7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Nonlinear Analysis of Wave Propagation in Liquids Containing Microbubbles
T. Kanagawa ,
T. Yano t, M. Watanabe and S. Fujikawa *
Division of Mechanical and Space Engineering, Hokkaido University, Sapporo 0608628, Japan
t Department of Mechanical Engineering, Osaka University, Suita 5650871, Japan
9 kanagawa@mechme.eng.hokudai.ac.jp
Keywords: Nonlinear dispersive wave, Bubbly liquid, Weak twodimensionality, KdV and KP equations
Abstract
Weakly nonlinear propagation of dispersive waves in an initially quiescent liquid containing a number of small
spherical gas bubbles is theoretically studied for the case of low frequency and long wavelength. Our preceding study
of onedimensional waves (Kanagawa et al. 2009) is extended to that of almost unidirectional waves propagating
mainly in one direction with a small perturbation to the transverse direction, i.e., weakly twodimensional waves.
On the basis of the method of multiple scales, a twodimensional KdV equation, i.e., the KadomtsevPetviashvili
(KP) equation with a dissipation term due to the liquid viscosity and compressibility can be derived from the basic
equations for bubbly flows.
Introduction
tively
Waves in bubbly liquids show fairly complex features as
compared with those in single phase fluids, and therefore
a number of theoretical studies have been carried out for
more than 50 years (e.g., Carstensen & Foldy 1947; Fox
et al. 1955; van Wijngaarden 1968, 1972; Kuznetsov
et al. 1978; Nigmatulin 1991; Gumerov 1992; Akha
tov et al. 1996; Khismatullin & Akhatov 2001; Gasenko
& Nakoryakov 2008). Especially, the dispersion in the
sense that waves of different frequencies propagate with
different phase velocities is an important property, which
is caused by bubble oscillations in the waves in bubbly
liquids.
In our preceding study (Kanagawa et al. 2009), we
have theoretically examined the onedimensional pro
gressive waves in a liquid with a number of small spher
ical gas bubbles. We have demonstrated that the appro
priate choices of scaling relations of physical parame
ters enable us to carry out the systematic derivations of
two types of equations for nonlinear wave propagation in
long ranges, i.e., the Kortewegde VriesBurgers (KdV
Burgers) and the nonlinear Schrodinger (NLS) equa
tions.
Here, let us show the scaling relations for the low
frequency and long wavelength band and the high fre
quency and short wavelength band in the dispersion re
lation of waves in bubbly liquids (see, Fig. 1), respec
(U* R* w*
CL L0
(o0L' L* B'
(o (), o(e), o(e)),
: (o (e4), 0(1), 0(1)),
for KdVBurgers
for NLS
where e2 (< 1) is a nondimensional amplitude of waves,
U* and L* are characteristic propagation speed and
/ NLS equation
dispersion
,, Stron E
T 4F KdVBurgers equation
O Wavenumber
Figure 1: The famous dispersion relation of waves in
a bubbly liquid (e.g., van Wijngaarden 1968, 1972).
Weakly nonlinear propagation of pressure waves in the
low frequency band and high frequency band are gov
erned by the KdVBurgers and NLS equations, respec
tively (Kanagawa et al. 2009). Here, wc denotes the
eigenfrequency of a single bubble.
length of waves, respectively, cLo and R0 are the speed
of sound of the liquid and bubble radius in an initially
unperturbed state, respectively, w* is a frequency of inci
dent waves, wB is the eigenfrequency of the single bub
ble, and the superscript asterisk denotes a dimen
sional quantity throughout the present paper. On the ba
sis of Eq. (1) and the method of multiple scales (e.g.,
Nayfeh 1973; Jeffrey & Kawahara 1982), the KdV
Burgers and NLS equations have been derived from ba
sic equations for bubbly flows, which have been pro
posed by our group and composed of the conservation
laws of mass and momentum for gas and liquid phases,
and Keller's bubble dynamics equation (see, our previ
ous papers: Egashira et al. 2004; Yano et al. 2006). The
liquid compressibility has been taken into account, and
this leads to the wave attenuation due to bubble oscil
lations. As a result, weakly nonlinear propagation of
waves in the low frequency band has been governed by
the KdVBurgers equation, and nonlinear modulation of
a quasimonochromatic wave train in the high frequency
band by the NLS equation, where the dissipation or at
tenuation terms are due to the liquid viscosity and com
pressibility.
Our aim of this paper is to extend our onedimensional
analysis (Kanagawa et al. 2009) to the weakly two
dimensional problem, where the waves concerned prop
agate mainly in one direction with a small perturbation
to the transverse direction, like a sound beam. We focus
on only the low frequency and long wavelength band
in Fig. 1 and derive the KadomtsevPetviashvili (KP)
equation (Kadomtsev & Petviashvili 1970) with a dis
sipation term, on the basis of the method of multiple
scales and the parameter scaling characterizing the low
frequency band. The KP equation corresponds to a two
dimensional version of the KdV equation and the lead
ing approximation to a system which governs weakly
nonlinear, weakly dispersive, and almost unidirectional
waves.
Formulation of the problem
Weakly nonlinear propagation of waves in an initially
quiescent liquid containing many small spherical gas
bubbles is analyzed. We focus on almost unidirectional
waves mainly propagate in the x* direction with a small
perturbation to the y* direction, i.e., all variations in y*
are slower than in x*, where x* (x*, y*) is the two
dimensional space coordinate. The liquid compressibil
ity is taken into account as in the gas phase, and this
leads to wave attenuation due to the acoustic radiation
from oscillating bubbles. For simplicity, however, the
viscosity in gas phase, the thermal conductivity in both
phases, the phase change across the bubbleliquid inter
face, and the Reynolds stress are neglected.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Governing equations The set of averaged equa
tions for bubbly flows is composed of the conserva
tion laws of mass and momentum for the gas and liquid
phases, the equation of motion for the bubble wall, the
equations of state for the gas and liquid phases, and so
on (see, Egashira et al. 2004; Yano et al. 2006).
Firstly, the conservation laws of the mass and momen
tum for the gas and liquid phases based on a twofluid
model (Egashira et al. 2004) are written as follows:
Op+ div* (aGp ) = 0, (2)
atLP
9LL + div* (aLPLU) 0, (3)
atGP
a9tpu + div* (cQGP;;U;U ) + cG grad*p;* F*
Ort*GGGG
'tL + div*(aLPUUL)
+ CL grad*pL + P*grad* ;G
F*, (5)
where t* is the time; u* (u*, v*) the fluid velocity, u*
and v* are the components of the x* and y* directions,
respectively; a the volume fraction; p* the density; p*
the pressure; and the subscripts G and L denote volume
averaged variables in gas and liquid phases, respectively.
In addition to the volumeaveraged pressures p* and pL,
the liquid pressure averaged on the bubbleliquid inter
face (Jones & Prosperetti 1985), P*, is introduced.
As the interfacial momentum transport F*, we em
ploy the following model of a virtual mass force in the
same manner as our previous studies (Yano et al. 2006;
Kanagawa et al. 2009):
F* 1iatG Dtu*
Dt*
DLU L
Dt* )
DGct DGp
2 aPL(UG Dt* _3aG(UG DL)tp
Dt* Dt*
where the values of coefficients, Bi, /2, and B3, may
be set as 1/2 for the spherical bubble. Equation (6) is
constructed on the basis of previous studies (see, Zhang
& Prosperetti 1994; Eames & Hunt 2004).
The Keller equation (Keller & Kolodner 1956; Keller
& Miksis 1980) for spherical oscillations of a bubble in
a compressible liquid is introduced:
S 1 DG1R* D/R*
CTo Dr* Dt*2
3 1 DR* Dc* (7)
2K 3cio Dt* Dt
S1 DGR* P* R* DG
LO I PLO POCLO D
Co Dt* Pso Pso0s0 Dt*
where R* is the averaged bubble radius and pLo is the
density in the initially unperturbed liquid. The second
term in the righthand side of Eq. (7) describes a damp
ing effect and is responsible for the wave attenuation
due to the acoustic radiation from oscillating bubbles;
the first term in the righthand side also results in the
wave attenuation due to the liquid viscosity p* through
Eq. (12) below.
The definition of operators Dc/Dt* and DL/Dt* are
DG a
D+ uG* grad*,
Dt* 9t*
DL a
D + ut grad*.
Dt* 9t*
Equations (2)(7) are closed by the following equa
tions: the constraint of the volume fraction, the poly
tropic equation of state for gas phase, the mass conserva
tion law inside the bubble, the Tait equation of state for
liquid phase, and the balance equation of normal stresses
at the bubbleliquid interface, as
OL 1 C,
PG = PGO R
2
SPLOCL0
PL PLO + 
/?
2au
P* = p p
R
( PLO 1
PLO!
4p* DGR
R* Dt*
where 7 is the polytropic exponent; /.*,, p*0, an
are the gas pressure, gas density, and bubble radii
the unperturbed state, respectively; n is the material
stant, e.g., n 7.15 for water; a* is the surface ten
and p* is the liquid viscosity which is assumed t
effective only at the bubbleliquid interface.
Method of multiple scales The time t* and s
coordinate x* (x*, y*) are, respectively, norma
by
t= = 1 (13)
T* L*' L*(13)
where T* and L* are the characteristic time and length,
respectively.
Let small parameter e2 (< 1) denotes a typical nondi
mensional amplitude of waves. Then we extend t, x, and
y to the following sets of independent variables:
to t, tl= c2t,
X = X1 62X,
Yo cy,
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
twodimensionality of waves, i.e., the spatial variation
of waves in the y direction is so small compared with
that in the x direction.
Accordingly, the dependent variables must now be re
garded as a function of these extended independent vari
ables. Thus the differential operators can be expanded
as follows:
0 09 + 2 0
a a a 2
0y 0ay
(15a)
(15b)
(15c)
The dependent variables are nondimensionalized and
expanded in a power series by using c:
a/ao = 1 + e2a1 + e4a2 + ..
R*/R = 1 + 2R1 + 4R2 +
6 tG1 + 4 UG2 +
6 2L1 + t4L2 + ,
63 G1 + [ 5tG2 + ,
63UL1 + 51UL2 + ,
0/ U:
1L /U
v0/U
0~L/U*
(16)
(17)
(18)
(19)
(20)
(21)
(11) where ao is the initial void fraction and U* is the charac
teristic propagation speed of waves. The characteristic
(12) propagation speed, length, and time of waves, U*, L*,
and T*, are related by L* = U T*.
i R0 As we consider almost unidirectional waves propagat
us in ing mainly in the x direction, the scaling of y in Eq. (14c)
con and the expansions of vc and VL in Eqs. (20) and (21)
sion; take the weak twodimensionality into account.
o be Furthermore, the expansion of the liquid density pL is
defined as
pace
lized
PL/PL 1+ PL1 + PL2 + ..
Here, the liquid compressibility is very small compared
with the gas compressibility.
The expansion of the liquid pressure pL is
PiU = PLO + 62PL1 + 4PL2 + ,
PLOw
where the expansion coefficients PLi (i
defined as
PL1
PL1
(14a)
(14b)
(14c)
1, 2, 3) are
PL2
PL2
PL3 n )p 1
PL3 2 2V2
where to and xo represent fast scales; whereas yo, tl,
and xz slow scales. Equation (14) implies the weak
Here, the definition of the parameter V is presented in
Eq. (28) below.
1 ,
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
The nondimensional pressure in the unperti
pGO and PLo are, respectively, introduced as
PGO *L,'2 = O(1), pLo
PLoU
PLo
PLoU
urbed state, (i.e., cL* > oo) corresponds to the limit of the incom
pressible liquid.
Equations of O(e2) The set of equations of lowest
0(1). (24) order is as follows: (i) the mass conservation law for gas
phase,
The ratio of initial densities of the gas and liquid is
defined as
 0(e4), (25)
PLO
so that the effect of this ratio is eliminated throughout
the present analysis. The wave motions concerned are
not affected substantially by this approximation.
We introduce the nondimensional liquid viscosity /t:
Pi 0(62) / 2. (26)
Incidentally, the eigenfrequency of linear spherical
symmetric oscillations of the single bubble, wc, is in
troduced as follows (Plesset & Prosperetti 1977):
aOl 3 OR1 UG1
Oto dto xo
0, (29)
(ii) the mass conservation law for liquid phase,
9ay
Dto
8 o0o
S UL1l
(1 ao) 0,
Oxo
(iii) the momentum conservation law for gas phase in the
x direction,
DUGl
;h57
D0UL
B1to
Q au
R3P
dxo
0, (31)
(iv) the momentum conservation law for liquid phase in
the x direction,
3y(p*O + 2*/R*)
PLO 0
2o*/Ro
SUL1 d UG1
(1 ao + pioo) loot0
"to Oto
DPL1
+(1 ao) D
oa
Finally, it may be noted that all the variables in the
initial unperturbed state, cLo, PLo, 0'.,, PLo, .* R* ,
and ao, are constants.
Derivation of KP equation
Now let us choose the scaling relations of physical pa
rameters, the characteristic velocity U*, length L*, and
time T*:
(v) the Keller equation,
R 2
R1 + Y2L1
We note that the lowest order of momentum conserva
tion laws in the y direction is of O(e3).
Equations (29)(33) reduce to
(28a)
(28b)
0(e) = Ve,
O(e) = Ae,
0(e) = Q6e,
where V, A, and Q, are of 0(1) (Q corresponds to a
normalized frequency of incident waves), w* = 1/T* is
a frequency of the sound source.
Equation (28) means that the wave motion concerned
is of a small propagation speed compared with the speed
of sound in the initial undisturbed liquid, of a large
wavelength compared with the initial bubble radius, and
of a low frequency compared with the eigenfrequency of
the bubble (see, Fig. 1).
It should be remarked that V denotes the magnitude
of the liquid compressibility which leads to the attenua
tion of waves due to the acoustic radiation. Here V > 0
D2R1
LDR1= t
0~t
2 D2 R1
'P
0, (34)
where the phase velocity v, is given by
(28c) c 3ao(1 ao + B1p)PGO + il(1
(28c) 3 io(1 ao)
ao)A2/Q2
(35)
Now, we can choose as vp = 1 and then the explicit
form of the characteristic velocity U* is determined. Ac
cordingly, the characteristic length L* is also explicitly
obtained since T* l/w* is specified, for example, at
the sound source.
Let us restrict ourselves to the rightrunning wave in
the leading order of approximation. The following inde
pendent variables are defined
900 to X0, XO X0.
0, (32)
Equation (34) is readily simplified into the following
equation for R1 = f(0o, Xo; tl, xi, yo):
Of Of Of
0o D e+ = 0. (37)
OX0o 00 ato ao
Rewriting Eqs. (29)(33) by 0o and integrating them,
we can express the firstorder variables in terms of the
function f (0o):
A2
PL1 f s4f,
(1 ao)[33o (1 0)s4]
ao(l ao +31)
UG1 (Si 3)f s2f,
UL1 f s3f
1 an
Equations of O(c3) The set of equations of O(e3)
is composed of only the momentum conservation laws
for gas and liquid phases in the y direction, as follows:
DvG1 OVL1 R1
B31 31 3P0nGO 0,
Oto Oto 7 ,
dVL1 DVG1
(1 ao + i3ao) D 1oiao
Oto ato
+(1 o)
By utilizing Eq. (38), we have
0VG1
000
Of
B1
)yon
OVL1
D0o
Of
B2
BO"f
where
3(1 ao + /3i) (1 O)2A2/Q2
B1 1 ao +i ao(l ao + 1)
3/3Bio + (1 ao)A2/Q2
1 o + B31
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
They are combined into the inhomogeneous equation:
1 DK1 1 DK2 1 ao + B1 iK3
C[R,]= +
3 d0o 3ao 0do 331(1 ao) do
1 OK4 A2 02K5
3ao(1 ao) 00o 3aon2 9 2
(51)
After some straightforward calculations, we obtain
(38) K = 2 o Df + A f + A9ilf
(38) a af Of f
0o0 ati axr i09o 0'0o
sif, (39) d2f D3 ) [B (1 a)B] 92f
+ A2 + A +
(40) (52)
where the coefficient of the derivative with respect to yo
in the above equation can easily be calculated by using
Eq. (45),
B1 (1 ao)B2
+ = 1. (53)
3 3ao
From the solvability or nonsecular condition for
Eq. (51), we impose
K(f; o, ti, t i, yo) = 0.
By using Eqs. (15) and (37), we can rewrite Eqs. (52)
(54) into
S[If i Af f62f
at [t x Dat alft
D2 f + 3f\f 1 D2f
+A + t3 2 55
Ji t3 \ 2 dy
Equation (55) finally reduces to
(42)
0. (43)
(44)
(45a)
O )Of Of OI2f Tf 3
(45b) 9 + Af + A2 + A3
ily ~ ~ a il iy l2 ilrS
1 D2f
S (56)
2 OT12
Equations of 0(e4)
the set of Eqs. (29)(33),
The set of 0(c4) is similar to
DOC2 3 R2 OuG2
o 3 + K1,
Dto Dto Daro
9to
/3Ou
?)70
DUL2
(1 ao) = K2,
SOL2 D OR2
B1 D 3PGOx
o)to ioano
DUL2
(1 o +/Bio) 
O)to
DUG2
B1ao OU
DOto
/Stono
+ a pL2
+(1 ~o)
R2
R2 + pL2 = K5.
a
K4, (49)
through the variable transformation
o = 62x, T = t (1 + c2Ao)x, j = cy. (57)
The coefficients are given by
(1 ao)A2V2
A2 = 6V
6c~o 4R t2j
6ao
and the explicit representation of the nonlinear coeffi
cient A1 (> 0) is so complex and hence not shown in
(50) present paper.
the present paper.
Equation (56) is the socalled KP equation (Kadomt
sev & Petviashvili 1970) with the dissipation term due to
the liquid viscosity and compressibility. The KP equa
tion governs the behavior of almost unidirectional waves
with the small transverse perturbation in the far field
characterized by a, r, and r1. When the dependence
of r direction of unknown function f may be vanished
in Eq. (56), the KdVBurgers equation derived by the
present authors Kanagawa et al. (2009) is retrieved.
Acknowledgements
This work was partly supported by the Grant in Aid
for Scientific Research of the Ministry of Health, La
bor and Welfare; Advanced Medical Technology, H19
nano010. The authors would like to express their deep
est gratitude to this grant.
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7th International Conference on Multiphase Flow,
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