Ideal liquid. Poiseuille's and Stokes' laws
Table of contents
1. Statement of the problem
2. Continuity equation
4. Steady laminar flow between parallel planes
5. Couette Current
6. Poiseuille Current
7. General case of flow between parallel walls
8. Example problem
Formulation of the problem
Laminar flows, some of which are discussed in this course project, are found in a variety of technical problems, in particular, in gaps and small cavities of machines. In particular, during the flow of such viscous liquids as oil, petroleum, and various fluids for hydraulic transmissions, stable laminar flows are formed, for the description of which the Navier–Stokes equations can serve as a reliable basis. The Hartmann flow, similar to the Poiseuille flow, is used, for example, in MHD pumps. In this case, a plane stationary flow of an electrically conductive fluid between two insulated plates in a transverse magnetic field is considered.
The objective of this course project is to consider and find the main characteristics of a flat stationary laminar flow of a viscous incompressible fluid with a parabolic velocity distribution (Poiseuille flow).
Continuity equation
The law of conservation of mass for a fluid moving in an arbitrary manner is expressed by the equation of continuity or continuity, which is one of the fundamental equations of fluid mechanics. To derive it, let us draw a closed surface S fixed in space in the liquid, limiting the volume W, and select an elementary area dS on it. Let n denote the unit vector of the external normal to S. Then the product cV n dS will represent the mass flowing out of the volume W or entering it per unit time, depending on the direction of the velocity on the site dS. Since n is the external normal, then V p > 0 on those sites dS where the liquid flows out of the volume W, and V p< 0 на той части поверхности S, через которую она втекает в этот объем. Следовательно, интеграл представляет собой разность масс жидкости, вытекшей из объема и поступившей в него за единицу времени.
This change in mass can be calculated in another way. To do this, we select the elementary volume dW. The mass of liquid in this volume may vary due to differences in inflow and outflow. The second change in mass in volume dW will be equal to 
and the second change in mass in volume W will be expressed by the integral.
The resulting expressions can be equated, since they give the same value. It should be taken into account that the first integral is positive if more fluid flows out through the surface S than flows in, and the second, under the same condition, is negative, since due to the continuity of the flow in the case under consideration, the density decreases with time.
According to the Ostrogradsky–Gauss theorem:
In vector analysis, the sum of partial derivatives of vector projections along the same coordinates is called divergence or vector divergence. In this case
therefore equation (1) can be rewritten as
Since the volume W is arbitrary, the integrand function is equal to zero, i.e.
(2)
Equation (2) is a continuity equation in differential form for arbitrary motion of a compressible fluid. Relationship (1) can be considered as an integral form of the continuity equation.
If we consider the condition of conservation of the mass of a moving liquid volume, we will also arrive at equation (2), which in this case can be given a different form.
Since c = c (x, y, z, t) and when the liquid volume moves x = x(t),
y = y (t), z = z (t), then
i.e. equation (2) will have the form
(3)
where dc/dt is the total derivative of density.
For steady motion of a compressible fluid, ∂c/∂t = 0 and. therefore, from equation (2) we obtain
For any motion of an incompressible fluid c = const and, therefore,
(5)
3. Equation of motion of a viscous fluid in the Navier-Stokes form
Equation of fluid motion in stresses:
According to Newton's law, viscous stresses at straight motion fluids are proportional to the rates of angular deformation. A generalization of this fact to the case of arbitrary motion is the hypothesis that tangential stresses, as well as parts of normal stresses that depend on the orientation of the areas, are proportional to the corresponding strain rates. In other words, in all cases of fluid motion, a linear relationship between viscous stresses and strain rates is assumed. In this case, the proportionality coefficient in the formulas expressing this relationship must be the dynamic viscosity coefficient m. Using the hypothesis that at a point in the liquid 
(it is indirectly confirmed in practice), we can write expressions for normal and shear stresses in a viscous fluid:
(7)
Introducing expressions (7) into equation (6), we obtain
Grouping terms with second derivatives, dividing by c and using the Laplace operator, we write:
These equations are called the Navier-Stokes equations; they are used to describe the movements of viscous compressible liquids and gases.
The equations of motion of inviscid liquids and gases can be easily obtained from the Navier-Stokes equations as a special case with m=const; for incompressible fluids, c = const should be taken.
The Navier-Stokes system of equations is not closed, since it contains six unknowns: V x, V y, V z, p, s and m. Another equation connecting these unknowns is the continuity equation (3).
As equations that close the system, the equations of state of the medium and the dependence of viscosity on state parameters are used. In many cases, it is also necessary to apply other thermodynamic relations.
For an incompressible fluid divV = 0, we obtain expressions that directly follow from system (8)
IN vector form The Navier-Stokes equation for an incompressible fluid will take the form:
Steady laminar flow between parallel planes
Let a viscous fluid flow in a channel formed by two parallel walls, one of which moves in its plane at a constant speed (see figure).
a – flow diagram; b – velocity distribution in the absence of a pressure gradient (Couette flow); c – velocity distribution in the case of stationary boundary planes (flow in a flat channel).
We consider the size of the channel in the direction normal to the drawing plane (along the z axis) to be large enough so that the influence of walls parallel to the xOy plane can be ignored. In addition, we assume that the movement is caused not only by the movement of one of the channel walls, but also by the pressure difference (or gradient) in the direction of the x axis. We neglect the influence of mass forces, because the Froude number is small due to the smallness of h, and we consider the streamlines to be straight, parallel to the x axis.
Then we express the initial conditions of the problem in the form:
From the continuity equation we immediately conclude that and since this will be true at all points, then Due to the absence of movement along the z axis, all derivatives along this coordinate will also vanish, and the Navier-Stokes equation in projection onto the z axis need not be written.
Then the system of equations of motion will be reduced to two equations:
The first is obtained from the projection of the Navier-Stokes equation onto the x coordinate axis, and the second of these equations indicates that the pressure depends only on x, i.e. p(y)=p(z)=0, and since then we can go from partial to total derivatives:
Let's denote and integrate this equation twice, we get:
Since, in accordance with the figure and the accepted assumptions, the pressure depends only on the x coordinate. To find the integration constants, we use the boundary conditions:
Thus, the law of velocity distribution in a flat channel will be written as:
(10)
Couette Current
Couette flow is a gradient-free flow. In this case the only reason movement is the movement of the plate. The flow is characterized by a linear velocity distribution law (Fig. b).
Shear (viscous) stress 
will be constant over the layer thickness, and the specific flow rate, i.e. flow rate through the living flow S=h·1, entrained by the moving plate, is equal to:
6. Poiseuille Current
This is the case of pressure flow in a flat channel with a parabolic velocity distribution (Fig. c). In accordance with equation (10) we obtain:
Maximum speed on the axis (at y=h/2) due to the parabolic speed distribution:
(12)
Dividing (11) by (12), we obtain the speed distribution law
It is not difficult to calculate other flow characteristics. Shear stress
On the walls, i.e. at y=0 and at y=h, takes maximum values
And on the axis at y=h/2 it becomes zero. As can be seen from these formulas, there is a linear law of distribution of tangential stresses over the thickness of the layer
The specific fluid consumption is determined by the formula
average speed
(13)
The average speed will be one and a half times less than the maximum.
Having integrated (13) over x, under the assumption that at x = 0 the pressure p = p 0 *, we obtain the required pressure difference:
It is also easy to calculate the intensity of the vortex component of the motion. Since in this case V y =V z =0 and V x =V, then
Considering that dp/dx<0, мы получи:
· at y< h/2, щ z < 0, т.е. частицы вращаются по часовой стрелке;
· for y > h/2, ы z > 0, i.e. the particles rotate counterclockwise (Fig. c).
Thus, the flow under consideration is vortex at all points; the ordered vortex lines represent straight, normal flow planes.
General case of flow between parallel walls
This case is typical 
The velocity distribution is determined by equation (10), where the pressure gradient dp/dx can be either negative or positive. In the first case, the pressure drops in the direction of the plate velocity V 0 , in the second case it increases. The presence of a positive pressure gradient can cause rip currents. Equation (10) can be conveniently represented in dimensionless form
which is graphically represented by a family of curves with one parameter
Dimensionless velocity profiles for the general case of flow between parallel walls.
Sample task
Let us consider the Poiseuille flow in relation to an MHD generator.
Magnetohydrodynamic generator, MHD generator - a power plant in which the energy of a working fluid (liquid or gaseous electrically conducting medium) moving in a magnetic field is converted directly into electrical energy. The speed of movement of a viscous medium can be either subsonic or supersonic; we choose a speed equal to V max = 300 m/s. Let the length of the linear channel be 10 meters. The distance between the plates in which the plasma flows is 1 meter. Let us take the maximum value of plasma viscosity to be 3·10 -4 Pa·Hs=8.3·10 -8 Pa·s.
Substituting the data into the formula for the pressure difference, taking into account that the average speed is one and a half times less than the maximum, we obtain:
This is the pressure loss when the working fluid passes through the linear channel of the MHD generator.
Bibliography
1. Beknev V.S., Pankov O.M., Yanson R.A. – M.: Mechanical Engineering, 1973. – 389 p.
2. Emtsev B.T. Technical hydromechanics. – M.: Mechanical Engineering, 1978. – 458 p.
3. Emtsev B.T. Technical hydromechanics. – M.: Mechanical Engineering, 1987. – 438 p.
4. http://ru-patent.info/21/20-24/2123228.html
5. http://ligis.ru/effects/science/83/index.htm
The flow in a long pipe of circular cross-section under the influence of a pressure difference at the ends of the pipe was studied by Hagen in 1839 and Poiseuille in 1840. We can assume that the flow, like the boundary conditions, has axial symmetry, so that - is a function only of the distance from the pipe axis . The corresponding solution to Equation (4.2.4) is:
In this solution there is an unrealistic feature (associated with a finite force acting on the fluid per unit
the length of the axis segment) if the constant A is not equal to zero; therefore, we choose exactly this value of A. Choosing a constant B such as to obtain at the pipe boundary at we find
Of practical interest is the volumetric flow of liquid through any section of the pipe, the value of which
where (modified) pressures in the initial and end sections of a pipe section of length Hagen and Poiseuille established in experiments with water that the flow depends on the first power of the pressure drop and the fourth power of the pipe radius (half of this power is obtained due to the dependence of the cross-sectional area of the pipe on its radius, and the other half is associated with an increase in speed and for a given resulting viscous force with increasing pipe radius). The accuracy with which the constancy of the ratio in the observations was obtained convincingly confirms the assumption that there is no sliding of liquid particles on the pipe wall, and also indirectly confirms the hypothesis about the linear dependence of viscous stress on the strain rate under these conditions.
The tangential stress on the pipe wall is equal to
so the total friction force in the direction of flow on a pipe section of length I is equal to
Such an expression for the total friction force on the pipe wall was to be expected, since all elements of the liquid inside this part of the pipe are at a given moment in time in a state of steady motion under the influence of normal forces at the two end sections and the friction force on the pipe wall. In addition, from expression (4.1.5) it is clear that the dissipation rate mechanical energy per unit mass of liquid under the influence of viscosity is determined in this case by the expression
Thus, the total dissipation rate in the liquid currently filling a section of a circular pipe of length I is equal to
In the case in which the medium in the pipe is a dripping liquid and there is atmospheric pressure at both ends of the pipe (as if the liquid were entering the pipe from a shallow open reservoir and flowing out the end of the pipe), a pressure gradient along the pipe is created by gravity. The absolute pressure in this case is the same at both ends and is therefore constant throughout the liquid, so the modified pressure is equal to a and
8.5. Viscosity. Poiseuille Current
So far we have not said anything about shear stress in a liquid or gas, limiting ourselves only to isotropic pressure within the framework of Pascal's law. It turns out, however, that Pascal’s law is exhaustive only in hydrostatics, and in the case of spatially inhomogeneous flows, the dissipative effect—viscosity—comes into play, as a result of which tangential stresses arise.
Let in a certain region of fluid flow two infinitely close layers of fluid, moving in the direction of the x axis, come into contact with each other on a horizontal surface with area S (Fig. 8.14). Experience shows that the friction force F between the layers on this site is greater, the larger the area S and the faster the flow velocity v changes in this place in the direction perpendicular to the site S, that is, in the direction of the y axis. The rate of change of speed v as a function of y is characterized by the derivative dv/dy.
Finally, the result obtained from the experiment can be written as:
F = ηS dv/dy. (8.27)
Here F is the force acting from the overlying layer on the underlying one, η is the proportionality coefficient, called the coefficient
fluid viscosity (abbreviated simply as fluid viscosity). Its dimension follows from formula (8.27) [η] = [m]/[l][t]; The unit of measurement is usually expressed as 1 Pa s. The direction of force F (to the right or left in Fig. 8.14) depends on whether the overlying layer is moving faster or slower relative to the underlying one. From (8.27) follows the expression for tangential stresses:
τ = η dv/dy.(8.28)
The viscosity coefficient η has different values for different liquids, and for a particular liquid it depends on external conditions, primarily on temperature. By their nature, friction forces in a liquid are forces of intermolecular interaction, that is, electromagnetic forces, just like the friction forces between solid bodies. Let us move on to consider the problem of calculating the flow rate of an incompressible fluid flowing in a horizontal round straight pipe with a constant cross-sectional area at a given pressure difference. Flow is the mass of liquid flowing per unit time through a pipe section. This task is extremely important
Rice. 8.15
practical significance: the organization of the operation of oil pipelines and even ordinary water supply certainly requires its solution. We will assume that we are given the length of the pipe l, its radius R, the pressures at the ends of the pipe P 1 and P 2 (P 1 >P 2), as well as the density of the liquid ρ and its viscosity η (Fig. 8.15).
The presence of friction forces leads to the fact that at different distances from the center of the pipe, liquid flows at different speeds. In particular, directly at the wall the liquid must be motionless, otherwise infinite tangential stresses would follow from (8.28). To calculate the mass of fluid flowing every second through the entire cross-section of the pipe, we divide this cross-section into infinitesimal annular areas with an internal radius r and an external r + dr and first calculate the fluid flow through each of these infinitesimal sections in which the speed
Mass of fluid dm flowing every second through an infinitesimal
cross section 2nrdr with speed v(r), is equal to
dm/dt = 2πr drρv(r). (8.29)
We obtain the total fluid flow Q by integrating expression (8.29)
by r from 0 to R:
Q = dm/dt = 2πρ 
rv(r) dr, (8.30)
where the constant value 2πρ is taken out of the integration sign. To calculate the integral in (8.30), it is necessary to know the dependence of the fluid velocity on the radius, that is, the specific form of the function v(r). To determine v(r), we will use the laws of mechanics already known to us. Let us consider at some point in time a cylindrical volume of liquid of some arbitrary radius r and length l (Fig. 8.15). The liquid filling this volume can be considered as a collection of infinitesimal liquid particles forming a system of interacting material points. During stationary fluid flow in a pipe, all these material points move at speeds independent of time. Consequently, the center of mass of this entire system also moves at a constant speed. The equation for the motion of the center of mass of a system of material points has the form (see Chapter 6)
where M is the total mass of the system, V cm - speed of the center of mass,
∑F BH is the sum of external forces applied at a selected moment in time to the system under consideration. Since in our case V cm = const, then from (8.31) we obtain
External forces are pressure forces F pressure acting on the bases of the selected cylindrical volume, and friction forces F tr acting on the side surface of the cylinder from the surrounding liquid - see (8.27):
As we have shown, the sum of these forces is zero, that is
This relationship after simple transformations can be written in the form
Integrating both sides of the equality written above, we obtain
The integration constant is determined from the condition that when r = Rsk-
the speed v must vanish. This gives
As we can see, the fluid speed is maximum on the axis of the pipe and, as it moves away from the axis, it changes according to a parabolic law (see Fig. 8.15).
Substituting (8.32) into (8.30), we find the required fluid flow
This expression for fluid flow is called Poiseuille's formula. A distinctive feature of relation (8.33) is the strong dependence of the flow rate on the radius of the pipe: the flow rate is proportional to the fourth power of the radius.
(Poiseuille himself did not derive a formula for flow rate, but investigated the problem only experimentally, studying the movement of liquid in capillaries). One of the experimental methods for determining the viscosity coefficients of liquids is based on the Poiseuille formula.
AND 
Liquids and gases are characterized by density.
- the density of the liquid depends in general on the coordinates and time
- density is a thermodynamic function and depends on pressure and temperature
The element of mass can be expressed from the definition of density
Through a selected area, you can determine the fluid flow vector as the amount of fluid passing through perpendicular to the area per unit time
Square vector.
In a certain elementary volume there are microparticles, and he himself is a macroparticle. 
Lines that can conventionally show the movement of a fluid are called current lines.
current function.
Laminar flow– a flow in which there is no mixing of the liquid and no overlap of flow functions, that is, a layered flow.
In Fig. laminar flow around an obstacle - in the form of a cylinder
Turbulent flow– a flow in which different layers mix. A typical example of a turbulent wake when flowing around an obstacle.
Almost on rice - current tube. For a stream tube, the streamlines do not have sharp deviations.
From the definition of density, the elementary mass is determined from the expression
elementary volume is calculated as the product of the cross-sectional area and the path traveled by the fluid
Then the elementary mass (mass of the liquid element) is found from the relation
dm = dV = VSdt
1) Continuity equation
In the most general case, the direction of the velocity vector may not coincide with the direction of the flow cross-sectional area vector
- the area vector has a direction
The volume occupied by a liquid per unit time is determined taking into account the rules of the scalar product of vectors
V Scos
Let us determine the liquid current density vector
j = V,j– flow density. - the amount of liquid flowing through a unit section per unit time
From the law of conservation of liquid mass
,
m thread = const
Since the change in mass of a liquid in a selected section is defined as the product of the change in volume and the density of the liquid, from the law of conservation of mass we obtain
VS = const VS = const
V 1 S 1 =V 2 S 2
those. the flow rate in different sections of the flow is the same
2) Ostrogradsky–Gauss theorem
Consider the fluid mass balance for a closed volume
the elementary flux through the site is equal to
where j is the flux density.
Laminar flow of a viscous incompressible fluid in a cylindrical pipe
Animation
Description
Due to the laminar (layered) nature of the flow of a viscous incompressible fluid in a cylindrical pipe, the flow velocity is in some way distributed over the cross section of the pipe (Fig. 1).
Velocity distribution at the entrance to the pipe during laminar flow
Rice. 1
L1 is the length of the initial section of the formation of a constant velocity profile.
Poiseuille's law (the mathematical expression of which is Poiseuille's formula) establishes the relationship between the volume of liquid flowing through a pipe per unit time (flow rate), the length and radius of the pipe, and the pressure drop in it.
Let the pipe axis coincide with the Oz axis of the rectangular Cartesian coordinate system. In laminar flow, the fluid velocity v at all points of the pipe is parallel to the Oz axis, i.e. v x = v y = 0, v z = v . From the continuity equation
dv /dt =F - (1/ r )grad p ,
where F is the field strength of mass forces;
p - pressure;
r - liquid density,
follows that
dv/dz = 0, i.e. v = f(x,y) .
From the equation of motion of a viscous incompressible fluid (Navier-Stokes) it follows:
dp/dx = dp/dy= 0,
dp/dz = dp/dz = h(d 2 v/dx 2 + d 2 v/dy 2 ) = const = -(D p/l) ,
where D p is the pressure drop over a pipe section of length l.
For a round cylindrical pipe, this equation can be represented as
(1/r)d(r(dv/dr))/dr = - D p/ h l ,
where r = sqr(x 2 + y 2) is the distance from the pipe axis.
The velocity distribution over the pipe cross section is parabolic and is expressed by the formula:
v(r) = (D p / 4 h l) (R 2 - r 2 ) ,
where R is the radius of the pipe;
r is the distance from the axis to the cross-section point under consideration;
h is the dynamic viscosity of the liquid;
D p - pressure drop over a pipe section of length l.
The second volumetric flow rate of liquid is determined by Poiseuille's formula:
Q c = [(p R 4 ) /8 h l] D p.
This formula is valid for laminar flows, the conditions of existence of which are characterized by the critical Reynolds number Re cr (Re = 2Q c /p R n, n - kinematic viscosity). At Re = Re cr, laminar flow becomes turbulent. For smooth round pipes Re cr » 2300.
Timing characteristics
Initiation time (log to -1 to 1);
Lifetime (log tc from -1 to 5);
Degradation time (log td from -1 to 1);
Optimal development time (log tk from 0 to 2).
Diagram:
Technical implementations of the effect
Poiseuille's law is applied to determine the coefficients of various liquids at various temperatures using capillary viscometers.
Technical implementation of the effect
Rice. 2
Designations:
1 - control section of the pipe;
2 - balloon;
3 - gearbox;
4 - pressure regulator;
5 - pressure gauge;
6 - valve;
7 - flow meter.
Poiseuille's equation plays an important role in the physiology of our circulation.
Applying an effect
The Poiseuille formula is used when calculating indicators for the transportation of liquids and gases in pipelines for various purposes. The laminar operating mode of oil and gas pipelines is the most energy-efficient. So, in particular, the friction coefficient in laminar mode is practically independent of the roughness of the inner surface of the pipe (smooth pipes).
Literature
1. Brekhovskikh L.M., Goncharov V.V. Introduction to continuum mechanics. - M.: Nauka, 1982.
2. Development and operation of oil, gas and gas condensate fields / Ed. Sh.K. Gimatudinova. - M.: Nedra, 1988.
Keywords
- viscosity
- pressure
- dynamic viscosity
- hydrodynamics
- viscous liquid
- laminar flow
- pressure
- pressure drop
- pipe
- Poiseuille's law
- Poiseuille formula
- Reynolds number
- Reynolds number is critical
Sections of natural sciences:
Formulation of the problem
The steady flow of an incompressible fluid with constant viscosity in a thin cylindrical tube of circular cross-section under the influence of a constant pressure difference is considered. If we assume that the flow will be laminar and one-dimensional (having only a velocity component directed along the channel), then the equation is solved analytically, and a parabolic profile (often called Poiseuille profile) - velocity distribution depending on the distance to the channel axis:
- v- fluid speed along the pipeline, m/s;
- r- distance from the pipeline axis, m;
- p 1 − p
- l- pipe length, m.
Since the same profile (in the appropriate notation) has a velocity when flowing between two infinite parallel planes, such a flow is also called Poiseuille flow.
Poiseuille's law (Hagen - Poiseuille)
The equation or Poiseuille's law(Hagen-Poiseuille law or Hagen-Poiseuille law) is a law that determines fluid flow during steady flow of a viscous incompressible fluid in a thin cylindrical pipe of circular cross-section.
Formulated for the first time by Gotthilf Hagen (German). Gotthilf Hagen, Sometimes Hagen) in 1839 and was soon re-bred by J. L. Poiseuille (English) (French. J. L. Poiseuille) in 1840. According to the law, the second volumetric flow rate of a liquid is proportional to the pressure drop per unit length of the tube and the fourth power of the pipe diameter:
- Q- liquid flow in the pipeline, m³/s;
- d- pipeline diameter, m;
- r- pipeline radius, m;
- p 1 − p 2 - pressure difference at the inlet and outlet of the pipe, Pa;
- μ - liquid viscosity, N s/m²;
- l- pipe length, m.
Poiseuille's law is applicable only for laminar flow and provided that the length of the tube exceeds the so-called length of the initial section necessary for the development of laminar flow in the tube.
Properties
- The Poiseuille flow is characterized by a parabolic velocity distribution along the radius of the tube.
- In each cross section of the tube, the average speed is half the maximum speed in this section.
see also
- Couette Current
- Couette-Taylor Current
Literature
- Kasatkin A. G. Basic processes and apparatuses of chemical technology. - M.: GHI, - 1961. - 831 p.
Wikimedia Foundation. 2010.
See what “Poiseuille Current” is in other dictionaries:
Parabolic velocity distribution in Poiseuille flow. The propellers show that this flow has non-zero vorticity. Poiseuille flow is a laminar flow of liquid through channels in the form of a straight circular cylinder or layer between ... ... Wikipedia
Continuum mechanics ... Wikipedia
Continuum mechanics Continuum Classical mechanics Law of conservation of mass Law of conservation of momentum ... Wikipedia
Translation and meaning of OFF in English and Russian What does off mean in English
What the Earth looks like from space - description, features and interesting facts
Modern Humanitarian Academy