Group Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Title: 9.1.4 - Nonlinear Analysis of Wave Propagation in Liquids Containing Microbubbles
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 Material Information
Title: 9.1.4 - Nonlinear Analysis of Wave Propagation in Liquids Containing Microbubbles Bubbly Flows
Series Title: 7th International Conference on Multiphase Flow - ICMF 2010 Proceedings
Physical Description: Conference Papers
Creator: Kanagawa, T.
Yano, T.
Watanabe, M.
Fujikawa, S.
Publisher: International Conference on Multiphase Flow (ICMF)
Publication Date: June 4, 2010
 Subjects
Subject: nonlinear dispersive wave
bubbly liquid
weak two-dimensionality
KdV and KP equations
 Notes
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 one-dimensional 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 two-dimensional waves. On the basis of the method of multiple scales, a two-dimensional KdV equation, i.e., the Kadomtsev–Petviashvili (KP) equation with a dissipation term due to the liquid viscosity and compressibility can be derived from the basic equations for bubbly flows.
General Note: The International Conference on Multiphase Flow (ICMF) first was held in Tsukuba, Japan in 1991 and the second ICMF took place in Kyoto, Japan in 1995. During this conference, it was decided to establish an International Governing Board which oversees the major aspects of the conference and makes decisions about future conference locations. Due to the great importance of the field, it was furthermore decided to hold the conference every three years successively in Asia including Australia, Europe including Africa, Russia and the Near East and America. Hence, ICMF 1998 was held in Lyon, France, ICMF 2001 in New Orleans, USA, ICMF 2004 in Yokohama, Japan, and ICMF 2007 in Leipzig, Germany. ICMF-2010 is devoted to all aspects of Multiphase Flow. Researchers from all over the world gathered in order to introduce their recent advances in the field and thereby promote the exchange of new ideas, results and techniques. The conference is a key event in Multiphase Flow and supports the advancement of science in this very important field. The major research topics relevant for the conference are as follows: Bio-Fluid Dynamics; Boiling; Bubbly Flows; Cavitation; Colloidal and Suspension Dynamics; Collision, Agglomeration and Breakup; Computational Techniques for Multiphase Flows; Droplet Flows; Environmental and Geophysical Flows; Experimental Methods for Multiphase Flows; Fluidized and Circulating Fluidized Beds; Fluid Structure Interactions; Granular Media; Industrial Applications; Instabilities; Interfacial Flows; Micro and Nano-Scale Multiphase Flows; Microgravity in Two-Phase Flow; Multiphase Flows with Heat and Mass Transfer; Non-Newtonian Multiphase Flows; Particle-Laden Flows; Particle, Bubble and Drop Dynamics; Reactive Multiphase Flows
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Volume ID: VID00217
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Holding Location: University of Florida
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Resource Identifier: 914-Kanagawa-ICMF2010.pdf

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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 060-8628, Japan
t Department of Mechanical Engineering, Osaka University, Suita 565-0871, Japan
9 kanagawa@mech-me.eng.hokudai.ac.jp
Keywords: Nonlinear dispersive wave, Bubbly liquid, Weak two-dimensionality, 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 one-dimensional 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 two-dimensional waves.
On the basis of the method of multiple scales, a two-dimensional KdV equation, i.e., the Kadomtsev-Petviashvili
(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 one-dimensional 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 Korteweg-de Vries-Burgers (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 KdV-Burgers
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 KdV-Burgers 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 KdV-Burgers 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 KdV-Burgers equation, and nonlinear modulation of
a quasi-monochromatic 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 one-dimensional
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 Kadomtsev-Petviashvili (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 bubble-liquid 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 two-fluid
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 volume-averaged pressures p* and pL,
the liquid pressure averaged on the bubble-liquid 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 right-hand 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 right-hand 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 bubble-liquid 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 bubble-liquid 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


two-dimensionality 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 two-dimensionality 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 o0-o


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,


3-y(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 right-running 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 a-o

Rewriting Eqs. (29)-(33) by 0o and integrating them,
we can express the first-order 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 non-secular 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 so-called 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 KdV-Burgers 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-
nano-010. 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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