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17.Сборник задач по методам вычислений. Учебное пособие / Под
ред. П.И.Монастырного. – 2-е изд. – Мн.: Университецкое, 2000. – 311 c.
Mengliyev Sh.A., Doctor of Philosophy in Technical Sciences
Termez State University (Ph.D)
Xamrayev A.B., student of Termez State University
TECHNOLOGY FOR CREATING A DEVICE FOR LAMINAR FLOW OF WATER
IN PIPES
Sh. Mengliyev A. Xamrayev
Abstract: The article discusses the mathematical modeling of the
movement of viscous incompressible fluids through a bundle of tubes
located inside the outer pipe. The laminar and turbulent modes of this
movement are considered, and the physical meaning of their occurrence is
also analyzed. The fluid flow through n tubes of length L and radius r located
inside the outer tube is considered. Calculation formulas are derived for
calculating the maximum velocity of this flow, the volume of fluid passing
through the cross section of the tube, the coefficient of resistance to friction
in the tube along the length of the flow, and also the maximum value of the
tangential stress. The results of the study of the relationship of the
coefficient of resistance to friction in the tube with the Reynolds number are
presented. A description is given of a device created according to the results
of a study that brings the disordered flow of liquids into a laminar state.
Keywords: Reynolds number, laminar flow, turbulent flow, parabolic
flow, friction force, integral, coordinate, pipe, viscosity, density, main flow
velocity, average speed, maximum speed, radius, Hooke, Gegin, Poiseuille,
Darcy-Weisbach, fluid volume, drag coefficient.
The motion of real fluids is often very different from that of laminar flow.
They have a special property called turbulence. As the Reynolds number
increases in real fluid flows in pipes, channels, and boundary layers, the
transformation of a laminar-shaped flow into a turbulent flow is clearly
observed. This transition of laminar flow to turbulent flow is sometimes
called turbulence, which is fundamental in the whole field of
hydrodynamics. Initially, such a transition was observed in the flow of
straight pipes and channels.
Information on the forces acting on a fluid for flow in a cylindrical tube
is given in the article [2; pp. 36-47].
Consider the motion of a tube of constant diameter along its entire
length and the flow of fluid through n tubes of length L and radius r placed
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inside the tube. In real liquids, the liquid sticks to the walls of the tubes and
exerts a repulsive stress on the flowing surface. This creates a force called
internal friction, which is the viscosity of liquids. Viscosity is the property of
gases and liquids to resist the action of external forces that cause the fluid to
move. The presence of experimental stresses and the adhesion of the fluid
to solid walls cause the real fluids in motion to be qualitatively different from
ideal fluids. We now calculate the forces acting on the fluid in the pipe, taking
into account the n tubes. Due to the adhesion, the velocity in the tube wall is
zero, and the velocity between the tubes reaches its maximum value. Some
concentric layers move so that the velocity is axial everywhere and the flow
is laminar. At a sufficiently long distance from the starting point of the tubes,
the velocity distribution of the flow in the tube does not depend on the
longitudinal coordinates along the radius.
The movement of the fluid in the tube is due to the fact that the pressure
decreases along the axis of the tube, but the pressure does not change in the
cross section perpendicular to the axis of each tube. The motion of each
element of the fluid accelerates due to the pressure drop and slows down
due to the shear stress caused by friction [3; pp. 36-38]. The pressure p is
assumed to be constant, i.e., across the entire tube section [4; pp. 59-60].
There are compressive forces and effects on the cylinder along the main
axis, which correspond to the inlet and outlet bases of the cylinder,
respectively, and there is an actuating force acting on the side surface of the
cylinder. It is necessary to determine the maximum flow rate in this cylinder,
the volume of liquid flowing through the cross section of the pipe, the
coefficient of frictional resistance of the pipe along the length of the pipe,
and the maximum value of the test voltage.
By balancing the forces acting on the fluid in the tube (Figure 1), we
obtain the following equation as a condition of equilibrium in the direction
of motion:
. (1)
Figure 1. N tubes are placed inside the tube
n
i
n
i
l
n
i
yL
y
p
y
p
1
1
2
1
2
0
2
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The projection of the internal friction force is obtained with a positive
sign because the velocity gradient is negative (the flow velocity of the layer
decreases with increasing coordinate).
From formula (1) we can determine the test voltage
(2)
In this case, the flow rate decreases with increasing coordinate and
becomes zero due to the viscosity. Therefore, we can say that according to
Newton's law of
friction.
Substituting this expression into (3), we get:
Henceforth
(4)
Now, assuming that when
, it is
, we integrate Equation
(4) with this initial condition to form the following equation
, (5)
To find the constant C in Equation (5), we use the condition that the
velocity is
when
i.e.
henceforth
(6)
we find that Set this value of the constant to (6)
equation and so on
(7)
we have the equation
Thus, we have a parabolic distribution of velocities along the radii of the
tubes (Figure 2). This velocity reaches its maximum value in the middle of
the tube ( y=0 ) and has the following maximum value:
0
2
l
p
p
y
L
dy
du
0
2
l
p
p
du
y
dy
L
0
2
l
p
p
du
y
dy
L
r
y
0
)
(
y
u
2
0
( )
4
l
p
p
u y
y
C
L
r
y
( )
0
u r
2
0
( )
4
l
p
p
u r
r
C
L
2
0
4
l
p
p
C
r
L
2
2
0
0
( )
4
4
l
l
p
p
p
p
u y
y
r
L
L
2
2
0
( )
(
)
4
l
p
p
u y
r
y
L
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185
(8)
Figure 2. Flow motion for a single tube
The total fluid flow (fluid flow) through the tube section is defined as
the circulating paraboloid volume (Figure 2) and is defined as
follows.Equation (7) we have the following formula:
(9)
Using the Gagen-Poiseuille formula for the total flow of fluid through a
tube with a circular cross section, we determine:
that is, we have a formula for the flow rate
(10)
Enter the average flow velocity across the cross section of the tube:
(11)
Given formula (10), we write (11) as follows
Comparing the function
with
defined by formula (8), the
average velocity for the motion
is half the maximum velocity.
We determine the pressure difference
.
2
0
max
4
l
p
p
u
r
L
2
2
2
2
0
max
2
2
( )
(
)
1
4
l
p
p
r
y
y
u y
r
u
L
r
r
r
r
r
r
y
y
u
dy
r
y
y
u
ydy
y
u
Q
0
2
4
2
max
0
2
3
max
0
4
2
2
2
2
)
(
4
0
(
)
8
l
p
p r
Q
L
2
r
Q
u
2
0
(
)
8
l
p
p r
u
L
)
(
y
u
max
u
max
2
1
)
(
u
y
u
0
(
)
l
p
p
0
2
8
l
Lu
p
p
r
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Henceforth
(12)
where
the diameter of the tube.
The pressure lost along the length of the flow is found by the Darcy-
Weisbach equation [5-9]. Summarizing the above formula for one tube for -
ta tubes, we obtain the following formula
(13)
Substituting the value of
from formula (4.34) into (13) we
obtain the following formula
or more
(14)
where is the number of tubes, the coefficient of resistance decreases
with increasing number of tubes
.
Above (14), we present the results obtained on the basis of the formula
(Fig.
3).
Figure 3. The dependence of the coefficient of resistance in a smooth pipe
on the number of tubes n and Re:
1) n=200; 2) n=300; 3) n=400; 4) n=500.
0
2
8
32
32
(
)
2
2
l
Lu
u
L
u
L
p
p
r
r
r
D
D
2
D
r
n
i
i
l
D
L
u
p
p
1
2
0
)
(
2
0
l
p
p
2
32
2
64
n
u L
D
D
D
n L
uD n
u
64
Re
n
n
Re
uD
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Figure 3 shows the results of calculations showing that the resistance
coefficient
for a smooth tube depends on the Rnolds number
. A
comparison of the results shows that the theoretical formula (14) is valid
for all values of the number
. At larger values of the number
, the
resistance decreases due to the active activation of the turbulence
mechanisms.
In the computational experiment, the characteristic parameters
Reynolds number Re and
of the coefficient of resistance are studied in
the following quantities:
,
. The figure shows that as the
number of tubes increases, the coefficient of resistance decreases.
Based on the above results, it is possible to create a device that allows
the regulation of water flows. The device can be implemented in the
manufacturing process, it can be used to regulate the flow in water
fountains.
The proposed device allows you to make the fountains look beautiful
and enrich them with different colors.
The main purpose of the device is to maintain the flow of water in the
form of a uniform laminar flow with the help of its working bodies.
The device can be explained by the following diagrams, its working
elements are shown, Figure 4 shows the internal section of the device, where
the base legs holding the device at an acute angle and the device is firmly
attached to it, 2 tube-shaped housing, 3 -cylindrical section with inlet and
outlet holes mounted vertically from top to bottom, 4 small-diameter tubes,
which serve to convert the tubular flow into laminar flow, and 5-distributor,
which serves to evenly distribute the flow to small-diameter tubes by
reducing the flow rate, 6 -network holding small-diameter tubes, which in
turn serves for the orderly movement of the flow.
Figure 4. Device interior
n
n
Re
Re
Re
n
Re
500 5000
0.0001 0.0007
n
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The device that converts the flow to a laminar view works as follows:
the flow from the inlet 3 is directed to the flow 5, which reduces the flow
rate and the flow is evenly distributed in the small diameter tubes, the
current passing through the small diameter tubes 4 becomes laminar and It
is shot in a laminar view through 3 outlet holes.
The experimental study found that the optimal length for a set of tubes
placed inside a tube was 12–16 cm, and that the resistance coefficient
decreased as the tube set increased, resulting in a device for converting
chaotic flows into laminar flows.
The proposed device is an in-depth analysis of laminar flow in the
movement of incompressible viscous fluids in the pipe, which is important
in technical applications, when it is necessary to convert turbulent (chaotic)
flows into laminar (layered, ordered) flows, including in parks. They can be
used when there is a need to effectively control the flow of water, that is, to
create a variety of pleasant and colorful streams of water in the installed
fountains, to give people an aesthetic pleasure. Laminarization of water
flows in the pipe is very useful and cost-effective, ie it saves water.
References:
1.Reynolds O. On the experimental investigation of the circumstances
which determine whether the motion of water shall be direct or sinuous, and
the law of resistance in parallel channels Phil. Trans.roy.soc. 1883. № 174.
P. 935-982.
2.Mengliyev Sh.A. Trubada qisilmaydigan yopishqoq suyuqliklar
harakatini laminar oqimga aylantirish Xorazm ma’mun akademiyasi
axborotnomasi, Xiva-2018. №. 2. Б. 105-110.
3.Гордин В.А. Дифференциальные и разностные уравнения
Изд.М.:«Высшая школа экономики», 2016. 517 с.
4.Горшков-Кантакузен В.А. К вопросу вычисления коэффициента
Дарси
методом
регрессионного
анализа
Материалы
XXI
Международного симпозиума "Динамические и технологические
проблемы механики конструкций и сплошных сред" имени А.Г.
Горшкова, 16-20 февраля 2015, Вятичи. Том 1./МАИ.:ООО "ТРП",2015. С.
59-60.
5.Кочен Н.Е., Кибель И.А., Розе Н.В.Теоретическая гидромеханика
М: Физматлиз, 1963. 728 с.
6.Лойцянский Л.Г. Ламинарный пограничный слой М: Физматлиз,
1962.479с.
7.Шлихтинг Г. Теория пограничного слоя. М.: Наука, 1974. 571 с.
8.Гольдштик М.А., Штерн В.Н. Гидродинамической устойчивость и
турбулентность. Hовосибирск: Наука, Сиб. Отд-ние, 1977. 366 с.
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9.Дразин Ф. Введение в теорию гидродинамической устойчивости.
М.:Физматлит, 2005. 88 с.
Baxodir Khamrokulov Independent researcher of UWED, PhD in law,
Tashkent, Uzbekistan
Compensation for moral damage caused by violation of the author's right
B. Khamrokulov
Abstract: Nowadays, there are many cases of using works protected by
copyright without obtaining the appropriate permission of the author or the
owner of such rights. The most common types of copyright infringement
include copying, distribution of the work, mass demonstration, mass
execution of works in concert halls, theatrical productions, translation of the
work into other languages, processing of the work (plagiarism), etc. As a
result of committing such an illegal act, not only material damage, but also
moral damage can be caused to the author of the work. In this article, the
issue of compensation for moral damage caused by violation of the rights of
the author has been studied.
Keywords: copyright law, intellectual property, moral damage, material
damage, mental calmness, anguish, feeling uncomfortable.
With the development of modern information and communication
technologies, the illegal users of literary works (literary-artistic, scientific,
educational, publicist and other works), dramatic and scenario works,
works of text and non-text music, musical-dramatic works, choreographic
works and pantomime, audiovisual works, painting, sculpture, graphics,
design works and other fine arts, works of landscape-applied and stage
decoration art, architecture, urban planning and garden-park development
works of art, photographic works and works created in similar ways to
photography are often encountered. As a result of this, personal non-
property and property rights of the author are violated.
We can say that our main laws aimed at protecting copyright are the
Civil Code of the Republic of Uzbekistan and the law of the Republic of
Uzbekistan "On copyright and related rights", the Criminal Code of the
Republic of Uzbekistan and other normative-legal acts. The result of the
reforms carried out in this area was the signing of the law of the Republic of
Uzbekistan "On the Accession of the Republic of Uzbekistan to the Treaty of
the World Intellectual Property Organization on Copyright (Geneva,
December 20, 1996)" on February 2, 2019.
