Propagation of Waves in a Fluid in a Thin Elastic Cylindrical Shell
DURDIMUROD DURDIYEV1, ISMOIL SAFAROV2, MUHSIN TESHAEV1
1Institute of Mathematics AS RUz, Bukhara Branch,
11, M. Iqbol str, Bukhara,
UZBEKISTAN
2Mathematics Department, Tashkent Institute of Chemical Technology,
32, Navoi str, Tashkent,
UZBEKISTAN
Abstract: - The oscillatory process of a viscoelastic shell of a cylindrical tipe filled with a liquid is considered.
Unlike other works, this paper focuses on the viscoelastic properties of a cylindrical shell and a liquid.
Differential equations for joint vibrations of a shell and liquid are obtained by the equations of a thin shell that
satisfies the Kirchhoff–Love hypotheses, and the equations of motion of a viscous liquid obey the Navier–
Stokes equation. After simple transformations, the integro-differential equations are reduced to ordinary
differential equations and solved using Godunov's orthogonal run method combined with Muller's
method. Based on the developed algorithm, natural frequencies and corresponding vibration modes were
obtained. For steady-state oscillations, all eigenvalues and eigenmodes turned out to be complex. For the first
time, it was found that the damping coefficient branches out after certain values of wave numbers. It was found
that the motion in a cylindrical shell is localized on the surface of the shell. At slow localization, starting
from a certain wave number, the natural oscillations become aperiodic.
Key-Words: - cylindrical shell, fluid viscosity, orthogonal sweep, eigenforms, complex arithmetic, steady-state
oscillations, damping coefficient, wave number, natural oscillations, cylindrical shell with
liquid.
Received: February 22, 2023. Revised: December 11, 2023. Accepted: February 12, 2024. Published: March 6, 2024.
1 Introduction
The investigation of the process of wave
propagation in a liquid located in an elastic
cylindrical shell has wide applications in various
fields, [1], [2]. This also applies to the problems of
the propagation of pulse pressure waves in blood
vessels and the stability of a blood vessel during the
propagation of pulse waves, [3], [4]. The task of the
process of propagation of normal waves in shells
filled with liquid was considered in several works,
a review of which is given in [5], where wave
propagation processes in the “cylindrical shell -
liquid” system, classical shell equations are used. In
[6], [7], the wave propagation process in an elastic
thick shell with an ideal liquid is investigated in a
linear formulation. The description of wave motion
is carried out based on a complete system of
equations of the dynamic theory of elasticity and
equations of motion of an ideal fluid. The
investigation of the dispersion equation in the large
wave numbers region and its spectrum showed the
existence of Stanley and Rayleigh surface waves,
[8], [9]. The solution to boundary value problems
in [10], was obtained based on the dynamic theory
of elasticity and equations of motion of an ideal
fluid, taking into account radiation into the
environment. In these works, the dispersion
characteristics of waves subject to damping were
obtained for the rigid material of the cylinder. It
was shown in [11], that for the rigid material of a
cylinder in a three-layer waveguide, there are
waves that are not subject to radiation damping. In
this paper, we consider the proses of the
propagation of waves in a thin shell with liquid.
The results of numerical calculations are analyzed
and discussed for torsional osculation of the shell
with a liquid.
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2 Methods
2.1 Problem Statement of Joint Oscillations
of a Cylindrical Shell and Liquid and
Methods of Solution
Consider an infinite viscoelastic cylindrical shell.
Let a non-axisymmetric harmonic wave propagates
along it. The shell has radius-R, thickness-2h,
density-
and Poisson's ratio-
0
, which is filled
with a viscous compressible liquid. In the study, we
use a cylindrical coordinate system
),,(
rx
by
aligning the x-axis with the axis of the shell. We
restrict ourselves to the consideration of linear shell
equations (Kirchhoff-Love type) and linearized
Navier-Stokes equations for viscous compressible
fluid, [12]. In the framework of the proposed
formulation, the system of equations has the form:
(1)
0)(
3
1
0
divgradPgrad
t
;
;0
1
0
div
t
(2)
;, 0
2
0constaa
p
;;;
;;;
rrrrrzx
rrxx
pqpqpq
uuu
(3)
.
1
;2
;
rrr
p
rrxr
pp
xr
p
x
r
rrxr
rr
rx
rx
(4)
Here L- differential operators matrix of
Kirchhoff–Love type shell theory:
;
11
2
1
2
1
141
2
1
2
1
2
1
2
1
2
2
2
22
2
23
3
22
3
2
3
3
22
3
22
2
2
22
2
2
2
22
2
Rx
a
RR
a
x
a
R
xR
R
a
x
a
R
a
x
a
x
R
xRx
RRx
L
zr uuuuu ,,
- the displacements vector;
qqqqq rx ,,
-the force vector of the external
load;
;12/22 Rha
tRE
– the relaxation core;
0
E
– the instantaneous modulus of elasticity;
,, rx
is the velocity vector of fluid
particles;
and
p
are the disturbances of
density and pressure in a liquid;
0
and
0
а
are
the density and speed of sound of the liquid;
,
are the viscosity coefficients;
3
2
;
rrrrx ppp ,,
are the components of the stress
tensor in a fluid.
The joint oscillations of the shell and liquid,
harmonic along the coordinate x and decaying in t,
or harmonic in t and decaying in x, are subject to
investigation.
Assume that the integral terms in (1) are small,
then
ti R
etrutru
,, 1
, where
tru ,
1
-is the
slowly varying vector function t. Further, applying
the freezing method, [13], equations (1) will be
reduced to the form:
,
11
2
2
0
2
0
2
t
u
E
p
hE
uL E
(5)
where
R
S
ER
С
EE i
1
;
0
)cos(
dR RR
C
E
,
0
)sin(
dR RR
S
E
- respectively, cosine and sine Fourier images,
R
-
real constant. The Rzhanitsyn-Koltunov kernel
1
/tAetR t
is taken as a viscoelastic
material model with parameters
,,A
, which are
determined from experiments, [14].
Let us replace the variable x by the
dimensionless coordinate:
Rx /
. Decomposing
the equations in coordinate form, it is easy to see
that equations (2) and (3) decompose into two
boundary value problems:
- torsional vibrations:
,
)(2
0
x
p
r
p
r
pxrr
(6)
,, r
p
rr
pxr
.0:0
.
)1(2
,0)(:
0
0
2
2
r
r
pr
v
E
Gpuh
x
u
GhRr
(7)
- longitudinal-transverse vibrations:
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,
,
0
0
x
xxrxrx
r
rxrr
rr
x
p
r
p
r
px
p
r
pp
r
р

(8)
,2
;2
;2
z
divkpp
r
divkpp
r
divkpp
x
xx
r
r
rr

(9)
󰇗 
󰇗  



 

(10)
.0,0,0
,0)(
,0:
0
0
2
2
00
4
4
rrz
zrz
rr
rrr
zrr
pr
uhp
R
u
R
v
x
u
C
uhp
z
u
R
u
R
C
z
u
DRr
(11)
The solution of the boundary value problem of
torsional vibrations (6)-(7) sought in the form, [15].
tiikx
m
xrxr emUVPPuрр
sin,,,,,,
1
, (12)
where
UVPP xr ,,,
-are the wave amplitudes.
The case m = 0 describes axisymmetric
vibrations of the shell with liquid. To clarify the
physical meaning of the complex wave number and
frequency, consider the following cases:
1) case
R
кк
;
IR i
. In this case,
the solution of (12) , will be sinusoidal along x, and
the amplitude decays as t proceeds;
2) case
IR iккк
;
R
. In this case
the oscillations will be stable, but along x damped.
For axisymmetric longitudinal-transverse
vibrations the conditions:
0 rzrрр
,
r
=0
must be met. By analogy with the indicated type of
motion, (12) describes natural or free vibrations. In
the case of a process that is stationary in time and
decaying along the coordinate, on the contrary, the
real frequency
R
is known, and the complex
wave number k -is the sought one. Unlike our own,
we will agree to call these oscillations steady. The
real parts of the values have the physical meaning
of the process frequencies, and the imaginary parts
- the damping rate of wave processes, [16]. The
value 1/Imk determines the propagation interval of
damped waves. The degree of attenuation is
characterized by the logarithmic decrement, [17]:
)(Re/)(Im2
c
(13)
similarly, the spatial decrement is equal to
kk
yRe/Im2
. (14)
Two types of oscillations (natural and steady)
can be given two different formulations of the
problem. In the non-stationary case, namely the
Cauchy problem for an infinite shell and the
boundary value problem for a semi-infinite interval
of variation of x. In both cases, the solution is
found using integral transformations from the
solutions of the corresponding stationary problems.
Thus, in the case of the Cauchy problem, the vector
of the main unknowns
c
Y
can be defined as a
superposition of waves:
The two types of oscillations (natural and
steady-state) can be given by two formulations. In
the unsteady case:
1) Cauchy problem for an infinite shell;
2) Boundary value problem for a semi-infinite
interval x.
In both cases, the solution is found using an
integral transformation from the solutions of the
corresponding stationary problems. In the case of
the Cauchy problem, the solution can be defined by
a superposition of waves:
n
n
c
n
cdktkkztkrYY )])((exp[),(
(15)
where
c
n
Y
- are eigenforms of the own oscillations,
spectrum of initial perturbation
)0,,(),( zrYzrf c
forms their linear combination:
󰇛󰇜
󰇛󰇜  󰇛󰇜
󰇛󰇜

. (16)
The main unknowns vector
y
Y
is calculated by
expression
n
y
k
ydtzikrYtzrY
])(exp[),(),,(
(17)
where
y
k
Y
-are the forms of steady-state
oscillations 󰇛󰇜󰇛󰇜
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󰇛󰇜 󰇛󰇜󰇛󰇜

󰇛󰇜

(18)
It is obvious that solutions (12) make sense only
when (15) and (17) exist. Substituting solutions
(12) into the system of differential equations (6)
and (7), obtained a system of ordinary differential
equations with complex coefficients, which is
solved by the Godunov orthogonal sweep method
in combination with the Muller method, [18], in
complex arithmetic.
Substituting the solutions of (12) into the
system (6) and (7), a system of ordinary
differential equations is obtained. The obtained
system is solved using the orthogonal run
method of Godunov, as well as Muller's
method, [18].
2.2 Torsional Vibrations
After substituting (12) into (6) and (7) we obtain a
system
.0:0
,0)(:
;
;
2
)(
2
0
2
1
222
0

r
r
r
rr
Pr
PUkGkhRr
P
i
r
V
dr
dV r
P
Vкi
dr
dP
(19)
Let us first examine the vibrations of the liquid
in the walls. Equations (12) can be transformed into
one equation for displacement v:
0
2
2
2
2
;0)
1
(
vV
rv
ik
rdr
dV
dr
Vd
. (20)
The solution (20), bounded at r = 0, is written in
the form
0)( 2
11
v
ikrJAV
, (21)
where J1 -is the Bessel function, A1 - an arbitrary
constant.
Taking into account the stationary state of the
shell, the dispersion equation takes the form
󰇛
󰇜 , (22)
where
)( 22
тnГkvi
, (23)
in case of natural vibrations
Г
 , (24)
here, Гm denotes the Bessel function roots related to
R. From (22), (23), the natural oscillations are
aperiodic, the nodal points are stationary, the
steady-state motions are oscillatory in nature, and
the nodal points move with velocity Cy.
Let us now consider relations (19) when the
fluid is inside. Then the dispersion equation
0)2
)(
)(
(
~
~1
0
23
2
2
2
zJ
zJ
z
Rhpa
v
a
k
, (25)
first obtained in [19]. In the notations:
ℎ
ℎ
 
 
where Jo-Bessel function of zero order.
Direct solution of equation (25) encounters
certain difficulties caused by the need to calculate
the Bessel function of a complex argument.
Therefore, we study (25) using asymptotic
representations of these functions for small and
large arguments z. For low-frequency oscillations,
this occurs for a small argument (z). According to
the known expansions J0 and J1, the power series
The analytical solution of (25) will lead to
certain difficulties, for example, calculating the
values of the Bessel function. Therefore, we
propose to investigate (25) by asymptotic method.
For low-frequency oscillations, this occurs for a
small argument (z). It is known that the power
series J0 , J1
...);
8
1(
2
)(...;
4
1
2
1
2
0 zz
zJ
z
J
(26)
Keeping only the first terms in expansions (26),
we obtain
0
2
2 a
k
In (26), leaving only the first two terms, we have
0)(
~
~
4
2
2
2
2
2
v
ik
hpa
v
i
a
k
, (27)
at steady-state oscillations roots of which is
defined as
󰇣󰇛

ℎ
󰇜󰇛 

ℎ
󰇜󰇤. (28)
Let us now consider the case when z is
sufficiently large, which refers to high-frequency
oscillations at low viscosity. Then for the Bessel
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function
)
4
sin()
2
()(),
4
cos()
2
()( 2/1
1
2/1
0
z
z
zJz
z
zJ
Based on (27) , (28) it can be shown that for a
sufficiently large positive imaginary part z, the
following holds:
.)(/)( 10 izJzJ
Then, taking into
account the smallness
compared to the value
2
/k
, we obtain an approximate dispersion
equation, which is also given in [19]:
0)
41.1
~
~
1(
3
2
2
il
Rh
р
v
a
k
, (29)
whence, as the viscosity coefficient
v
tends to
zero (to infinity), we have
0
k
, which is
obtained for small
from (27). This example
indicates to inconsistency of asymptotic estimates
in the mid-frequency region.
Thus, for investigating wave processes by
asymptotic methods, it is not possible to show the
limits of applicability of the results obtained and
their estimates. To solve spectral problems, a
numerical solution of relations (19) is proposed
using the orthogonal sweep method. It has been
established that the problem of natural vibrations
(19) allows no more than one complex value of ω,
corresponding to vibrations of the shell together
with the liquid. The remaining eigenvalues are
purely imaginary. They correspond to aperiodic
movements of the liquid with an almost motionless
shell. Eigenforms corresponding to complex
eigenvalues are also complex. For steady-state
oscillations, all eigenvalues k and eigenmodes are
complex.
3 Results and Analysis
Let us consider the own osculations of a shell filled
with liquid. In Figure 1 shows the dispersion curves
of Imω depending on the wave number k.
Fig. 1: Change of Imω as a function of wave
number k
The curves were obtained with coefficient
values η: 1) 0.0009: 2) 0.0018 3) 0.15 4) 0.018.
Table 1 shows the values of the real parts of
frequencies and relative phase velocities Re(C/Vs),
corresponding to surface waves of the Stoneley and
Rayleigh type for a steel cylinder filled with water
R=0.25 and 0.75 (RE=0) [20].
The phase velocity of the lower wave
approaches the velocity of the Stoneley wave from
below. The phase velocity of the composite
waveguide tends to the Rayleigh wave velocity
from below with increasing wave number (Table
1).
Table 1. Dependence of the real part of the phase
velocity as a function of the wave number r
R1
k
ReC/Vs
0.25
10
0.48633
0.25
20
0.49756
0.25
30
0.53472
0.25
40
0.62175
0.25
50
0.92761
0.25
70
0.94129
0.25
80
0.94453
0.25
100
0.94749
Fig. 2: Change of Re as a function of wave
number r (
=2, η=0,018;б)
=2, η=0,0018).
Analysis of the dependence of energy
dissipation on wave number r shows that there are
two opposite trends. With increasing wave number
at a fixed amplitude
, according to (12),
tangential stresses
z
p
increase linearly; on the
other time, as evidenced by Figure 2 and Figure 3,
at the same time the amplitudes of fluid motion
near the shell are localized.
3,
4
1
2
1
0
r
Re
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The difference between curves 1.2 and 3.4 is
such that some trend prevails. At small wave
numbers, the proper shape v of the radius will
depend linearly, i.e., the entire mass of the liquid is
involved in the movement. If localization of
amplitudes occurs slowly, then starting from a
certain k (due to an increase in stress) the own
oscillations become aperiodic (curves 3, 4).
Fig. 3: Dependence of the amplitude of
displacement on the radius r
It has been established that the critical value of
the viscosity coefficient ηk is in the interval
0.0125 ; 0.0120
. If the amplitude of fluid vibrations
decreases quickly enough, then the movement will
be oscillatory (curves 1, 2). Then stresses related to
large wave numbers dominate over other stresses
and increase as localization increases. Therefore,
the attenuation coefficient increases with increasing
k.
In paper [20], the problem of determining the
relaxation kernel of the obtained integro-
differential viscoelastic equation is considered.
4 Conclusions
The problem statement is formulated and a
technique is developed for solving the problem of
propagation of natural waves on a cylindrical shell
with a viscous liquid. In the case of steady-state
oscillations, all calculated eigenvalues and
eigenmodes turned out to be complex. With slow
localization, natural oscillations become aperiodic
(starting from a certain wave number).
References:
[1] Bao X.L., Raju P.K., Überall H.
Circumferential waves on an immersed,
fluid-filled elastic cylindrical shell, Journal
of Acoustic Society of America. 105(5), 1999,
2704-2709.
[2] Shmakov V.P. Selected works on
hydroelasticity and dynamics of elastic
structures. Moscow, Bauman Moscow State
Technical University, 2011.
[3] Bert C.W., Baker J.L., Egle D.M. Free
vibrations of multilayer anisotropic
cylindrical shells. Journal of Composite
Materials. Vol.3 July, 1969, 480-499.
[4] Safarov I. and Teshaev M. Control of
resonant oscillations of viscoelastic systems.
Theoretical and applied mechanics, 2023,
https://doi.org/10.2298/TAM220510007S.
[5] Farshidianfar A.P., Oliazadeh, M.H., Farshidi
anfar A. Exact Analysis of Resonance
Frequencies of Simply Supported Cylindrical
Shells. International Scholarly and Scientific
Research & Innovation. 7(4), 2013, 335-
341.
[6] Flugge W. Stresses in Shells. Springer, New
York, 1973.
[7] Mirzaev, I.M., Nikiforovskii, V.S. Plane
wave propagation and fracture in elastic and
imperfectly elastic jointed structures. Soviet
Mining Science, 9, 1973, 161–165,
https://doi.org/10.1007/BF02506181.
[8] Mirzaev, I.M. Interaction between the bit and
the rock. Soviet Mining Science, 11, 1975,
70–73, https:// doi.org/10.1007/BF02501021.
[9] Karimov K., Khudjaev M., Akhmedov A.
Modeling fluid outflow from a channel
consisting of three different segments / E3S
Web of Conferences. 258, 2021, 08021, DOI:
10.1051/e3sconf/202125808021.
[10] Karimov K., Khudjaev M., Akhmedov A.
Modeling fluid outflow from a channel
consisting of three different segments / E3S
Web of Conferences. 258, 2021, 08021, DOI:
10.1051/e3sconf/202125808021.
[11] Jeong Ho You, K. Inaba. Fluid-structure
interaction in water-filled thin pipes of
anisotropic composite materials. Journal of
Fluids and Structures. 36(0), 2013, 1, 162 –
173.
[12] Klehchev A.A. Against Phase Veloсities of
Elastic Waves in Thin Orthotropic
Cylindrical Shell. Advances in Signal
Processing. 1(3), 2013, 44-47.
[13] Kunte M.V., A. Sarkar, V.R. Sonti.
Generalized asymptotic expansions for
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coupled wavenumbers in fluid- filled
cylindrical shells. Journal of Sound and
Vibration, 329, 2010, 5356-5374.
[14] Honarvar F., Enjilela E., Sinclair A.N. An
alternative method for plotting dispersion
curves, Ultrasonics, 49(1), 2009 Jan, 15-18.
[15] Jeong Ho You, K. Inaba. Fluid-structure
interaction in water-filled thin pipes of
anisotropic composite materials, Journal of
Fluids and Structures, 36(0), 2013(1), 162 –
173.
[16] Pavlovsky V.S., Borisenko V.I. Study of
vibrations of a cylindrical shell containing
liquid, Appl. fur. 5(6), 1969, 87-91.
[17] Novichkov Yu. N. Study of natural vibration
frequency spectra of cylindrical shells
containing compressible fluid. Moscow,
Nauka, 1966.
[18] Ter-Akopyants G.L. Mathematical modeling
of elastic deformation waves in pipes
interacting with liquid. Abstract of
candidate's dissertation. St. Petersburg.
2015.
[19] Guz A. N., Kubenko V. D., Babaev A. E.
Hydroelasticity of shell systems, Kiev,
Vishcha Shkola, 1984.
[20] Durdiev, Durdimurod Kalandarovich and
Totieva, Zhanna Dmitrievna, “The problem
of determining the one-dimensional matrix
kernel of the system of viscoelasticity
equations”, Math Methods in Appl Sciences,
41:17(2018), 8019-8032.
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Durdimurod Durdiev and Ismoil Safarov posed
the problem mathematically. Methodology
development or design; creation of models.
- Muhsin Teshaev implemented Ideas; formulation
or evolution of overarching research goals and
objectives.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
No conflicts of interests
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
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