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Derivation of the RayleighPlesset
Equation in Terms of Volume
T. G. Leighton
Abstract
The most common nonlinear equations of motion for the pulsation of a spherical gas
bubble in an infinite body of liquid arise in the various forms of the RayleighPlesset
equation, expressed in terms of the dependency of the bubble radius on the conditions
pertaining in the gas and liquid. However over the past few decades several important
analysis have begun with a heuristicallyderived form of the RayleighPlesset equation
which considers the bubble volume, instead of the radius, as the parameter of interest,
and for which the dissipation term is not derived from first principles. The predictions
of these two sets of equations can differ in important ways, largely through differences
between the methods chosen to incorporate damping. As a result this report derives
the RayleighPlesset equation in terms of the bubble volume from first principles in
such a way that it has the same physics for dissipation (viscous shear) as is used in the
radius frame.
Contents
1. Introduction
4
2. Derivation of the RayleighPlesset equation in terms of the bubble radius 6
2.1. Derivation of the RayleighPlesset equation in terms of the bubble
radius using an energy balance . . . . . . . . . . . . . . . . . . . . . . .
6
2.2. Derivation of the RayleighPlesset equation in terms of the bubble
radius using the NavierStokes equation . . . . . . . . . . . . . . . . . .
8
3. Derivation of the RayleighPlesset equations in terms of bubble volume
from the energy balance
12
4. The liquid pressure
4.1. The liquid pressure assuming no dissipation . . . .
4.2. The inclusion of viscous losses . . . . . . . . . . . .
4.2.1. Calculation of the losses in the radius frame
4.2.2. Calculation of the losses in the volume frame
14
14
16
17
18
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5. Conclusions
20
A. Derivation of equation (2.2)
21
B. Derivation of equation (2.4)
22
C. Derivation of equation (2.24)
23
D. Derivation of equation (3.2)
24
E. Derivation of equation (3.4)
26
List of Symbols
27
Literature
29
3
1. Introduction
The most popular nonlinear equation for description of the nonlinear response of a
gas bubble (see figure 1.1) in liquid to a driving pressure field is the RayleighPlesset
equation. This equation can be derived from first principles using the bubble radius
R(t) as the dynamic parameter. However there exist heuristic formulations based
on a form of the RayleighPlesset equation in which the bubble volume V is used as
the dynamic parameter, where the damping is not derived from first principles. The
predictions of the two approaches do not always agree, and this study was undertaken
to derive a form of the RayleighPlesset equation in which the bubble volume V is
used as the dynamic parameter, and where the physics describing the dissipation is
identical to that used when the RayleighPlesset equation is cited in the radius frame.
This report will proceed by using the following common assumptions:
• The bubble exists in an infinite medium
• The bubble stays spherical at all times during the pulsation
• Spatially uniform conditions exist within the bubble
• The bubble radius is much smaller than the wavelength of the driving sound
field
• There are no body forces acting (e.g. gravity)
• Bulk viscous effects can be ignored
• The density of the surrounding fluid is much greater than that of the internal gas
• The gas content is constant
4
Figure 1.1.
Schematic drawing of a sphere with radius R(t)
5
2. Derivation of the RayleighPlesset
equation in terms of the bubble
radius
The RayleighPlesset equation is usually derived by considering the dynamics of the
bubble radius, R. This can be done in two basic ways:
• Differentiation with respect to R of the balance between the kinetic energy in the
liquid and the potential energy in the gas
• Integration of the NavierStokes equation
These two methods will be shown in section 2.1 and section 2.2. The chapter 3 will
discuss ways of deriving the RayleighPlesset equation in terms of the dynamics of
the bubble volume.
2.1. Derivation of the RayleighPlesset equation in
terms of the bubble radius using an energy
balance
Consider a spherical gas bubble which pulsates in an incompressible liquid as a result
of an insonifying field (the long wavelength limit being assumed throughout this
report). The fluid velocity u (r, t) falls off as an inverse square law with range r as a
result of the assumption of liquid incompressibility, which implies that:
u (r, t) =
R 2 (t)
· Ṙ (t)
r 2 (t)
(2.1)
where the bubble has the radius R (t) and the wall velocity Ṙ (t). As the bubble radius
changes, for example from an equilibrium value R0 to some other, work is done on the
bubble by that pressure which would exist at the location of the center of the bubble
were the bubble will not to be present. If the spatial scales over which this pressure
changes are much greater than the bubble radius, this almost equals the liquid pressure
far1 from the bubble, p∞ = p0 + P (t), which includes a dynamic component P (t). The
1
If tb is the timescale over which the boundary moves, then the condition r & c∞ tb describes what
is commonly understood to be ’far from the bubble’.
6
difference between this work, and that done by the pressure pL at the bubble wall
equals the kinetic energy in the liquid, ΦKE (see Appendix A):
Z r=∞
1
4πr 2 u 2 dr = 2πρ0 R 3 Ṙ 2
(2.2)
φKE = ρ0 ·
2
r=R
This balance can be expressed as follows:
Z R
(pL − p∞ ) · 4πR 2 dR = 2πρ0 R 3 Ṙ 2
(2.3)
R0
Differentiation of this with respect to R, noting that
∂ Ṙ 2
1 ∂ Ṙ 2
= ·
= 2R̈
∂R
Ṙ ∂t
(2.4)
gives
pL (t) − p∞ = ρ0 ·
3Ṙ 2
RR̈ +
2
!
+ O Ṙ/c .
(2.5)
The term on the left arises from the difference in work done at the bubble wall and
remote from the bubble. The terms on the right arises from the kinetic energy imparted
to the liquid. If the pressure far from the bubble p∞ comprises both a static component
p0 and an applied driving pressure P (t), then this can be expressed in equation (2.5)
to give the socalled RayleighPlesset equation of motion [8]:
RR̈ +
1
3Ṙ 2
=
· (pL (t) − p0 − P (t)) + O Ṙ/c
2
ρ0
7
(2.6)
2.2. Derivation of the RayleighPlesset equation in
terms of the bubble radius using the
NavierStokes equation
In the following derivation, the use of the dot notation in this, and the subsequent
equations of motion, indicates the use of the material derivative [8], i.e.:
∂
D
#»
#»
=
+ u ·∇
Dt
∂t
where #»
u is the liquid particle velocity and
(2.7)
∂
∂
∂
#»
∇ = x̂
+ ŷ
+ ẑ
∂x
∂y
∂z
is the Nabla operator, where {x̂, ŷ, ẑ} are the unit vectors in their respective directions.
For the discussion of the pulsation of a single bubble whose center remains fixed in
space, as occurs in this report, the convective term (the second term on the right) is
zero. Before applying the equations of this book, critical evaluation should be made of
their applicability, given this restriction. Models of translating bubbles need careful
evaluation. Even where bubbles are assumed to pulsate only, if they exist in a dense
cloud then the convective term may be significant [4].
The following derivation assumes, that the material outside the gas bubble wall is
incompressible, and assumes that spatially uniform conditions are given to exist within
the bubble.
When these assumptions are applied for the case of a gas bubble in a liquid, the
equations for the conservation of energy within the liquid can be coupled to that of
the diffusion of dissolved gas within it, and to the equation for conservation of mass
in the liquid:
1 Dρ #» #»
+ ∇ · u = 0 (Continuity Equation)
ρ Dt
(2.8)
which yields, after multiplying both side of equation (2.8) with ρ and remembering
equation (2.7)
∂ρ #»
+ ∇ · (ρ #»
u) = 0
(2.9)
∂t
where #»
u is the liquid particle velocity and ρ is the liquid density. For the equation of
conservation of momentum in the liquid, we have:
D #»
u
ρ
=ρ ·
Dt
#»
∂u
#» #»
#»
+ u ·∇ u
∂t
X #»
4η
#»
#» #»
#» #»
=ρ ·
F ext − ∇p +
+ ηB · ∇ ∇ · #»
u − η ∇ × ∇ × #»
u ,
3
8
(2.10)
where p represents the sum of all steady and unsteady pressures.
Equation (2.10) is called the NavierStokesEquation. It simplifies in a number of
ways for limits which are often appropriate to gas bubbles in water [8]. The term
#» #»
η ∇ × ∇ × #»
u encompasses the dissipation of acoustic energy associated with, amongst
other things, vorticity, and hence is zero in conditions of irrotational
flow (required for
#» #» #»
4η
+ η · ∇ ∇ · u represents the
the definition of a velocity potential). The term
3
B
product of viscous effects (through the shear η and bulk ηB viscosities of the liquid),
#»
with the gradient of ∇ · #»
u (which, from equation (2.9), represents in turn the liquid
compressibility). As an interaction term, it is generally small. Note that setting it to
zero does not imply that all viscous effects are neglected, but simply that they appear
P #»
only through the boundary condition. Lastly, the term
F ext represents the vector
summation of all body forces which are neglected in the formulations of this report.
If it is then assumed that the bubble remains spherical at all times and pulsates in an
infinite body of liquid, then because of spherical symmetry, the particle velocity in
the liquid #»
u is always radial and of magnitude u(r, t), and equations (2.9) and (2.10)
reduce, respectively, to:
1 ∂ (r 2 ρu)
∂ρ
+ 2
=0
∂t r
∂r
(2.11)
∂u
∂u 1 ∂p
+u
+
=0 .
∂t
∂r ρ ∂r
(2.12)
and to Euler’s Equation
The bubble radius R(t) oscillates about some equilibrium radius R0 with the bubble
wall velocity Ṙ(t).
Approximations of this sort are required because the solution of the conservation
equations of continuum mechanics both within and outside the bubble, with suitable
boundary conditions at the wall, is not simple [1–3, 5–13, 15, 16].
The derivation of the RayleighPlesset equation given previously (section 2.1) was
based on equating the kinetic and potential energies. To follow the approach discussed
above, equation (2.12) is combined with the linearized wave equation for a velocity
potential Φ,
∇2 Φ −
1 ∂ 2Φ
=0 ,
c 2 ∂t 2
(2.13)
given2
#»
#»
u = ∇Φ ,
(2.14)
where in the spherically symmetric conditions assumed above, equation (2.13) has the
solution
2
#»
u = −∇Φ.
Note that some authors use the alternative convention #»
9
ψ · (t − r/c)
,
(2.15)
r
and the amplitude function ψ has yet to be determined. At first sight it may seem odd
to apply a linearized wave equation, given that the intended equation of dynamics
is nonlinear, but this is justified provided that the nonlinearity introduced into the
system by the bubble we are describing is much greater than the nonlinearity of the
system when our bubble is removed (which is usually the case for single bubbles
in an otherwise bubblefree medium). In the long wavelength limit, equation (2.13)
simplifies into the equation of mass conservation in an incompressible liquid,
Φ=
∇2 Φ = 0 ,
(2.16)
because [17]
1 ∂ 2Φ
c 2 ∂t 2
2
R(t)
∼
1 ,
(2.17)
∇2 Φ
λ
where λ is the wavelength of the insonifying field. In this limit, both equation (2.15)
and (2.16) give
ψ(t)
,
(2.18)
r(t)
where ψ can be evaluated by application to equation (2.16) of the boundary condition
Φ(r, t) =
∂Φ
∂r
r=R
= Ṙ (which follows from equation (2.14)). This gives ψ = −R 2 Ṙ and
R 2 Ṙ
(2.19)
r
which recovers the incompressible relation (2.1). Another consequence is, that equation
(2.12) becomes
Φ(r, t) = −
∂ 2Φ
1 ∂
+
∂t∂r 2 ∂r
Z
r
r1
∂ 2Φ 0 1
dr +
∂t∂r0
2
Z
r
r1
∂Φ
∂r
2
1 ∂p
=0 ⇒
ρ0 ∂r
R r1 λ ⇒
2
Z r1
∂
∂Φ
1 ∂p 0
0
dr = −
dr ,
0
0
∂r ∂r
ρ0 ∂r0
r
+
(2.20)
where the use of the invariant density ρ0 reflects the assumed liquid incompressibility.
Using the independence of the variables r and t, and given that r1 is sufficiently far
from the bubble that the liquid pressure there is dominated, not by the presence of the
bubble but rather by the static pressure p0 and the applied one P (t), then (2.20) gives
the pressure at a location r in the liquid, in terms of the velocity potential Φ:
2
∂Φ(r, t) 1 ∂Φ(r, t)
p(r, t) − p0 − P (t)
+
=−
.
(2.21)
∂t
2
∂r
ρ0
10
The RayleighPlesset equation (2.6) is obtained simply by evaluating equation (2.21) at
the bubble wall, where application of equation (2.19) implies:
Φ(r, t)
r=R
= − R Ṙ ,
2RṘ 2 + R 2 R̈
∂Φ(r, t)
∂Φ(r, t)
=−
⇒
= −RR̈ − 2Ṙ 2 ,
∂t
r
∂t
r=R
∂Φ(r, t)
R 2 Ṙ
∂Φ(r, t) R 2 Ṙ
= 2 ⇒
= 2
= Ṙ ,
∂r
r
∂r
r r=R
r=R
(2.22)
the last of these being a logical consequence of equation (2.14). Applications of these
relationships at the bubble wall (where p(r = R, t) = pL (t)) to equation (2.21) gives:
∂Φ(r, t)
∂t
1
+
2
r=R
∂Φ(r, t)
∂r
2
RR̈ +
=−
r=R
p(r = R, t) − p0 − P (t)
⇒
ρ0
3Ṙ 2 pL (t) − p0 − P (t)
=
2
ρ0
(2.23)
the RayleighPlesset equation in terms of the bubble radius.
In similar vein, application of equation (2.22) to the body of the liquid gives the pressure field radiated by the bubble, where the retarded time tr = t − r/c is recovered
in place of t to impose a finite sound speed on this ’incompressible’ medium (see
Appendix C):
4 !
2
Ṙ (tr )
R(tr )
R(tr )
Pb,r (r) = ρ0 ·
· R(tr )R̈(tr ) + 2Ṙ 2 (tr ) −
·
(2.24)
r
2
r
The radiated pressure comprises a component which decays to as r −4 [8], such that far
from the bubble wall the radiated field is dominated by the r −2 term, and
ρ V̈ (t )
ρ0 R(tr )
0
r
2
Pb,r (r) ≈
· R(tr )R̈(tr ) + 2Ṙ (tr ) =
r
4πr
since V = 4 π R 3 /3 ⇒ V̇ = 4 π R 2 Ṙ ⇒ V̈ = d 4 π R 2 Ṙ /dt.
11
(2.25)
3. Derivation of the RayleighPlesset
equations in terms of bubble
volume from the energy balance
Consider a spherical gas bubble which pulsates in an incompressible liquid1 as a result
of an insonifying field (the long wavelength limit being assumed throughout this
report). The fluid velocity u(r, t) falls off as an inverse square law with range r as a
result of the assumption of liquid incompressibility. The mass of liquid which moves
at the bubble wall, in time ∆t, is ρ0 V̇ (t)∆t, whilst that at range r is ρ0 4 π r 2 u(t)∆t,
which implies that:
u(r, t) =
V̇ (t)
4π r2
(3.1)
where the bubble has the volume V (t) and the wall volume velocity V̇ (t). As the
bubble volume changes, for example from an equilibrium value V0 to some other,
work is done on the bubble by that pressure which would exist at the location of
the center of the bubble were the bubble not to be present. If the spatial scales over
which this pressure changes are much greater than the bubble radius, this almost
equals the liquid pressure far2 from the bubble, p∞ = p0 + P (t), which includes a
dynamic component P (t). The difference between this work, and that done by the
pressure pL at the bubble wall, equals the kinetic energy in the liquid (see Appendix D):
φKE
1
1
= ρ0
2
Z
r=∞
4πr 2 u 2 dr =
r=R
(3.2)
In most situations of general interest, the flow of a conventional liquid, such as water or air, is
incompressible to a high degree of accuracy. Now, a fluid is said to be incompressible when the
mass density of a comoving volume element does not change appreciably as the element moves
through regions of varying pressure. In other words, for an incompressible fluid, the rate of
change of ρ following the motion is zero: i.e.,
Dρ
= 0.
Dt
In this case, the continuity equation (2.8) reduces to
#»
∇ · #»
u = 0.
2
ρ0 V̇ 2 (t)
.
8πR
See footnote on page 6.
12
This balance can be expressed as follows:
Z
V
V0
ρ0 V̇ 2 (t)
ρ0 V̇ 2 (t)
(pL − p∞ ) dV =
=
8πR
8π
4π
3V
13
.
(3.3)
Differentiation of this with respect to V , noting that (see Appendix E)
∂ V̇ 2
1 ∂ V̇ 2
=
= 2V̈
∂V
V̇ ∂t
(3.4)
1
· (pg + pV − pσ − p0 − P (t)) + O Ṙ/c .
ρ0
(3.5)
gives
1
8π
4π
3V
31
V̇ 2
2V̈ −
3V
!
=
13
4. The liquid pressure
Having obtained the RayleighPlesset equation in terms of the pressure in the liquid
at the bubble wall (pL ) in radius (equations (2.6) and (2.23)) or volume (equation
(3.5)) form, the next stage in generating a formulation suitable for predicting bubble
dynamics is to find appropriate expressions for pL . To do this in a way which accounts
for all loss mechanisms in a nonlinear and timedependent manner is very complicated,
and most simulations rely on assumptions to simplify the problem. The simplest
solution assumes that no dissipation occurs, and this solution is derived in section 4.1.
4.1. The liquid pressure assuming no dissipation
Initially this was done by neglecting all forms of dissipation. Evaluation of pL for use
in the equations of motion then becomes a question of calculating pi , the pressure
inside the bubble. After the initial step of expressing the internal pressure in the bubble
in terms of the sum of the gas (pg ) and vapor (pV ) pressures, correcting for the Laplace
pressure introduced through the effect of surface tension (pσ ):
pi = p g + pV = pL + pσ ⇒
pL = p g + p V − pσ
(4.1)
where
pσ = 2 σ/R = 2 σ
3V
4π
31
(4.2)
.
Evaluation of equation (4.1) when the bubble is at equilibrium size and the pressures
take values appropriate for that size in the absence of any driving pressure
(R = R0 ; V = V0 ; pi = pi,e ; pg = pg,e see [8]) gives:
pi,e
2σ
= pg,e + pV = p0 +
= p0 + 2 σ ·
R0
4π
3V0
31
.
(4.3)
This requires an understanding of thermal losses from the gas.
By far the most common way of calculating pg is to appeal to a polytropic law. It
involves calculating the pressure in the gas at a given bubble size by comparing it with
the pressure at equilibrium. From equation (4.3) we know, that the gas pressure at
equilibrium (pg,e ) is equal to the sum of the static pressure in the liquid just outside
14
the bubble wall (p0 ), plus the Laplace pressure at equilibrium 2 σ/R0 (where σ is the
surface tension [8]) minus that component due to vapor (pV ). Hence when the bubble
has the radius R, the pressure in the gas will be:
3 κ
3 κ
R0
2σ
R0
pg = pg,e ·
= p0 +
− pV ·
(4.4)
R
R0
R
in term of the bubble radius, and
pg = pg,e ·
V0
V
κ
=
p0 + 2 σ ·
4π
3V0
31
!
κ
V0
− pV ·
V
(4.5)
in volume terms. The use of the polytropic index κ adjusts the relationship between
bubble volume and gas pressure (effectively, the ’spring constant’ of the bubble) to
account for heat flow across the bubble wall, but crucially it ignores net thermal losses
from the bubble (see below). Therefore if the RayleighPlesset equation is evaluated
using a polytropic law, the result would, without correction, ignore two of the major
sources of dissipation: (i) net thermal losses and, through the incompressible assumption, (ii) radiation losses. Approximate corrections, which artificially enhance the
viscosity to account for thermal and radiation damping, are available through use
of enhancements to the viscosity, although these are only partially effective. These
enhancements are based on the same physics as the ’linear’ damping coefficients.
The pressure in the liquid in the radius frame is therefore found by substitution of
equation (4.4) into (4.1) which gives
3 κ
2σ
R0
− pV ·
pL = pg + pV − pσ = p 0 +
+ pV − pσ
(4.6)
R0
R
and substitution of this into equation (2.6) or (2.23) gives
3Ṙ 2
1
RR̈ +
= · (pg + pV − pσ − p0 − P (t)) + O Ṙ/c ⇒
2
ρ0
#
"
3 κ
2
1
R0
2σ
3Ṙ
2σ
= ·
+ pV −
− p0 − P (t)
RR̈ +
− pV ·
p0 +
2
ρ0
R0
R
R
+ O Ṙ/c
(4.7)
(4.8)
which is the RayleighPlesset equation expressed in terms of the bubble radius with
no dissipation included.
Similarly, substitution of equation (4.5) into (4.1) gives
"
#
1
13
κ
4π 3
V0
4π
pL = p g + p V − pσ = p0 + 2 σ ·
− pV ·
+ pV − 2 σ ·
(4.9)
3V0
V
3V
15
and substitution of this into equation (3.5) gives the polytropic version of the RayleighPlesset equation in terms of bubble volume, with no dissipation:
1
·
8π
1
·
8π
4π
3V
13
4π
3V
13
·
·
V̇ 2
2V̈ −
3V
!
1
· (pg + pV − pσ − p0 − P (t))
ρ0
+ O Ṙ/c ⇒
!
"
!
1
V̇ 2
1
4π 3
2V̈ −
= ·
p0 + 2 σ ·
− pV
3V
ρ0
3V0
#
κ
13
V0
4π
·
− p0 − P (t)
+ pV − 2 σ ·
V
3V
=
(4.10)
+O Ṙ/c
which is the RayleighPlesset equation expressed in terms of the bubble volume with
no dissipation included.
4.2. The inclusion of viscous losses
The most common approach in the radius form of the equation is to neglect all forms
of loss except viscous losses (an assumption which is often not valid), which enables
an expression for the liquid pressure to be obtained by dynamically matching normal
stresses across at the bubble wall.
2 σ 4 η Ṙ
−
(4.11)
R
R
which for η = 0, is what could be obtained by substituting equation (4.4) into (4.1).
This approach explicitly introduces viscous damping. It has been derived for the
radius frame, [14] and can also be derived for the volume frame.
The relationships of section 4.1 were derived for an inviscid liquid, but when the shear
viscosity of the liquid is finite. Finite shear modifies the liquid pressure at the bubble
wall (pL ) such that it differs from the pressure at a boundary with in the fluid (p0 ) by an
amount proportional to the principle rate of strain in the radial direction (ε0r = ∂u/∂r)
as follows, [8] :
∂u
pL = p0 − 2ηε0r = p0 − 2η ·
.
(4.12)
∂r
This can readily be converted into forms relevant to the radius and volume frames,
using their respective incompressibility relations, i.e.
pL = pi −
u(r, t) =
R 2 (t)
∂u(r, t)
2R 2 Ṙ
·
Ṙ(t)
⇒
=
−
r 2 (t)
∂r
r3
16
(4.13)
from equation (2.1)(for the radius frame) and (from equation (3.1) for the volume
frame):
V̇ (t)
∂u(r, t)
=−
∂r
2πr 3
Substitution of equation (4.13) into (4.12) gives, for the radius frame:
(4.14)
4ηR 2 Ṙ
(4.15)
r3
whilst substitution of equation (4.14) into (4.12) gives the equivalent expression for the
volume frame:
η V̇ (t)
pL = p0 +
.
(4.16)
πr 3
These expressions ((4.15) and (4.16)) can now be used to derive equation (4.11) and an
equivalent expression for the volume frame.
pL = p 0 +
4.2.1. Calculation of the losses in the radius frame
Within the body of the liquid, Bernoulli’s equation follows from the integration of a
suitably reduced form of the NavierStokes equation (equation (2.10)), [8]:
p0
p∞ ∂Φ u 2
−
(Bernoulli’s equation)
=
−
ρ0
ρ0
∂t
2
(4.17)
Substitution for the velocity potential from equation (2.19) and for the velocity from
the incompressibility relation (2.1) into (2.12), and evaluation of the result at the bubble
wall (r = R), gives:
p∞ ∂Φ(r, t)
p0
=
−
ρ0
ρ0
∂t
1
− ·
2
r=R
R 2 (t)
· Ṙ(t)
r 2 (t)
2
r=R
(4.18)
which is readily evaluated using equation (2.22):
p∞
3Ṙ 2
p0
=
+ RR̈ +
.
ρ0
ρ0
2
(4.19)
Eliminating p0 from equation (4.19) and (4.15) (evaluated at the bubble wall) gives:
!
4η Ṙ
3Ṙ 2
1
RR̈ +
=
· pL −
− p∞ ,
(4.20)
2
ρ0
R
17
which, when equation (4.6) is used to substitute for pL , gives
3Ṙ 2 1
RR̈ +
= ·
2
ρ
−
"
2σ
p0 +
− pV
R0
3 κ
R0
·
+ pV .
R
#
2σ 4η Ṙ
−
− p0 − P (t)
R
R
(4.21)
,
where the assumption is made that the pressure in the liquid far from the bubble (p∞ ) is
a summation of steady (p0 ) and timevarying (P (t)) components (i.e. p∞ = p0 + P (t)).
4.2.2. Calculation of the losses in the volume frame
To incorporate shear into the RayleighPlesset equation in the volume frame, substitutions into equation (4.17) can be made for the liquid velocity using equation (3.1) and,
an appropriate velocity potential (2.14) can be calculated from equation (3.1) using the
relevant boundary conditions:
V̇ (t)
.
4πr
In this way equation (4.17) can be evaluated at the bubble wall (r = R):
Φ(r, t) =
p∞ ∂Φ(r, t)
p0
=
−
ρ0
ρ0
∂t
1
− ·
2
r=R
V̇ (t)
4πr 2
(4.22)
!2
r=R
⇒ (Bernoulli’s Equation)
(4.23)
p∞
V̈
V̇ 2 (t)
p0
=
+
−
.
ρ0
ρ0
4πR 2 · (4πR 2 )2
Elimination of p0 from equation (4.16) and (4.23) gives:
V̈
V̇ 2 (t)
1
−
= ·
2
4πR 2 · (4πR 2 )
ρ0
!
31
1
V̇ 2 (t)
4π
1
· V̈ −
·
= ·
4π
3V
6V
ρ0
18
4η V̇ (t)
pL − p∞ −
3V
!
4η V̇ (t)
pL − p∞ −
3V
!
⇒
.
(4.24)
Substituting for pL from equation (4.9) gives:
!
V̇ 2 (t)
4η V̇ (t)
V̈
1
−
⇒
(4.25)
= · pL − p∞ −
4πR 2 · (4πR 2 )2 ρ0
3V
!
!
"
31
1
κ
V0
1
V̇ 2 (t)
4π
1
4π 3
· V̈ −
− pV ·
·
= ·
p0 + 2σ ·
+ pV
4π
3V
6V
ρ0
3V0
V
#
13
4π
− p0 − P (t) ,
−2σ ·
3V
which is the RayleighPlesset equation in terms of the bubble volume, including
viscous damping.
19
5. Conclusions
It has been possible to derive a form of the RayleighPlesset equation from first
principles which incorporates viscous damping.
20
A. Derivation of equation (2.2)
Given is the kinetic energy according to φKE = 12 · m · u 2 with the substitution m = ρ · V
and with the surface of a sphere O = 4π · r 2 :
1
= ρ0 ·
2
φKE
Z
r(t)=∞
4πr 2 (t)u 2 (r, t) dr(t) .
(A.1)
R 4 (t)
· Ṙ 2 (t) dr(t) ,
r 4 (t)
(A.2)
r(t)=R(t)
Substituting (2.1) in (A.1) yields
Z
r(t)=∞
r 2 (t) ·
2π · ρ0 ·
r(t)=R(t)
which simplifies, while bearing in mind that
b
Z
b
f (x) = F (x)
a
a
Z
= F (b) − F (a) and
1
1
dx = −
2
x
x
to the following equation
2π · ρ0 · R 4 (t) Ṙ 2 (t) · −
1
r(t)
r(t)=∞
.
r(t)=R(t)
(A.3)
Calculating
1
−
r(t)
r(t)=∞
r(t)=R(t)
1
1
=− − −
∞
R(t)
ones gets finally
R 4 (t) 2
Ṙ (t)
R(t)
= 2π · ρ0 · R 3 (t) Ṙ 2 (t) .
2π · ρ0 ·
21
(A.4)
B. Derivation of equation (2.4)
Expanding equation (2.4) with
∂t
∂t
gives
∂ Ṙ(t)2
∂t
·
,
∂t
∂R(t)
(B.1)
which yields, after writing
1
=
∂R(t)
∂t
1
Ṙ(t)
and performing the differentiation
Ṙ(t) ·
∂ Ṙ(t) ∂ Ṙ(t)
1
+
· Ṙ(t) ·
∂t
∂t
Ṙ(t)
(B.2)
equation (2.4) as
∂ Ṙ(t)
∂t
= 2 · R̈(t)
2·
22
(B.3)
C. Derivation of equation (2.24)
Writing equation (2.21) as
"
2 #
∂Φ(r, t) 1 ∂Φ(r, t)
+
= − (p(r, t) − p0 − P (t))
ρ0 ·
∂t
2
∂r
(C.1)
and knowing, that
∂Φ(r, t)
2RṘ 2 + R 2 R̈
=−
∂t
r
R
= · −2Ṙ 2 − RR̈
r
∂Φ(r, t) R 2 Ṙ
= 2 ,
∂r
r
and substituting these into equation (C.1) yields
!2
2
R
1
R Ṙ
= − (p(r, t) − p0 − P (t)) .
ρ0 · · −2Ṙ 2 − RR̈ + ·
r
2
r2
(C.2)
Multiplying equation (C.2) with (−1) and rearranging this equation finally gives
"
#
Ṙ 2 R 4
R
· RR̈ + 2Ṙ 2 −
·
ρ0 ·
= (p(r, t) − p0 − P (t)) .
(C.3)
r
2
r
23
D. Derivation of equation (3.2)
Given is the equation
φKE
1
= ρ0
2
which simplifies to
Z
r=∞
Z
4πr 2 u 2 dr
(D.1)
r=R
r=∞
r 2 u 2 dr .
2πρ0 ·
(D.2)
r=R
Performing the integration, knowing that
Z
Z
0
uv dx = uv −
Z
u0 v dx
b
f (x) dx = F (b) − F (a).
a
and setting v 0 = r 2 and u = u 2 yields
Z r=∞
1
r 2 u 2 dr = r 3 u 2
3
r=R
Z
r=∞
r=∞
−
r=R
r=R
2 3
r uu̇ dr .
3
(D.3)
Writing the integral of the right hand side of equation (D.3) with help of equation (3.1)
(and noting that V̇ is independent of r) in the form
!
!
Z
Z
2 r=∞ 3
V̇
d
V̇
2 r=∞ 3
r uu̇ dr =
r ·
·
dr
3 r=R
3 r=R
4πr 2
dr 4πr 2
and simplifying yields
1
· V̇ 2 ·
24π 2
Z
r=∞
r=R
d
r·
dr
1
r2
dr .
Performing the differentiation and simplifying gives
Z r=∞
1
1
2
−
· V̇ ·
dr .
2
2
12π
r=R r
(D.4)
(D.5)
Integration yields
1
1
· V̇ 2 ·
2
12π
r
24
(D.6)
and combining equation (D.3) with (D.6) gives
Z r=∞
r=∞
1
1
1
−
r 2 u 2 dr = r 3 u 2
· V̇ 2 ·
2
3
12π
r
r=R
r=R
r=∞
.
r=R
(D.7)
Simplifying this by setting u as given in equation (3.1) and considering the domain of
integration yields
1
1
1
1
1
1
2
2
2 1
2 1
−
−
.
(D.8)
· V̇ ·
· V̇ ·
· V̇ · −
· V̇ ·
48π 2
∞ 12π 2
∞
48π 2
R 12π 2
R
Combining this result with equation (D.2) finally gives
2
φKE = 2πρ0 ·
φKE =
ρ0 V̇ 2
.
8πR
25
V̇
16π 2 R
!
(D.9)
E. Derivation of equation (3.4)
Expanding the left hand side of equation (3.4) with ∂t/∂t yields
∂ V̇ 2 1
·
∂V ∂t
∂t
(E.1)
∂ V̇ 2 1
=
∂t ∂V
∂t
(E.2)
∂ V̇ 2 1
·
=
.
∂t V̇
(E.3)
∂ 2
V̇
= 2 V̇ · V̈ ,
∂t
(E.4)
Differentiation of the left term gives
which, multiplied with 1/V̇ yields finally
2 V̇ · V̈ ·
∂ V̇ 2
= 2 V̈ .
∂V
1
V̇
(E.5)
c D p p̂ pg pg,e pi pi,e pV pL p0 p∞ (t) r R (t) R0 t ~u V (t) V0 κ γ λ r φKE ρ0 Ψ σ η ηB Φ
26
List of Symbols
c
D
p
p̂
pg
pg,e
pi
pi,e
pV
pL
p0
p∞ (t)
r
R (t)
R0
t
~u
V (t)
V0
κ
γ
λ
r
φKE
ρ0
Ψ
σ
η
ηB
Φ
sound speed in the liquid
the material derivative
the sum of all steady and unsteady pressures outside the bubble wall
the pressure at a boundary within the fluid
the sum of all steady and unsteady pressures in the gas
the value of pg when R = R0
the sum of all steady and unsteady pressures in the bubble interior
the value of pi when R = R0
vapor pressure
pressure in the liquid at the bubble wall
the static pressure in the liquid just outside the bubble wall
the value of p very far from the bubble
range from the bubble center
bubble radius
equilibrium bubble radius
time
the liquid particle velocity
bubble volume
equilibrium bubble volume
Polytropic index of the gas
ratio of specific heats for the gas
the wavelength of the insonifying field
Principle component of strain tensor in the radial direction
the kinetic energy in the liquid
liquid density (assumed to be constant)
amplitude function
surface tension
shear viscosity of the liquid
bulk viscosity of the liquid
velocity potential
27
m/s
Pa
Pa
Pa
Pa
Pa
Pa
Pa
Pa
Pa
Pa
m
m
m
s
m/s
m3
m3
m
Pa
J
kg/m3
N/m
Pa s
Pa s

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29
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