Full text of DOI:10.1134/1.1327326

Doklady Physics, Vol. 45, No. 10, 2000, pp. 539–542. Translated from Doklady Akademii Nauk, Vol. 374, No. 4, 2000, pp. 489–492.
Original Russian Text Copyright © 2000 by Nigmatulin, Gubaœdullin, Beregova.
MECHANICS
Method of Resonance Overcompression in a Bubble Liquid
by a Moderate Aperiodic Action
Academician R. I. Nigmatulin, A. A. Gubaœdullin, and O. Sh. Beregova
Received July 3, 2000
Several years ago, the phenomenon of sonolumines- carrier liquid with a large quantity of bubbles dispersed
cence, that is, the luminescence of gas bubbles in an in it.
acoustic field, was discovered [1, 2]. This phenomenon
In this study, we propose a method for processing a
is of interest not only from the scientific point of view;
limited volume of the bubble liquid by an aperiodic
it also has a number of important applications for prac-
moderate-amplitude wave action, as a result of which
tice. For example, the appearance of such a direction in
waves arise with amplitudes exceeding that of the initi-
chemical technology as sonochemistry is associated
ating action by several orders of magnitude. This
with the discovery of sonoluminescence. By virtue of
method is illustrated by the results obtained from a
the appearance of high temperatures in bubbles, the
direct numerical simulation.
acoustic field can initiate certain chemical reactions
We consider a cylindrical volume of bubble liquid,
which has the length L bounded by solid walls and a
mobile piston (Fig. 1).
which are impossible under other conditions. But the
most impressive fact is that a nuclear-fusion reaction
can be initiated in bubbles at superhigh temperatures.
Deuterium bubbles in heavy water at overcompressions
The basketball mode for the excitation of the gas–
can release thermonuclear energy (“bubble nuclear liquid-mixture is realized by means of specifying the
fusion”), but a routine ultrasound is not sufficient to following boundary condition at the piston:
make this take place.
In recent years, a number of studies [3–11] were
devoted to the theoretical description of the behavior of
an individual gas bubble vibrating in a liquid under a
wave-field action provided that the pressure and tem-
perature in the gas can reach extremely high values.
The principal idea of the new approach [7], referred to
as the “basketball” mode, is the coordination of the pro-
cess of varying the pressure in a liquid with the forced
vibrations of a bubble and the use of a nonlinear reso-
nance during an aperiodic action of an external field of
 pmax
,
v
p
≥ 0
pp =

pmin
,
v < 0,

p
where p and v are the pressure and velocity of the
p
p
medium at the piston. In such a situation, the waves
traveling from the piston to the wall, the waves
reflected from the wall and traveling back to the piston
reflected from it, etc. propagate in the bubble mixture.
For the numerical investigation of the problem for-
a moderate-amplitude pressure. To realize this idea, we mulated, we use the model of the dynamic behavior of
formulated and solved the problem of spherically sym- a bubble liquid and the method of its computer realiza-
metric vibrations of a gas bubble in a compressible liq- tion outlined in [12].
uid [8–10]. On the basis of the analytical solution
In Fig. 2, we show the time dependences for the
obtained, we developed an efficient computer code for
pressure at the piston, the piston velocity, and the gas
the mathematical simulation of the bubble collapse
pressure in bubbles in the middle of the volume (x =
with allowance for various dissipative mechanisms,
L/2) calculated for the case of the basketball and wave
such as viscosity, heat conduction, radiation, ioniza-
(
pp = pmax) modes of excitation of the hydrogen–gly-
tion, wave processes around and inside the bubble, and
heat-and-mass exchange between the bubble and the
ambient liquid under overcompressions of the bubble.
cerin bubble mixture with the parameters a = 1 mm,
0
The investigation of the processes taking place in an
individual collapsing bubble is obviously an important
and necessary stage; however, the above applications
are associated with a bubble liquid, i.e., a mixture of
L
Piston
Rigid
Bubble liquid
wall
x
Institute of Theoretical and Applied Mechanics
(
Tyumen Branch), Siberian Division, Russian Academy
Fig. 1. Schematic of the piston excitation for a bubble
liquid.
of Sciences, Tumen, Russia
1
028-3358/00/4510-0539$20.00 © 2000 MAIK “Nauka/Interperiodica”

5
40
NIGMATULIN et al.
pp/p0
(a)
(b)
1
0
v/v
*
x = 0
x = 0
0
–
0.1
pp/p0
1
0
x = 10 cm
x = 10 cm
10
20
30
40 0
10
20
30
40
t, ms
Fig. 2. Profiles of liquid and gas pressures and piston velocities for (a) basketball mode and (b) steady-state mode of bubble-liquid
excitation.
5
α = 2%, T = 293 K, L = 20 cm, p = 10 Pa, p
=
equilibrium approximation from the formulas
0
0
0
max
1
.2p , p = p , and v = p /ρ , where a, α, ρ, T, and
0 min 0
0 l
∗
p
α ρ
max
p
R
D = -------- -- , D = ----------,
S
R
p are the radius and volume content of the bubbles, the
density, temperature, and pressure, respectively. The
subscripts l and g denote the liquid and gas parameters,
and 0 implies the initial values of the parameters.
α ρ
S l
0
l
2
max
p
α p
0 0
p = ---------, α = --------- -- ,
R
S
p
p
max
0
It can be seen (Fig. 2) that the maximum gas pres-
sure in the bubbles grows for the basketball mode of the
piston motion during each subsequent travel of a wave,
whereas for a constant pressure at the piston (a step-like
wave), the gas pressure in the bubbles tends to the pres-
sure at the piston.
where α is the volume gas content behind the incident
S
wave and p is the pressure behind the wave reflected
R
from the wall.
The eigenfrequency ω of the bubble vibrations in
R
the mixture can be determined from the formula
1
0
/3
Figure 2 confirms the fundamental possibility for
excitation of overcompression in a bubble mixture by
means of an aperiodic moderate-amplitude action.
1
3γ pmax
a (1 – ϕ)ρ
1.1α
– α
0
ω = -- ---------------------, ϕ = --------------------------- -- ,
R
1 – α
0
l
The mechanical system under consideration exhib- where γ is the gas adiabatic index and ϕ is the correc-
its a resonance. Comparing the eigenfrequencies of this tion taking into account the fact that the bubbles are not
vibrating system and those of the bubble vibrations in single.
the mixture, it is possible to estimate the system param-
eters for which the excitation mode is resonant.
Equalizing the frequencies ω and ω , we can find
S R
the resonance values of the parameters. As can be seen
from the formulas, one of the parameters (a , α ,
The eigenfrequency ω of the system for the wave
0
0
S
L, pmax) is determined by the resonance condition from
the given values of the remaining parameters (for the
chosen liquid and gas).
In Fig. 3, we display the calculated oscillograms for
pressure in the liquid, pressure and temperature in the
excitation can be estimated as
2
T
π
L
L
ω = ---- -- , T = ---- -- + ------,
S
D
D
S
R
where D and D are the velocities for the wave travel- gas bubbles, the radius, and the volume content of the
S
R
ing from the piston and the wave reflected from the bubbles situated in the middle of the volume (x = L/2)
wall, respectively. Their values can be calculated in the for the resonance excitation of a water–air gas–liquid
DOKLADY PHYSICS Vol. 45 No. 10
2000

METHOD OF RESONANCE OVERCOMPRESSION
541
p/p0
000
a/a0
3
1
p
pg
1
00
2
1
0
1
0
1
0
Tg/T0
α
7
6
5
4
3
2
1
0.010
0.001
0
0
1
2
3
4
0
1
2
3
4
t, ms
Fig. 3. Profiles of liquid and gas pressures, radius, temperature, and volume concentration of bubbles in the case of the resonance
excitation.
mixture with the parameters a = 1 mm, α = 0.1%, T = tures can be achieved for a gas in bubbles. This fact is
0
0
0
2
93 K, L = 5 cm, p = 0.1 MPa, and p = 1.5p . It can qualitatively illustrated by the profiles of the pressure,
0 max 0
temperature, bubble radius, etc. calculated in terms of
the single-velocity two-temperature model with two
pressures in the bubble mixture with an incompressible
liquid phase. In order to obtain more exact quantitative
information associated with particular applications,
e.g., with the problem of bubble nuclear fusion, it is
necessary to develop more complicated models of bub-
ble liquids, which take into account various dissipative
mechanisms, such as the compressibility of the liquids,
radiation, ionization, wave processes around and inside
individual bubbles, etc.
be seen that, in this case, the pressure amplitude is two
orders of magnitude higher than that of the initiating
pulse, whose amplitude is only 0.05 MPa. As this takes
place, the gas temperature increases to a value higher
than 2000 K for a short time. The time dependences for
the bubble radius a, radial velocity w, and volume con-
centration α show that the amplitudes of their vibra-
tions increase with time.
The wave properties of the bubble liquid have been
relatively well studied. The behavior of shock waves
depends on the choice of carrier liquid phase (its den-
sity and viscosity), but is to a greater extent determined
by the dispersed phase, even when it is small not only
in mass, but also in volume. The gas properties in bub-
bles, their size, and the character of the interphase heat
exchange can radically influence the structure of the
shock wave.
ACKNOWLEDGMENTS
This work was supported by the Council of the Pro-
gram for Support of Leading Scientific Schools (project
no. 00-15-96157).
The numerical analysis carried out showed that with
decreasing the radius of the bubbles the response of the
bubble mixture is enhanced in the resonance-excitation
mode. This fact agrees with the conclusion obtained
in [10] for an individual bubble.
REFERENCES
1
. K. S. Suslik, Sci. Am., No. 2, 80 (1989).
2
. D. F. Gaitan, L. A. Crum, C. C. Church, and R. A. Roy,
Thus, we have proposed the method of the reso-
nance excitation of a bounded volume of a bubble liq-
uid by an aperiodic moderate-amplitude action, as a
result of which extremely high pressures and tempera-
J. Acoust. Soc. Am. 91, 3166 (1992).
3. V. Kamath, A. Prosperetti, and F. N. Egolfopulos,
J. Acoust. Soc. Am. 94, 248 (1993).
DOKLADY PHYSICS Vol. 45 No. 10 2000

5
42
NIGMATULIN et al.
4
5
6
7
. L. A. Crum and R. A. Roy, Sonolumin. Sci. 266 (14), 233
9. I. Sh. Akhatov, N. K. Vakhitova, G. Ya. Galeeva, et al.,
(
1994).
. C. C. Wu and P. H. Roberts, Phys. Rev. Lett. 70, 3424
1993).
Prikl. Mat. Mekh. 61, 952 (1997).
1
0. R. I. Nigmatulin, I. Sh. Akhatov, and N. K. Vakhitova,
(
Prikl. Mekh. Tekh. Fiz. 40 (2), 111 (1999).
. R. I. Nigmatulin, I. Sh. Akhatov, N. K. Vakhitova, and
R. T. Lahey, Chem. Eng. Commun. 168, 145 (1998).
1
1. A. A. Aganin and M. A. Il’gamov, Prikl. Mekh. Tekh.
Fiz. 40 (2), 101 (1999).
. R. I. Nigmatulin, V. Sh. Shagapov, N. K. Vakhitova, and
R. T. Lekhi, Jr., Dokl. Akad. Nauk 341, 37 (1995) [Phys. 12. A. A. Gubaœdullin, A. I. Ivandaev, and R. I. Nigmatulin,
Dokl. 40, 122 (1995)].
Prikl. Mekh. Tekh. Fiz., No. 2, 78 (1978).
8
. R. I. Nigmatulin, I. Sh. Akhatov, and N. K. Vakhitova,
Dokl. Akad. Nauk 348, 768 (1996) [Phys. Dokl. 41, 276
(1996)].
Translated by V. Bukhanov
DOKLADY PHYSICS Vol. 45
No. 10
2000