I. BERNYK et al.: EFFECT OF RHEOLOGICAL PROPERTIES OF MATERIALS ON THEIR TREATMENT ...
465–468
EFFECT OF RHEOLOGICAL PROPERTIES OF MATERIALS ON
THEIR TREATMENT WITH ULTRASONIC CAVITATION
VPLIV REOLO[KIH LASTNOSTI MATERIALOV NA NJIHOVO
OBDELAVO Z ULTRAZVO^NO KAVITACIJO
Iryna Bernyk
1
, Oleksandr Luhovskyi
1
, Ivan Nazarenko
2
1
National Technical University (KPI), Department of Applied Hydro-Aeromechanics and Mechatronics, Institute of Mechanical Engineering,
Prosp. Peremohy 37, Solomyanskyi district, 03056 Kyiv, Ukraine
2
Kyiv National University of Construction and Architecture, Department of Machines and Equipment of Technological Processes, Faculty of
Automation and Information Technologies, Povitroflotsky Avenue 31, 03037, Kyiv, Ukraine
i_nazar@i.ua
Prejem rokopisa – received: 2017-02-10; sprejem za objavo – accepted for publication: 2018-02-02
doi:10.17222/mit.2017.021
In the research, we proved that the optimum level of energy is the basis for an effective treatment of materials using ultrasonic
cavitation. The key energy parameters are the pressure at the contact zone of the cavitator and material, and the speed of the
contact zone movement. The main objectives of our work were to investigate the changes in the pressure components and
determine their values considering the rheological parameters and the parameters of the dynamic material and cavitator. To
achieve the objectives, we researched the elastic and dissipative components of rheological properties and determined their
functions depending on the parameters of ultrasonic vibrations. The research methodology was based on the use of the
fundamentals of the classical theory of acoustics. The mathematical description of the process was done using a model system
including an acoustic apparatus and environment; this model was made by the authors. We calculated dissipation with the
equations for environment motion considering two different laws of dissipation-factor changes; this was the requirement of the
new model system. We proposed a new mathematical model; the researched system of the acoustic apparatus and the
environment was considered as a whole. So, the elastic and dissipative parameters of the system were regulated among
themselves. The selected mechanism of the regulation parameters was a system in resonance, achieved with a high-quality
process of cavitation with a minimum energy consumption. We found the analytical dependence for determining the dynamic
pressure on the environment with the acoustic apparatus. These dependencies provided the basis for assessing the influence of
rheological properties on the treatment process using ultrasonic cavitation.
Keywords: material, rheological properties, ultrasonic cavitation, energy, pressure
V pri~ujo~i raziskavi so avtorji dokazali, da je optimalni energijski nivo osnova za u~inkovito obdelavo materialov z ultrazvo~no
kavitacijo. Klju~na energijska parametra sta tlak v kontaktni coni kavitatorja in materiala, ter hitrost gibanja kontaktne cone.
Glavni cilj raziskave je bil dolo~iti njune vrednosti ob spreminjanju tla~nih komponent, z upo{tevanjem reolo{kih parametrov in
parametrov dinamike materiala in kavitatorja. Da bi dosegli cilje raziskave, so avtorji raziskovali elasti~ne in disipacijske
(izgubne) komponente reolo{kih lastnosti in dolo~ili njihove funkcijsko odvisne parametre ultrazvo~nih vibracij. Raziskovalna
metodologija je temeljila na uporabi temeljev klasi~ne teorije akustike. Za matemati~ni opis procesa so avtorji uporabili lastni
modelni sistem akusti~ni aparat - okolje. Izra~unali so disipacijo v ena~bah za gibanje okolja z dvema razli~nima zakonoma
sprememb faktorjev disipacije; to je bila zahteva novega modelnega sistema. Avtorji so predlagali nov matemati~ni model;
raziskovani sistem akusti~ni aparat – okolje, ki so ga obravnavali kot celoto. Tako elasti~ni kot disipacijski parametri sistema so
med seboj samoregulirani. Izbrani mehanizem parametrov reguliranja je bil sistem v resonanci. To so dosegli z visoko
kvalitetnim procesom kavitacije ob najmanj{i porabi energije. Dobili so analiti~no odvisnost za dolo~itev dinami~nega tlaka na
okolje z akusti~nim aparatom. Ta odvisnost jim je slu`ila za oceno vpliva reolo{kih lastnosti na proces obdelave z ultrazvo~no
kavitacijo.
Klju~ne besede: material, reolo{ke lastnosti, ultrazvo~na kavitacija, energija, tlak
1 INTRODUCTION
Today’s technologies use cavitation as the major
method for the materials treatment in various fields.
1–12
The ultrasonic technology can intensify technological
processes, increase the degree of raw-material utiliza-
tion, change the material properties, create new sub-
stances and environments, ensure environmental and
production safety.
5
The existing research methods for the
cavitation process are usually used separately for finding
adequate machine constructions;
13
they have to generate
their vibrations with the resonance frequency, deter-
mined by various conditions. The technological aspect of
the cavitation bubble process was also investigated.
Scientific works
1,2
researched the processes of bubble
formation and its oscillation, increasing the size with the
next imploding bubbles; these works determined the fac-
tors such as the change in the external pressure on a
separate bubble and the change in the pressure in its
center,
2
the sonoluminescence availability,
8–10
the che-
mical effects of ultrasound,
11,12
the temperature increase,
2
the evaluation of a possible loss of the bubble spherical
shape,
2
the influence of viscosity, and the spatial and
temporal dynamics of multiple bubbles.
7
It was proved
7
that the threshold amplitude of the sound pressure
needed to perform a cavitation process depends on many
parameters; these include the static pressure, the sound
frequency, the type of fluid, and the amount of dissolved
Materiali in tehnologije / Materials and technology 52 (2018) 4, 465–468 465
UDK 66.017:665.7.035.6:620.193.16:620.179.16 ISSN 1580-2949
Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 52(4)465(2018)

gas and impurities. We should mention that a cavitation
process is characterized by a number of parameters
(Figure 1).
The above-mentioned studies made significant contri-
butions to the understanding of the physics of a cavi-
tation process, and the identification of the modes and
parameters. However, there is the issue of the interaction
of the cavitation machine and the treated material. How
does the process of moving this system because of the
energy transfer from the cavitator to the material due to
the complex transfer of mass and heat in the current
field, with a high cavitation transformation of the energy
and shock waves take place? What model should des-
cribe the motion? The exact description of this process is
too complicated, but it is obvious that this pressure is the
key parameter of the evolution of gas and air bubbles in
an acoustic field formed at the boundaries of the cavi-
tator-environment system. Both the pressure and the
energy used for the technological process of cavitation
should be researched, obtaining the knowledge of the
rheological properties of materials, and the acoustic
parameters and characteristics of a cavitator. The re-
search method was formed by assessing the impact of the
rheological properties of the material on the resonance of
the machine-environment system, considered for deter-
mining the parameters of the cavitation process as a
single vibro-acoustical system.
2 EXPERIMENTAL PART
The methodological techniques for the research and
for selecting the physical and mathematical models of
the technological environments under the influence of
cavitation require the knowledge of the rheological pro-
perty changes such as elasticity, toughness and ductility.
We researched the influence of rheological properties in
the process of their treatment in terms of dispersed envi-
ronments; their viscous and plastic properties were
evaluated considering the laws of dynamic-viscosity-
coefficient change. The influence of the elastic and
inertial properties of the technological environment in
terms of cavitation treatment was considered when
calculating the speed of wave propagation c
K
in Equation
(1):
6
c
E
K
=
 (1)
Where E is the elastic modulus of the environment
with the density  . The speed of the wave propagation
determines the ratio of the elastic (E) and mass ( ) cha-
racteristics of the environment, having a certain influ-
ence on the process of cavitation.
Equation (1) is mostly used for elastic environments;
we should use the following equation for gas-saturated
liquids:
2
c
K ac
=  (2)
where  ac
is the adiabatic compressibility. Comparing
Equations (1) and (2), it should be noted that parameters
E and  ac
are correlated as  ac
=1/ E.
The parameter c
K
has a great influence on the process
of cavitation. That is why it was researched in other
scientific works.
3,4
The speed of the sound in gas-liquid
technological environments depends on the ratio between
gas and liquid components; for example, water with gas
bubbles has the range of change c
K
, which spans quite
widely from 20 m/s to 100 m/s. As we can see from
source,
4
numeric values of the cavitating liquid change in
a narrow range: c
K
= 25 ... 30 m/s; reference
3
includes
the following numeric values: c
K
= 10…12 m/s.
When choosing a dynamic model to determine the
parameters of a cavitation process, the conditions of the
interaction between the cavitation machine and techno-
logical environment were considered as well as the
transferred and recorded energy of the contact zone in
terms of their mutual oscillations. Under such condi-
tions, the contact zone works for one oscillation period:
AF x t t
K K
d =
∫
 sin cos
/


0
2
(3)
Where F
k
is the contact force with phase-shift angle
 and the WRT amplitude of the contact zone;
x is the
speed of the contact-zone oscillation; is the oscillation
frequency and t is the time.
It was decided that F
k
= P
k
·S, where P
k
is the pressure
in the contact zone with an area of S.
The theory of the contact-pressure determination was
based on the model of the cavitation machine-technolo-
gical environment system; it was a discrete-continuum
model following the laws of the changes in the fre-
quency-independent and frequency-dependent dissipa-
tion coefficients.
I. BERNYK et al.: EFFECT OF RHEOLOGICAL PROPERTIES OF MATERIALS ON THEIR TREATMENT ...
466 Materiali in tehnologije / Materials and technology 52 (2018) 4, 465–468
Figure 1: Parameters of a cavitation process for the treatment of a
technological environment

The equations of the acoustic-wave propagation for
the frequency-independent model of the energy dissipa-
tion in the technological environment is the following:
∂
∂
∂
∂
2
2
2
2
u
z E
u
t
=
 *
(4)
where  is the environment density and E* is the com-
plex module of elasticity.
The solution of Equation (4), according to the Fourier
method, is presented as a complex wave function in
Equation (5):
() uzt A e A e e
n
kz
n
kz
n
in t
nn
(,)=+
−
=−∞
+∞
∑
12
 (5)
Displacement u is calculated with the product of two
functions: one function is dependent on the argument
zz Ae A e
n
kz
n
kz
nn
()=+
−
12
; the other function is only de-
pendent on argument T
n
(t)= e
in t  . We have: n – the num-
ber of harmonics, k =  /c, – the oscillation frequency,
c – the speed of the waves propagation in the
environment, i – the imaginary unit, z – coordinate and t
– the time. In Equation (5), there are A
1n
and A
2n
– the
constants determined from the boundary conditions.
Then, comparing the coefficients with the same
harmonics after calculating the equations for constants
A
1n
and A
2n
, we have the following equation for calcul-
ating the environment pressure on the apparatus oscilla-
tion (the contact area):
ptl A
nn
(,) 0
1
2
1
2
2
2
=+  (6)
where A
1
is the amplitude of the contact-zone oscilla-
tion, and 1n
and 2n
are the wave coefficients:
 
  
1 22
22
22
n
nnnn
nn n n
ll
lll
=
+
++
sh
ch
sin
() (c o s )
(7)
 
  
2 22
22
22
n
nnnn
nn n
ll
ll
=
−
++
sin sh
ch () (c o s
n
l)
(8)
The wave coefficients take into consideration line
option l, and the acoustic wave extends along it. Coeffi-
cients  n
,  n
take into consideration the influence of
dissipation on the shape and length of the wave of the
n-th harmonic.
The wave equation for the frequency-dependent
model of energy dissipation in the technological environ-
ment is the following:
∂
∂
∂
∂
∂
3∂
∂
∂
2
2
2
2
2
2
4 u
t
c
u
z t
u
z
=+
⎛ ⎝ ⎜ ⎞ ⎠ ⎟   (9)
where c is the speed of the wave propagation; and  is
the coefficient of viscosity.
The solution of Equation (9) is as follows:
ux A kx A kx () c o s s i n =+
12 pp
(10)
Where k
p
is the complex constant of the wave propa-
gation, calculated with the following equation:
kki
c
i
c
p
=−=−
⎛ ⎝ ⎜ ⎞ ⎠ ⎟   
 2
3
2
3
That is why the amplitude is calculated with the
following equation:
uAkxAkx t =+ ( cos sin )sin
12 pp
 (11)
We have the equation to calculate the pressure in the
contact zone:
p
kl l kl l
kk l
=
−
+


  0
22
2
(sin cos ) (cos )
() ( c o s
sh
ch[] sh  lk ll )( s i n )
22
+
(12)
where 0
is the speed of the contact-zone oscillation. We
can calculate the amplitude of the ultrasonic pressure
wave based on coefficient k, resistance coefficient and
the layer thickness for various technological environ-
ments due to this equation. The speed of wave propa-
gation c and density  are very important parameters of
Equation (12); the product of wave propagation c and
density is the wave impedance; its calculation depends
on the rheological properties of the technological envi-
ronment. Resistance coefficient  is dependent on the
viscosity; it also limits the amplitude of the system
oscillation in the resonant mode. It influences the
amplitude, formation and development of the cavitation
field.
Taking into consideration the adopted approach, the
coefficient of viscosity of technological environment h
was calculated with Equation (12):
  =
R
a
(13)
where R
a
is the active component of support; and  is
the speed of the system oscillation.
The input impedance of the oscillating system at its
resonance frequency is Z = Z
H
+ Z
K
, so its equivalent
resistance is equal to the load resistance Z
H
and internal
resistance of technological environment Z
C
.
The calculation of the impedance of the cavitator-
environment system Z was based on the measurement of
I. BERNYK et al.: EFFECT OF RHEOLOGICAL PROPERTIES OF MATERIALS ON THEIR TREATMENT ...
Materiali in tehnologije / Materials and technology 52 (2018) 4, 465–468 467
Figure 2: Measuring piezoceramic hydrophone

the sound pressure using piezoceramic hydrophones
(Figure 2).
3 RESULTS
When the plates of the hydrophone piezo element are
in the reception mode, the potential difference occurs; its
value is proportional to the sound pressure. Taking into
consideration the peak value of the sound pressure on the
ultrasonic transducer axis:
Px P e
x
() () =⋅
−
0
 (14)
we could draw a graph of normalized pressure P(x) from
the coordinates of point measurements (Figure 3).
Px
Px
P
e
x
()
()
()
==
−
0
 (15)
4 DISCUSSION
It was observed that the most significant change in
the sound pressure is at a distance of 10 mm from the
radiating surface; its value is 20 dB. The change in the
shape of the sound field is no longer significant. Based
on an overview of the impedance,
6
we can conclude that
the active part of the impedance represents the part of the
energy spent for the formation of the cavitation process;
the reactive component reflects the energy spent for the
weight fluctuations of the technological environment.
The viscosity action is essential for small values of the
amplitudes at the initial stage of the bubble formation,
when bubbles have small radii. The viscosity does not
influence the bubbles with large initial radii. The reason
for this is the fact that bubbles with small radii are
constrained in their rise because of the viscous forces at
the initial stage of the bubble formation. This conclusion
is logical; to intensify the process of cavitation, the
handling process should be used with the increasing
acoustic pressure. Thus, both the viscosity coefficient of
the environment and the elasticity coefficient are
important parameters, influencing the formation of
cavitation bubbles, their development and the final stage
of the process – the compression and slamming.
5 CONCLUSIONS
The mathematical description of the motion of the
system of cavitator-machining material is based on a
model that was considered as a single, subdued wave
process; the parameters of the mathematical model of the
machine and the environment were coordinated among
themselves.
The analytical dependence for calculating the contact
pressure took into account the rheological properties of
the material and adequately reflects the real conditions of
the treatment with the ultrasonic cavitation.
The coefficients of elasticity and viscosity are im-
portant parameters, affecting the formation of cavitation
bubbles, their development and the effectiveness of the
final stage of the ultrasonic treatment of various techno-
logical materials.
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I. BERNYK et al.: EFFECT OF RHEOLOGICAL PROPERTIES OF MATERIALS ON THEIR TREATMENT ...
468 Materiali in tehnologije / Materials and technology 52 (2018) 4, 465–468
Figure 3: Dependence of the normalized sound pressure on the
distance to the source of ultrasound