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Perturbation analysis of flow about spherically pulsating bubble at the velocity node and the antinode of a standing wave with different boundary conditions
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Perturbation analysis of flow about spherically pulsating bubble at the velocity node and the antinode of a standing wave with different boundary conditions
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Content
Perturbation
analysis
of
flow
about
spherically
pulsating
bubble
at
the
velocity
node
and
the
antinode
of
a
standing
wave
with
different
boundary
conditions
By
Mohammad
K.
AlHamli
-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐-‐
A
Dissertation
Presented
to
the
FACULTY
OF
THE
GRADUATE
SCHOOL
UNIVERSITY
OF
SOUTHERN
CALIFORNIA
In
Partial
Fulfillment
of
the
Requirements
for
the
Degree
DOCTOR
OF
PHILOSOPHY
(Mechanical
Engineering)
December
2015
Copyright
2015
Mohammad
K.
AlHamli
i
Abstract
We have examined the steady streaming phenomenon with regard to a pulsating bubble
levitated in a standing wave, positioned at the velocity node and antinode. The bubble
undergoes two types of oscillations when placed in a standing wave. The first mode of the
oscillation is a lateral one and the second is radial. We used the singular perturbation
method to analytically study this problem.
We considered a spherical cavity with radius 𝑎 oscillates with frequency 𝜔 and velocity
𝑈
!
𝑒
!"#
in a viscous fluid. If the amplitude of the oscillation is small compared with the
particle’s radius (𝜀=𝑈
!
/𝜔𝑎≪ 1) and the frequency parameter is large i.e. |𝑀|
!
=
𝜔𝑎
!
/𝜈≫ 1, which is the ratio of the bubble radius, 𝑎, to the viscous length; here 𝜔 is the
frequency and ν is the kinematic viscosity, then a thin boundary layer (also called the
Stokes layer) will form around the sphere. This Stokes layer, which is thin in comparison
to the radius, has vorticity and can be described to the leading order in 𝜀 as purely
oscillatory flow. However, at higher order there are both steady and non-steady parts.
The steady part will drive the bulk of the fluid to form the steady streaming.
We solved the pulsating bubble for both free shear and non-slip boundary conditions in
the velocity node and antinode of a standing wave. For the velocity node it was found, for
the shear free boundary condition, that the streaming does exist and it is of the order of
𝑂 𝜀𝜀
!
and 𝑂 𝑘𝑎 where 𝑘 is the wavenumber and 𝜀
!
is the ratio of the radial amplitude
to the radius of the bubble.. The streaming has a quadrupole nature, i.e., the flow has a
symmetrical pattern about the equatorial plane of the bubble. The difference is in the
direction of the flow between the upper and the lower plane. The phase shift between the
two modes of oscillation, 𝜙, and the frequency parameter, |M|, both play a major role in
the behavior of the flow field. Changing the phase shift or the frequency parameter would
not only alter the streaming intensity but also, at specific values, the shape of the flow
streamlines. For the non-slip boundary condition the steady streaming was of a greater
value. It’s actually ten times greater than the free shear boundary. Nonetheless, for the
large values of |M|, the steady streaming flow pattern and behavior is similar. For the
velocity antinode it was found, the streaming flow has both a dipole and a quadrupole
structure. It also depends on the frequency parameter |M| and the phase shift 𝜙. When
ii
|M| is small the flow has vortices in the third and the fourth quadrants. As the |M|
increases the vortices vanishes and the flow becomes purely dipole in nature. The phase
shift 𝜙 increases and decreases the flow intensity at certain values.
iii
Contents
Abstract
_______________________________________________________________________________
i
1. Introduction
_________________________________________________________________________
1
2. Streaming
at
the
velocity
node
of
a
standing
wave
__________________________
13
2.1.
The
dimensionless
parameters
and
the
far
field
conditions
_________
13
2.2. Equations
of
motion,
dimensionless
scaling,
and
the
boundary
conditions
_____________________________________________________________________
15
2.3. Solution
for
the
pulsating
bubble
at
the
node:
shear
free
boundary
condition
______________________________________________________________________
17
2.3.1. The
leading
order
solution,
𝑂 1
______________________________________
18
2.3.1.(a) The
boundary
layer
____________________________________________
20
2.3.2. The
first
order
solution,
𝑂 𝜀
__________________________________________
23
2.3.3. Discussion
_______________________________________________________________
29
2.4.
Solution
for
the
pulsating
bubble
at
the
node:
no
slip
boundary
condition
______________________________________________________________________
34
2.4.1. Discussion
_______________________________________________________________
37
2.5. Remarks
_______________________________________________________________________
41
3. Streaming
at
the
velocity
antinode
of
a
standing
wave
_____________________
42
3.1. The
dimensionless
parameter
and
the
far
field
conditions
___________
42
3.2. Equations
of
motion,
dimensionless
scaling,
and
the
boundary
conditions
_____________________________________________________________________
43
3.3. Solution
for
the
pulsating
bubble
at
the
antinode:
shear
free
boundary
condition
______________________________________________________________________
45
3.3.1. The
leading
order
solution,
𝑂 1
______________________________________
45
3.3.1.(a) The
boundary
layer
____________________________________________
46
3.3.2. The
first
order
solution,
𝑂 𝜀
__________________________________________
49
iv
3.3.3. Discussion
_______________________________________________________________
52
3.4.
Solution
for
the
pulsating
bubble
at
the
antinode:
no
slip
boundary
condition
______________________________________________________________________
57
3.4.1. Discussion
_______________________________________________________________
59
3.5. Remarks
_______________________________________________________________________
64
4. Final
remarks
______________________________________________________________________
65
5. References
__________________________________________________________________________
71
1
1. Introduction
Interest in microstreaming has increased in recent years due to lab on a chip
applications and as a drug delivery mechanism. Scientists use microstreaming in
microfluidics to: micromix, micropump, filtration etc., where a rotating mechanical
system is hard to manufacture (Hashmi, Yu, Reilly-Collette, Heiman, and Xu, 2012;
Wang, Jalikop, and Hilgenfeldt, 2012). In the medical field they started using
microbubbles as a contrast agent to enhance ultrasound imaging. Currently, however, the
microbubbles are used to deliver drugs where the understanding of the microstreaming
flow patterns is considered to be essential because it plays an important part of the system
(Lentacker, Smedt, and Sanders, 2009).
We have examined the steady streaming phenomenon with regard to a pulsating
bubble levitated in a standing wave, positioned at the velocity node and antinode with
different boundary conditions. The bubble undergoes two types of oscillations when
placed in a standing wave. The first mode of oscillations is lateral and the second is
radial. We used the singular perturbation method to analytically study this problem.
Cavities or bubbles in a sound field can be divided into two categories: 1) Stable and
2) Transient. Stable bubbles are bubbles oscillate around the equilibrium size and will
continue their oscillation over many cycles of the acoustic pressure. Transient bubbles,
however, will expand way above their original size, and then they will collapse violently
(Neppiras 1980, Crum 1982, Young 1989). (Figure 1.1)
Figure 1.1: Pressure wave and bubble response to it. (a) Stable bubble; the response is linear with some
phase lag depending on the frequency, (b) transit bubble; the response is not linear and the bubble will
break after some cycles, and (c) Sonoluminescence, when the pressure wave amplitude is much higher (not
to scale).
2
Our focus here will be on a stable bubble, which is usually filled with a gas i.e., air.
However, under special circumstances, especially with low boiling point fluids, stable
vapor bubbles do exist (Neppiras, 1980). The existence of the gas in the bubble will make
it analogous to a damped forced oscillator (Leighton, 1994; Young, 1989) because of the
ability of the gas to absorb acoustic energy and reverse its direction as the pressure eases.
The bubble could radially oscillate below, at, or above resonance. If a steady-state
response is assumed, then the pulsation could be in phase or out of phase with the driving
pressure. Bubble radial oscillations below resonance take place in phase (stiffness
control). As the frequency increases the phase lag between the driving pressure and the
bubble oscillation will increase. It reaches a value of 𝜋/2 when the frequency reaches the
resonance. As the frequency increases further, the phase lag will increase until it reaches
a maximum value of 𝜋 (inertia control) (Leighton, 1994; Young, 1989; Leighton, Walton,
and Pickworth, 1990). See (Figure 1.2)
When a standing wave forms in a fluid medium, nodes and antinodes of both velocity
and pressure would develop. The pressure antinode is the velocity node and visa versa.
The pressure undergoes its maxima at its antinodes and at these points the velocity has its
minima (Rossing, 2007) (Figure 1.3). A bubble located in a sound field would experience
a net force due to pressure fluctuation. This net force is called the primary Bjerknes force,
which affects the bubble location in the standing wave. To understand quantitatively how
Figure 1.2: Phase lag as a function of the frequency to the resonance for different damping
values (Leighton, 1994).
3
the acoustic force affects the bubble we look at the formulation from (Eller, 1968). The
acoustic force on a bubble that is smaller than the sound wavelength will be
𝐹=− 𝑉 𝑡 ∇𝑝 𝐫,𝑡 , (1.1)
where 𝑝(𝐫,𝑡) is the acoustic pressure at position r and time t. Making the sound wave
symmetric along the z-axis, the acoustic pressure will be
𝑝 𝑧,𝑡 =𝑃
!
sin 𝑘𝑧 sin 𝜔𝑡 (1.2)
where k is the wavenumber, P
A
is the pressure wave amplitude, and 𝜔 is the angular
frequency. The gradient of the acoustic pressure is
𝛁𝑝 𝑧,𝑡 =𝑃
!
𝑘cos 𝑘𝑧 sin 𝜔𝑡 𝑧. (1.3)
The bubble pulsation is proportional to the pressure amplitude. For small pressure
amplitude the bubble pulsates radially according to
𝑅 𝑡 =𝑎 1+𝜀
!
sin 𝜔𝑡 , (1.4)
where 𝑎 is the equilibrium radius and 𝜀
!
is the radial oscillation amplitude. Here 𝜀
!
is
found from the Rayleigh–Plesset equation after substituting both equation (1.2) and
equation (1.4) and considering the requirement of stable bubble while neglecting 𝜀
!
!
terms and assuming isothermal process (Brennen, 1995). In terms of 𝑃
!
and 𝜔, 𝜀
!
has the
following relationship
𝜀
!
=
𝑃
!
sin 𝑘𝑧
𝜌𝑎 𝜔
!
−𝜔
!
!
. (1.5)
4
Here 𝜔
!
is the resonance frequency and 𝜌 the fluid density. Substituting equations (1.3),
(1.3) and (1.4) in equation (1.1), and taking into consideration that the volume is
𝑉 𝑡 =
!
!
𝜋𝑅
!
, the Bjerknes force is found to be
𝐹=−
𝜋𝑘𝑎𝑃
!
!
𝜌 𝜔
!
−𝜔
!
!
sin 2𝑘𝑧 . (1.6)
The above equation shows that the force is zero at the node and the antinode.
Also, giving special attention to the sin 2𝑘𝑧 in the force equation (1.6), and sin 𝑘𝑧 in
the pressure equation (1.2), and as we can see from Figure (1.2), we can conclude the
following: when the resonance frequency of the bubble is larger than the driving
frequency (positive force), the bubble will move to the velocity node (pressure antinode)
in the standing sound wave (Figure 1.4). And when the resonance frequency is smaller
than the driving frequency (negative force), the bubble will move to the velocity antinode
(pressure node). For a bubble at the velocity node means that the equilibrium radius 𝑎 of
the bubble, is smaller than the size of the bubble at the imposed frequency, 𝑅
!
. In the case
of a bubble located at the velocity antinode means, 𝑎> 𝑅
!
(Brennen, 1995) and
(Leighton, Walton, and Pickworth, 1990).
Figure 1.3: Pressure in a standing wave where the node and antinode were labeled
and the location of a bubble with respect to frequency and radius.
5
The location of the bubble at either the node or the antinode requires different
treatment when we start solving the equations of motion. We have to consider the
compressibility effect when solving for the flow around the bubble, especially for the
leading terms, that is located at the velocity node. But we assume incompressible flow
when we solve for the bubble located at the velocity antinode. To fully understand why
such treatment is necessary we have to examine the velocity equation for a standing wave
which has the following form
𝑢
!
=𝑈
!
cos 𝑘 𝑧+𝑧
!
sin 𝜔𝑡 (1.7)
Where at the velocity node 𝑘𝑧
!
=
!
!
, the velocity becomes
𝑢
!
=−𝑈
!
sin 𝑘𝑧 sin 𝜔𝑡 (1.8)
Expanding sin 𝑘𝑧 around 𝑘𝑧= 0 we have
𝑢
!
=−𝑈
!
𝑘𝑧−
!
!
!
!
!!
+⋯ sin 𝜔𝑡 . (1.9)
Figure 1.4: Pressure, velocity, and force graphs for frequency greater than the
resonance. The force is directed toward the velocity antinode.
6
For small particle we can take only the first term of the expansion
𝑢
!
~−𝑈
!
𝑘𝑧sin 𝜔𝑡 . (1.10)
On the other hand, at the velocity antinode where 𝑘𝑧
!
= 0, the velocity becomes
𝑢
!
=𝑈
!
cos 𝑘𝑧 sin 𝜔𝑡 . (1.11)
Again expanding around 𝑘𝑧= 0
𝑢
!
=𝑈
!
1−
!
!
!
!
!!
+⋯ sin 𝜔𝑡 . (1.12)
For small particle only the first term of the expansion is needed, and
𝑢
!
~𝑈
!
sin 𝜔𝑡 . (1.13)
Comparing equation (1.10), which represents velocity at the node, to equation (1.13) that
represents velocity at the antinode, we notice the presence of 𝑘𝑧. This extra term makes
the treatment between the node and the antinode quite different. This makes complete
sense because at the velocity node the pressure has the maximum fluctuation.
Acoustic streaming or steady streaming is the net mean flow caused by a sound wave.
Naming the phenomenon steady streaming rather than acoustic streaming is sometimes
preferred by scientists, as pointed out by (Riley, 2001). He states that the term “acoustic
streaming” implies a degree of compressibility; however many of the streaming
phenomenon that researchers are interested in can be described under incompressible
flow approximation. (Lighthill, 1978) pointed out that the streaming is driven by
Reynolds stresses, which he defined as the mean value of the acoustic momentum flux.
He wrote: “[…] the dissipation of acoustic energy flux that permits gradients in
momentum flux to force acoustic streaming motions”. There are two types of situations
that can induce the dissipation of acoustic energy flux, which in turn force streaming
(Sadhal, 2012). The first happens in the main body of the fluid, e.g. the quartz wind. The
7
second type arises from vibrating fluid medium interacting with a solid wall or a
vibrating solid object in a static fluid. The second type is what we focus on in this
research. It was first studied by (Rayleigh, 1884). He studied the phenomenon as it
occurred in the Kundt's tube. It was further investigated in connection with the boundary
layer theory by Schlichting, where he introduced the concept of the inner streaming
(Schlichting, 1932). However, he stopped short from connecting the boundary layer
streaming, or the inner streaming, with the outer streaming, or streaming in the main
body of the fluid. (Stuart, 1966) outlined that the inner streaming (streaming within the
boundary layer) will persist into the region outside the boundary layer to form outer
streaming. When a body is placed in a standing wave or a body is oscillated in fluid at
rest, the fluid within the viscous boundary layer (Stokes layer) is of rotational character.
This rotationality causes nonlinear effects which induce the steady secondary flow that is
theoretically interpretable by taking the time-average of the overall flow. The bulk of the
fluid, though it is purely periodic, will be generally irrotational (Figure 1.5). This is why
the fluid in such an analysis is usually divided into two regions: the inner layer, the
Stokes layer, and the outer region or the outer flow. The vortices in the Stokes layer will
persist and drive the flow in the outer region which is generally in the Stokes regime. The
Stokes layer has a thickness of 𝛿∼
!
!
where ν is the kinematic viscosity. Of course this
is assuming a large value for the frequency parameter, i.e. |𝑀|
!
=
!!
!
!
≫ 1, which is the
ratio of the bubble radius, 𝑎, to the viscous length. However; if |𝑀|
!
≪ 1, then the first
order vorticity will diffuse over a wider region and wouldn’t be confined to the Stokes
layer (Riley, 1966). As an example for steady streaming, we can consider a spherical
particle with radius 𝑎 that oscillates with frequency 𝜔 and velocity 𝑈
!
𝑒
!"#
in a viscous
fluid. If the amplitude of the oscillation is small compared with the particle’s radius
(𝜀=𝑈
!
/𝜔𝑎≪ 1) and the frequency parameter is large i.e. |𝑀|
!
≫ 1, then a thin
boundary layer (the Stokes layer) will form around the sphere. This Stokes layer, which is
thin in comparison to the radius, has vorticity and can be described to the leading order in
𝜀 as purely oscillatory flow. However, at higher order there are both steady and non-
steady parts. The steady part will drive the bulk of the fluid to form the steady streaming.
8
Axisymmetric flow analysis can be useful in finding the steady streaming in the case
of spherical particle placed in an oscillatory medium with large |𝑀|
!
. There are different
approaches to try and solve these types of problems using perturbation. 1) Perturbation of
the equations of motion without any restriction on |𝑀|. This was tried by us where we
found the leading order solution. However the solution was too long and complicated that
a time average of the first order solution would be nearly impossible to find. 2) Longuet-
Higgins (Longuet-Higgins, 1998) type of expansion with 𝜀 and 𝜀
!
, 3) Davidson and Riley
(Davidson and Riley, 1971) who expanded for |𝑀|≫ 1 and perturbed with both 𝜀 and
|𝑀|, and 4) expansion with 𝜀 and neglecting all the terms of 𝑂(𝑀
!!
) and smaller (Zhao
and Sadhal, 1999).
The last approach was pioneered by (Riley, 1966) who pointed out, in the same paper,
that Oseen-type approximation would not work for |𝑀|
!
≫ 1. He cited (Andres and
Ingard, 1953) where they have used Oseen-type approximation to find the streaming
around a cylinder placed in low Reynolds number flow. We have built up on (Riley,
1966) work and that of (Zhao et al., 1999). In these papers drops and bubbles were
studied as vibrating spheres that have either a continuous tangential stress or a no-slip
condition at the boundary. The case of a bubble was considered by (Davidson et al.,
1971) who assumed a shear-free boundary condition for a non-contaminated bubble.
Figure 1.5: The different regions and flows in the steady streaming analysis (Sadhal, 2012).
9
(Longuet-Higgins, 1998) considered the two modes of oscillations, the lateral and radial,
around a clean bubble located at the velocity antinode (as were many of the previous
works) of a standing wave. Although a solution was formulated for a vibrating solid
sphere at the velocity node (Zhao et al., 1999), no solution exists which considers both
modes of oscillations. Additionally, a solution for a contaminated bubble (non-slip
boundary condition) that exhibits these two modes of oscillations does not exist for a
bubble trapped at the velocity antinode. To fill this gap, we formulated solutions for the
pulsating bubble at both the velocity node and at the antinode of a standing wave.
Although we have adopted an expansion similar to that of (Zhoa et al., 1999), our
approach was slightly different. The difference arises from the treatment of the frequency
parameter in the governing equations. It will not be neglected at the order of 𝑂(𝑀
!!
) in
the governing equations and the expansions. We kept this order of |𝑀| because the
streaming will not be captured for order 𝑂(1) in the case of clean bubble. It is well
known that high frequency oscillation generally relates to a large frequency parameter.
Therefore, by not neglecting terms in the order of 𝑂(𝑀
!!
), we assumed an intermediate
range value for |𝑀|, i.e., |𝑀|> 1. Again, as it will be seen later, in the case of free shear
boundary condition, this assumption makes sense. However, the value of |𝑀|
!
will be
much larger than 1, i.e. |𝑀|
!
≫ 1. Therefore, all terms in the order of 𝑂(𝑀
!!
) will be
neglected. Other assumptions include:
• The bubble will stay spherical in shape; the surface tension forces are much higher
than the inertial forces;
• The motion within the cavity will be ignored; that is to say the viscosity in the
surrounding fluid is much larger than that of the gas in the bubble;
• The wavelength is much larger than the bubble radius; one wave length considered
to be the far field;
• No fluid transport takes place from the bubble to the surrounding fluid and vice-
versa;
• The lateral and the radial oscillation modes have the same order of amplitude i.e.
𝜀≈ 𝜀
!
≪ 1; this is true when the bubble radius is small compared to the field
wavelength (Plesset and Prosperetti, 1977), and the effects of the gravity are
ignored.
10
Before we proceed with the solution we ought to look at some realistic numbers
and how exactly the flow will behave in a general picture. We have assumed small
amplitude of radial oscillation. However this assumption is not valid when the pressure
wave amplitude is high and the excited frequency is close to the resonance frequency of
the bubble. This is evident from table (1.1) and table (1.2). For a ratio of the excited to
the resonance frequencies, 𝜔 𝜔
!
= 0.97 the ratio of the radial amplitude to the bubble
radius, 𝜀
!
is the highest. As 𝜔 𝜔
!
decreases’ so does 𝜀
!
, to the point where it plateaues at
a minimum value for small 𝜔 𝜔
!
. The resonance frequency, 𝜔
!
, decreases as bubble
radius increases. The same can be said about the excited frequency 𝜔. For small bubble
but high frequency the bubble will be located at the velocity antinode in the standing
wave as the ratio 𝜔 𝜔
!
is larger than unity - this is obvious from table (1.2). The
boundary layer thickness is constant throughout the different values of the bubble radius
but it changes as we change the frequency parameter. When the frequency parameter
decreases so does the boundary layer thickness. However with both cases, the boundary
layer is indeed small compared to the bubble radius, and the bubble is also small
compared to the wavelength.
11
Bubble
Radius,
a
Resonance
frequency,
ω
o
Frequency,
ω,
at
|M|=10
Ratio
of
the
excited
to
the
resonance
frequencies
Boundary
layer
thickness,
𝛿
Radial
Amplitude
of
oscillation, 𝜀
!
μm
Hz
MHz
Hz
MHz
ω/ω
o
m
Non-‐
dimensional
Non-‐
dimensional
5
4.12E+06
4.12E+00
4.02E+06
4.02E+00
0.97
7.07E-‐07
0.141
4.60E-‐04
10
2.06E+06
2.06E+00
1.00E+06
1.00E+00
0.49
1.41E-‐06
0.141
3.09E-‐05
15
1.37E+06
1.37E+00
4.46E+05
4.46E-‐01
0.32
2.12E-‐06
0.141
2.64E-‐05
20
1.03E+06
1.03E+00
2.51E+05
2.51E-‐01
0.24
2.83E-‐06
0.141
2.51E-‐05
25
8.25E+05
8.25E-‐01
1.61E+05
1.61E-‐01
0.19
3.54E-‐06
0.141
2.45E-‐05
30
6.87E+05
6.87E-‐01
1.12E+05
1.12E-‐01
0.16
4.24E-‐06
0.141
2.42E-‐05
35
5.89E+05
5.89E-‐01
8.20E+04
8.20E-‐02
0.14
4.95E-‐06
0.141
2.40E-‐05
40
5.15E+05
5.15E-‐01
6.28E+04
6.28E-‐02
0.12
5.66E-‐06
0.141
2.39E-‐05
45
4.58E+05
4.58E-‐01
4.96E+04
4.96E-‐02
0.11
6.36E-‐06
0.141
2.39E-‐05
50
4.12E+05
4.12E-‐01
4.02E+04
4.02E-‐02
0.10
7.07E-‐06
0.141
2.38E-‐05
55
3.75E+05
3.75E-‐01
3.32E+04
3.32E-‐02
0.09
7.78E-‐06
0.141
2.38E-‐05
60
3.44E+05
3.44E-‐01
2.79E+04
2.79E-‐02
0.08
8.49E-‐06
0.141
2.37E-‐05
65
3.17E+05
3.17E-‐01
2.38E+04
2.38E-‐02
0.07
9.19E-‐06
0.141
2.37E-‐05
70
2.94E+05
2.94E-‐01
2.05E+04
2.05E-‐02
0.07
9.90E-‐06
0.141
2.37E-‐05
75
2.75E+05
2.75E-‐01
1.78E+04
1.78E-‐02
0.06
1.06E-‐05
0.141
2.37E-‐05
80
2.58E+05
2.58E-‐01
1.57E+04
1.57E-‐02
0.06
1.13E-‐05
0.141
2.37E-‐05
85
2.43E+05
2.43E-‐01
1.39E+04
1.39E-‐02
0.06
1.20E-‐05
0.141
2.37E-‐05
90
2.29E+05
2.29E-‐01
1.24E+04
1.24E-‐02
0.05
1.27E-‐05
0.141
2.36E-‐05
95
2.17E+05
2.17E-‐01
1.11E+04
1.11E-‐02
0.05
1.34E-‐05
0.141
2.36E-‐05
100
2.06E+05
2.06E-‐01
1.00E+04
1.00E-‐02
0.05
1.41E-‐05
0.141
2.36E-‐05
Table 1.1: Air bubble in water for frequency parameter of, |M|=10 and pressure amplitude of 10.
12
Bubble
Radius
Resonance
frequency,
ω
o
Frequency,
ω,
at
|M|=20
Ratio
of
the
excited
to
the
resonance
frequencies
Boundary
layer
thickness,
𝛿
Radial
Amplitude
of
oscillation, 𝜀
!
μm
Hz
MHz
Hz
MHz
ω/ω
o
m
Non-‐
dimensional
Non-‐
dimensional
5
4.12E+06
4.12E+00
1.61E+07
1.61E+01
3.90
3.54E-‐07
0.071
-‐1.66E-‐06
10
2.06E+06
2.06E+00
4.02E+06
4.02E+00
1.95
7.07E-‐07
0.071
-‐8.43E-‐06
15
1.37E+06
1.37E+00
1.78E+06
1.78E+00
1.30
1.06E-‐06
0.071
-‐3.43E-‐05
20
1.03E+06
1.03E+00
1.00E+06
1.00E+00
0.97
1.41E-‐06
0.071
4.60E-‐04
25
8.25E+05
8.25E-‐01
6.43E+05
6.43E-‐01
0.78
1.77E-‐06
0.071
6.00E-‐05
30
6.87E+05
6.87E-‐01
4.46E+05
4.46E-‐01
0.65
2.12E-‐06
0.071
4.08E-‐05
35
5.89E+05
5.89E-‐01
3.28E+05
3.28E-‐01
0.56
2.47E-‐06
0.071
3.42E-‐05
40
5.15E+05
5.15E-‐01
2.51E+05
2.51E-‐01
0.49
2.83E-‐06
0.071
3.09E-‐05
45
4.58E+05
4.58E-‐01
1.98E+05
1.98E-‐01
0.43
3.18E-‐06
0.071
2.90E-‐05
50
4.12E+05
4.12E-‐01
1.61E+05
1.61E-‐01
0.39
3.54E-‐06
0.071
2.78E-‐05
55
3.75E+05
3.75E-‐01
1.33E+05
1.33E-‐01
0.35
3.89E-‐06
0.071
2.70E-‐05
60
3.44E+05
3.44E-‐01
1.12E+05
1.12E-‐01
0.32
4.24E-‐06
0.071
2.64E-‐05
65
3.17E+05
3.17E-‐01
9.51E+04
9.51E-‐02
0.30
4.60E-‐06
0.071
2.59E-‐05
70
2.94E+05
2.94E-‐01
8.20E+04
8.20E-‐02
0.28
4.95E-‐06
0.071
2.56E-‐05
75
2.75E+05
2.75E-‐01
7.14E+04
7.14E-‐02
0.26
5.30E-‐06
0.071
2.53E-‐05
80
2.58E+05
2.58E-‐01
6.28E+04
6.28E-‐02
0.24
5.66E-‐06
0.071
2.51E-‐05
85
2.43E+05
2.43E-‐01
5.56E+04
5.56E-‐02
0.23
6.01E-‐06
0.071
2.49E-‐05
90
2.29E+05
2.29E-‐01
4.96E+04
4.96E-‐02
0.22
6.36E-‐06
0.071
2.47E-‐05
95
2.17E+05
2.17E-‐01
4.45E+04
4.45E-‐02
0.21
6.72E-‐06
0.071
2.46E-‐05
100
2.06E+05
2.06E-‐01
4.02E+04
4.02E-‐02
0.19
7.07E-‐06
0.071
2.45E-‐05
Table 1.2: Air bubble in water for frequency parameter of, |M|=20 and pressure amplitude of 10.
13
2. Streaming at the velocity node of a standing wave
It was pointed out in section 1, the introduction, that the streaming flow around the
velocity node is considered to be compressible for the leading order. This is why the use of
the stream function is not possible. It is later in the analysis that the stream function can be
introduced.
We will solve the problem by introducing the nondimensional parameters in the equations
of motion. Then we perturb these equations and solve for the pressure at the far field and just
outside the boundary layer. To deal with the problem of singularity within the boundary layer,
we scale the radial parameters. We then solve for the streaming within the boundary layer and
finally match the solution with the second order time average Stokes flow outside the
boundary layer.
This section is divided into five subsections. We start with the dimensionless parameters
and the far field conditions followed by the equations of motion. After that, a solution for a
clean bubble and a contaminated bubble will be presented and finally we will end up this
section with remarks.
2.1. The dimensionless parameters and the far field conditions
Spherical polar coordinates with the origin at the center of the bubble have been adopted.
The z-axis passes through the center of the sphere and points along the direction of the lateral
vibration (see Figure 2.1).
Figure 2.1: The two modes of oscillation and the notation of the coordinate
system of the bubble.
14
The following dimensionless parameters are used:
𝑅𝑒=
𝑈
!
𝑎
𝜈
; 𝑀
!
=
𝑖𝜔𝑎
!
𝜈
; 𝜀=
𝑈
!
𝜔𝑎
=
𝑅𝑒
𝑀
!
≪ 1; 𝜀
!
=
𝑃
!
sin 𝑘𝑧
𝜌𝑎
!
𝜔
!
−𝜔
!
!
≪ 1 , (2.1)
where 𝑈
!
,𝜈,𝑎,𝑅𝑒,and 𝑀 are the characteristic velocity for the lateral oscillation, the
kinematic viscosity of the medium, the radius of the bubble, the Reynolds number, and the
frequency parameter, respectively. The parameters 𝜀 and 𝜀
!
are the ratios of the lateral
displacement amplitude and the radial displacement amplitude to the radius of the bubble,
respectively. Here the assumption of a high frequency standing wave is made so that
𝑀
!
≫ 1. The local velocity in the neighborhood of the velocity node of a standing wave is
𝑢
!
=−𝑈
!
𝑘𝑧𝑒
!"#
, (2.2)
where 𝑘=𝜔/𝑐 is the wavenumber.
Away from the bubble, the flow is irrotational; therefore a potential function can be defined,
i.e.,
𝒖=𝛁𝜑 . (2.3)
The far-field potential function is
𝜑
!
=
𝑈
!
𝑘
cos 𝑘𝑧 𝑒
!"#
=
𝑈
!
𝑘
1−
𝑘
!
𝑟
!
cos
!
𝜃
2
+⋯ 𝑒
!"#
. (2.4)
Defines the Stokes stream function (Kundu, and Cohen, 2004),
𝑢
!
=
1
𝑟
!
sin𝜃
𝜕𝜓
𝜕𝜃
(2.5)
𝑢
!
=
−1
𝑟sin𝜃
𝜕𝜓
𝜕𝑟
(2.6)
and
equation
(2.2)
with
its
relation
to
𝑢
!
and
𝑢
!
,
we
can
find
the
expression
for
𝜓
the
at
the
far
field,
15
𝜓
!
=−
𝑈
!
𝑘𝑟
!
3
cos𝜃sin
!
𝜃𝑒
!"#
. (2.7)
2.2. Equations of motion, dimensionless scaling, and the boundary conditions
The flow parameters are scaled as follows:
𝒖=
𝒖
∗
𝑈
!
, 𝜓=
𝜓
∗
𝑈
!
𝑎
!
,𝒙=
𝒙
∗
𝑎
, 𝜑=
𝜑
∗
𝑈
!
𝑎
, 𝜏=𝜔𝑡, 𝑝=
𝑝
∗
𝜌
!
𝑈
!
𝜔𝑎
,
𝜌=
𝜌
∗
𝑐
!
𝜌
!
𝑈
!
𝜔𝑎
,and 𝛁=𝑎𝛁
∗
, (2.8)
where the asterisks denote the dimensioned quantities, 𝜓 is the stream function, a is the
bubble nominal radius, t is time, 𝜌
!
is the constant medium density, and p and 𝜌 are the
acoustic pressure and density. Using the adiabatic relation, 𝜌=𝑝/𝑐
!
, the dimensionless
parameter p and 𝜌 are equal. This fact will become useful later in the analysis.
The dimensionless equations of continuity and momentum after using the scaled flow
parameters above are
𝑘𝑎
!
𝜕𝜌
𝜕𝜏
+𝛁 ∙𝒖+𝜀 𝑘𝑎
!
𝛁 ∙ 𝜌𝒖 = 0 , (2.9)
and
1+𝜌𝜀 𝑘𝑎
!
𝜕𝒖
𝜕𝜏
+𝜀 1+𝜌𝜀 𝑘𝑎
!
𝒖 ∙𝛁𝒖=−∇𝑝+
1
𝑀
!
∇
!
𝒖 . (2.10)
We notice in the above equations the presence of the dimensionless parameters 𝜀 and 𝑀
!
,
we use the small quantity 𝜀 as a perturbation parameter, and in the course of expansion the
convective terms will be eliminated for the first order. The frequency parameter, 𝑀
!
, is
found in front of the viscous term in a fashion similar to the Reynolds number in the
incompressible momentum equation for a Newtonian fluid. It is apparent that large 𝑀
!
will
cause the singularity in the solution of the momentum equation.
16
Although the equations of motion will be the same for all of the problems to be solved,
the boundary conditions will differ. For the pulsating bubble the boundary condition at the
surface of the bubble that corresponds to the radial oscillation is,
𝑢
!
=
𝜕𝑅
𝜕𝜏
= 𝜀
!
𝑒
! !!!
at 𝑟=𝑅 𝜏 . (2.11)
where 𝑅
∗
𝑡 =𝑎(1+𝜀
!
sin 𝜔𝑡 ), and in dimensionless form with radial phase shift 𝜙
included,
𝑅 𝜏 = 1− 𝑖𝜀
!
𝑒
! !!!
. (2.12)
The second boundary condition depends on whether we will approach the problem as a
tangential less solid body or a free surface bubble. At the interface of the clean bubble the
tangential shear stress will be continuous,
𝜏
!"(!"!!#$)
= 𝜏
!"(!)
at 𝑟=𝑅 𝜏 (2.13)
where the 𝜏
!"(!"!!#$)
is the tangential stress due to the viscosity inside the bubble and 𝜏
!"(!)
due to the surrounding fluid. Assuming air inside the bubble, and water as the fluid
surrounding the free-surface bubble, the boundary condition is
𝜏
!"
= 0 at 𝑟=𝑅 𝜏 . (2.14)
Knowing that the radial velocity does not change with angle at the boundary, i.e.
!!
!
!"
=
0 𝑎𝑡 𝑟=𝑅(𝜏), The shear-free surface condition may also take the form,
𝜕
𝜕𝑟
𝑢
!
𝑟
= 0 at 𝑟=𝑅 𝜏 . (2.15)
In the case of a contaminated bubble, however, we will adopt the no slip boundary condition,
which is
17
𝑢
!
= 0 at 𝑟=𝑅 𝜏 . (2.16)
At the far field, i.e. 𝑟→∞, the non-dimensional velocity, stream function, and the potential
function were found to be,
𝑢
!
=−𝑘𝑎𝑟cos𝜃𝑒
!"
, (2.17)
𝜓
!
=−𝑘𝑎
𝑟
!
3
cos 𝜃 sin
!
𝜃 𝑒
!"
, (2.18)
and,
𝜑
!
=
1
𝑘𝑎
−
𝑘𝑎𝑟
!
6
−
𝑘𝑎𝑟
!
3
𝑃
!
𝜇 𝑒
!"
. (2.19)
where 𝑃
!
𝜇 =
!
!
3𝜇
!
−1 , (Abramowitz and Stegun, 2013) is the Legendre polynomial, and
𝜇= cosθ.
2.3. Solution for the pulsating bubble at the node: shear free boundary
Using the fact that 𝜀≪ 1 and is found in the dimensionless equation of motion, we apply
the perturbation method to expand the velocity, acoustic pressure and density,
𝒖=𝒖
!
+𝜀𝒖
!
+𝑂 𝜀
!
, (2.20)
𝑝=𝑝
!
+𝜀𝑝
!
+𝑂 𝜀
!
, (2.21)
and,
𝜌=𝜌
!
+𝜀𝜌
!
+𝑂 𝜀
!
. (2.22)
After inserting (2.20), (2.21), and (2.22) into the continuity and the momentum equations,
(2.9) and (2.10), these equations respectively take the following form
18
𝑘𝑎
!
𝜕𝜌
!
𝜕𝜏
+𝜀
𝜕𝜌
!
𝜕𝜏
+⋯ +∇ ∙ 𝒖
!
+𝜀𝒖
!
+⋯ +𝜀 𝑘𝑎
!
∇
∙ 𝜌
!
+𝜀𝜌
!
+⋯ 𝒖
!
+𝜀𝒖
!
+⋯ = 0 (2.23)
and
1+𝜌
!
𝜀 𝑘𝑎
!
+𝜌
!
𝜀
!
𝑘𝑎
!
+⋯
𝜕𝒖
!
𝜕𝜏
+𝜀
𝜕𝒖
!
𝜕𝜏
+⋯
+𝜀 1+𝜌
!
𝜀 𝑘𝑎
!
+𝜌
!
𝜀
!
𝑘𝑎
!
+⋯ 𝒖
!
+𝜀𝒖
!
+⋯ ∙∇ 𝒖
!
+𝜀𝒖
!
+⋯
=−∇ 𝑝
!
+𝜀𝑝
!
+⋯ +
1
𝑀
!
∇
!
𝒖
!
+𝜀𝒖
!
+⋯ (2.24)
2.3.1 The Leading Order Solution:
From the momentum equation (2.24), the leading order velocity is described by
𝜕𝒖
!
𝜕𝜏
=−𝛁𝑝
!
+
1
𝑀
!
∇
!
𝒖
!
, (2.25)
where the viscous term,
!
!
!
∇
!
𝒖
!
, can be neglected because of the assumption that 𝑀
!
≫ 1.
Then equation (2.25) becomes,
𝜕𝒖
!
𝜕𝜏
=−𝛁𝑝
!
. (2.26)
Equation (2.26) indicates that the leading order solution is irrotational. Therefore, the leading
velocity 𝒖
!
can be represented as a gradient of the potential function 𝜑
!
so that
𝒖
!
=𝛁𝜑
!
, (2.27)
and equation (2.26) integrates to
𝜕𝜑
𝟎
𝜕𝜏
=−𝑝
!
. (2.28)
19
For irrotational flow we found a relation between the potential function and the pressure.
Since the potential function in the far field is known, the pressure and density can be found by
applying equation (2.28) to 𝜑
!
in equation (2.19), i.e.,
𝑝
!
=𝜌
!
=−𝑖
1
𝑘𝑎
−
𝑘𝑎𝑟
!
6
−
𝑘𝑎𝑟
!
3
𝑃
!
𝜇 𝑒
!"
. (2.29)
The leading order for the perturbed continuity equation (2.23), is
𝑘𝑎
!
𝜕𝜌
!
𝜕𝜏
+𝛁 ∙𝒖
!
= 0 . (2.30)
With small (ka), only the term −𝑖
!
!"
𝑒
!"
will be needed from equation (42). We took only the
first part of equation (2.29), because after multiplying by 𝑘𝑎
!
the other terms will be in the
order of 𝑂 𝑘𝑎
!
and this is considered to be a small negligible quantity. Then the continuity
equation becomes
𝑘𝑎𝑒
!"
+𝛁 ∙𝒖
!
= 0 . (2.31)
Writing equation (2.31) in terms of potential function, 𝜑
!
, we have
∇
!
𝜑
!
+𝑘𝑎𝑒
!"
= 0 . (2.32)
Solving the above equation and using the far field condition, equation (2.19), and the normal
velocity at the surface of the bubble i.e. 𝑢
!!
=
!!
!
!"
= 𝜀
!
𝑒
!(!!!)
at r=R, the solution takes the
form,
𝜑
!
=
1
𝑘𝑎
−
𝑘𝑎
3
𝑟
!
2
+
𝑅
!
𝑟
−
𝑘𝑎
3
𝑟
!
+
2
3
𝑅
!
𝑟
!
𝑃
!
𝜇 𝑒
!"
−𝜀
!
𝑅
!
𝑟
𝑒
! !!!
, (2.33)
and the leading order acoustic pressure and density are
20
𝑝
!
=𝜌
!
=−𝑖
1
𝑘𝑎
−
𝑘𝑎
3
𝑟
!
2
+
𝑅
!
𝑟
−
𝑘𝑎
3
𝑟
!
+
2
3
𝑅
!
𝑟
!
𝑃
!
𝜇 𝑒
!"
+ 𝑖𝜀
!
𝑅
!
𝑟
𝑒
! !!!
. (2.34)
So far we found the leading order solution for the flow outside the boundary layer. However,
in order to theoretically see streaming we have to find the flow field within the boundary layer
(where the vortices are generated) and match it with the outer flow.
2.3.1(a) The Boundary Layer:
In the boundary layer, the velocity can be expressed as
𝒖
!
=𝑢
!
!
𝒓+𝑢
!
!
𝜽 . (2.35)
Where the superscript “𝑏” indicates the boundary layer, 𝑢
!
!
the normal velocity, and 𝑢
!
!
the
tangential velocity. When 𝑀
!
≫ 1, the vorticity will be confined to the Stokes layer that has
a thickness of the order 𝛿∼𝑎 𝑀 (Riley, 1966). Therefore the normal velocity inside the
Stokes layer can be scaled to
𝑢
!
!
=
𝑀
2
𝑢
!
!
, (2.36)
and the inner variable within the boundary layer
𝜂=
𝑀
2
𝑟−𝑅 . (2.37)
Here when 𝑟=𝑅, 𝜂= 0. Next, we expand the velocity, acoustic pressure, and acoustic
density in the boundary layer using 𝜀 to find the perturbation solution, i.e.,
𝒖
!
=𝒖
!
!
+𝜀𝒖
!
!
+𝑂 𝜀
!
, (2.38)
𝑝
!
=𝑝
!
!
+𝜀𝑝
!
!
+𝑂 𝜀
!
, (2.39)
21
and
𝜌
!
=𝜌
!
!
+𝜀𝜌
!
!
+𝑂 𝜀
!
. (2.40)
Inserting equations (2.38), (2.39), and (2.40) into the momentum equation (2.10) after scaling
the radial parameters using equations (2.36) and (2.37) while keeping the expansion to the
order of |M|
-1
and (ka) results in
𝜕𝑝
!
!
𝜕𝜂
= 0 (2.41)
and
𝜕𝑢
!!
!
𝜕𝜏
=
1
𝑅
−
2
𝑅
!
𝑀
𝜂 1−𝜇
!
𝜕𝑝
!
!
𝜕𝜇
+
1
2
𝜕
!
𝑢
!!
!
𝜕𝜂
!
+
2
𝑅𝑀
𝜕𝑢
!!
!
𝜕𝜂
. (2.42)
From equation (2.41), we can conclude that the leading order pressure in the Stokes layer is a
function of 𝜏 and 𝜃 only. This is a result that is consistent with the boundary layer theory that
the boundary layer is thin in character for large frequency. Then from equation (2.34) where
we substitute 𝑟=𝑅 we have,
𝑝
!
!
=𝜌
!
!
=𝑝
!
|
!!!
=−𝑖
1
𝑘𝑎
−
𝑘𝑎
2
𝑅
!
−
5
9
𝑘𝑎𝑅
!
𝑃
!
𝜇 𝑒
!"
+ 𝑖𝜀
!
𝑅𝑒
! !!!
. (2.43)
The shear-free surface boundary condition that is given by equation (2.15) after changing the
radial parameters and expanding using equation (2.38), to the leading order is
𝜕𝑢
!!
!
𝜕𝜂
|
!!!
=
2
𝑀 𝑅
𝑢
!!
!
|
!!!
. (2.44)
In equation (2.44) the term that was multiplied by
!
!
will not be omitted because
!!
!!
!
!"
|
!!!
is
really small and comparable to
!
! !
𝑢
!!
!
|
!!!
. This is the reason why the expansion was kept to
22
the order of
!
!
. We solve equation (2.42) after substituting for the pressure term from
equation (2.43) and using the boundary condition from equation (2.44), The solution for the
leading order tangential velocity was found to be
𝑢
!!
!
=
5
3
𝑘𝑎𝑅𝜇 1−𝜇
!
1−
2
𝑅𝑀
𝜂 1−
2 2
2 2+𝑅|𝑀|(1+ 𝑖)
𝑒
! !!! !
𝑒
!"
. (2.45)
Again, inserting equations (2.38), (2.39), and (2.40) into the continuity equation (2.9) after
scaling the radial parameters using equations (2.36) and (2.37) to the order of |M|
-1
, results in
an equation for the leading order normal velocity,
𝑘𝑎
!
𝜕𝜌
!
!
𝜕𝜏
+
𝜕𝑢
!!
!
𝜕𝜂
+
2 2
𝑅𝑀
𝑢
!!
!
− 1−𝜇
!
1
𝑅
−
2
𝑅
!
𝑀
𝜂
𝜕𝑢
!!
!
𝜕𝜇
+
𝜇
1−𝜇
!
1
𝑅
−
2
𝑅
!
𝑀
𝜂 𝑢
!!
!
= 0 . (2.46)
Solving the above equation using the scaled boundary condition for the normal velocity,
𝑢
!!
!
|
!!!
=
!
!
𝑢
!!
!
|
!!!
=
!
!
𝜀
!
𝑒
!(!!!)
, the solution for the leading order normal velocity was
found to be
23
𝑢
!!
!
=−
10
3
𝑘𝑎𝑃
!
𝜇
𝑅𝑀
2
1−𝑒
!
! !
!!
!
−𝜂+
2𝑖𝑀 𝑅
𝑀
!
𝑅
!
+4𝑖
𝑒
!
! !
!!
!
−𝑒
! !!! !
+
4𝑖
𝑀
!
𝑅
!
+4𝑖
𝜂𝑒
! !!! !
−
2+2𝑖 𝑀 𝑅
𝑀
!
𝑅
!
+4𝑖 𝑀 𝑅+ 1+ 𝑖 2
𝑒
!
! !
!!
!
−𝑒
! !!! !
𝑒
!"
−𝑘𝑎
𝑅𝑀
2 2
1−𝑒
!
! !
!!
!
𝑒
!"
+
𝑀
2
𝜀
!
𝑒
! !!!
𝑒
!
! !
!!
!
. (2.47)
2.3.2 The First Order Solution, 𝑂 𝜀 :
For the first-order solution, the interest is in the acoustic streaming which is
independent of time. Therefore we consider only the steady state solution. The continuity
equation inside the boundary layer to the order of 𝜀 is
𝑘𝑎
!
𝜕𝜌
!
!
𝜕𝜏
+𝛁 ∙𝒖
!
!
+ 𝑘𝑎
!
𝛁 ∙ 𝜌
!
!
𝒖
!
!
= 0 . (2.48)
For steady streaming, we take the time average of the above equation, to obtain
𝛁 ∙𝒖
!
!
+ 𝑘𝑎
!
𝛁 ∙ 𝜌
!
!
𝒖
!
!
= 0 , (2.49)
where the time averaged quantity 𝑘𝑎
!
𝛁 ∙ 𝜌
!
!
𝒖
!
!
= 0. Therefore,
𝛁 ∙𝒖
!
!
= 0 . (2.50)
24
Equation (2.50) indicates that the first order time average flow in the boundary layer is
incompressible. The momentum equation within the boundary layer in the radial direction, 𝜂
and in the order of 𝜀 and 𝑂 𝑘𝑎 after taking the time average is
𝜕𝑝
!
!
𝜕𝜂
= 0 . (2.51)
The momentum equation in the tangential direction, 𝜃, is
1
2
𝜕
!
𝑢
!!
!
𝜕𝜂
!
+
2
𝑅𝑀
𝜕𝑢
!!
!
𝜕𝜂
=−
1
𝑅
−
2
𝑅
!
𝑀
𝜂 1−𝜇
!
𝜕𝑝
!
!
𝜕𝜇
+ 𝑘𝑎
!
𝜌
!
!
𝜕𝑢
!!
!
𝜕𝜏
+𝑢
!!
!
𝜕𝑢
!!
!
𝜕𝜂
−
1
𝑅
−
2
𝑅
!
𝑀
𝜂 1−𝜇
!
𝑢
!!
!
𝜕𝑢
!!
!
𝜕𝜇
+
2
𝑅𝑀
𝑢
!!
!
𝑢
!!
!
. (2.52)
knowing 𝑢
!!
!
and 𝑢
!!
!
from equations (2.45) and (2.47), we keep equation (2.52) to the order
of 𝑂 𝑘𝑎 . Then we take the derivative of (2.52) twice with respect to 𝜂, while keeping in
mind (2.51), so we can eliminate the pressure term 𝑝
!
!
. The result is a fourth order differential
equation for 𝑢
!!
!
. We take the time average of this fourth order differential equation to 𝑂 𝜀
!
to obtain,
1
2
𝜕
!
𝑢
!!
!
𝜕𝜂
!
+
2
𝑀
𝜕
!
𝑢
!!
!
𝜕𝜂
!
=
𝜕
!
𝜕𝜂
!
𝑢
!!
!
𝜕𝑢
!!
!
𝜕𝜂
+
𝜕
!
𝜕𝜂
!
2
𝑀
𝑢
!!
!
𝑢
!!
!
. (2.53)
The general solution of the above equation after taking the limit of 𝑢
!!
!
= 𝑜 𝜂 𝑎𝑠 𝜂→∞ is,
25
𝑢
!!
!
=𝐶
!
+
1
4
2𝑀 𝜂𝐶
!
−
1
8
𝑀
!
𝜂𝐶
!
−
1
4
2𝑀 𝑒
!
! !
!
!
𝐶
!
+𝑔
𝑀
!
32
𝑒
!
! !
!
!
cos𝜙
+𝑔 𝜇
2 𝑀
16
𝜂𝑒
!
! !
!
!
cos𝜙+𝑔 𝜇
1
8
𝜂
!
𝑒
!
! !
!
!
cos𝜙
+𝑔 𝜇
1
2
𝑒
! !!
! !
!
!
cos𝜂
2
𝑀
𝜂+
3 2
𝑀
cos𝜙+
2
𝑀
−1 sin𝜙
−𝑔 𝜇
1
2
𝑒
! !!
! !
!
!
sin𝜂 1−
2
𝑀
cos𝜙+
2
𝑀
𝜂+
3 2
𝑀
sin𝜙 ,(2.54)
where:
𝑔 𝜇 =
10
3
𝜀
!
𝑘𝑎𝜇 1−𝜇
!
,
and 𝐶
!
,𝐶
!
,𝐶
!
,and 𝐶
!
are constants of integration to be determined from the boundary
conditions and matching with the far field. The terms in the order of 𝑂(𝑘𝑎)
!
are neglected
(see e.g., Rednikov,
Zhao,
Sadhal,
and
Trinh, 2006). Before we proceed, we express the
velocity as a stream function knowing that the time average first order streaming flow is
incompressible from equation (2.50), i.e.,
𝑢
!!
!
=−
𝑀
2
1
𝑟 1−𝜇
!
𝜕𝜓
!
!
𝜕𝜂
(2.55)
and,
𝑢
!!
!
=−
𝑀
2
1
𝑟
!
𝜕𝜓
!
!
𝜕𝜇
. (2.56)
The far field and the boundary condition for the stream function is
26
𝜓
!
!
= 𝜊 𝜂
!
(2.57)
this was chosen in order to make the far field streaming velocity vanish as 𝜂→∞. From the
free-surface condition at the interface of the bubble and the fluid around it,
𝜕
!
𝜓
!
!
𝜕𝜂
!
|
!!!
=
2
𝑀
𝜕𝜓
!
!
𝜕𝜂
|
!!!
(2.58)
and the zero normal velocity,
𝜕𝜓
!
!
𝜕𝜇
|
!!!
= 0 . (2.59)
After expressing the velocity components in terms of the stream function and invoking the
boundary conditions above, we obtain
27
𝜓
!
!
=
2
𝑀
𝑘𝑎𝜀
!
𝜇 1
−𝜇
!
5
6
−5
2
𝑀
−
5
32
𝑀
!
2
+
5
3
𝜂+
85
3
2
𝑀
𝜂+
5
3
𝑀
2
𝜂+
5
48
𝑀
!
𝜂 cos𝜙
+
5
32
𝑀
!
2
+
5
24
𝑀
!
𝜂+
5
24
𝑀
2
𝜂
!
𝑒
!
! !
!
!
cos𝜙
+
5
6
+
5
2
2
𝑀
+
5
3
𝜂+
35
3
2
𝑀
𝜂−
5
3
𝑀
2
𝜂 sin𝜙
−
5
6
−
5
6
2
𝑀
𝜂−5
2
𝑀
cos𝜙
+
5
6
+
5
6
2
𝑀
𝜂+
5
2
2
𝑀
sin𝜙 𝑒
! !!
! !
!
!
cos𝜂
−
5
6
+
5
6
2
𝑀
𝜂+
5
2
2
𝑀
cos𝜙
−
5
6
−
5
6
2
𝑀
𝜂−5
2
𝑀
sin𝜙 𝑒
! !!
! !
!
!
sin𝜂
+
2
𝑀
1−𝜇
!
𝐶
!
𝑀
!
8
1−𝑒
!
! !
!
!
−
3
2
𝑀
!
2
𝜂 −𝐶
!
𝑀
!
4
𝜂+
1
4
𝑀
2
𝜂
!
+𝐶
!
𝑀
!
16
𝜂
!
+
𝑀
!
8 2
𝜂 . (2.60)
To find the constant 𝐶
!
and 𝐶
!
in equation (2.60) we match with the streaming from the flow
outside the boundary layer. The streaming flow outside the boundary layer is considered to be
incompressible. Therefore, the stream function, 𝜓
!
, can be introduced (Zhao et al., 1999),
𝒟
!
𝜓
!
= 0 . (2.61)
The above correspond to Stokes flow. Here 𝒟
!
=
!
!
!!
!
+
!!!
!
!
!
!
!
!!
!
.
Taking into account the matching requirement of the behavior of the standing wave and
knowing the condition 𝜓
!
= 𝑜 𝑟
!
as 𝑟→∞, the general solution is found to be
28
𝜓
!
= 𝑏
!
+
𝐷
!
𝑟
!
−
3𝑏
!
𝑟
!
−
3𝐷
!
𝑟
!
𝜇 1−𝜇
!
+
7𝑏
!
𝑟
!
+
7𝐷
!
𝑟
!
𝜇
!
1−𝜇
!
. (2.62)
By introducing the 𝜂 variable in equation (2.62) and letting 𝜂→ 0, then matching it with
equation (2.60) while letting 𝜂→∞ gives the unknown constants. The solutions for 𝜓
!
and
𝜓
!
!
turn out to be,
𝜓
!
= 𝑘𝑎𝜀
!
𝜇 1−𝜇
!
1−
1
𝑟
!
5
6
2
|𝑀|
+
85
3|𝑀|
!
+
5
6
+
5
48
|𝑀|
2
cos𝜙
+
5
6
2
|𝑀|
+
35
3|𝑀|
!
−
5
6
sin𝜙 −
5
16
|𝑀|
2
cos𝜙 (2.63)
and,
𝜓
!
!
= 𝑘𝑎𝜀
!
𝜇 1−𝜇
!
×
5
6
2
𝑀
−
10
𝑀
!
−
5
32
𝑀
!
+
5
3
𝜂
2
𝑀
+
170
3𝑀
!
𝜂+
5
3
𝜂+
5
24
|𝑀|
2
𝜂 cos𝜙
+
5
32
𝑀
!
+
5
12
|𝑀|
2
𝜂+
5
24
𝜂
!
𝑒
!
! !
|!|
!
cos𝜙
+
5
6
2
|𝑀|
+
5
3
2
|𝑀|
𝜂+
70
3𝑀
!
𝜂−
5
3
𝜂 sin𝜙
−
5
6
2
𝑀
−
10
𝑀
!
cos𝜙+
5
6
2
𝑀
sin𝜙 𝑒
! !!
! !
!
!
cos𝜂
−
5
6
2
𝑀
cos𝜙−
5
6
2
𝑀
−
10
𝑀
!
sin𝜙 𝑒
! !!
! !
!
!
sin𝜂 . (2.64)
As is usually seen with streaming around particles, the inner streaming extends to the outer
region to 𝑂 𝜀 . Furthermore, the composite solution can be found by combining the boundary
layer and the outer streaming solutions and subtracting the common term. We found the
composite solution to be,
29
𝜓
!
!
= 𝑘𝑎𝜀𝜇 1−𝜇
!
−
5
16
𝑀
2
+
5
16
𝑀
2
𝑒
!
! !
!
!
+
5
12
𝜂𝑒
!
! !
!
!
cos𝜙
+ 1−
1
𝑟
!
5
6
2
𝑀
+
85
3𝑀
!
+
5
6
+
5
48
𝑀
2
cos𝜙
+
5
6
2
𝑀
+
35
3𝑀
!
−
5
6
sin𝜙 . (2.65)
2.3.3 Discussion
From equations (2.63), (2.64), and (2.65), steady streaming is found to exist for both
the outer and the inner flow fields. As can be seen from Figure (2.2), which is a graph of the
acoustic streaming for phase shift 𝜙= 20° and frequency parameter 𝑀 = 10, the flow has a
symmetrical pattern about the equatorial plane of the bubble. The difference is in the direction
of the flow between the upper and the lower plane. The phase shift, 𝜙, and the frequency
parameter, |M|, both play a major role in the behavior of the flow field. Changing the phase
shift or the frequency parameter would not only alter the intensity of the flow but also, at
specific values, the shape of the flow streamlines. In the first quadrant of the 𝑟𝑧-plane and for
a value of 𝑀 = 10, which is considered to be an intermediate value, two circulating regions
develop at 𝜙= 0° (Figure 2.4). Near the bubble surface the circulating region has a clockwise
flow and far from the bubble surface the flow has an opposite flow pattern. As we increase the
phase shift, the vortices close to the surface start to diminish and intensity of the outer vortices
increases and get closer to the bubble boundary. The farther vortices start to open up initially
from the outer layers. At 𝜙= 60° the flow has the highest intensity and the vortices next to
the bubble surface totally disappear. The streaming flow consists of an open flow streamlines
that runs from the poles to the equator. After this peak value, at 𝜙= 60°, the flow start to
decrease in intensity, however, the stream lines get even closer to the bubble surface. This
behavior will continue all the way to phase shift 𝜙= 90° which has the lowest streaming flow
intensity. To look at the behavior of the streaming flow field for different frequency
parameter, we increased the value of the frequency parameter to 𝑀 = 30 and start to
examine the flow field for a range of values from 𝜙= 0° to 𝜙= 90° (Figure 2.3 and Figure
2.5). The intensity of the flow becomes higher as we increase the frequency parameter and the
30
flow direction is from the poles to the equator. Also, the vortices next to the bubble surface
disappear. As 𝜙 further increased the intensity reached a peak value and did not change from
𝜙= 5° to 𝜙= 20°. The intensity start to decline after reaching the peak value, as 𝜙 was
additionally increased until it reached its lowest value at 𝜙= 90°. The smallest intensity was
consistent for both cases, and in fact in all of the cases of different values of |M|. This
consistency is the result of the contribution of the sin𝜙 term in equation (2.65) to the
streaming.
An increase in |M| value will increase the intensity of the flow. As the intensity
increases, the inner vortices (or circulation regions close the bubble surface) vanish. This
shows that for an intermediate value of |M|, which results in a weak flow, the vortices would
diffuse over large distances. This phenomenon can be captured only with the composite
solution. Therefore, the composite solution gives a much better interpretation of the flow. It
should be noted that the large 𝑀 approximation in the current analysis significantly restricts
the validity to small velocity amplitude so that the perturbed field to 𝑂 𝜀𝜀
!
remain indeed
small but at the same time useful in many small-scale applications.
31
Figure 2.2: The Composite solution streamlines of the steady streaming at phase shift
20° and |M| = 10
Figure 2.3: The composite solution streamlines of the steady streaming at phase shift
20° and |M| = 30
32
Figure 2.4: The streamlines for the composite solution with 𝑀 = 10 and varying the phase shift; a) 𝜙= 0°, b) 𝜙= 30°, c)
𝜙= 45°, d) 𝜙= 60°, e) 𝜙= 75°, and f) 𝜙= 90°. The color bar scaling was set to ±1𝐸−3.
(a)
(b)
(c)
(d)
(e)
(f)
33
Figure 2.5: The streamlines for the composite solution with 𝑀 = 30 and varying the phase shift; a) 𝜙= 0°, b) 𝜙= 30°, c)
𝜙= 45°, d) 𝜙= 60°, e) 𝜙= 75°, and f) 𝜙= 90°. The color bar scaling was set to ±1𝐸−3.
(a)
(b)
(c)
(d)
(e)
(f)
34
2.4. Solution for the pulsating bubble: no slip boundary
The
difference
between
the
solution
of
the
pulsating
bubble
with
rigid
boundary
and
the
pulsating
bubble
with
a
shear
free
boundary
is
the
boundary
conditions
only.
The
no-‐slip
situation
arises
from
contaminants
on
the
bubble
surface
(Leighton,
1995;
Clift,
Grace,
and
Weber,
2005).
Rather
than
solving
using
the
boundary
condition
in
equation
(2.15),
the
shear-‐free
boundary
condition,
we
used
the
boundary
condition
in
equation
(2.16),
the
no
slip
boundary
condition.
The
rest
of
the
procedure
was
the
same.
Therefore
we
present
the
solution
without
going
though
the
details.
The
leading
order
solution
for
the
tangential
velocity
to
O(1),
was
found
to
be
𝑢
!!
!
=
5
3
𝑘𝑎𝑅𝜇 1−𝜇
!
1−
2
𝑅𝑀
𝜂 1−𝑒
! !!! !
𝑒
!"
(2.66)
and
for
the
radial
velocity,
𝑢
!!
!
=−
10
3
𝑘𝑎𝑃
!
𝜇
𝑅𝑀
2
𝑒
!
! !
!!
!
− 1−𝑒
! !!! !
𝜂+
𝑅𝑀
2
1−𝑒
! !!! !
𝑒
!"
−𝑘𝑎
𝑅𝑀
2 2
1−𝑒
!
! !
!!
!
𝑒
!"
+
𝑀
2
𝜀
!
𝑒
! !!!
𝑒
!
! !
!!
!
. (2.67)
The
general
solution
to
the
first
order,
𝑂 𝜀 , for the tangential velocity is
35
𝑢
!!
!
=𝐶
!
+
1
4
2𝑀 𝜂𝐶
!
−
1
8
𝑀
!
𝜂𝐶
!
−
1
4
2𝑀 𝑒
!
! !
!
!
𝐶
!
+𝑦 𝜇
𝑀
!
32
+
2 𝑀
16
𝜂+
1
8
𝜂
!
𝑒
!
! !
!
!
cos𝜙
+
1
2
𝑒
! !!
! !
!
!
cos𝜂
𝑀
2 2
−
5 2
2𝑀
−
𝜂
2
cos𝜙
+
1
2
−
2
𝑀
−
𝑀
2 2
−
2
𝑀
𝜂+
𝜂
2
sin𝜙
+
1
2
𝑒
! !!
! !
!
!
sin𝜂 2−
𝑀
2 2
−
3 2
𝑀
−
2
𝑀
𝜂+
𝜂
2
cos𝜙
+
2
2𝑀
𝜂+
3 2
2𝑀
−
𝑀
2 2
sin𝜙 2.68
where ,
𝑦 𝜇 =
10
3
𝜀
!
𝑘𝑎𝜇 1−𝜇
!
.
We
changed
the
first
order
tangential
velocity,
𝑢
!!
!
,
to
stream
function
using
the
knowledge
that
the
first
order
time
independent
flow
within
the
boundary
layer
is
incompressible.
Then
we
applied
the
boundary
conditions,
and
the
stream
function
within
the
boundary
layer
was
found
to
be
36
𝜓
!
!
= 𝑘𝑎𝜀
!
𝜇 1−𝜇
!
× −
35
12
2
𝑀
+
145
6𝑀
!
−
5
32
𝑀
!
+
5
3
2
𝑀
𝜂−
25
2𝑀
!
𝜂+
5
6
𝜂
−
65
24
𝑀
2
𝜂 cos𝜙+
5
32
𝑀
!
+
5
12
𝑀
2
𝜂+
5
24
𝜂
!
𝑒
!
! !
!
!
cos𝜙
+
5
6
−
35
12
2
𝑀
+
25
12
2
𝑀
𝜂−
5
6
𝜂 sin𝜙
−
145
6𝑀
!
−
35
12
2
𝑀
− cos𝜙
+
5
6
−
35
12
2
𝑀
−
5
6
2
𝑀
𝜂 sin𝜙 𝑒
! !!
! !
!
!
cos𝜂
−
5
6
−
5
4
2
𝑀
cos𝜙+
5
3
2
𝑀
−
25
2𝑀
!
sin𝜙 𝑒
! !!
! !
!
!
sin𝜂 .(2.69)
Using
the
same
assumption
that
the
streaming
flow
outside
the
boundary
layer
is
consider
to
be
a
Stokes
flow
and
matching
it
with
the
boundary
layer
solution,
the
outer
solution
was
found
to
be,
𝜓
!
= 𝑘𝑎𝜀
!
𝜇 1−𝜇
!
5
6
2
|𝑀|
sin𝜙−
5
16
𝑀
2
cos𝜙
+ 1−
1
𝑟
!
5
6
2
|𝑀|
+
5
12
−
65
48
|𝑀|
2
cos𝜙+
25
24
2
|𝑀|
−
5
12
sin𝜙 ,(2.70)
and
the
composite
solution
𝜓
!
!
= 𝑘𝑎𝜀𝜇 1−𝜇
!
5
16
𝑀
2
𝑒
!
! !
|!|
!
+
5
12
𝜂𝑒
!
! !
|!|
!
−
5
16
𝑀
2
−
5
6
2
|𝑀|
𝑒
! !!
! !
!
!
sin𝜂 cos𝜙+
5
6
2
|𝑀|
−
5
6
2
|𝑀|
𝑒
! !!
! !
!
!
cos𝜂 sin𝜙
+ 1−
1
𝑟
!
5
6
2
|𝑀|
+
5
12
−
65
48
|𝑀|
2
cos𝜙+
25
24
2
|𝑀|
−
5
12
sin𝜙 . (2.71)
37
2.4. 1
Discussion:
The
first
detail
that
can
be
noticed
with
the
no
slip
boundary
condition
is
the
stronger
intensity
of
the
streaming
that
was
induced.
This
is
apparent
from
Figure
(2.6)
and
Figure
(2.7)
for
frequency
parameter
𝑀 = 10 and
𝑀 = 30 with
a
phase
shift
𝜙= 20°.
The
streaming
flow
maximum
intensity
for
the
no-‐slip
boundary
condition
was
10
times
higher
than
the
streaming
in
the
case
of
the
shear-‐free
boundary
condition.
The
flow
pattern
was
different
though.
The
open
flow
persists
throughout
the
range
0°≤𝜙< 90°
of
the
phase
shift,
except
at
𝜙= 90°,
taking
into
account
that
|M|
was
held
constant
at
an
intermediate
value
of
10,
Figure
(2.8).
To
fully
understand
the
flow
behavior,
we
graphed
the
streaming for different value of the phase shift and held the
frequency parameter constant. When 𝑀 = 10, at 𝜙= 0°
the
flow
direction
is
from
the
poles
to
the
equator
and
has
the
maximum
intensity.
As
we
increase
the
value
of
the
phase
shift
the
flow
intensity
becomes
weaker
until
it
reaches
a
minimum
value
at
𝜙= 90°.
At
this
minimum
value,
𝜙= 90°,
the
streaming
has
two
regions
one
close
to
the
bubble
and
has
a
clockwise
rotation
and
away
from
the
bubble
where
the
flow
persist
with
from
the
poles
to
the
equator
behavior.
The solution is also sensitive to |M|; as |M|
increases the flow intensity increases. Figure (2.9) shows 𝑀 = 30 for different values of 𝜙.
The flow indeed has higher intensity than the lower frequency parameter flow. But, has the
same peaking point at 𝜙= 0° and minimum streaming at 𝜙= 90°.
38
Figure 2.6: The composite solution streamlines of the steady streaming at phase shift
20° and |M| = 10 for the no slip boundary condition
Figure 2.7: The composite solution streamlines of the steady streaming at phase shift
20° and |M| =30 for the no slip boundary condition
39
(a)
(b)
(c)
(d)
(e)
(f)
Figure 2.8: The streamlines for the composite solution with |𝑀|= 10 and varying the phase shift; a) 𝜙= 0°, b)
𝜙= 30°, c) 𝜙= 45°, d) 𝜙= 60°, e) 𝜙= 75°, and f) 𝜙= 90°. The color bar scaling was set to ±1𝐸−2.
40
Figure 2.9: The streamlines for the composite solution with |𝑀|= 30 and varying the phase shift; a) 𝜙= 0°, b)
𝜙= 30°, c) 𝜙= 45°, d) 𝜙= 60°, e) 𝜙= 75°, and f) 𝜙= 90°. The color bar scaling was set to ±1𝐸−2.
(a)
(b)
(c)
(d)
(e)
(f)
41
2.5
Remarks
For a pulsating bubble both the frequency parameter, |M|, and the phase shift, 𝜙,
play a major role in defining the shape and the intensity of the streaming flow. While the
intensity of the streaming is proportional to |M|, the effect of 𝜙 on the flow depends on
the boundary condition. For shear-free boundary condition, increasing 𝜙 would increase
the intensity of the streaming flow until a maximum value is reached. For |𝑀|= 10 the
maximum value is at 𝜙= 60° and when |𝑀|= 30 the maximum streaming intensity has
a range of phase shift values, 5°≤𝜙≤ 20°. After the flow intensity reaches its
maximum value, increasing 𝜙 would decrease the flow intensity. The minimum value for
both 𝑀 = 10 and 𝑀 = 30 is at 𝜙= 90°. For the no slip boundary condition, the
streaming has a maximum value at 𝜙= 0° for both 𝑀 values. As the value of the phase
shift increases from 𝜙= 0°, the streaming intensity decreases until the flow intensity is
minimum at 𝜙= 90°. For an intermediate value of |M| when the flow is weak (shear-free
boundary condition) the vortices close to the bubble surface diffuse over a wider region.
This can be seen only with the composite solution. Therefore, the composite solution
gives a better interpretation of the flow field. Although the vortices close to the bubble
disappear when |M| is large, they are not a function of |M| alone but also depend on 𝜙.
The circulating regions do exist when 𝜙= 45° but not for 𝜙= 90°, although the flow
has lower intensity when 𝜙= 90°. Of course this happens because in equation (2.65) all
the cos𝜙 terms will disappear at 𝜙= 90° and the contribution will be only from sin𝜙
term. The flow in general has the same flow direction regardless of the change in either
|M| or 𝜙, where the direction is from the poles to the equator. The no-slip boundary
produces stronger streaming than the shear-free boundary. However, the streaming in
general is small. It is on the 𝑂 𝜀𝜀
!
.
42
3. Streaming at the velocity antinode of a standing wave
The bubble locates at the velocity antinode when the driving frequency is smaller than the
resonance frequency (equation 1.6), which also means that the bubble radius is larger than the
radius at the resonance frequency 𝑎> 𝑅
!
. Moreover, for a bubble located at the velocity
antinode the ratio
!
!
!
> 1, which makes the phase lag values to be between 𝜙= 90° and
𝜙= 180° (Figure 1.2). The location of the bubble at the velocity antinode will require a
different treatment to it than the case at the velocity node location. From equation (1.13), the
streaming flow around a bubble is considered to be incompressible. Therefore, we can
introduce the stream function early in the analysis. We will solve for the pulsating bubble at
the velocity antinode using two boundary conditions. The first boundary condition is the
shear-free and the second is the no-slip.
The solution will be constructed in the same manner as section 2 but the nondimensional
momentum equation will be presented in terms of the Stokes stream function. Here, the
momentum equation will be perturbed using the small ratio of the lateral displacement
amplitude, 𝜀. The momentum equation will be solved for outside the boundary layer flow
first. Then, to take care of the singular character and all of the boundary conditions, we scale
the problem within the boundary layer and solve for the flow within it. The time averaged
flow to the first order within the boundary layer (the streaming flow) will be matched with the
first order solution for the outer flow.
3.1. The dimensionless parameters and the far field conditions
We adopt the same spherical polar coordinates with the origin at the center of the bubble
as the flow around the velocity node. Moreover, the dimensionless parameters in equation
(2.14) will be adopted. The far field conditions, however, will be different. The velocity
around the velocity antinode region is
𝑢
!
=𝑈
!
𝑒
!"#
. (3.1)
43
The flow is considered to be incompressible, therefore we can introduce the stream function.
For spherical polar coordinates the relations between 𝑢
!
,
𝑢
!
,
and
the
Stokes stream
function
𝜓
are (Kundu et al., 2004)
𝑢
!
=
1
𝑟
!
sin𝜃
𝜕𝜓
𝜕𝜃
(3.2)
𝑢
!
=
−1
𝑟sin𝜃
𝜕𝜓
𝜕𝑟
. (3.3)
Moreover,
knowing
the
relation
between
𝑢
!
,
𝑢
!
, and
𝑢
!
(namely
𝑢
!
=𝑢
!
cos𝜃
and
𝑢
!
=−𝑢
!
sin𝜃)
we
can
solve
for
the
far
field
stream
function
𝜓
!
=
𝑈
!
𝑟
!
2
sin
!
𝜃𝑒
!"#
. (3.4)
3.2. Equations of motion, dimensionless scaling, and the boundary conditions
The flow parameters are scaled using equation (2.21). Where after applying the scaled
parameters the momentum equation in terms of Stokes stream function is
𝜕
𝜕𝜏
𝐷
!
𝜓 +
𝜀
𝑟
!
𝜕 𝜓, 𝐷
!
𝜓
𝜕 𝑟, 𝜇
+2 𝐷
!
𝜓 𝐿𝜓 =
1
𝑀
!
𝐷
!
𝜓 (3.5)
where
𝐷
!
=
𝜕
!
𝜕𝑟
!
+
1− 𝜇
!
𝑟
!
𝜕
!
𝜕 𝜇
!
,
𝐿=
𝜇
1− 𝜇
!
𝜕
𝜕𝑟
+
1
𝑟
𝜕
𝜕 𝜇
,
and
44
𝜇= cos𝜃 .
As usual, the introduction of the Stokes stream function into the momentum equation
has the advantage of eliminating the pressure term. Two important dimensionless parameters
show up in equation (3.5); the small ratio of the lateral displacement amplitude to the bubble
radius, 𝜀 , and the frequency parameter, 𝑀
!
. The presence of 𝑀
!
in equation (3.5) is
analogous to the Reynolds number in the creeping flow around a sphere (Johnson, 2004).
However, with the creeping flow the Reynolds number is small and in our case the frequency
parameter is large. Therefore, the viscous term in equation (3.5) will be small and
insignificant if we compare it to time dependent term and the nonlinear terms.
The boundary conditions for the problems to be solved are,
𝜕𝜓
𝜕 𝜇
=−𝜀
!
𝑒
! !!!
+𝑂 𝜀
! !
at 𝑟=𝑅 𝜏 , (3.6)
which corresponds to the bubble radial oscillations at its surface [equation (2.23)]. The 𝑂 𝜀
!"
terms are deemed to be very small and negligible. The bubble radius 𝑅 𝜏 is the same as in
equation (2.25). The second boundary condition assumed for clean bubble, which means that
at the interface the tangential shear stress will be zero. From equation (2.28) the shear-free
boundary condition is
𝜕
!
𝜓
𝜕𝑟
!
=
2
𝑅
𝜕𝜓
𝜕𝑟
at 𝑟=𝑅 𝜏 . (3.7)
For a contaminated bubble (no-slip case), instead of equation (3.7), the boundary condition is
𝜕𝜓
𝜕𝑟
= 0 at 𝑟=𝑅 𝜏 . (3.8)
At the far field, i.e. 𝑟→∞, the nondimensional Stokes stream function was found to be,
𝜓
!
=
𝑟
!
2
sin
!
𝜃 𝑒
!"
=
𝑟
!
2
1− 𝜇
!
𝑒
!"
(3.9)
45
3.3. Solution for the pulsating bubble at the velocity antinode: shear free boundary
We start the solution by perturbing the Stokes stream function relying on the assumption
that 𝜀≪ 1,
𝜓=𝜓
!
+𝜀𝜓
!
+𝑂 𝜀
!
. (3.10)
After substituting equation (3.10) into momentum equation (3.5) we obtain,
𝜕
𝜕𝜏
𝐷
!
𝜓
!
+𝜀𝜓
!
+⋯
+
𝜀
𝑟
!
𝜕 𝜓
!
+𝜀𝜓
!
+⋯ , 𝐷
!
𝜓
!
+𝜀𝜓
!
+⋯
𝜕 𝑟, 𝜇
+2 𝐷
!
𝜓
!
+𝜀𝜓
!
+⋯ 𝐿 𝜓
!
+𝜀𝜓
!
+⋯
=
1
𝑀
!
𝐷
!
𝜓
!
+𝜀𝜓
!
+⋯ . (3.11)
3.3.1 The Leading Order Solution:
From equation (3.11) the leading order equation, 𝑂 𝜀
!
, is
𝜕
𝜕𝜏
𝐷
!
𝜓
!
=
1
𝑀
!
𝐷
!
𝜓
!
. (3.12)
With the assumption that the frequency parameter 𝑀
!
≫ 1, the
!
!
!
𝐷
!
𝜓
!
term may be
neglected, and equation (3.12) becomes
𝜕
𝜕𝜏
𝐷
!
𝜓
!
= 0 . (3.13)
This has the general solution,
46
𝜓
!
= 𝐴𝑟
!
+
𝐵
𝑟
1− 𝜇
!
𝑒
!"
. (3.14)
Requiring the far-field 𝑟→∞ behavior as giving by equation (3.9), we can conclude that
𝐴=
!
!
. Then equation (3.14) reduces to,
𝜓
!
=
𝑟
!
2
+
𝐵
𝑟
1− 𝜇
!
𝑒
!"
. (3.15)
The constant 𝐵 in equation (3.15) can be found by matching with the solution within the
boundary layer.
3.3.1(a). The Boundary Layer:
To overcome the singularity inherited in equation (3.12) for the case when 𝑀
!
≫ 1
and satisfy the boundary conditions, we must scale the Stokes stream function. For 𝑀
!
≫ 1,
the vorticity is confined to the boundary layer which has an order of 𝛿∼𝑎 𝑀 (Riley,
1966). This scaling arises from the fact that 𝛿
∗
=
!
!
𝜈 𝜔 and 𝑀 =
!
!
𝜔 𝜈. Then the
stream function within the boundary layer is scaled as follows
𝜓
!
=
!
!
𝜓 . (3.16)
where the superscript “𝑏” indicates the boundary layer and 2 in the denominator leads to
𝛿=
!
!
, where the 𝛿 is the nondimensional boundary layer thickness. The radial distance
within the boundary layer according to the same scaling will be,
𝜂=
𝑀
2
𝑟−𝑅 . (3.17)
47
Once again, this was chosen so that when 𝑟=𝑅, 𝜂= 0. Using the above scaling equation
(3.16) and equation (3.17) to transform equation (3.11) in terms o the inner variables, we have
𝜕
𝜕𝜏
𝐷
!
𝜓
!
=
𝜕
𝜕𝜏
𝑀
2
𝜕
!
𝜓
!
𝜕𝜂
!
+
2
𝑀
1− 𝜇
!
𝑅
!
𝜕
!
𝜓
!
𝜕 𝜇
!
+𝑂
1
𝑀
!
(3.18)
𝜕 𝜓
!
, 𝐷
!
𝜓
!
𝜕 𝜂, 𝜇
=
𝑀
2
𝜕𝜓
!
𝜕𝜂
𝜕
!
𝜓
!
𝜕𝜇𝜕𝜂
!
−
𝑀
2
𝜕𝜓
!
𝜕𝜇
𝜕
!
𝜓
!
𝜕𝜂
!
+
2
𝑀
1− 𝜇
!
𝑅
!
𝜕𝜓
!
𝜕𝜂
𝜕
!
𝜓
!
𝜕 𝜇
!
−
2
𝑀
1− 𝜇
!
𝑅
!
𝜕𝜓
!
𝜕𝜇
𝜕
!
𝜓
!
𝜕 𝜇
!
𝜕𝜂
+𝑂
1
𝑀
!
(3.19)
with
𝐿𝜓
!
=
𝜇
1− 𝜇
!
𝜕𝜓
!
𝜕𝜂
+
2
𝑀 𝑅
𝜕𝜓
!
𝜕𝜇
+𝑂
1
𝑀
!
, (3.20)
𝐷
!
𝜓
!
=
𝑀
2
𝜕
!
𝜓
!
𝜕𝜂
!
+
2
𝑀
1− 𝜇
!
𝑅
!
𝜕
!
𝜓
!
𝜕 𝜇
!
+𝑂
1
𝑀
!
, (3.21)
and
1
𝑀
!
𝐷
!
𝜓
!
=
𝑀
2 2
𝜕
!
𝜓
!
𝜕𝜂
!
+
2
𝑀 𝑅
!
𝜕
!
𝜓
!
𝜕𝜂
!
𝜕 𝜇
!
+𝑂
1
𝑀
!
. (3.22)
The boundary layer Stokes stream function is perturbed with 𝜀,
𝜓
!
=𝜓
!
!
+𝜀𝜓
!
!
+𝑂 𝜀
!
. (3.23)
After inserting the expansion, equation (3.23), into the scaled momentum equation, we have to
the order of 𝑂(𝜀
!
),
48
𝜕
𝜕𝜏
𝑀
2
𝜕
!
𝜓
!
!
𝜕𝜂
!
+
2
𝑀
1− 𝜇
!
𝑅
!
𝜕
!
𝜓
!
!
𝜕 𝜇
!
=
𝑀
2 2
𝜕
!
𝜓
!
!
𝜕𝜂
!
+
2
𝑀 𝑅
!
𝜕
!
𝜓
!
!
𝜕𝜂
!
𝜕 𝜇
!
+𝑂
1
𝑀
!
. (3.24)
Multiplying both sides of the above equation with
!
!
, we obtain
𝜕
𝜕𝜏
𝜕
!
𝜓
!
!
𝜕𝜂
!
=
1
2
𝜕
!
𝜓
!
!
𝜕𝜂
!
+𝑂
1
𝑀
!
(3.25)
Equation (3.25) has the general solution,
𝜓
!
!
= 𝐶
!
+𝐶
!
𝜂+𝐶
!
𝑒
! !!! !
+𝐶
!
𝑒
!!! !
𝑒
!"
. (3.26)
The solution to equation (3.26) has to be bounded, so we set 𝐶
!
= 0. Before using the
boundary conditions we have to scale them using the same scaled variables, namely equation
(3.16) and equation (3.17). In terms of these scaled variables, the radial pulsation condition is
𝜕𝜓
!
!
𝜕 𝜇
=−
𝑀
2
𝜀
!
𝑒
! !!!
at 𝜂= 0 . (3.27)
For the shear-free boundary condition,
𝜕
!
𝜓
!
!
𝜕𝜂
!
=
2 2
𝑀 𝑅
𝜕𝜓
!
!
𝜕𝜂
at 𝜂= 0 . (3.28)
Solving equation (3.26) using equations (3.27) and (3.28), we found
𝜓
!
!
=𝐶
!
1+ 𝑖+
𝑖𝑀 𝑅
2
𝜂+𝑒
! !!! !
−1 𝑒
!"
−
𝑀
2
𝜇𝜀
!
𝑒
! !!!
. (3.29)
To find the constant 𝐶
!
we do the usual outer and inner matching of equation (3.29) with
equation (3.15). This entails changing the 𝑟 variable to 𝜂 and taking 𝜂→ 0 (inner) and 𝜂→∞
49
(outer). The solution for the Stokes stream function to the order 𝑂(𝜀
!
) within the boundary
layer then was found to be,
𝜓
!
!
=
3
2
𝑅𝜂 1− 𝜇
!
𝑒
!"
−
3
2
𝑅
1+ 𝑖+
𝑖𝑀 𝑅
2
1−𝑒
! !!! !
1− 𝜇
!
𝑒
!"
−
𝑀
2
𝜇𝜀
!
𝑒
! !!!
, (3.30)
and the outer solution is
𝜓
!
=
𝑟
!
2
+
𝑅
!
2𝑟
1− 𝜇
!
𝑒
!"
. (3.31)
3.3.2. The First Order Solution, 𝑂 𝜀 :
To the order of 𝑂 𝜀 within the boundary layer the momentum equation is,
𝜕
𝜕𝜏
𝜕
!
𝜓
!
!
𝜕𝜂
!
+
1
𝑅
!
𝜕𝜓
!
!
𝜕𝜂
𝜕
!
𝜓
!
!
𝜕𝜇𝜕𝜂
!
−
2 2
𝑀 𝑅
!
𝜂
𝜕𝜓
!
!
𝜕𝜂
𝜕
!
𝜓
!
!
𝜕𝜇𝜕𝜂
!
−
1
𝑅
!
𝜕𝜓
!
!
𝜕𝜇
𝜕
!
𝜓
!
!
𝜕𝜂
!
+
2 2
𝑀 𝑅
!
𝜂
𝜕𝜓
!
!
𝜕𝜇
𝜕
!
𝜓
!
!
𝜕𝜂
!
+
2
𝑅
!
𝜇
1− 𝜇
!
𝜕𝜓
!
!
𝜕𝜂
𝜕
!
𝜓
!
!
𝜕𝜂
!
−
4 2
𝑀 𝑅
!
𝜇
1− 𝜇
!
𝜂
𝜕𝜓
!
!
𝜕𝜂
𝜕
!
𝜓
!
!
𝜕𝜂
!
=
1
2
𝜕
!
𝜓
!
!
𝜕𝜂
!
+𝑂
1
𝑀
!
. (3.32)
All the nonlinear terms are known because we have a solution for 𝜓
!
!
. These nonlinear terms
are responsible for the streaming. The streaming flow is a phenomena that is independent of
time, and therefore only the time average of equation (3.32) will be needed. Therefore, the
time average of equation (3.32) is,
50
1
2
𝜕
!
𝜓
!
!
𝜕𝜂
!
=
1
𝑅
!
𝜕𝜓
!
!
𝜕𝜂
𝜕
!
𝜓
!
!
𝜕𝜇𝜕𝜂
!
−
2 2
𝑀 𝑅
!
𝜂
𝜕𝜓
!
!
𝜕𝜂
𝜕
!
𝜓
!
!
𝜕𝜇𝜕𝜂
!
−
1
𝑅
!
𝜕𝜓
!
!
𝜕𝜇
𝜕
!
𝜓
!
!
𝜕𝜂
!
+
2 2
𝑀 𝑅
!
𝜂
𝜕𝜓
!
!
𝜕𝜇
𝜕
!
𝜓
!
!
𝜕𝜂
!
+
2
𝑅
!
𝜇
1− 𝜇
!
𝜕𝜓
!
!
𝜕𝜂
𝜕
!
𝜓
!
!
𝜕𝜂
!
−
4 2
𝑀 𝑅
!
𝜇
1− 𝜇
!
𝜂
𝜕𝜓
!
!
𝜕𝜂
𝜕
!
𝜓
!
!
𝜕𝜂
!
. (3.33)
Equation (3.33) has the following general solution,
𝜓
!
!
=𝐶
!
+𝐶
!
𝜂+𝐶
!
𝜂
!
+𝐶
!
𝜂
!
+
9
4
𝜇 1− 𝜇
!
2
𝑀
𝑒
!!
cos𝜂+
2
𝑀
𝜂𝑒
!!
sin𝜂
−
3
4
𝜀
!
1− 𝜇
!
𝑒
!!
2 2
𝑀
+
2 2
𝑀
3+𝜂 −1 cos 𝜂+𝜙
+
2 2
𝑀
−
2 2
𝑀
1−𝜂 −1 sin 𝜂+𝜙 . (3.34)
To find the particular solution for 𝜓
!
!
, we use the following boundary conditions on equation
(3.34),
𝜓
!
!
= 𝑜 𝜂
!
as 𝜂→∞, (3.35)
𝜕𝜓
!
!
𝜕 𝜇
= 0 at 𝜂= 0, (3.36)
and,
𝜕
!
𝜓
!
!
𝜕𝜂
!
=
2
𝑀
𝜕𝜓
!
!
𝜕𝜂
at 𝜂= 0 . (3.37)
Then, the particular solution for equation (3.34), after using equations (3.35), (3.36), and
(3.37) was found to be
51
𝜓
!
!
=−
9
4
2
𝑀
𝜇 1− 𝜇
!
1−𝑒
!!
cos𝜂+𝜂sin𝜂 − 1+
2𝑀
2
𝜂
−
3
4
𝜀
!
1− 𝜇
!
1−
8 2
𝑀
cos𝜙+ sin𝜙
+
2
𝑀
+
2𝑀
2
cos𝜙+
4 2
𝑀
+7−
2𝑀
2
sin𝜙 𝜂
− 1−
2 2
𝑀
4+𝜂 cos 𝜂+𝜙 + 1−
2 2
𝑀
𝜂 sin 𝜂+𝜙 𝑒
!!
.(3.38)
Equation (3.38) represents the streaming flow within the boundary layer. To find the
streaming flow outside the boundary layer, we assume a similar approach as in equation
(2.74). The streaming flow outside the boundary layer is considered to be a Stokes flow i.e.
𝒟
!
𝜓
!
= 0. Solving for Stokes flow, taking into consideration the matching requirement from
equation (3.38) and knowing the fact that the streaming flow 𝜓
!
= 𝑜 𝑟
!
as 𝑟→∞, the
general solution for the streaming flow outside the boundary layer is
𝜓
!
= 𝑏
!
𝑟+
𝐷
!
𝑟
1−𝜇
!
+ 𝑏
!
+
𝐷
!
𝑟
!
𝜇 1−𝜇
!
. (3.39)
By introducing the inner variable 𝜂 in equation (3.39) and letting 𝜂→ 0, then matching it with
equation (3.38) while letting 𝜂→∞ gives the unknown constants. The solutions for 𝜓
!
was
found to be,
𝜓
!
=
9
8
2
𝑀
1+
2𝑀
2
1−
1
𝑟
!
𝜇 1− 𝜇
!
+
3
8
𝜀
!
2
𝑀
+
2𝑀
2
cos𝜙+
4 2
𝑀
+7−
2𝑀
2
sin𝜙
1
𝑟
−𝑟
× 1− 𝜇
!
. (3.40)
52
3.3.3. Discussion:
Although the phase shift, 𝜙, and the frequency parameter, 𝑀 , play a role in defining
the shape and the intensity of the streaming flow, the frequency parameter, 𝑀 , has a more
interesting contribution. To examine this contribution we fix the value of the phase shift to
𝜙= 135° and vary values of the frequency parameter 𝑀 (figure 3.1). At a value of
𝑀 = 10 the streaming flow has a direction from the north pole to the south pole with a
circulating vortex the equatorial plane. The flow is symmetric about the 𝑧 -axis and
asymmetric about the radial axis. As we increase 𝑀 , the circulation regions start to vanish
and the streaming flow further tend to a dipolar shape. Where for large value of 𝑀 the flow
is totally dipolar in shape. This behavior is the result of the two trigonometric terms in
equation (3.40). The first term is 1− 𝜇
!
and the second term is 𝜇 1− 𝜇
!
. The first term
arises from the pulsation and lateral vibration of the bubble and tends to make the streaming
flow dipolar. The second term arises from the lateral vibration only and leans the streaming
flow toward a quadrupole nature. Because of the existence of the frequency parameter 𝑀 in
the nominator in the first term and the denominator in the second term the flow becomes
dipolar for large frequency. Moreover, the first term has 𝑂(1) radial dependence and the
second term has dependency of 𝑂(𝑟) which means the second term will dominate as 𝑟→∞.
To examine the effect of the phase shift, 𝜙, on the flow, we set the frequency
parameter to 𝑀 = 10 and we vary the phase shift from 𝜙= 90° to 𝜙= 180° (figure 3.2).
The flow is symmetric about the 𝑧-axis and asymmetric across the radial axis. The streaming
flow from the north pole to the south pole with prominent circulation regions below the
equator that has anticlockwise circulation in the third quadrant and clockwise circulation in
the fourth quadrant of the plane figure. As the value of the phase shift increases the streaming
intensity starts to increase as well. At the same time the circulation regions start to open up
from the outer rings and merge with the flow. The streaming intensity reaches a maximum
value at 𝜙= 150° and then starts to decrease as we further increase the phase shift. The
minimum value occurs at 𝜙= 90°. Although the circulation regions shrink as the phase shift
increases, they do not disappear. We further increase the frequency parameter to 𝑀 = 40
and vary the phase shift, 𝜙 (figure 3.3). As 𝑀 increases so does the flow intensity and as a
result the flow is almost dipole. The flow does not wrap around the bubble in a symmetric
53
pattern around the 𝑧-axis as it passes the bubble. The streaming flow streamlines do not
change as we vary the phase shift, however the intensity of the flow changes. The streaming
intensity starts to increase as we increase the value of the phase shift from 𝜙= 90° until
𝜙= 140°. At 𝜙= 140° the intensity reaches its maximum value. As the value of 𝜙 increases
further, the intensity of the flow starts to decrease.
54
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3.1: Streamlines of the outer solution with 𝜙= 100° and varying frequency parameter;
a)|𝑀|= 10, b) |𝑀|=20, c) |𝑀|=30, d) |𝑀|= 40, e) |𝑀|=50, and f) |𝑀|=60.
55
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3.2: Streamlines of the outer solution with |𝑀|= 10 and varying the phase shift; a) 𝜙= 90°, b)
𝜙= 105°, c) 𝜙= 120°, d) 𝜙= 135°, e) 𝜙= 150°, and f) 𝜙= 180°.
56
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3.3: Streamlines of the outer solution with |𝑀|= 40 and varying the phase shift; a) 𝜙= 90°, b)
𝜙= 105°, c) 𝜙= 120°, d) 𝜙= 135°, e) 𝜙= 150°, and f) 𝜙= 180°.
57
3.4. Solution for the pulsating bubble at the velocity antinode: no slip boundary
We will present the solution to the no slip boundary condition in this section. The no slip
boundary is really important in two cases: the first case is when the bubble is contaminated
and the second case is in the medical field where surfactants are used to stabilize the micro
bubble for drug delivery or as ultrasound contrast agents. The solution procedure to the no slip
boundary condition will be the same as that for free shear boundary. However, instead of
using equation (3.7) as the boundary condition we will use equation (3.8). Moreover, because
of the no slip boundary the expansion to the order of 𝑀
!!
in the equations of motion is not
required. Therefore, the momentum equation to O(1)
is
𝜕
𝜕𝜏
𝐷
!
𝜓
!
=
1
𝑀
!
𝐷
!
𝜓
!
, (3.41)
and
within
the
boundary
layer
it
is
𝜕
𝜕𝜏
𝜕
!
𝜓
!
!
𝜕𝜂
!
=
1
2
𝜕
!
𝜓
!
!
𝜕𝜂
!
+𝑂
1
𝑀
. (3.42)
From
equation
(3.41)
above
and
using
the
radial
velocity
boundary
condition
that
correspond
to
the
pulsation
of
the
bubble,
equation
(3.6),
and
the
no-‐slip
boundary
condition,
equation
(3.8),
as
will
as
the
boundedness
condition
as
𝑟→∞,
equation
(3.9),
we
found
a
solution
to
the
first
order
for
𝑂 𝜀
the
flow
outside
the
boundary
layer,
𝜓
!
=
𝑟
!
2
+
𝑅
!
2𝑟
1− 𝜇
!
𝑒
!"
. (3.43)
Within
the
boundary
layer,
using
equation
(3.42),
the
solution
was
found
to
be,
𝜓
!
!
=
3
4
1− 𝑖 𝑅 1+ 𝑖 𝜂−1+𝑒
! !!! !
1− 𝜇
!
𝑒
!"
−
𝑀
2
𝜇𝜀
!
𝑒
! !!!
. (3.44)
58
We
notice
here
that
the
first
order
flow
outside
the
boundary
layer
is
the
same
as
the
shear-‐free
boundary
condition
solution,
equation
(3.31).
However,
the
solution
within
the
boundary
layer,
as
expected,
is
different
than
that
of
the
shear
free
boundary
condition.
As
a
result
this
will
contribute
to
the
difference
in
the
final
solution
of
the
streaming,
as
the
first
order
solution
will
induce
nonlinear
effects.
The
momentum
equation
within
the
boundary
layer
to
𝑂 𝜀
is,
𝜕
𝜕𝜏
𝐷
!
𝜓
!
!
+
1
𝑅
!
−
2 2
𝑀 𝑅
!
𝜂
𝜕 𝜓
!
!
, 𝐷
!
𝜓
!
!
𝜕 𝜂, 𝜇
+2
1
𝑅
!
−
2 2
𝑀 𝑅
!
𝜂 𝐷
!
𝜓
!
!
𝐿𝜓
!
!
=
1
𝑀
!
𝐷
!
𝜓
!
!
. (3.45)
We expand the above momentum equation to the order of 𝑀
!!
, where we have
1
2
𝜕
!
𝜓
!
!
𝜕𝜂
!
−
𝜕
!
𝜓
!
!
𝜕𝜏𝜕𝜂
!
=
1
𝑅
!
𝜕𝜓
!
!
𝜕𝜂
𝜕
!
𝜓
!
!
𝜕𝜇𝜕𝜂
!
−
𝜕𝜓
!
!
𝜕𝜇
𝜕
!
𝜓
!
!
𝜕𝜂
!
+
2𝜇
1− 𝜇
!
𝜕
!
𝜓
!
!
𝜕𝜂
!
𝜕𝜓
!
!
𝜕𝜂
+𝑂
1
𝑀
. (3.46)
Because we are interested in the streaming phenomena, we take the time average of equation
(3.46),
𝜕
!
𝜓
!
!
𝜕𝜂
!
= 2
𝜕𝜓
!
!
𝜕𝜂
𝜕
!
𝜓
!
!
𝜕𝜇𝜕𝜂
!
−2
𝜕𝜓
!
!
𝜕𝜇
𝜕
!
𝜓
!
!
𝜕𝜂
!
+
4𝜇
1− 𝜇
!
𝜕
!
𝜓
!
!
𝜕𝜂
!
𝜕𝜓
!
!
𝜕𝜂
. (3.47)
Upon solving equation (3.47) using the appropriate boundary conditions to obtain,
𝜓
!
!
=
9
2
𝜇 1− 𝜇
!
𝑒
!!
5
2
cos𝜂+
3
2
sin𝜂+𝜂sin𝜂+
𝑒
!!
8
+
5
4
𝜂−
21
8
+
3
2
𝑀
2
𝜀
!
1− 𝜇
!
𝑒
!!
sin𝜂−𝜂 cos𝜙
+ 𝑒
!!
cos𝜂+𝜂−1 sin𝜙 . (3.48)
59
Equation (3.48) is the steady streaming within the boundary layer. The streaming flow outside
the boundary layer consider to be Stokes flow, and therefore we use the momentum equation,
𝒟
!
𝜓
!
= 0 . (3.49)
The general solution for equation (3.49), taking into account the requirement that as 𝑟→∞,
𝜓
!
= 𝑜 𝑟
!
, and the matching requirement with equation (3.48), is
𝜓
!
= 𝑏
!
𝑟+𝐷
!
𝑟
!!
1− 𝜇
!
+ 𝑏
!
+𝐷
!
𝑟
!!
𝜇 1− 𝜇
!
. (3.50)
To determine the integration constants, we change the 𝑟 variable to 𝜂 in the equation above
and take the limit as 𝜂→ 0, then match it with equation (3.48), after taking the limit as 𝜂→
∞. The final solution to the streaming flow was found to be,
𝜓
!
=
3
4
𝑀
2
𝜀
!
sin𝜙−cos𝜙 𝑟−𝑟
!!
1− 𝜇
!
+
45
16
1−𝑟
!!
𝜇 1− 𝜇
!
. (3.51)
3.4.1. Discussion:
The streaming flow has the same dependency on both the frequency parameter, 𝑀 ,
and the phase shift, 𝜙, as in the other sections and solutions. However, as expected, the flow
intensity is larger in the case of the no-slip boundary condition than that of the shear-free case.
If we look closely at equation (3.40), the streaming flow stream function for the case of shear-
free boundary condition, and we eliminate all the 𝑀
!!
terms, we will end up with
𝜓
!
=
9
4
1−𝑟
!!
𝜇 1− 𝜇
!
+
3
4
𝑀
2
𝜀
!
sin𝜙−cos𝜙 𝑟−𝑟
!!
1− 𝜇
!
−
21
8
𝜀
!
sin𝜙 𝑟−𝑟
!!
1− 𝜇
!
+𝑂
1
𝑀
. (3.52)
The last negative term in the above equation is the difference between the no-slip boundary
and the shear-free boundary. For large 𝑀 , where the value of 𝑀 will dominate the flow,
60
both equation (3.52) and equation (3.51) will be equal. Also, for the case where sin𝜙= 0,
which is at 𝜙= 180°, the flow will have the same intensity between the two cases. We
graphed the streaming flow streamlines for values of frequency parameter from 𝑀 = 50 to
𝑀 = 100. We chose large values for 𝑀 because we have neglected 𝑂 𝑀
!!
in this part.
We particularly notice the dominance of the pulsating part on the flow and the almost dipole
in pattern with no circulating regions. As we increase the frequency parameter even further,
the flow becomes dipole with symmetry across the 𝑧-axis and the equatorial plane. Moreover,
increasing the frequency parameter did increase the flow intensity.
We graph the streaming flow for 𝑀 = 100 and range of phase shift from 𝜙= 90° to
𝜙= 180° (figure 3.5). The flow is at its minimum value when 𝜙= 90° and as we increase
the value of 𝜙 the streaming flow intensity increases. It reaches maximum value at 𝜙= 135°,
after that it starts to decrease again as we further increase the phase shift. It reaches the same
minimum value as 𝜙= 90° when 𝜙= 180°. When we reduce the value of the frequency
parameter to 𝑀 = 40 and change the value of the phase shift, the maximum flow intensity
was found to be at 𝜙= 135° and the minimum intensity was at a value of 𝜙= 90° and
𝜙= 180° (figure 3.6). The flow is less intense than for the value of 𝑀 = 100, as expected.
The circulation regions don’t exist anymore as the value of the frequency parameter is high.
The dependence on the phase shift arises from the term sin𝜙−cos𝜙 in equation (3.51). It
is easy to see that at 𝜙= 135° is term is maximum and at 𝜙= 90° when 𝜙= 180° is
minimum.
61
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3.4: Streamlines of the outer solution for no slip boundary condition with 𝜙= 100° and
varying frequency parameter; a)|𝑀|= 50, b) |𝑀|= 60, c) |𝑀|= 70, d) |𝑀|=80, e) |𝑀|=90,
and f) |𝑀|= 100.
62
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3.5: Streamlines of the outer solution with |𝑀|= 100 and varying the phase shift; a) 𝜙= 90°, b)
𝜙= 105°, c) 𝜙= 120°, d) 𝜙= 135°, e) 𝜙= 150°, and f) 𝜙= 180°.
63
(a)
(b)
(c)
(d)
(e)
(f)
Figure 3.6: Streamlines of the outer solution with |𝑀|= 40 and varying the phase shift; a) 𝜙= 90°, b)
𝜙= 105°, c) 𝜙= 120°, d) 𝜙= 135°, e) 𝜙= 150°, and f) 𝜙= 180°.
64
3.5. Remarks
The intensity of the steady streaming for the no-slip boundary condition is larger than
that for the shear-free boundary condition. This is expected, as the solid boundary would
generate more strength to the vorticities. However, at specific values of the phase shift, 𝜙, the
difference is higher. This is due to the sin𝜙 term in equation (3.52). From equations (3.51)
and (3.52), the streaming has two contributions, one from the lateral oscillation and the other
from the radial oscillation. The radial oscillation at high frequency parameter, 𝑀 , would
dominate the flow where the flow pattern is a diploe due to the present of the 1− 𝜇
!
i.e.
sin
!
𝜃 term in equation (3.51) and equation (3.52). Although the phase shift plays a major part
in the flow pattern, yet its effect is not as dominate as the frequency parameter effect.
65
4. Final remarks
Using the singular perturbation theory, we obtained the steady streaming flow field
for a pulsating bubble trapped, first at the velocity node then at the velocity antinode of a
standing acoustic wave.
In section 2, we have showed that the streaming pattern was quadrupole in nature.
Moreover, for the shear-free boundary condition, at mid-range values of the frequency
parameter (𝑀 ), and phase shift values from 𝜙= 0° to 𝜙= 45°, circulating regions
manifest near the bubble surface develop. These circulations would vanish once the phase
shift increases, or at high frequency parameter (high flow intensity). The streaming flow
therefore depends on both the frequency parameter and the phase shift. Higher frequency
parameter means higher intensity and because of the relationship 𝛿=
!
!
, corresponding
to smaller boundary layer thickness. So, for mid-range frequency parameter, the vortices
that were developed inside the boundary layer would diffuse over a larger distance than
the case of the large frequency parameter.
At the velocity antinode (section 3), the flow also was found to be of quadrupole
character for the shear-free boundary condition and at mid-range frequency parameter.
However, the streaming flow was not symmetric about the equatorial plane, and
circulating regions develop below the equatorial plane with the flow direction from the
north pole to the south. This asymmetry almost disappears and the flow becomes more
dipole at large values of the frequency parameter. For the no-slip boundary, the streaming
flow was dipole because we assumed large values for the frequency parameter and did
not account for the mid-range values. The streaming also depends on the phase shift to
lesser extent.
If we compare our work to the only previous analytical work that was carried out for
a solid particle that was trapped at the velocity node (Zhao et al., 1999), we can see that
that flow in the previous study has octupole pattern (figure 4.1). Additionally, the flow
was of 𝑂 𝑘𝑎
!
which is smaller than 𝑂 𝑘𝑎 for the case with the radial oscillation.
Therefore, the bubble radial oscillations coupled with lateral oscillations seem to generate
more intense streaming. Also, the fact that we have a combination of two types of
oscillations, the streaming occurs at lower orders of the perturbation parameters.
66
(Longuet-Higgins, 1997) explain that the reason for the greater streaming is the fact that
“the radial component […] brings about stronger stretching of the vortex lines in the
neighborhood of the sphere”. The inclusion of the 𝑂 𝑀
!!
in our expansion for the
momentum equation made us visualize the circulating regions that would develop at mid-
range values for the frequency parameter and showed us the streaming flow depends on
the driving frequency.
For the velocity antinode, we can compare our work with that of (Davidson et. al.,
1971) for lateral oscillating bubble and (Longuet-Higgins, 1997) for the combined lateral
and radial oscillation bubble. The streaming flow in the lateral oscillation case has a
quadrupole streamlines with symmetric patterns across the 𝑧-axis and the equatorial plane
(Figure 4.2) and the streaming flow in the radial and lateral oscillations has a dipole
pattern and is symmetric about the 𝑧-axis (Figure 4.3). The streaming is also stronger in
the pulsating case. In our study (section 3.3) the frequency parameter to a large extent
defines the streaming flow pattern. For low 𝑀 the streaming is asymmetric about the
equatorial plane with a circulating vortex region. For large 𝑀 the streaming is dipole
and symmetric about the 𝑧-axis. This extra finding is the result of our inclusion of the
𝑂 𝑀
!!
terms, which has been deemed negligible in the previous studies.
In the literature, an example of such dependency on the frequency parameter can be
found in the paper by Andres et al. (1953). Although their study was for a cylinder in a
sound field, it can give an insight in how the streaming flow would behave for different
values of 𝑀 . From Figure 4.4, the flow has a circulation for modified Reynolds number
of unity (modified Reynolds number is 𝑅
!
=
!
!
∗
which in our study is
!
!
). As the
modified Reynolds number increases the center of the circulation moves toward the
cylinder and eventually vanish. The modified Reynolds number in their case is analogous
to the frequency parameter in our case. We can draw a conclusion that the frequency
parameter 𝑀 does define the streamlines shape, and at low or mid-range values for the
frequency parameter, the circulating regions do exist.
The time-average steady streaming is small in general and it is in the order of 𝑂 𝜀𝜀
!
for the pulsating case. But, it has many practical applications in both engineering and the
medical fields. Ryu, Chung, & Cho (2010) ran an experiment where they used
microstreaming from a pulsating bubble to pump water through a capillary tube. The tube
67
was placed vertically on top of the bubble and as they acoustically start to excite the
bubble, water started to be pumped through the tube due to steady streaming. In the small
scale of lab-on-a-chip, the mixing is governed by diffusion, which is a slow process
(Hashmi et. al., 2012). Using an acoustically pulsating bubble as a micromixer would
increase the mixing rate. Ahmed, Mao, Juluri, & Huang (2009) trapped a bubble inside a
groove at the side of a micro-channel and acoustically drove the bubble to pulsate and
generate steady streaming. This method increased the mixing rate inside the micro-
channel significantly. In the biological field, microstreaming that was generated from a
single pulsating bubble has enough shear force to manipulate, deform, or rapture a
vesicles that is located in the flow field of the steady streaming (Marmottant &
Hilgenfeldt, 2003). This enables scientists to perform the cell lysis without any intrusive
processes that may impact the accuracy of the results.
68
Figure 4.1: Steady streaming at the velocity node. The fluid motion is
clockwise in the upper vortex, and anticlockwise in the lower one (Zhao et. Al.,
1999).
Figure 4.2: Steady streaming at the velocity antinode for |𝑴|
𝟐
≫𝟏. The direction of
the flow indicated by the arrows (Davidson et al., 1971).
69
Figure 4.3: steady streaming at the velocity antinode with both lateral and radial
oscillations (longuet-Higgins, 1997).
Figure 4.4(a): streamlines for a cylinder in an acoustic wave. Small
modified Reynolds number (Andres et al., 1953).
70
Figure 4.4(b): streamlines for a cylinder in an acoustic wave. Small
modified Reynolds number;
𝒂
𝜹
= 𝟏 (Andres et al., 1953).
Figure 4.4(c): streamlines for a cylinder in an acoustic wave. Small
modified Reynolds number;
𝒂
𝜹
= 𝟑. No more circulation (Andres et al.,
1953).
71
References
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bubbles: a versatile tool for lab on a chip applications. Lab Chip, 12, 4216-4227.
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microparticles in bubble streaming flows. Biomicofluidics, 6, 12801(1) –
12801(11).
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forces. Eur. J. phys., 11, 47-50.
9. Rossing, T. D. (2007) Springer Handbook of Acoustics. New York: Springer
Science+Business Media.
10. Eller, A. (1968). Force on a bubble in a standing acoustic wave. J. Acoust. Soc.
Am. 43, 170 – 171.
11. Brennen, C. E. (1995) Cavitation and Bubble Dynamics. New York: Oxford
University Press.
12. Riley, N. (2001). Steady Streaming. Annu. Rev. Fluid Mech., 33, 43-65.
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15. Lord Rayleigh, (1884). On the Circulation of Air observed in Kundt's Tubes and
some Allied Acoustical Problems. Philos. Trans. R. Soc. London, 175, 1-21.
16. Schlichting, H. (1932). Phys. Z., 33, 327-335.
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18. Riley, N. (1966). On a sphere oscillating in a viscous fluid. J. Mech. And Applied
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Oscillating Bubble for Automated Implantable Microfluidic Devices. Journal of
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73
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Abstract (if available)
Abstract
We have examined the steady streaming phenomenon with regard to a pulsating bubble levitated in a standing wave, positioned at the velocity node and antinode. The bubble undergoes two types of oscillations when placed in a standing wave. The first mode of the oscillation is a lateral one and the second is radial. We used the singular perturbation method to analytically study this problem.
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Creator
AlHamli, Mohammad K. (author)
Core Title
Perturbation analysis of flow about spherically pulsating bubble at the velocity node and the antinode of a standing wave with different boundary conditions
School
Viterbi School of Engineering
Degree
Doctor of Philosophy
Degree Program
Mechanical Engineering
Publication Date
09/30/2016
Defense Date
08/06/2015
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), Eliasson, Veronica (
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), Lee, Vincent W. (
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), Redekopp, Larry G. (
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