divergence is a scalar function of the coordinates that determine the position of the point P in space.
Let's find the expression for divergence in Cartesian coordinates. Let's consider a small volume in the form of a parallelepiped with edges parallel to the coordinate axes in the vicinity of the point P (x, y, z) (Fig. 7.3). In view of the smallness of the volume (we will tend to zero), the values within each of the six faces of the parallelepiped can be considered unchanged. The flow through the entire closed surface is formed from flows flowing through each of the six faces separately.
Let's find the flow through a pair of faces perpendicular to the rest 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 flow through face 2 is equal to . The total flow in the X direction is . The difference is the increment when offset along the X axis by . Due to the smallness, this increment can be represented as . Then we get . Similarly, through pairs of faces perpendicular to the Y and Z axes, the flows are equal to and . Total flow through a closed surface. Dividing this expression by , we find the divergence of the vector at the 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 bounded by the surface S into an infinitely large number of infinitely small elements (Fig. 7.4).
For any element, the vector flow through the surface of this element is . Summing over all elements , we obtain the flow through the surface S, which bounds the volume V: , integration is performed over the volume V, or
This is the Ostrogradsky-Gauss theorem. Here , is the unit normal to the surface dS at the given point.
Let us return to the flow of an incompressible fluid. Let's build a contour. Let's imagine that we have somehow frozen the liquid instantly in the entire volume, except for 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 contour length, . This value is called the circulation of the vector along the contour (since the channel has a constant cross section and the velocity modulus does not change). At the moment of solidification of the walls, for each fluid particle in the channel, the velocity component perpendicular to the wall will be extinguished, and only the component tangential to the contour will remain. This component is associated with the momentum , 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, is the channel cross section. The fluid is ideal - there is no friction, so the action of the walls can only change the direction, its value will remain constant. The interaction between fluid particles will cause such a redistribution of momentum between them, which will equalize the speeds of all particles. In this case, the algebraic sum of the impulses is preserved, therefore, where is the circulation velocity, 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 properties of the field, averaged over a region with dimensions on the order of the contour diameter . To obtain the field characteristic at the point P, it is necessary to reduce the size of the contour, contracting it to the point P. In this case, the limit of the ratio of the circulation of the vector along the flat contour, contracting to the point P, to the value of the contour plane S: is taken as the field characteristic. The value 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 space, which can be set by the direction of the positive normal to the plane of the contour (positive is the normal associated with the direction of bypassing the contour by the rule of the right screw). Defining this limit for different directions, we get different values, and for the opposite directions of the normal, these values differ in sign. For some direction of the normal, the limit value will be maximum. Thus, the limit value behaves as a projection of some vector onto the direction of the normal to the plane of the contour along which the circulation is taken. Maximum value limit determines the modulus of this vector, and the direction of the positive normal at which the maximum is reached gives the direction of the vector. This vector is called the rotor or vortex of the vector : .
To find the 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 area S, for which the normal to the area coincides with one of axes X,Y,Z. If, for example, direct along the X axis, we find . The contour is located in this case in a plane parallel to YZ, let's take the contour in the form of a rectangle with sides and . At , the values 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 circuit, we obtain the value: . The difference is the increment when you move along Y by . Due to the smallness, this increment can be represented as . Similarly, the difference . Then the circulation along the considered contour ,
where is the area of the contour. Dividing the circulation by , we find the projection of the rotor on the X axis: . Similarly, , . Then the rotor of the vector is determined by the expression: + ,
Knowing the curl of the vector at each point of some 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 the Stokes theorem.
The part of the fluid bounded by streamlines is called the streamtube. The vector , being tangent to the streamline 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.
Consider the section of the current tube S (Fig. 7.8.) perpendicular to the direction of velocity. We will assume that the velocity of fluid particles is the same at all points of this section. In time, all particles will pass through the section S, the distance of which at the initial moment does not exceed the value . Therefore, in time, a volume of liquid will pass through the section S, equal to , and per unit time, a volume of liquid will pass through the section S, equal to .. We assume that the stream tube is so thin that the speed of particles in each of its sections 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 liquid between the sections and (Fig. 7.9.) will remain unchanged. Then the volumes of fluid flowing per unit time through the sections and must be the same:
Thus, for an incompressible fluid, the value in any section of the same stream tube must be the same:
This statement is called the jet continuity theorem.
The motion of an ideal fluid is described by the Navier-Stokes equation:
where t is the 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 makes it possible to determine the projections of the velocity of a medium particle as a function of coordinates and time. To close the system, a continuity equation is added to the Navier-Stokes equation, which is a consequence of the jet continuity theorem:
To integrate these equations, it is required to set the initial (if the motion is not stationary) and boundary conditions.
7.2. Pressure in a flowing fluid. Bernoulli's equation and its corollary
Considering the motion 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 single out a stream tube of small cross section in a stationary flowing ideal fluid (Fig. 7.10). Let us consider the volume of liquid bounded by the walls of the stream tube and cross sections perpendicular to the stream lines and . Over time, this volume will move along the stream tube, and the section will move to position , having passed the path , the section will move to 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 a time in any of the points of the unshaded part of the considered volume (for example, point O in Fig. 7.10) has the same speed (and the same kinetic energy), which had a particle that was at the same point at the initial moment of time. Therefore, the energy increment of the entire considered volume is equal to the difference between the energies of the shaded volumes and .
In an ideal fluid, there are no friction forces, so the energy increment (7.1) is equal to the work done on the selected volume by the pressure forces. The pressure forces on the side surface are perpendicular at each point to the direction of movement of the particles and no work is performed. The work of forces applied to the sections and is equal to
Equating (7.1) and (7.2), we obtain
Since the sections and were taken arbitrarily, it can be argued that the expression remains constant in any section of the current tube, i.e. in a stationary ideal fluid flowing along any streamline, the condition
This is the Bernoulli equation. For a horizontal streamline, equation (7.3) takes the form:
7.3. OUTPUT OF LIQUID FROM THE HOLE
Let us apply the Bernoulli equation to the case of liquid outflow from a small hole in a wide open vessel. Let's 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 some initial level can be considered the same, the pressures in both sections are equal to atmospheric and also the same, and 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. Forces of internal friction
An ideal fluid, i.e. fluid without friction, is an abstraction. All real liquids and gases, to a greater or lesser extent, have viscosity or internal friction.
Viscosity is manifested in the fact that the movement that has arisen in a liquid or gas after the termination of the action of the forces that caused it, gradually stops.
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 bottom plate is held in place, the top plate is set in motion relative to the bottom 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 well-defined 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 fluid. Let us denote it, and the part of the fluid lying under the plane acts on the part of the fluid lying above the plane with the force . In this case, and are determined by formula (7.4). Thus, this formula expresses the force between the fluid layers in contact.
It has been experimentally proven that the velocity of fluid particles changes in the direction z, perpendicular to the plates (Fig. 7.6) according to a linear law
Liquid particles that are 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 sign of the modulus in this formula is set for the following reason. When the direction of movement is changed, the derivative of the velocity will change sign, while the ratio is always positive. In view of what has been said, expression (7.4) takes the form
The SI unit of viscosity is such a viscosity at which the velocity gradient with modulus , leads to the appearance of an internal friction force of 1 N per 1 m of the contact surface of the layers. This unit is called the Pascal second (Pa s).
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