If the action of forces is uniformly distributed over the entire surface of the corresponding face, then in any section parallel to these faces, a tangential stress arises
. Under the action of stresses, the body is deformed in such a way that one face is displaced relative to the other by a certain distance a. If the body is mentally divided into elementary layers parallel to the considered faces, then each layer will be shifted relative to the layers adjacent to it.

Under shear deformation, any straight line initially perpendicular to the layers will deviate by some angle φ. whose tangent is called the relative shift

, (2.61)

where b is the height of the face. Under elastic deformations, the angle φ is very small, so we can assume that
and
.

Experience shows that the relative shear is proportional to the tangential stress

, (2.62)

where G is the shear modulus.

Shear modulus depends only on the properties of the material and is equal to the tangential stress at an angle φ = 45˚. The shear modulus, like Young's modulus, is measured in pascals (Pa). Shearing a rod at an angle causes force

=GSφ, (2.63)

where G S is the coefficient of elasticity of the rod under shear deformation.

Viscosity(internal friction) ( English. viscosity) - one of the transfer phenomena, the property of fluid bodies (liquids and gases) to resist the movement of one of their parts relative to another. Mechanism internal friction in liquids and gases is that randomly moving molecules transfer momentum from one layer to another, which leads to the alignment of velocities - this is described by the introduction of a friction force. The viscosity of solids has a number of specific features and is usually considered separately. The basic law of viscous flow was established by I. Newton (1687): As applied to liquids, viscosity is distinguished:

• Dynamic (absolute) viscosity µ - the force acting on a unit area of ​​a flat surface, which moves at a unit speed relative to another flat surface located at a unit distance from the first. In the SI system, dynamic viscosity is expressed as Pa×s(pascal second), off-system unit P (poise).
• Kinematic viscosity ν is the ratio of dynamic viscosity µ to the density of the liquid ρ .

• ν , m 2 /s - kinematic viscosity;
• μ , Pa×s – dynamic viscosity;
• ρ , kg / m 3 - the density of the liquid.

Force of viscous friction

This is the phenomenon of the occurrence of tangential forces that prevent the movement of parts of a liquid or gas in relation to each other. Lubrication between two solids replaces dry sliding friction with sliding friction of liquid or gas layers against each other. The speed of the particles of the medium smoothly changes from the speed of one body to the speed of another body.

The force of viscous friction is proportional to the speed of relative motion V bodies, proportional to the area S and inversely proportional to the distance between the planes h.

The coefficient of proportionality, depending on the type of liquid or gas, is called dynamic viscosity coefficient. The most important thing in the nature of viscous friction forces is that in the presence of any arbitrarily small force, the bodies will begin to move, that is, there is no static friction. Qualitatively significant difference of forces viscous friction from dry friction

If a moving body is completely immersed in a viscous medium and the distances from the body to the boundaries of the medium are much greater than the dimensions of the body itself, then in this case we speak of friction or environment resistance. In this case, the sections of the medium (liquid or gas) immediately adjacent to the moving body move at the same speed as the body itself, and as the distance from the body increases, the speed of the corresponding sections of the medium decreases, turning to zero at infinity.

The resistance force of the medium depends on:

• its viscosity
• from body shape
• on the speed of the body relative to the medium.

For example, when a ball moves slowly in a viscous fluid, the friction force can be found using the Stokes formula:

A qualitatively significant difference between the forces of viscous friction and dry friction, among other things, the fact that the body in the presence of only viscous friction and an arbitrarily small external force will necessarily begin to move, that is, for viscous friction there is no static friction, and vice versa - under the influence of only viscous friction, the body, which initially moved, never (in macroscopic approximation that neglects Brownian motion) will not stop completely, although the motion will slow down indefinitely.

Viscosity of gases

The viscosity of gases (the phenomenon of internal friction) is the appearance of friction forces between gas layers moving relative to each other in parallel and at different speeds. The viscosity of gases increases with increasing temperature

The interaction of two layers of gas is considered as a process during which momentum is transferred from one layer to another. The force of friction per unit area between two layers of gas, equal to the momentum transferred per second from layer to layer through unit area, is determined by Newton's law:

where:
dv / dz- velocity gradient in the direction perpendicular to the direction of motion of the gas layers.
The minus sign indicates that momentum is carried in the direction of decreasing velocity.
η - dynamic viscosity.

ρ is the density of the gas,
(ν) - arithmetic mean speed of molecules
λ is the mean free path of the molecules.

Viscosity of some gases (at 0°C)

Fluid Viscosity

Fluid Viscosity- this is a property that manifests itself only when the fluid is in motion, and does not affect fluids at rest. Viscous friction in liquids obeys the law of friction, which is fundamentally different from the law of friction of solids, because depends on the area of ​​friction and the velocity of the fluid.
Viscosity- the property of a liquid to resist the relative shear of its layers. Viscosity is manifested in the fact that with the relative movement of fluid layers on the surfaces of their contact, shear resistance forces arise, called internal friction forces, or viscosity forces. If we consider how the velocities of different layers of the liquid are distributed over the cross section of the flow, then we can easily see that the farther from the walls of the flow, the greater the speed of the particles. At the walls of the flow, the fluid velocity is zero. An illustration of this is the drawing of the so-called jet flow model.

A slowly moving fluid layer "slows down" the adjacent fluid layer moving faster, and vice versa, a layer moving at a higher speed drags (pulls) a layer moving at a lower speed. Forces of internal friction appear due to the presence of intermolecular bonds between the moving layers. If a certain area is allocated between adjacent layers of the liquid S, then according to Newton's hypothesis:

• μ - coefficient of viscous friction;
• S is the area of ​​friction;
• du/dy- speed gradient

Value μ in this expression is dynamic viscosity coefficient, equal to:

• τ - shear stress in the liquid (depends on the type of liquid).

The physical meaning of the coefficient of viscous friction- a number equal to the friction force developing on a unit surface with a unit velocity gradient.

In practice, it is more often used kinematic viscosity coefficient, so named because its dimension lacks a force notation. This coefficient is the ratio of the dynamic coefficient of viscosity of the liquid to its density:

Units of measurement of the coefficient of viscous friction:

• N·s/m 2 ;
• kgf s / m 2
• Pz (Poiseuille) 1 (Pz) \u003d 0.1 (N s / m 2).

Analysis of the Viscosity Property of a Fluid

For dropping liquids, the viscosity depends on the temperature t and pressure R, however, the latter dependence manifests itself only at large pressure changes, on the order of several tens of MPa.

The dependence of the dynamic viscosity coefficient on temperature is expressed by a formula of the form:

• µt - coefficient of dynamic viscosity at a given temperature;
• μ 0 - coefficient of dynamic viscosity at a known temperature;
• T - set temperature;
• T 0 - temperature at which the value is measured μ 0 ;
• e

The dependence of the relative coefficient of dynamic viscosity on pressure is described by the formula:

• μ P - coefficient of dynamic viscosity at a given pressure,
• μ 0 - coefficient of dynamic viscosity at a known pressure (most often under normal conditions),
• R - set pressure,;
• P 0 - pressure at which the value is measured μ 0 ;
• e - the base of the natural logarithm is 2.718282.

The influence of pressure on the viscosity of a liquid appears only at high pressures.

Newtonian and non-Newtonian fluids

Newtonian liquids are liquids for which the viscosity does not depend on the strain rate. In the Navier-Stokes equation for a Newtonian fluid, there is a viscosity law similar to the above (in fact, a generalization of Newton's law, or Navier's law).

1. Internal friction (viscosity) of a liquid. Newton's equation.

2. Newtonian and non-Newtonian fluids. Blood.

3. Laminar and turbulent flows, Reynolds number.

4. Poiseuille's formula, hydraulic resistance.

5. Pressure distribution during the flow of a real liquid through pipes of various sections.

6. Methods for determining the viscosity of liquids.

7. Effect of viscosity on some medical procedures. Laminarity and turbulence of the gas flow during anesthesia. The introduction of fluids through a dropper and syringe. Rhinomanometry. Photohemotherapy.

8. Basic concepts and formulas.

9. Tasks.

Hydrodynamics- a branch of physics that studies the motion of incompressible fluids and their interaction with surrounding bodies.

8.1. Internal friction (viscosity) of a fluid. Newton's equation

In a real liquid, due to the mutual attraction and thermal motion of molecules, internal friction, or viscosity, takes place. Consider this phenomenon in the following experiment (Fig. 8.1).

Rice. 8.1. The flow of a viscous fluid between plates

Let us place a liquid layer between two parallel solid plates. The "bottom" plate is fixed. If we move the “upper” plate at a constant speed v 1, then the “upper” 1st layer of liquid, which we consider to be “adhered” to the upper plate, will move with the same speed. This layer affects the underlying 2nd layer directly below it, forcing it to move at a speed v 2 , and v 2< v 1 . Каждый слой (выделим n layers) transmits motion to the underlying layer at a lower speed. The layer directly "adhered" to the "bottom" plate remains motionless.

Layers interact with each other: nth layer speeds up the (n+1)th layer, but slows down the (n-1)th layer. Thus, there is a change in the fluid flow velocity in the direction perpendicular to the layer surface (x axis). Such a change is characterized by the derivative dv/dx, called speed gradient.

The forces acting between the layers and directed tangentially to the surface of the layers are called forces of internal friction or viscosity. These forces are proportional to the area of ​​the interacting layers S and to the velocity gradient. For many liquids, internal friction forces obey Newton's equation:

The coefficient of proportionality η is called the coefficient of internal friction or dynamic viscosity(dimension η in SI: Pas).

8.2. Newtonian and non-Newtonian fluids.

Blood

Newtonian fluid

A fluid that obeys Newton's equation (8.1) is called Newtonian. The coefficient of internal friction of a Newtonian fluid depends on its structure, temperature and pressure, but does not depend on the velocity gradient.

A Newtonian fluid is a fluid whose viscosity does not depend on the velocity gradient.

Most liquids (water, solutions, low molecular weight organic liquids) and all gases have the properties of a Newtonian fluid.

Viscosity is determined using special instruments - viscometers. The values ​​of the viscosity coefficient η for some liquids are presented in the table.

The value of blood viscosity presented in the table refers to a healthy person in a calm state. With heavy physical work, blood viscosity increases. Some diseases also affect the value of blood viscosity. Yes, at diabetes blood viscosity increases to 23?10 -3 Pas, and in case of tuberculosis it decreases to 1*10 -3 Pas. Viscosity affects such a clinical parameter as the erythrocyte sedimentation rate (ESR).

non-newtonian fluid

non-newtonian fluid A fluid whose viscosity depends on the velocity gradient.

Structured disperse systems (suspensions, emulsions), solutions and melts of some polymers, many organic liquids, etc. have the properties of a non-Newtonian liquid.

Other things being equal, the viscosity of such liquids is much greater than that of Newtonian liquids. This is due to the fact that due to the adhesion of molecules or particles in a non-Newtonian fluid, spatial structures are formed, the destruction of which requires additional energy.

Blood

Whole blood (a suspension of red blood cells in a protein solution - plasma) is a non-Newtonian fluid due to the aggregation of red blood cells.

The erythrocyte normally has the shape of a biconcave disc with a diameter of about 8 microns. It can significantly change its shape, for example, with different osmolarity of the medium (Fig. 8.2).

In immobile blood, erythrocytes aggregate, forming the so-called "coin columns", consisting of 6-8 erythrocytes. An electron microscopic examination of the thinnest sections of coin columns revealed the parallelism of the surfaces of adjacent erythrocytes and a constant intererythrocyte distance during aggregation (Fig. 8.3).

Figure 8.4 shows (drawing) the aggregation of whole blood in wet smears, which are large conglomerates composed of many rouleaux. When the blood is stirred, the aggregates are destroyed, and after the mixing is stopped, they are restored again.

When blood flows through the capillaries, the erythrocyte aggregates disintegrate and the viscosity decreases.

The implantation of special transparent windows into the skin folds made it possible to photograph the flow of blood in the capillaries. Figure 8.5, made from such a photograph, clearly shows the deformation of blood cells.

Rice. 8.2. Average cross-section of an erythrocyte at different medium osmolarity

Rice. 8.3. Scheme of the electron diffraction pattern of the aggregate from normal erythrocytes

Rice. 8.4. Whole blood aggregation

Rice. 8.5. Deformation of erythrocytes in capillaries

Deforming, erythrocytes can move one after another in capillaries with a diameter of only 3 microns. It is in such thin capillary vessels that gas exchange occurs between blood and tissues.

A very thin layer of plasma is formed near the capillary wall, which plays the role of a lubricant. Due to this, the resistance to the movement of red blood cells is reduced.

8.3. Laminar and turbulent flows, Reynolds number

In a liquid, the flow can be laminar or turbulent. Figure 8.6 shows this for one colored jet of liquid flowing into another.

In case (a), the jet of colored liquid retains an unchanged shape and does not mix with the rest of the liquid. In case (b), the colored jet is torn apart by random eddies, the pattern of which changes with time. The concept of "stream tube" is inapplicable to turbulent flow.

Rice. 8.6. Laminar (a) and turbulent (b) flow of a liquid jet

Laminar (layered) flow- a flow in which layers of liquid flow without mixing, sliding relative to each other. Laminar flow is stationary - the flow velocity at each point in space remains constant.

Consider the laminar flow of a Newtonian fluid in a pipe of radius R and length L, the pressures at the ends of which are constant (P 1 and P 2). Let's single out a cylindrical current tube of radius r (Fig. 8.7).

The liquid inside this tube is affected by the pressure force F d \u003d πg 2 (P 1 - P 2) and the viscous friction force F tr \u003d 2πrLηdv / dr (2πrL - flat

Rice. 8.7. Stream tube and friction force acting on it

spare side surface). Since the flow is stationary, the sum of these forces is zero:

In accordance with the above expression, there is a parabolic dependence of the velocity v liquid layers from the distance from them to the pipe axis r (the envelope of all velocity vectors is a parabola) (Fig. 8.8).

The current layer has the highest speed along the pipe axis(r = 0), the layer "adhered" to the wall (r = R) is immobile.

Rice. 8.8. The velocities of the layers of fluid flowing through the tube are distributed along a parabola

Turbulent (vortex) flow- a flow in which the velocities of fluid particles at each point randomly change. This movement is accompanied by the appearance of sound. Turbulent flow is a chaotic, highly irregular, disordered flow of a fluid. Fluid elements move along complex disordered trajectories, which leads to mixing of layers and the formation of local eddies.

The structure of a turbulent flow is an unsteady set of very a large number small eddies superimposed on the main "middle flow".

At the same time, it is possible to talk about a flow in one direction or another only on average over a certain period of time.

Turbulent flow is associated with additional energy consumption during fluid movement: part of the energy is spent on random movement, the direction of which differs from the main direction of flow, which in the case of blood leads to additional work hearts. The noise generated by the turbulent flow of blood can be used to diagnose the disease. This noise is heard, for example, on the brachial artery when measuring blood pressure.

Turbulent blood flow may occur due to uneven narrowing of the vessel lumen (or local bulging). Turbulent flow creates conditions for platelet sedimentation and formation of aggregates. This process is often the starting

in thrombus formation. In addition, if the thrombus is weakly connected to the vessel wall, then under the action of a sharp pressure drop along it, due to turbulence, it can begin to move.

Reynolds number

The concepts of laminarity and turbulence are applicable both to the flow of fluid through pipes and to the flow around various bodies. In both cases, the nature of the flow depends on the flow velocity, the properties of the fluid, and the characteristic linear size of the pipe or streamlined body.

The English physicist and engineer Osborne Reynolds (1842-1912) made up a dimensionless combination, the value of which determines the nature of the flow. Subsequently, this combination was called the Reynolds number (Re):

The Reynolds number is used in modeling hydro- and aerodynamic systems, in particular the circulatory system. The model must have the same Reynolds number as the object itself, otherwise there will be no match between them.

An important property of turbulent flow (compared to laminar flow) is high flow resistance. If it were possible to "extinguish" the turbulence, then it would be possible to achieve a huge saving in the power of the engines of ships, submarines, and aircraft.

8.4. Poiseuille formula, hydraulic resistance

Consider what factors determine the volume of fluid flowing through horizontal pipe.

Poiseuille formula

With a laminar flow of fluid through a pipe of radius R and length L, the volume Q of fluid flowing through a horizontal pipe in one second can be calculated as follows. We select a thin cylindrical layer of radius r and thickness dr (Fig. 8.9).

Rice. 8.9. Cross section of a pipe with a selected liquid layer

Its cross-sectional area is dS = 2πrdr. Since a thin layer is selected, the liquid in it moves at the same speed v. In one second, the layer will transfer the volume of liquid

Substituting here the formula for the velocity of a cylindrical liquid layer (8.4), we obtain

This relation is valid for the laminar flow of a Newtonian fluid.

The Poiseuille formula can be written in a form that is valid for pipes of variable cross section. Let us replace the expression (P 1 - P 2) / L with the pressure gradient dP / d /, then we get

As can be seen from (8.8), under given external conditions, the volume of liquid flowing through the pipe is proportional to fourth degree its radius. This is a very strong addiction. So, for example, if the radius of the vessels decreases by 2 times during atherosclerosis, then in order to maintain normal blood flow, the pressure drop must be increased by 16 times, which is practically impossible. As a result, oxygen starvation of the corresponding tissues occurs. This explains the occurrence of "angina pectoris". Relief can be achieved by administering a drug that relaxes the muscles of the arterial walls and allows an increase in the lumen of the vessel and, therefore, blood flow.

The flow of blood passing through the vessels is regulated by special muscles surrounding the vessel. With their contraction, the lumen of the vessel decreases and, accordingly, the blood flow decreases. Thus, a slight contraction of these muscles very accurately controls the flow of blood into the tissues.

In the body, by changing the radius of the vessels (narrowing or expansion), by changing the volumetric blood flow velocity, the blood supply to tissues and heat exchange with the environment are regulated.

Causes of the movement of blood through the vessels

home driving force blood flow - the pressure difference at the beginning and at the end of the vascular system: in the systemic circulation - the pressure difference in the aorta and the right atrium, in the small circle - in the pulmonary artery and the left atrium.

Additional factors that contribute to the movement of blood through the veins towards the heart:

1) semilunar valves of the veins of the extremities, which open under the pressure of blood only towards the heart;

2) suction action chest associated with negative pressure in it during inspiration;

3) contraction of the muscles of the limbs, for example, when walking. In this case, pressure is applied to the walls of the veins, and the blood, thanks to the valves and the suction action of the chest during inhalation, is squeezed out into areas located closer to the heart.

Hydraulic resistance

Let's draw an analogy between the Poiseuille formula and the formula of Ohm's law for a section of the current circuit: I = ΔU/R. To do this, we rewrite formula (8.8) in the following form: Q = (P 1 - Р 2)/. If we compare this formula with Ohm's law for electric current, then the volume of liquid flowing through the section of the pipe in one second corresponds to the strength of the current; the pressure difference at the ends of the pipe corresponds to the potential difference; and the value 8ηL /(πR 4) corresponds to electrical resistance. They call her hydraulic resistance:

The hydraulic resistance of a pipe is directly proportional to its length and inversely proportionalfourth degreeradius.

If the change in the kinetic energy of the fluid in a certain area can be neglected, then the considered analogy is also applicable to a flow of variable cross section:

The hydraulic resistance of a section is the ratio of the pressure drop to the volume of liquid flowing in 1 second:

The presence of hydraulic resistance is associated with overcoming the forces of internal friction.

The laws of hydrodynamics are much more complicated than the laws of direct current, therefore the laws of connecting pipes (blood vessels) are more complicated than the laws of connecting conductors. So, for example, places of a sharp narrowing of the flow (even with a small length) have a large own hydraulic resistance. This explains the significant increase in the hydraulic resistance of the blood vessel during the formation of a small plaque.

The presence of own resistance at places of a sharp narrowing of the flow must be taken into account when calculating the resistance of a section consisting

Rice. 8.10. Pipes connected in series (a) and in parallel (b)

from pipes of various diameters. On fig. 8.10a shows the series resistance of three pipes. Narrowing points have their own resistance X 12 and X 23 . Therefore, the resistance of the section is

The electrical analogue (8.13) of the formula for calculating the hydrodynamic resistance of a parallel connection (Figure 8.10, b) also requires taking into account the resistance of the pipe junctions.

8.5. Pressure distribution during the flow of a real liquid through pipes of various sections

When a real fluid flows through a horizontal pipe, the work of external forces is spent on overcoming internal friction. Therefore, the static pressure along the pipe gradually decreases. This effect can be demonstrated by a simple experiment. We install manometric tubes in different places of a horizontal pipe through which a viscous liquid flows (Fig. 8.11).

Rice. 8.11. Pressure drop of a viscous liquid in pipes of various sections

It can be seen from the figure that with a constant pipe cross section, the pressure drops in proportion to the length. At the same time, the rate of pressure drop (dP/d l) increases as the pipe section decreases. This is due to the increase in hydraulic resistance with decreasing radius.

In the human circulatory system, capillaries account for up to 70% of the pressure drop.

8.6. Methods for determining the viscosity of liquids

The set of methods for measuring the viscosity of a liquid is called viscometry. The device for measuring viscosity is called viscometer. Depending on the method of measuring viscosity, the following types of viscometers are used.

1. The Ostwald capillary viscometer is based on the Poiseuille formula. Viscosity is determined by measuring the time it takes for a liquid of known mass to flow through a capillary under the action of gravity at a certain pressure drop.

2. Medical Hess viscometer with two capillaries in which two liquids move (for example, distilled water and blood). The viscosity of a single liquid must be known. Considering that the movement of liquids in the same time is inversely proportional to their viscosity, the viscosity of the second liquid is calculated.

3. Viscometer based on the Stokes method, according to which, when a ball of radius R moves in a liquid with viscosity η at low speed v the drag force is proportional to the viscosity of this fluid: F = 6πηRv (Stokes formula). Red blood cells move in a viscous fluid - blood plasma. Since erythrocytes are disk-shaped and settle in a viscous liquid, their sedimentation rate (ESR) can be determined approximately by the Stokes formula. The sedimentation rate is judged by the amount of plasma above the settled erythrocytes. Normally, the erythrocyte sedimentation rate is 7-12 mm/h for women and 3-9 mm/h for men.

4. Viscometer rotary(Fig. 8.12) consists of two coaxial (coaxial) cylinders. The radius of the inner cylinder is R, the radius of the outer cylinder is R+ΔR (ΔR<< R). Пространство между цилин-

Rice. 8.12. Rotational viscometer (sections along and perpendicular to the axis)

cores are filled with the investigated liquid up to a certain height h. Then the inner cylinder is brought into rotation, applying a certain moment of forces M, and the steady-state speed ν is measured.

The viscosity of a liquid is calculated by the formula

Using a rotational viscometer, it is possible to measure the viscosity at different angular speeds of rotation of the rotor. This method makes it possible to establish the relationship between viscosity and velocity gradient, which is important for non-Newtonian fluids.

8.7. The effect of viscosity on some medical

procedures

anesthesia

Some medical procedures use anesthesia. At the same time, it is necessary, if possible, to reduce the efforts expended by the patient on breathing through the endotracheal and other respiratory tubes, through which the respiratory mixture is supplied from the anesthesia machines (Fig. 8.13).

Smoothly curved connecting tubes are used to ensure smooth gas flow. Irregularities in the inner walls of the tube, sharp bends and changes in the inner diameter of the tubes

Rice.8.13. Patient breathing through an endotracheal tube

Rice. 8.14. The occurrence of gas flow turbulence in a tube with sharp cross-sectional inhomogeneities

and compounds are often the reasons for the transition from laminar flow to turbulent (Fig. 8.14), which makes it difficult for the patient to breathe.

Figure 8.15 is an x-ray of the patient's head showing that the endotracheal tube is kinked in the throat. In this case, the patient will definitely have difficulty breathing.

Administering liquids through a syringe and drip

A syringe is a very simple device (Fig. 8.16) that is used for injections. Nevertheless, when describing its operation, an error is often made related to finding the pressure drop (ΔP) on the needle, which leads to an incorrect result. Think that

Rice. 8.15. X-ray showing a kink in the breathing tube

Rice. 8.16. Syringe operation

ΔP = F/S, where F is the force acting on the piston and S is its area. In this case, the following considerations are taken into account: the piston moves slowly and the dynamic pressure of the liquid in the cylinder can

neglect. This is not true - at the entrance to the needle, the streamlines thicken and the velocity of the liquid increases sharply.

A rigorous calculation (see problem 8.12) leads to the following result. Pressure drop across the needle (ΔP) is the solution to the quadratic equation

The values ​​of all quantities are substituted in SI.

Below are the results of calculations for two needles 4 cm long, the diameters of which differ by 1.5 times.

From the results in the table below, it can be seen that AP is not at all equal to F/S! In this case, an increase in the diameter of the needle by 1.5 times leads to an increase in the volumetric velocity by only 3.5 times, and not by 5 times (1.5 4 = 5.06), as might be expected. The laminar nature of the flow takes place in both cases.

Another device for intravenous infusion is a dropper (Fig. 8.17), which allows you to inject liquid by gravity due to the pressure difference created when the chamber with the drug is raised to a certain height (~ 60 cm).

Formulas 8.14, 8.15 are also applicable here if we replace the value of F/S with the hydrostatic pressure of the liquid column pgh. In this case, S is the cross-sectional area of ​​the tube, and u is the velocity of the fluid in it. Below are the results of calculations for h = 60 cm.

The values ​​obtained are correct, but do not correspond to what is actually happening. In this case, an overestimated value for the volumetric injection rate of the drug is obtained - 0.827 cm 3 /s. Real speed Q = 0.278 cm 3 / s (based on 500 ml in 30 minutes). The discrepancy is obtained due to the fact that the hydraulic resistance created by the device that clamps the tube is not taken into account.

Rhinomanometry

Full nasal breathing is a necessary prerequisite for the normal function of the auditory tube, which largely depends on the degree of aeration of the nasopharynx and the correct passage of air flows in the nasal cavity. The cause of nasal breathing disorders is often some congenital pathology, such as a cleft lip and palate. Often in the treatment of this pathology

Rice. 8.17. The introduction of the drug through a dropper

surgical methods are used, for example, reconstructive rhinoplasty (rhinoplasty - nose reconstruction operations). For an objective characterization of the results of surgical intervention, rhinomanometry is used - a method for determining the volume of nasal breathing and resistance. The airflow rate is characterized by the Poiseuille formula, taking into account the pressure gradient due to pressure changes in the nasopharyngeal space; diameter and length of the nasal cavity; characteristics of the air flow in the nasopharynx (laminarity or turbulence). This method is implemented using the device - rhinomanometer, which allows you to register pressure in one half of the nose while the patient breathes through the other. This is done using a catheter, which is specially attached to the nose. The computer scheme of the rhinomanometer allows you to automatically measure the total volume and resistance of air during inhalation and exhalation, separately analyze the flow and air resistance in each half of the nose and calculate their ratio. This allows you to determine nasal breathing before and after surgery and assess the degree of restoration of nasal breathing.

Photohemotherapy

In diseases accompanied by an increase in blood viscosity, the method of photohemotherapy is used to reduce blood viscosity. It consists in the fact that a small amount of blood (about 2 ml / kg of weight) is taken from the patient, subjected to UV irradiation and injected back into the bloodstream. Approximately 5 minutes after the administration of 100-200 ml of irradiated blood to patients, a significant decrease in viscosity is observed in the entire volume (about 5 liters) of circulating blood. Studies of the dependence of viscosity on the speed of blood movement have shown that during photohemotherapy, viscosity decreases the most (by about 30%) in slow moving blood and does not change at all in fast moving blood. UV irradiation causes a decrease in the ability of erythrocytes to aggregate and increases the deformability of erythrocytes. In addition, there is a decrease in the formation of blood clots. All these phenomena lead to a significant improvement in both macro- and microcirculation of the blood.

8.8. Basic concepts and formulas

End of table

8.9. Tasks

1. Derive a formula for determining the viscosity of a rotational viscometer. Given: R, ΔR, h, ν, M.

2. Determine the time of blood flow through the viscometer capillary if water flows through it in 10 s. The volumes of water and blood are the same. The density of water and blood are p 1 = 1 g/cm 3 , ρ 2 = 1.06 g/cm 3 . The viscosity of blood relative to water is 5 (η 2 /η 1 = 5).

3. Let us assume that the pressure gradient in two blood vessels is the same, and the blood flow (volume flow) in the second vessel is 80% less than in the first. Find the ratio of their diameters.

4. What should be the pressure difference AR at the ends of a capillary with a radius r = 1 mm and a length L = 10 cm, so that in a time t = 5 s a volume V = 1 cm 3 of water can be passed through it (viscosity coefficient η 1 = 10 -3 Pas ) or glycerin (η 2 = 0.85 Pas)?

5. The pressure drop in a blood vessel of length L = 55 mm and radius r = 1.5 mm is 365 Pa. Determine how many milliliters of blood flows through the vessel in 1 minute. Blood viscosity coefficient η = 4.5 mPa-s.

6. In atherosclerosis, due to the formation of plaques on the walls of the vessel, the critical value of the Reynolds number can decrease to 1160. For this case, determine the rate at which the transition from laminar to turbulent blood flow is possible in a vessel with a diameter of 2.5 mm. The density of the blood is ρ = 1050 kg/m 3 , the viscosity of the blood is η = 5x10 -3 Pas.

7. The average blood velocity in the aorta with a radius of 1 cm is 30 cm/s. Find out if the flow is laminar? Blood density ρ = 1.05x10 3 kg / m 3.

η \u003d 4x10 -3 Pa-s; Re cr = 2300.

8. With heavy physical exertion, the blood flow rate sometimes doubles. Using the data of the example of problem (7), determine the nature of the flow in this case.

Solution

Re = 2x1575 = 3150. The flow is turbulent.

Answer: the Reynolds number is greater than the critical value, so the flow can become turbulent.

10. Determine the maximum mass of blood that can pass through the aorta in 1 s while maintaining the laminar nature of the flow. Aortic diameter D = 2 cm, blood viscosity η = 4x10 -3 Pa-s.

11. Determine the maximum volumetric flow rate of liquid through the needle of a syringe with an inner diameter D = 0.3 mm, at which the laminar nature of the flow is preserved.

12. Find the volumetric velocity of the liquid in the syringe needle. Liquid density - ρ; its viscosity is η; diameter and length of the needle D and L, respectively; force acting on the piston - F; piston area - S.

Integrating over r, we get:

Let the piston of the syringe move under the action of a force F with a speed u. Then the power of the external force N F = Fu.

The total work of all forces is equal to the change in kinetic energy. Hence,

Substituting the found value A P into the second equation, we get all the quantities of interest to us: the piston speed and, the volumetric blood flow rate Q, the fluid velocity in the needle v.

Objective: study of the phenomenon of viscous friction and one of the methods for determining the viscosity of liquids.

Instruments and accessories: balls of various diameters, micrometer, caliper, ruler.

Elements of the theory and method of experiment

All real liquids and gases have internal friction, also called viscosity. Viscosity is manifested, in particular, in the fact that the movement that has arisen in a liquid or gas after the cessation of the causes that caused it, gradually stops. From everyday experience, for example, it is known that in order to create and maintain a constant flow of fluid in a pipe, it is necessary to have a pressure difference between the ends of the pipe. Since, in a steady flow, the fluid moves without acceleration, the need for the action of pressure forces indicates that these forces are balanced by some forces that slow down the movement. These forces are internal friction forces.

Two main modes of liquid or gas flow can be distinguished:

1) laminar;

2) turbulent.

In a laminar flow regime, a liquid (gas) flow can be divided into thin layers, each of which moves in the general flow at its own speed and does not mix with other layers. The laminar flow is stationary.

In a turbulent regime, the flow becomes unsteady - the speed of particles at each point in space changes randomly all the time. In this case, intensive mixing of the liquid (gas) takes place in the flow.

Let us consider the laminar flow regime. Let us single out two layers in the flow with area S, located at a distance ∆ Z apart and moving at different speeds. V 1 and V 2 (Fig. 1). Then a viscous friction force arises between them, proportional to the velocity gradient D V/D Z in a direction perpendicular to the direction of flow:

Where the coefficient μ is by definition called the viscosity or coefficient of internal friction, D V=V 2-V 1.

From (1) it can be seen that the viscosity is measured in pascal seconds (Pa s).

It should be noted that the viscosity depends on the nature and state of the liquid (gas). In particular, the value of viscosity can significantly depend on temperature, which is observed, for example, in water (see Appendix 2). Failure to take this dependence into account in practice in some cases can lead to significant discrepancies between theoretical calculations and experimental data.

In gases, viscosity is due to the collision of molecules (see Appendix 1), in liquids, it is due to intermolecular interactions that limit the mobility of molecules.

Viscosity values ​​for some liquid and gaseous substances are given in Annex 2.

As already noted, the flow of a liquid or gas can take place in one of two modes - laminar or turbulent. The English physicist Osborne Reynolds found that the nature of the flow is determined by the value of the dimensionless quantity

Where is a quantity called kinematic viscosity, V is the velocity of the fluid (or the body in the fluid), D is some characteristic size. In the case of fluid flow in a pipe under D understand the characteristic size of the cross section of this pipe (for example, diameter or radius). When a body moves in a fluid D understand the characteristic size of this body, for example, the diameter of a ball. For values Re< 1000 the flow is considered laminar, at Re> 1000 the flow becomes turbulent.

One of the methods for measuring the viscosity of substances (viscometry) is the falling ball method, or the Stokes method. Stokes showed that a ball moving at a speed V in a viscous medium, there is a viscous friction force equal to , where D is the diameter of the ball.

Consider the motion of the ball as it falls. According to Newton's second law (Fig. 2)

Where F— force of viscous friction, — force of Archimedes, — force of gravity, ρ F And ρ are the densities of the liquid and the material of the balls, respectively. The solution to this differential equation will be the following dependence of the speed of the ball on time:

Where V 0 is the initial speed of the ball, and

Is the speed of steady motion (at T>>τ). The quantity is the relaxation time. This value shows how quickly the stationary mode of motion is established. It is usually considered that T≈3τ the motion practically does not differ from the stationary one. Thus, by measuring the speed VAt, the viscosity of the liquid can be calculated. Note that the Stokes formula is applicable at Reynolds numbers less than 1000, that is, in the laminar regime of fluid flow around the ball.

A laboratory apparatus for measuring the viscosity of liquids using the Stokes method is a glass vessel filled with the liquid under study. Balls are thrown from above, along the axis of the cylinder. There are horizontal marks in the upper and lower parts of the vessel. By measuring the time of movement of the ball between the marks with a stopwatch and knowing the distance between them, the speed of the steady movement of the ball is found. If the cylinder is narrow, then the calculation formula must be corrected for the influence of the walls.

Taking into account these corrections, the formula for calculating the viscosity will take the form:

Where L - distance between marks, D is the diameter of the inside of the vessel.

Work order

1. Use a caliper to measure the inner diameter of the vessel, use a ruler to measure the distance between the horizontal marks on the vessel, and use a micrometer to measure the diameters of all the balls used in the experiment. The acceleration due to gravity is assumed to be 9.8 m/s2. The density of the liquid and the density of the substance of the balls are indicated on the laboratory setup.

2. Lowering the balls one by one into the liquid, measure the time it takes for each of them to travel between the marks. Record the results in a table. The table shows the number of the experiment, the diameter of the ball and the time of its passage, as well as the result of calculating the viscosity for each experiment.