EXPLOSION INITIATION OF A SINGULAR CHEMICALLY ACTIVE GAS BUBBLE IN
ACOUSTIC FIELD OF OTHER BUBBLE
D.V.Voronin
M.A. Lavrentyev Institute of Hydrodynamics, Siberian Branch of the Russian Academy of
Sciences, Novosibirsk 630090, Russia
A two-dimensional unsteady mathematical model of compression and rarefaction waves
propagation through two-phase liquid / gas bubbles medium (inert liquid – chemically reactive gas)
is formulated. The dynamics of a single bubble was investigated numerically with special attention
on non- spherical mode of its form. The influence ob bubble jet deformation on possibility of its
self-ignition and propagation of self-supported detonation wave along a bubble chain was
researched as well.
Introduction
Experimental researches [1] of shock wave (SW) propagation along a column of the liquid,
containing vertically located bubble chain with chemically active gas mixture, have shown that at
the certain initial conditions self-supported wave is moving along the chain. Explosion of one
bubble as a result of chemical reaction is accompanied by radiation of a secondary shock wave. The
process of its interaction with the subsequent bubble repeated the previous stage. This effect has
received the name of bubble detonations.
In the subsequent experiments [2 - 4] it was revealed, that in reactive bubbly systems
(bubbles filled in all cross section of a shock tube) exists self-supported mode of wave generation as
a single wave package which speed of propagation D exceeds one of shock wave in passive bubble
systems at the same volumetric concentration. The structure of a wave represents soliton, moving
with quasi-constant velocity, which is less than frozen sound speed in a mix.
The first attempts of calculation of parameters of bubble detonations [5-7] with the help of
the classical Jouguet scheme under the assumption of equality of pressure of phases have resulted in
simple formulas for the velocity of the ideal detonation. However obtained values D were 10 - 30 %
below experimental ones. So it is impossible to recognize the model (with equilibrium of pressure
of phases) satisfactory and the account of real kinetics of relaxation processes is required at the
description of the wave structure.
The subsequent mathematical models [7-14] take into account the non-equilibrium
character of phase pressure and are based on the model of Jordanski-Кogarko-Wijngaarden [15] of a
liquid with gas bubbles motion. The two-phase medium thus is actually supposed homogeneous, but
having special properties. In works [7-9] it is assumed, that dissipative losses of energy are absent
in the environment, and the liquid phase is incompressible. In [7] the model with energy dissipation
is considered also (due to heat conductivity). Compressibility of a liquid phase is taken into account
in models [10-14] where dissipative losses are caused by: viscosity and heat conductivity [10],
viscosity [11], viscosity and acoustic radiation of bubbles [12]. In [13] it is shown, that the account
of energy dissipation due to acoustic radiation of bubbles is necessary for the existence of the
stationary decision as a lonely wave. It is also specified that the numerical decision [11] of nonstationary problem on initiation of bubble detonations was inaccurate. The two-dimensional nonstationary problem on detonation wave (DW) propagation in a cylindrical column of chemically
active bubbly medium was solved in [14]. The wave structure of a zone of reaction and the wave
velocity were calculated.
Using of Jordanski-Кogarko-Wijngaarden model in works [7 - 14] has allowed to model
waves with numerous bubbles and to get a good conformity with experiments. However in some
cases its using is problematic. For example, in systems with large bubbles which size is more than
sizes of a numerical grid cell, and also at the description of processes of coalescence, bubble
crushing and their not spherical collapse which may play determining influence on the parameters
of the flow in a bubble vicinity. Here it is necessary to use model with obvious allocation of borders
between phases.
The problem on DW propagation in a bubbly media includes two basic problems: (1)
determination of conditions of a single bubble detonation under influence of an external initiating
wave, (2) description of the mechanism of wave interaction of the blown up bubble with next ones
(with their possible subsequent detonation). The estimation of critical energies of initiation of a
single bubble explosion at its adiabatic compression is made in work [16]. Here bubble dynamics is
determined by Rayleigh equation in view of viscosity in acoustic approximation. However, as it is
specified in the review [17], the whole of the problem remained unsolved.
Mathematical formulation of the problem
We shall consider the current of an ideal compressible liquid (water) in horizontally located
channel having diameter L0. Originally spherical bubble having diameter d0 and filled with a gas
mix 2H2+O2 is placed at the axis of symmetry of the channel. The system is in a condition of
balance at initial pressure in a liquid p0 = 1 bar and initial temperature T0 = 298K. At the instant t0 =
0, the acoustic pulse consisting of a phase of compression and a phase of rarefaction starts to move
from the left closed end of the tube to the right one (N - wave). The structure of pressure in the
wave has sinusoidal character. In some cases there is a time delay between the phases of
compression and rarefaction Dt*. The case when the pulse consists only of the phase of compression
or the phase of rarefaction was also investigated. At passage of a pulse through the bubble,
essentially non-stationary current appeared in the tube with formation of secondary waves.
The two-phase flow was modeled within the framework of hydrodynamic approximation.
The basic equations of motion of two-phase flow are based on laws of conservation of mass, pulse
and energy for two-dimensional non-stationary current of the compressible environment without
taking into account effects of dissipation for the case of axial symmetry:
¶s
¶a
¶b
1
+
+
= f ,
¶t
¶z
¶r
r
(1)
Here z, r are the spatial cartesian coordinates directed along an axis of symmetry (z) and across the
channel (r), respectively; t - time.
s , a, b, f are defined by formulas:
s = ( r , ru, rv, rE , rm , rY ),
For a gas phase vector functions
a = ( ru, ru 2 + p, ruv, u ( rE + p), rmu, rYu ),
b = ( rv, ruv, rv 2 + p, v( rE + p), rmv, rYv),
f = (0, 0, 0, 0, rWm , rWY ),
(
(2)
)
E = U + u 2 + v 2 / 2.
Here r is the density; u, v - components of a vector of speed in a direction of axes z and r,
respectively; E - full energy per unit of gas mass; p is the pressure; Y - the period of the induction,
at the beginning of the zone of induction Y = 1 and at its end Y = 0; m is the mean molecular mass
of gas; Wm and WY - speeds of change of m and Y, respectively.
The propagation of shock and detonation waves through chemically active gas was modeled
in the paper. Therefore for the description of possible chemical reactions the two-stage model of
chemical kinetics [18] was used, when chemical reactions occur at a point of the environment after
the end of a chemical ignition delay tig, counted after the passage of a leading shock wave front.
If 1 > Y > 0, then
WY = -
1
, Wm = 0,
tig
æ 1
1 p
1
+ E Д çç
U=
g 0 -1 r
è m 0 m min
If Y = 0, then
(3)
ö
÷÷,
ø
WY = 0, Wm = Wm (T , m , r ),
(4)
U = U (T , m ).
Here mmin - molecular mass of gas in a dissociated state, Eд - average energy of dissociation of
products of chemical reaction, g is the parameter of an adiabatic curve. The index zero corresponds
to the initial condition of substance. The chemical ignition delay is defined according to the
experimental data [19]:
æ 16328 ö
t ig [O2 ] = 6,89 ×10 -11 expç
÷,
è RT ø
(5)
where [O2] is a concentration of oxygen in a mix (mol per liter).
After the expiration of the delay Y = 0, and the speed Wm is defined according to the model
[20 - 22]:
2
m ö
1æ
÷ Wm (T, m, r) = 4K+ r 2 çç1m è mmax ÷ø
3/ 2
ö
æ EД öæ m
æ
æ qöö
A2T r çç1- expç- ÷ ÷÷ expçç- ÷÷çç
-1÷÷
è Tøø
è
è RT øè mmin ø
(6)
3/ 2
Where mmax - molecular mass of gas in a recombined state, T - temperature of gas, R - a universal
gas constant, q is the effective temperature of excitation of oscillatory degrees of freedom; K+, A2 constants. For internal energy of gas the caloric equation of state is valid [23]:
ö RT
æ1
ö 3æ m
ö
1 ö æç 3 æ m
q /T
÷
÷÷ + çç + 1÷÷ + çç - 1÷÷
U (T , m ) = E Д çç ç4 m
÷ m
exp(
/
)
1
2
T
m
m
m
q
è
min ø è è a
ø
è a ø
ø
(7)
Where ma is molecular mass of gas in an atomic state.
The system of basic equations becomes is completed by the thermal equation of a state of
ideal gas:
p RT
=
r
m
(8)
Further it is supposed, that chemical reactions occur only in a gas phase. Then for a liquid phase
s , a, b, f
s = ( r , ru, rv),
(condensed or c-phase) vector functions
in the equation (1) are defined by ratios:
a = ( ru , ru 2 + p, ruv),
b = ( rv, ruv, rv 2 + p ),
(9)
f = (0, 0, 0).
To complete the system (1) - (9) the ratios specifying a shock adiabatic curve of water were
used:
D = C + LU* , r ( D - U* ) = r0 D,
p = r0 DU* + p0 ,
E=
p + p0
2
(10)
æ 1 1ö
çç - ÷÷ + E0 ,
è r0 r ø
Here D is the velocity of a shock wave; U* - mass speed behind the front of SW; L and C are
constants.
Boundary conditions. We believe, that interface borders represent contact discontinuity
surfaces. At each of them boundary conditions of a continuity of a normal component of tension
tensor and normal to the surface component of a vector of speed are valid:
nk s ik(1) - nk s ik( 2 ) = F ni ,
(V
(1)
-V
(2)
)n = 0
(11)
Here sik - components of tension tensor, and V = (u , v) - values of a vector of speed of the flow
taken at the different sides of the contact surface; ni - components of the vector normal to the
surface, F - value of the force of a superficial tension. The influence of a gravity, a magnetic field,
phase transitions and the phenomena of viscosity, heat conductivity and diffusion of substance are
not taken into account in the model. Therefore first of conditions (11) has the following kind
(Laplas formula [24]):
æ 1
1 ö
÷÷,
p1 - p 2 = a çç +
R
R
è 1
2 ø
where p1, p2 - values of pressure at the different sides of the contact surface; R1, R2 -. the main
radiuses of curvature of the surface, a - the coefficient of superficial tension. The bottom border of
computed area is the axis of symmetry; top one is a wall of the tube. They are considered closed,
the condition of non-penetration (the flow speed v = 0) is valid here; the left border is closed (u =
0); the right border is open, its boundary conditions correspond to conditions on a free boundary.
Numerical algorithm. The system of the equations (1) - (11) with the stated boundary
conditions was solved numerically with the help of the method of individual particles which is
updating of Harlow method of particles in cells [25]. The non-uniform computational grid with a
condensation near to an axis of symmetry was used. The numerical algorithm provided a
condensation of the grid in areas with the great gradients of the flow parameters. Number of
particles in cells is variable. The numerical algorithm provides an opportunity of association and
crushing of the individual particles belonging to the same body, depending on the flow parameters.
The maximal number of particles in a cell is equal to seven. Eventually gas bubble may be
deformed, broken down, collapsed and stuck together to others.
Numerical results
The values of parameters of a shock adiabatic curve of water were taken according to the
data [26]. The following values of characteristic constants were used: for a liquid r01 = 1 g/cm3, C =
1,7 km/s, L = 1,7, a = 0,072 N/m; for gas r = 0,0004909 g/cm3, g = 1,4.
As calculations have shown, an opportunity of explosion of a single bubble critically
depends on the parameters of an initiating pulse. If the length of a wave in a phase of compression
l* > > d0, then quasi spherical collapse of the bubble occurs, and after achievement of the ignition
temperature Tig (Tig ~1200K) gas in the bubble is ignited practically instantly. If l* < d0, the surface
of the bubble may have significant distortions of the form due to the loss of stability of its border
with a liquid. The Rayleigh-Taylor instability develops at the great enough values of the pressure in
a phase of compression (at Bond numbers greater critical Bo = 40), at sliding phases KelvinHelmholtz instability takes place (for Weber numbers We > 10). That results in destruction of the
bubble. As a rule, the pulse, initiating bubble fluctuations, is long enough in experiments, whether it
is an initial wave of compression or soliton at bubble detonations. So l* > d0, and the conditions for
development of mentioned above instabilities do not arise. But at lengths of the waves comparable
with d0 the third type of instability - jet deformation of bubble (see, for example, the data of
experiment [27]) is realized, owing to a gradient of pressure in an initiating pulse.
Consider the flowfield in a bubble vicinity t = 43ms at passage of a wave of compression
through the system. The initial data here correspond to one of the experiment [1]. The bubble
having the initial diameter d0 = 10 mm and filling with a gas mix 2H2 + O2 is located in the center
of the channel (having diameter of 50 mm and filled with water). The bubble pulsations are raised
by the wave of compression with amplitude of 100 bar, moving from the left to the right. In a wave
of compression the bubble begins to shrink. The numerical analysis shows, that bubble compression
is non-spherical. At instant t = 43ms the average diameter of the bubble is approximately 0,73 %
from the initial value. The velocity u at the left border of the bubble (at the point of crossing of
profiles) is equal to 157 m/s, the maximal speed of the left border achieves 175 m/s at the point
slightly above the axis of symmetry. Transverse velocity v is equal at this point to -110 m/s. On the
opposite side of the bubble modules of speeds are much lower. For example, at the axis of
symmetry u = -36m/s. Therefore, the bubble is deformed more intensively from the left side. It
results in the occurrence of the cumulative water jet directed inside the bubble. The jet generates
SW inside the bubble. At the big gradients of pressure in the initial SW, this jet achieves the
opposite bubble wall, and it gets thus toroidal form. In acoustic waves of compression with smooth
increase of pressure the jet may be weak and does not achieve the opposite bubble wall, or
practically is not visible. The form of the deformed bubble is close to ellipsoid in this case.
At the interaction of the bubble with a single rarefaction wave of the same amplitude, the jet
of gas directed from the bubble in a liquid towards to the wave is formed. Consider dynamics of a
single bubble in the rarefaction wave, d0 = 1mm. Readout of time here is conducted from the
moment of the wave approach to the bubble. By the instant t = 15,6 ms the jet is formed on the
bubble surface. By the instant t = 19,6 ms it breaks down, micro bubble is separated from it and goes
downwards (t = 22,6 ms). At t = 25,6 ms the collapse of the micro bubble takes place.
As a whole, the numerical simulations of the problem have shown, that a single bubble
dynamics essentially varies depending on initial parameters of the flow. At d0 less than critical
value, the bubble is quickly compressed in a falling wave of compression till the sizes of one
particle in a numerical cell without appreciable distortion of the form (quasi-spherical collapse). At
increase of d0, a cumulative water jet is formed on the left bubble wall, directed inside the bubble; it
reaches opposite wall of the bubble, the latter one gets the toroidal form with subsequent collapse.
At the further growth of d0 the cumulative jet stops inside the bubble, subsequent unloading comes
in a falling rarefaction wave at the left bubble wall, so the gas bubble arises on the left wall,
directed towards to the wave of rarefaction. Then the jet fragmentation occurs, generating ob micro
bubbles with significantly smaller (the order of magnitude) size, than the initial bubble. The last two
cases we shall name as jet deformation of a bubble. As calculations confirm, a transitive zone
between different scripts of a bubble deformation is very narrow. In fig. 1 the calculated changes of
a bubble dynamics is submitted depending on values of initial flow parameters. The curve 1 in fig. 1
divides areas with spherical and non-spherical deformations of a bubble in an initial stage after
passage of a compression wave, the curve 2 separates the area appropriate to a bubble collapse in a
falling wave, from the area of its deformation without the collapse. Thus, area I corresponds to
quasi-spherical collapse, II - to spherical deformation without a collapse, III - to jet deformation of a
bubble without a collapse, IV - to jet deformation with the subsequent collapse.
At the separation of the micro bubble, the local area with the increased values of pressure
and temperature appears in gas. So, for the air bubble the local pressure may achieve 0,8 GPa, and
temperature is up to 11000 K [28]. For chemically reacting bubbles it is equivalent to
microexplosion in this area. Therefore it is necessary to find out, whether the microexplosion of gas
inside a jet is capable to cause explosion of all the gas in the bubble.
At the passage of an acoustic pulse through the group of bubbles, the complex flow with
secondary waves of compression and rarefaction appears. It results in even more essential
deformations of their form, especially for the bubbles, located inside the group.
If the acoustic complex passes through the group of nearby located bubbles their mutual
influence generates more complicated wave structure and facilitates the occurrence of jet
deformation. In fig. 2 the field of pressure in a vicinity of originally spherical bubbles (their initial
diameter d0 = 800 microns, the channel diameter L0 = 4 mm) is submitted. More dark of tone in
figure corresponds to waves of compression, light ones- to waves of rarefaction. In the top figure
(t=9,2 us) it is visible, that the first bubble from the left side is deformed with formation of gas get,
the second one- with formation of cumulative water jet inside the bubble, the third is compressed
and gets a plate form, the fourth is punched by the cumulative jet and has the toroidal form. At the
subsequent instant (the bottom figure, t=12,0 us) the first bubble is fragmented, micro bubbles are
appears; at the second bubble a gas jet arises; the third keeps its form; the fourth bubble is in a
condition of a collapse and is not distinct in the figure. In course of time the distance between the
bubbles enlarges.
Thus, in a bubble cluster a jet deformation of bubbles and their crushing takes place. If for
explosion of a spherical bubble (taking place at initial pressure of 1 bar), reduction of its diameter
more than in 3 times is necessary, at jet deformation the ignition begins in jets or nearby micro
bubbles at more great values of average bubble diameter. Therefore the explosion of a single bubble
is possible in a weak acoustic wave and even in the single waves of rarefaction providing jet
deformation of a bubble.
Consider the influence of the micro bubble explosion (a fragment of the jet) at its collapse
on a closely located big bubble. The field of pressure in a vicinity of the large bubble is considered
after the passage of rarefaction wave through group of bubbles and the subsequent fragmentation of
a gas jet. Time between the cadres is Dt = 2,0 ms. In the first cadre a big bubble is deformed and
micro bubble located from the right hand side. The distance from the centers of bubbles is 2,4 mm.
In the second cadre the collapse of the micro bubble with its subsequent explosion is taken place. In
the third cadre the intensive process of expansion of products of explosion with formation of the jet
directed to the big bubble is occured. In the fourth cadre at interaction of the jet with the big bubble,
the latter is compressed and gets the toroidal form. As calculation [16] show, the critical energy of
initiation E* of explosion of gas in the bubble is approximately equal to 6 J at p0 = 1 bar. With
growth of p0 the energy decreases under the law: E* ~ 1/p2. Our computations show, that energy of
a cumulative jet is enough to excite the explosion in the big bubble.
Consider conditions of propagation of self-supported wave along a bubble chain at the initial
conditions appropriate to the experiment [1]. The distance between bubble centers is equal to 50
mm. It is visible that the first from above bubble decreases in a wave of compression and loses the
spherical form. At its surface so-called "nose" is formed, directed downwards. The collapse of the
"nose" results in ignition of gas in the bubble (a luminescent point in the first bubble at t = 61 ms).
Though the most part of gas in the bubble quickly burns down, generating a secondary wave of
compression, still it is possible to see areas of the not burned gas (so-called "pockets") at t = 64 ms.
The secondary wave of compression goes to the second bubble, which is partially compressed by
the initial SW. Burning of the second bubble occurs similarly: ignition of a mix in collapsing "nose"
takes place at t = 109 ms, and then there is an explosion of gas in all the bubble. After the explosion
gas bubble continues to be compressed on inertia, and then it extends, and it size may exceed the
initial value essentially. The form of bubbles is far from spherical. The velocity of the wave of
explosion propagation along a bubble chain (~500 m/s), a way of the distortion of their form
coincides with experimental data [1].
Micro explosions of not spherical bubbles may result in propagation of a of self-supported
detonation wave sliding along a bubble chain. In fig. 3 a bubble detonation wave velocity is shown
depending on initial volumetric concentration of a gas phase in a mix b0. A shaped curve in the
figure –is the data of experiment [2], continuous curve are the results of calculations. It is visible,
that the model describes adequately enough the parameters of a detonation in bubble media.
Conclusion
Explosion of a single chemically active bubble in an inert liquid is possible at the absence of
its collapse. If the length of a falling wave is comparable with the initial bubble diameter, then there
is its jet deformation. The subsequent crushing of a jet and a collapse of formed micro bubbles is
capable to initiate a detonation of the basic bubble, even if the initial wave was the wave of
rarefaction. The opportunity of propagation of self-supported waves along a chain of large bubbles
in an inert liquid is numerically confirmed. That corresponds to available experimental data.
References
1.
Hasegawa T., Fujiwara T. Detonation in Oxyhydrogen Bubbled Liquids//Proc. 19th
Intern. Symp. on Combustion. Hafia, 1982, 675-683.
2.
Sychev A.I., Pinaev A.V. Self-supported detonation in liquids with explosive gas
bubbles. - J. Appl. Mech. Techn. Physics, 1986, N 1, P. 133-138.
3.
Pinaev A.V., Sychev A.I. Influence of physical-chemical properties of gas and a
liquid on parameters and a condition of existence of a wave of a detonation in systems a liquid-gas
bubbles. – Physics of Combustion and Explosion, 1987, v. 23, N 6, P. 76-84.
4.
Sychev A.I. Influence of the bubble size on characteristics of waves of a detonation.
- Physics of Combustion and Explosion, 1995, v. 31, N 5, P. 83 - 91.
5.
Mitrofanov V.V. Detonation waves in heterogeneous environments: the Manual.
Novosibirsk: NGU, 1988.
6.
Kuznetsov N.,М., Kopotev V.A. Structure of a wave and condition of ChapmanJouguet at a heterogeneous detonation in liquids with gas bubbles. - Docklady AN SSSR. 1989. v.
304, N 4. P. 850 - 853.
7.
Shagapov V.S., Vahitova N.K. Waves in a bubbly system at presence of chemical
reactions in a gas phase. - Physics of Combustion and Explosion, 1989. V. 25, N 6. P. 14 - 22.
8.
Kedrinskii V.K., Mader Ch. L. Accidential detonation in bubble liquids. - Proc. 16th
Intern. Symp. on Shock Tube and Waves / H.Groenig (Eds.). 1987. P. 371 - 376.
9.
Ljapidevskij V.J. About speed of a bubble detonations. - Physics of Combustion and
Explosion, 1990. V. 26, N 4. P. 137 - 140.
10.
Zamaraev F.N., Kedrinskij V.K., Mejder Ch. Waves in chemically active bubble
medium. - J. Appl. Mech. Techn. Physics, 1990, N 2, P. 20 - 26.
11.
Trotsjuk A.V., Fomin P.A. Model of a bubble detonation. - Physics of Combustion
and Explosion, 1992. V. 28, N 4. P. 129 - 136.
12.
Taratuta S.P. A detonation and heat exchange in two-phase bubble environments. –
Dissertation Philosophy Dr. Novosibirsk, 1999.
13.
Zhdan S.A. About a stationary detonation in a bubble media. - Physics of
Combustion and Explosion, 2002. V. 38, N 3. P. 85 - 95.
14.
Zhdan S.A. A detonation of a column of chemically active bubble medium. Physics of Combustion and Explosion, 2003. V. 39, N 4. P. 107 - 112.
15.
Jordanskii S.V. About the equation of movement of the liquid containing gas
bubbles. - J. Appl. Mech. Techn. Physics, 1960, N 3. P. 102 - 110.
16.
Vasil'ev A.A., Kedrinskij V.K., Taratuta S.P. Dynamics of a single bubble with
chemically active gas. - Physics of Combustion and Explosion, 1998. V. 34, N 2. P. 121 - 124.
17.
Kedrinskij V.K. Hydrodynamics of explosion: experiment and models. Novosibirsk: Siberian Branch of the Russian Academy of Science Publ., 2000.
18.
Levin V.A., Korobejnikov V.P. Strong explosion in a gas mixture of gases – Izvestia
AN SSSR. The mechanics of a liquid and gas. 1969. N 6. P. 48 - 51.
19.
Strehlow R.A., Mauer R.E., Rajan S. Transverse waves in detonation: I.Spacing in
hydrogen - oxygen systems. - AIAA Journal, 1969, v.7. N 2, p. 323 - 328.
20.
Nikolaev J.А. Model of kinetics of chemical reactions at high temperatures. Physics of Combustion and Explosion. 1978. V. 14, N 4. P. 73 - 76.
21.
Nikolaev J.А., Fomin P.A. About calculation of equilibrium flows of chemically
reacting gases. - Physics of Combustion and Explosion. 1982. V. 18, N 1. P. 66 - 72.
22.
Nikolaev J.А., Fomin P.A. The approximate kinetics equation in heterogeneous
systems of gas - condensed phase type. - Physics of Combustion and Explosion. 1983. V. 19, N 6.
P. 49 - 58.
23.
Nikolaev J.А., Zak D.V. The coordination of models of chemical reactions with the
second beginning of thermodynamics. - Physics of Combustion and Explosion. 1988. V. 24, N 4. P.
87 - 90.
24.
Landau L.D., Lifshits E.M. Theoretical physics. Т. 6. Hydrodynamics. M.: Nauka.
1988. 736 p.
25.
Agurejkin V.A., Kriukov B.P. A method of individual particles for calculation of
currents of multicomponent environments with the big deformations. - Numerical methods of the
mechanics of the continuous environment. 1986. V. 17, N 1. P. 17-31.
26.
Trunin R.F. Compression of the condensed substances by high pressures of shock
waves (laboratory researches). - Successes of physical sciences, 2001, V. 171, N 4, P. 387 - 414.
27.
Voronin D.V., Sankin G.N., Teslenko V.S., Mettin R., Lauterborn W. Secondary
acoustic waves in polydisperse bubbly media. - J. Appl. Mech. Techn. Physics. 2003. V. 44, N 1, P.
22 - 32.
28.
Voronin D.V., Sankin G.N., Teslenko V.S., Mettin R., Lauterborn W. Bimodal
bubble cluster as a result of bubble fragmentation in a bipolar acoustic pulse. - Nonlinear acoustics
at the Beginning of the 21st Century ". Proceedings of the 16th International Symposium on
Nonlinear Acoustics (ISNA-16.) 19-23 August 2002, Moscow, Russia. Eds: O.V. Rudenko, O.A.
Sapozhnikov. Faculty of Physics, Moscow State University, 2002, vol. 2, pp. 931-934.
Fig. 1. Deformation of a single bubble in a wave of compression.
Fig. 2. Generation of cumulative jet on a bubble surface.
Fig. 3. Numerical and experimental dependences of bubble detonation velocity on concentration of
gas phase in a medium.