What is called the force of internal friction of the liquid. Open Library - open library of educational information

1.13. HYDRODYNAMICS OF A VISCOUS LIQUID

The concept of viscosity. Force of internal friction. Laminar and turbulent fluid flow. Reynolds number. O determination of viscosity by the Stokes method, the Poiseuille method. Movement of bodies in liquids and gases. Similarity methods in physics.

An ideal fluid is a physical model that allows us to understand the essence of the phenomenon in some approximation. Viscosity or internal friction is inherent in all real liquids, which leads to the appearance of fundamentally new properties in them. In particular, the movement that has arisen in the liquid after the termination of the action of the causes that caused it, gradually slows down. Therefore, the liquid in its movement in the pipe experiences resistance. This kind of resistance is called viscous, thus emphasizing the difference from the resistance in solids. Viscosity - this is the property of real liquids to resist the movement of one part of the liquid relative to another. When moving some layers real liquid relative to others arise strengthinternal friction directed by tangent to the surface of the layers.

In solids, in the case of an attempt to change their shape (for example, when one part of the body is shifted relative to another), an elastic shear deformation force arises that is proportional to the displacement of atoms located at the nodes of the crystal lattice of neighboring atomic layers. In a liquid, this force is proportional to the change in velocity observed during the transition between adjacent layers of interacting molecules. Consider the following experience. We place the liquid between two solid parallel plates equal area S, located at a distance d. Let's try to move one of the plates relative to the other. Experience shows that in order to maintain a constant relative speed of movement of these plates  one of them needs to apply a constant force F, directed along the surface of the plate and proportional to the area of the plate S.

|F| = η·|  | S/d, (13.1)

where η is a constant value for a given liquid, called viscosity.

The need for such a force is due to the “sticking” of the near-boundary liquid molecules to the plates, which in turn causes the molecules in the liquid volume to move at different speeds. The magnitude of the force F depends on the properties of the fluid and is due to the interaction between the fluid layers sliding relative to each other. This interaction characterizes internal friction.

Rice. 13.1. Interaction of liquid molecules located in adjacent layers.

Let us consider the interaction of liquid layers moving parallel to each other and to the walls of the pipe in which this liquid is enclosed. On fig. 13.1 shows adjacent layers of liquid, located at a distance Δz from each other. The area of contacting layers S is essentially more sizes molecules. The upper and lower layers of the selected volume move parallel to the pipe axis and have different speeds:  1 and  2, respectively. To maintain the constancy of these velocities, it is necessary to apply forces of constant magnitude to the surfaces of the selected volume F 1 and F 2 , which must balance the forces of internal friction F tr1 and F tr2 acting between adjacent layers of the selected liquid volume.

In accordance with Newton's third law, the forces of internal friction are equal in magnitude and opposite in direction, so the top layer slows down the movement of the bottom, and the bottom one accelerates the movement of the top (see Fig. 13.1). The value of the internal friction force is given Newton's formula:

F tr = η·|Δ  /Δz| S, or

(13.2)

where η is the viscosity coefficient;

|Δ/Δz| is the modulus of the velocity gradient, showing how quickly the value of the velocity vector changes in the direction perpendicular to the fluid flow. Velocity gradient ∆ v /∆x shows how quickly the speed changes when moving from layer to layer in the direction x perpendicular to the direction of motion of the layers.

S is the surface area of the contacting liquid layers.

Proportionality factor η , which depends on the nature of the liquid and temperature, is called dynamic viscosity (or simply viscosity ). The physical meaning of the viscosity coefficient follows from expression (13.2):

viscosity coefficient numerically equal to strength internal friction, acting per unit area of the surface of the contacting layers, at a unit velocity gradient.

In the SI system, viscosity is measured in Pa s, and in CGS it is measured in poise (Pz): 1 Pa s \u003d 10 Ps. The viscosity coefficient of a liquid depends on the nature of the liquid (in particular, its density) and temperature, decreasing with an increase in the latter according to an exponential law. For a more objective account of the nature of the interaction of molecules in continuous media with different densities, for example, in liquids and gases, the concept of the kinematic viscosity coefficient is introduced.

The kinematic viscosity coefficient is equal to the ratio of the coefficientη to the density of the medium.

To explain the temperature dependence of the viscosity coefficient of liquids, it is necessary to take into account the nature of the thermal motion of their constituent molecules. It basically reduces to mechanical vibrations of molecules around equilibrium positions, which, unlike those in a solid, change with time due to transitions of molecules to neighboring positions with a local minimum of potential energy. In order for a liquid molecule to jump from one temporary equilibrium position to another, it must break bonds with its neighbors, that is, overcome a potential barrier with a height of W. The value of W is called the activation energy. The reciprocal of the bond breaking probability is determined by the ratio of the activation energy to the thermal energy, which is equal to the product of the Boltzmann constant k and absolute temperature T. On the other hand, the molecules of a liquid are most of the time near the equilibrium position, and the moving mass of the liquid entrains neighboring layers mainly due to the forces of intermolecular interaction, which decrease with increasing temperature, and, therefore, the viscosity also decreases with increasing temperature.

Ya. I. Frenkel, based on the character thermal motion molecules in liquids, showed that the temperature dependence of the viscosity of a liquid has an activation character and is described by the expression:

η = C e  W /(k T) , (13.3)

where W - activation energy;

T is the absolute temperature;

C is a constant value;

k- Boltzmann's constant, k = 1.38 10 -23 J/K;

e is the base of the natural logarithm.

Applying Newton's formula (13.2) to solve problems related to fluid flow, one can obtain certain quantitative patterns that are used to experimentally determine the viscosity coefficient. The most accurate and common methods for determining viscosity are:

Rice. 13.2. Velocity of liquid layers in a horizontal pipe under laminar flow.

The flow of real liquids and gases. The flow of a viscous fluid through pipes, depending on a number of conditions, can be laminar (or layered) and turbulent (or vortex).

In the case of laminar flow, all fluid molecules move parallel to the axis of the pipe and, being at the same distance from the axial center of the pipe, have equal velocities (see Fig. 13.2). The current is called laminar (layered) , if along the flow each selected thin layer slides relative to the neighboring ones, without mixing with them.

The current is called turbulent (vortex) if the liquid particles go over from layer to layer (have velocity components perpendicular to the flow). Turbulent motion is characterized by the presence of a normal (perpendicular to the direction of fluid flow) component of the velocity of molecular motion and a sharp drop in the flow velocity when approaching the boundaries. The trajectory of the movement of molecules is a complex curved line.

The nature of the flow can be established using the dimensionless quantity - Reynolds number: (13.4)

γ = η / ρ - kinematic viscosity; ρ is the density of the liquid; v is the fluid velocity averaged over the pipe section; d - characteristic linear dimension, for example, the diameter of the pipe. At Re ≤ 1000 observed laminar flow, transition from laminar currents to turbulent takes place in the area 1000 ≤ Re ≤ 2000 , and when Re=2300 (for smooth pipes) the flow is turbulent.

Frontal pressure and lift force. Consider movement solid body relative to a fluid at rest in some IFR. Based on the principle of relativity, this problem is equivalent to a stationary fluid flow around a stationary body.

The force acting on motionless body in the direction of flow is called drag, and the force acting on it in the perpendicular direction is called lift.

A stationary flow of an ideal fluid around a solid body does not cause the appearance of lift and drag. Let us show this by the example of a symmetrical body at rest with respect to the observer. In this case, the streamlines relative to the vertical axis passing through the center of mass of the body perpendicular to the direction of fluid flow are symmetrical. Consequently, for symmetric elementary spatial regions, the values of the velocities in the current tube are equal in magnitude. Then, based on the Bernoulli equation, the pressures in these areas are pairwise equal and there is no drag.

In view of the symmetry of the problem (but with respect to the axis parallel to the flow), the lift force is also equal to zero.

Rice. 13.3. Lift force acting on a rotating body placed in a gas flow.

Magnus effect. The situation is different for a viscous liquid or gas. Let a body rotating about its center of mass be immersed in a gas stream (see Fig. 13.3). The layers of molecules adjacent to the body participate in two movements: rotational, due to the presence of viscous friction between the body and gas, and translational, associated with the movement of gas along the axis of the pipe. Based on the vector law of velocity transformation, a pattern of streamlines is obtained, shown in Fig. 13.3, i.e., the speed of the flow of gas molecules above a solid body is higher than below it. Therefore, in accordance with the Bernoulli equation, the pressure above the body will be lower than below it, and a lifting force appears.

The emergence of a lifting force as a result of air circulation around a solid body is called the Magnus effect.

Rice. 13.4. The movement of air molecules around an airplane wing.

The most typical example is the presence of lift at the wing of an aircraft as it moves relative to the air. Due to the characteristic shape of the wing near its sharp trailing edge, vortex air flows arise in the nearby air layers, and the direction of rotation of the molecules is counterclockwise (see Fig. 13.4). These vortex flows gradually grow and break away from the wing, but due to the presence of viscous friction, they cause the air molecules adjacent to it to rotate clockwise around the wing surface. The presence of circulation due to viscous friction leads to the emergence of a lift force.

The law of similarity.

Geometric, kinematic, dynamic similarity.

The stage of studying the dependence of the quantity of interest on the system of selected determining factors can be performed in two ways: analytical and experimental. The first way is applicable only for a limited number of problems and, moreover, usually only for simplified models of phenomena.

Another way, experimental, in principle can take into account many factors, but it requires scientifically based experiments, planning of the experiment, limiting its scope. necessary minimum and systematization of the results of experiments. In this case, the modeling of phenomena should be justified.

These problems can be solved by the so-called similarity theory, i.e., the similarity of incompressible fluid flows.

Hydrodynamic similarity consists of three components: geometric similarity, kinematic and dynamic.

Geometric similarity, as is known from geometry, is the proportionality of similar sizes and the equality of the corresponding angles. Geometric similarity is understood as the similarity of those surfaces that limit flows, i.e., the similarity of channels (or channels).

The ratio of two similar sizes of similar channels will be called a linear scale and denote this value by . This value is the same for similar channels I and II.

Kinematic similarity means the proportionality of local velocities at similar points and the equality of the angles characterizing the direction of these velocities:

Where is the scale of velocities, which is the same for kinematic similarity.

Since (where T is time, – time scale).

The geometric similarity of streamlines follows from the kinematic similarity. It is obvious that for the kinematic similarity, the geometrical similarity of the channels is required.

Dynamic similarity is the proportionality of the forces acting on similar volumes in similar kinematic flows and the equality of the angles characterizing the direction of these forces.

In liquid flows, there are usually different forces: pressure forces, viscosity (friction), gravity, etc. Compliance with their proportionality means complete hydrodynamic similarity. The implementation in practice of a complete hydrodynamic similarity turns out to be very difficult, therefore, they usually deal with partial (incomplete) similarity, in which the proportionality of only the main, main forces is observed.

1.13. HYDRODYNAMICS OF A VISCOUS LIQUID

In solids, in the case of an attempt to change their shape (for example, when one part of the body is shifted relative to another), an elastic shear deformation force arises that is proportional to the displacement of atoms located at the nodes of the crystal lattice of neighboring atomic layers. In a liquid, this force is proportional to the change in velocity observed during the transition between adjacent layers of interacting molecules. Consider the following experience. Let us place the liquid between two solid parallel plates of equal area S, located at a distance d. Let's try to move one of the plates relative to the other. Experience shows that in order to maintain a constant relative speed of movement of these plates  one of them needs to apply a constant force F, directed along the surface of the plate and proportional to the area of the plate S.

|F| = η·|  | S/d, (13.1)

where η is a constant value for a given liquid, called viscosity.

Rice. 13.1. Interaction of liquid molecules located in adjacent layers.

Let us consider the interaction of liquid layers moving parallel to each other and to the walls of the pipe in which this liquid is enclosed. On fig. 13.1 shows adjacent layers of liquid, located at a distance Δz from each other. The area of the contacting layers S is much larger than the dimensions of the molecules. The upper and lower layers of the selected volume move parallel to the pipe axis and have different speeds:  1 and  2, respectively. To maintain the constancy of these velocities, it is necessary to apply forces of constant magnitude to the surfaces of the selected volume F 1 and F 2 , which must balance the forces of internal friction F tr1 and F tr2 acting between adjacent layers of the selected liquid volume.

F tr = η·|Δ  /Δz| S, or

(13.2)

where η is the viscosity coefficient;

S is the surface area of the contacting liquid layers.

the viscosity coefficient is numerically equal to the force of internal friction acting per unit area of the surface of the contacting layers, with a unit velocity gradient.

The kinematic viscosity coefficient is equal to the ratio of the coefficientη to the density of the medium.

To explain the temperature dependence of the viscosity coefficient of liquids, it is necessary to take into account the nature of the thermal motion of their constituent molecules. It basically reduces to mechanical vibrations of molecules around equilibrium positions, which, unlike those in a solid, change with time due to transitions of molecules to neighboring positions with a local minimum of potential energy. In order for a liquid molecule to jump from one temporary equilibrium position to another, it must break bonds with its neighbors, that is, overcome a potential barrier with a height of W. The value of W is called the activation energy. The reciprocal of the bond breaking probability is determined by the ratio of the activation energy to the thermal energy equal to the product of the Boltzmann constant k and the absolute temperature T. On the other hand, the molecules of a liquid are most of the time near the equilibrium position, and the moving mass of the liquid entrains neighboring layers mainly due to forces of intermolecular interaction, which decrease with increasing temperature, and, consequently, the viscosity also decreases with increasing temperature.

Ya. I. Frenkel, based on the nature of the thermal motion of molecules in liquids, showed that the temperature dependence of the viscosity of a liquid has an activation character and is described by the expression:

η = C e  W /(k T) , (13.3)

where W - activation energy;

T is the absolute temperature;

C is a constant value;

k - Boltzmann's constant, k = 1.38 10 -23 J/K;

e is the base of the natural logarithm.

Rice. 13.2. Velocity of liquid layers in a horizontal pipe under laminar flow.

The flow of real liquids and gases. The flow of a viscous fluid through pipes, depending on a number of conditions, can be laminar (or layered) and turbulent (or vortex).

The nature of the flow can be established using the dimensionless quantity - Reynolds number: (13.4)

Frontal pressure and lift force. Consider the motion of a rigid body relative to a fluid at rest in some IFR. Based on the principle of relativity, this problem is equivalent to a stationary fluid flow around a stationary body.

The force acting on a stationary body in the direction of flow is called drag, and the force acting on it in the perpendicular direction is called lift.

In view of the symmetry of the problem (but with respect to the axis parallel to the flow), the lift force is also equal to zero.

Rice. 13.3. Lift force acting on a rotating body placed in a gas flow.

The emergence of a lifting force as a result of air circulation around a solid body is called the Magnus effect.

Rice. 13.4. The movement of air molecules around an airplane wing.

The law of similarity.

Geometric, kinematic, dynamic similarity.

The other way, the experimental one, can in principle take into account many factors, but it requires scientifically substantiated setting up of experiments, planning the experiment, limiting its volume to the necessary minimum, and systematizing the results of the experiments. In this case, the modeling of phenomena should be justified.

These problems can be solved by the so-called similarity theory, i.e., the similarity of incompressible fluid flows.

Hydrodynamic similarity consists of three components: geometric similarity, kinematic and dynamic.

The ratio of two similar sizes of similar channels will be called a linear scale and denote this value by . This value is the same for similar channels I and II.

Kinematic similarity means the proportionality of local velocities at similar points and the equality of the angles characterizing the direction of these velocities:

Where is the scale of velocities, which is the same for kinematic similarity.

Since (where T is time, – time scale).

The geometric similarity of streamlines follows from the kinematic similarity. It is obvious that for the kinematic similarity, the geometrical similarity of the channels is required.

Dynamic similarity is the proportionality of the forces acting on similar volumes in similar kinematic flows and the equality of the angles characterizing the direction of these forces.

Different forces usually act in fluid flows: pressure forces, viscosity (friction), gravity, etc. Observance of their proportionality means complete hydrodynamic similarity. The implementation in practice of a complete hydrodynamic similarity turns out to be very difficult, therefore, they usually deal with partial (incomplete) similarity, in which the proportionality of only the main, main forces is observed.

Ideal Fluid, t

An ideal fluid, that is, a 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 cessation of the causes that caused it, gradually stops.

To clarify the patterns that the forces of internal friction obey, consider the following experiment. Two parallel plates are immersed in a liquid (Fig. 153), the linear dimensions of which significantly exceed the distance between them d. The bottom plate is held in place, the top one is driven relative to the bottom one at a certain speed. Experience gives that to move the top plate with constant speed it is necessary to act on it with a well-defined constant force f. Since the plate does not receive acceleration, it means that the action of this force is balanced by an oppositely directed force equal in magnitude to it, which, obviously, is the friction force acting

on the plate as it moves through the liquid. Let's denote it f tr.

By varying the speed of the plate, the area of the plates S and the distance between them d, we can obtain that

(58.1 )

where is the coefficient of proportionality, which depends on the nature and state (for example, temperature) of the liquid and is called the coefficient of internal friction or the coefficient of viscosity, or simply the viscosity of the liquid (gas).

The lower plate, when the upper plate moves, is also subject to the action of a force equal in magnitude to . In order for the bottom plate to remain stationary, the force must be balanced by the force .

Thus, when two plates immersed in a liquid move relative to each other, an interaction occurs between them, characterized by the force (58.1). The impact of the plates on each other is carried out, obviously, through the liquid enclosed between the plates, being transferred from one layer of liquid to another. If you mentally draw a plane parallel to the plates anywhere in the gap (see the dotted line in Fig. 153), then you can assert. That the part of the fluid lying above this plane acts on the part of the fluid lying under the plane with the force , and the part of the fluid lying under the plane, in turn, acts on the part of the fluid lying above the plane with the force , and the value of and is determined by the formula ( 58.1). Thus, formula (58.1) determines not only the friction force acting on the plates, but also the friction force between the contacting parts of the fluid.

If we examine the velocity of fluid particles in different layers, it turns out that it changes in the direction z perpendicular to the plates (Fig. 153), according to the linear law

Using equality (58.3), formula (58.1) for the internal friction force can be given the form

(58.4 )

The value shows how fast the speed changes in the direction of the z-axis, and is called the velocity gradient (more precisely, this is the modulus of the velocity gradient; the gradient itself is a vector).

Formula (58.4) was obtained by us for the case when the speed changes according to a linear law (in this case, the speed gradient is constant). It turns out that this formula remains valid for any other law of velocity change during the transition from layer to layer. In this case, to determine the force of friction between two layers adjacent to each other, it is necessary to take the value of the gradient in the place where the imaginary interface between the layers passes. So, for example, when a liquid moves in a round pipe, the velocity is equal to zero near the pipe walls, is maximum on the pipe axis, and, as can be shown, at not too high flow velocities, it changes along any radius according to the law

(58.5 )

where R is the pipe radius, is the speed along the pipe axis, is the speed at a distance z from the pipe axis (Fig. 154). Let us mentally draw a cylindrical surface of radius r in the liquid Parts of the liquid lying along different sides from this surface, act on each other with a force, the value of which per unit surface is equal to

m, i.e., increases in proportion to the distance of the interface from the axis of the pipe (we omitted the “-” sign obtained by differentiating (58.5) with respect to r, since formula (58.4) gives only the modulus of the internal friction force).

Everything said in this paragraph applies not only to liquids, but also to gases.

The SI unit of viscosity is the viscosity at which a velocity gradient of 1 m/s per 1 m results in an internal friction force of 1 n per 1 m 2 of the contact surface of the layers. This unit is designated n * sec / m 2.

In the CGS system, the unit of viscosity is poise (pz), equal to such a viscosity at which a velocity gradient of 1 cm/sec per 1 cm leads to the appearance of an internal friction force of 1 dyne per 1 cm2 of the contact surface of the layers. A unit equal to a poise is called a micropoise (mkpz).

Between the poise and the unit of viscosity in SI there is a relation

The viscosity coefficient depends on temperature, and the nature of this dependence is significantly different for liquids and gases. In liquids, the viscosity coefficient decreases strongly with increasing temperature. In gases, on the contrary, the viscosity coefficient increases with temperature. The difference in the nature of behavior with temperature changes indicates the difference in the mechanism of internal friction in liquids and gases.

The phenomenon of internal friction from a macroscopic point of view is associated with the emergence of friction forces between layers of gas or liquid moving parallel to each other with different velocities. From the side of the layer moving faster, an accelerating force acts on the slower moving layer. Conversely, a slowly moving layer slows down faster moving layers of gas. The friction forces that arise in this case are directed tangentially to the contact surface of the layers.

Consider famous experience Newton. Let there be two parallel plates (Fig. 1), between which there is a gas (liquid).

The constant a is determined from the condition that for x = h u = u 0 , i.e. u 0 = ah. Whence a = u 0 /h. Then expression (3.3.1) takes the form

where - constant factor proportionality, which is called the coefficient of viscous friction. Taking into account that the force of viscous friction , we rewrite equality (3.3.3) in the form

This is Newton's law of internal viscous friction, which established it experimentally. The law states: in the stationary (laminar) motion of layers of liquid or gas with various speeds between them there are tangential forces proportional to the velocity gradient of the layers and the area of their contact. The physical meaning of the viscosity coefficient lies in the fact that it is numerically equal to the force acting per unit area of the surface, parallel to the flow velocity of a gas or liquid, with a velocity gradient.

According to Newton's second law, , where K is the momentum of the elementary mass of the gas layer. Therefore, (3.3.5) can be represented in the form of infinitesimals:

Then Newton's law (3.3.6) states: the momentum transferred during the time dt through the area dS perpendicular to the X axis is proportional to the time dt, the size of the area dS and the velocity gradient . The minus sign means that the momentum is transferred in the direction of decreasing layer velocity.

From the molecular kinetic point of view, the cause of internal friction is the superposition of the ordered motion of gas layers with different hydrodynamic velocities u and the chaotic thermal motion of molecules. As a result of thermal motion, molecules from the faster layer carry with them a greater ordered momentum and, colliding, transfer it to the molecules of the slower moving layer, as a result of which it increases the speed. On the contrary, when molecules pass from a slowly moving layer to a faster layer, they bring a smaller ordered momentum into it, which leads to a decrease in the ordered velocity of this layer. An increase or decrease in the hydrodynamic velocity of a gas layer, according to the second law of dynamics, indicates the presence of an internal friction force acting between the layers. Consequently, due to the thermal chaotic motion, the velocities of the layers will equalize, unless, of course, external forces do not support layer velocity differences.

Thus, from the point of view of the molecular-kinetic theory, each molecule transfers an ordered momentum to the process of internal friction, thereby causing a change in the momentum of the layer. Substituting in general equation transfer (4.4.7) and , we get: at its ends. Having measured all the indicated quantities in the experiment, the viscosity coefficient is found from the Poiseuille formula.

real liquid viscosity is inherent, which manifests itself in the fact that any movement of liquid and gas spontaneously stops in the absence of the causes that caused it. Let us consider an experiment in which a liquid layer is located above a fixed surface, and a plate floating on it with a surface moves from above it with a speed S(Fig. 5.3). Experience shows that in order to move the plate at a constant speed, it is extremely important to act on it with a force . Since the plate does not receive acceleration, it means that the action of this force is balanced by another force equal to it in magnitude and oppositely directed, which is the force of friction . Newton showed that the force of friction

, (5.7)

where d is the thickness of the liquid layer, h is the viscosity coefficient or coefficient of friction of the liquid, the minus sign takes into account different direction vectors F tr and v o. If we examine the velocity of fluid particles in different places of the layer, it turns out that it changes according to a linear law (Fig. 5.3):

v(z) = (v 0 /d) z.

Differentiating this equality, we get dv/dz= v 0 /d. With this in mind

formula (5.7) takes the form

F tr=- h(dv/dz)S , (5.8)

where h- dynamic viscosity coefficient. Value dv/dz is called the velocity gradient. It shows how fast the speed changes in the direction of the axis z. At dv/dz= const velocity gradient is numerically equal to velocity change v when it changes z per unit. We put numerically in formula (5.8) dv/dz =-1 and S= 1, we get h = F. this implies physical meaning h: the viscosity coefficient is numerically equal to the force that acts on a liquid layer of unit area with a velocity gradient, equal to one. The SI unit of viscosity is called the pascal second (denoted Pas). In the CGS system, the unit of viscosity is 1 poise (P), with 1 Pas = 10P.

- Thus, the modulus of the force of internal friction

, (6.8) where the coefficient of proportionality h, depending on the nature of the liquid, is called dynamic viscosity (or simply viscosity). The unit of viscosity is Pascal second (Pa×s): 1 Pa×s is equal to the dynamic viscosity of the medium in which, at laminar flow in a gradient...

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