I n g e n i e u r - A r c h i v 48 (t 979) 8 5 - - 95
IngenieurArchiv 9 by Springer-Verlag 1979
T h e C o l l a p s e of a B u b b l e A t t a c h e d to a S o l i d W a l l * A. Shima, Sendai and Y. Sato, Tokyo
Summary: T h e p r o b l e m of t h e b e h a v i o r of a b u b b l e a t t a c h e d to a solid wall w a s solved n u m e r i c a l l y b y t h e v a r i a t i o n a l m e t h o d , in w h i c h t h e v i s c o s i t y a n d c o m p r e s s i b i l i t y in t h e liquid are n e g l e c t e d . T h e effects of t h e s u r f a c e t e n s i o n a n d t h e c o n t a c t a n g l e on t h e collapsing b u b b l e were t a k e n into a c c o u n t . As a result, it w a s clarified t h a t for t h e c o n t a c t a n g l e c~ < 90 ~ t h e p r o j e c t i n g p a r t of liquid a d v a n c i n g a g a i n s t a solid wall was f o r m e d in t h e collapsing process r e g a r d l e s s of t h e a m o u n t of g a s in t h e b u b b l e a n d t h e b u b b l e size, a n d t h a t t h e effects of t h e W e b e r n u m b e r a n d t h e c o n t a c t angle on tile b u b b l e s h a p e were large, while t h e effect of a m o u n t of g a s in t h e b u b b l e w a s small.
Obersicht: Mit HiKe der V a r i a t i o n s r e c h n u n g wird der Kollaps einer a n einer f e s t e n W a n d h a f t e n d e n Blase a n a l y s i e r t . D u t c h n u m e r i s c h e R e c h n u n g e n fflr v e r s c h i e d e n e W e r t e des iRandwinkels wird dessert EinfluB a u f die B l a s e n g e s t a l t b e i m K o l l a p s u n t e r s n c h t . E s zeigt sich, d a b sich bei B l a s e n m i t e i n e m R a n d w i n k e l c~ % 90 ~ u n a b h ~ n g i g v o m G a s i n h a l t der Blase u n d y o n der B l a s e n g r 6 B e ein a u f die W a n d g e r i c h t e t e r F l f l s s i g k e i t s s t r a h l bildet. W~thrend die B l a s e n g e s t a l t n u r w e n i g v o m G a s i n h a l t a b h ~ n g t , b e e i n f l u s s e n die W e b e r s c h e Z a h l u n d d e r R a n d w i n k e l die G e s t a l t der B l a s e erheblich.
1 Introduction The problem of the collapse of a bubble attached to a solid wall was first treated by Naud6 and Ellis [t]. By the use of the linearized perturbation theory of Plesset and Mitchell [2], they analysed the microiet formation in an non-hemispherical bubble. Further, high-speed motion pictures of bubbles generated by spark methods were used to test their theory experimentally. Experimental results on the shape of bubbles collapsing in contact with a solid wall confirmed their theory. Based on these results, they offered a partial explanation of the increase in the damage potential of bubbles when the initial perturbation of the hemispherical shape were reduced and pointed out that the impact velocity of the jet, another parameter, possibly the radius of the tip of the jet, seemed to influence the damage. Chapman and Plesset [3] analysed two cases of initially nonspherical bubbles. For the first of these, the initial bubble shape was roughly that of a prolate ellipsoid. The other case had an oblate initial shape. Their results show that a jet was formed when the bubble collapsed. In their study, however ,the surface tension of liquid and the existence of gas inside the bubble have been ignored. Recently Shims and Nakajima [4] solved these problems by expanding the variational method for the dynamics of non-spherical bubbles by Hsieh [5]. They clarified that the surface tension accelerates the collapse of the bubble and promotes the deformation of the bubble, and that the internal gas acts in braking for the collapse of the bubble. In these studies, the interfacial tensions at the solid-gas interface and at the solid-liquid interface were * T h e a u t h o r s are v e r y m u c h i n d e b t e d to Mr. N. Miura, Mrs. M. T s u j i n o a n d Miss S. T a k a h a s h i for t h e i r assistance throughout the investigation
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Ingenieur-Archiv 48 (1979)
ignored, t h a t is, the angle of contact was assumed to be 90 ~. However, the contact angle differs with the bubble size and the state of solid wall. Therefore, the effect of contact angle on the bubble behavior is very important. In this paper we analysed the behavior of a bubble attached to solid wall b y considering the effect of the contact angle. The contact angle was assumed to be constant through the motion of the bubble.
2
Basis Equations
Let us consider an axially symmetric collapse of the bubble attached to a solid wall. The spherical co-ordinate system is shown in Fig. t. In the figure 0 denotes the origin of the coordinate system, and R the distance from the origin to the bubble surface. The angle 0 is t a k e n as shown in the figure. The following assumption will be made: (i) (if) (iii) (iv)
The The The The
liquid is incompressible and invicid. gas inside the bubble is uniform and its flow is ignored. gravity, heat transfer and gas diffusion are ignored. contact angle c~ is constant independently of time.
Under the above assumptions, the motion of the bubble m a y be determined b y solving Laplaee's equation about the velocity potential @: r ~ ~r
~r/ ~- r2sin0 O0 sin 0 ~
r2
= 0,
(t)
where r is the distance from origin 0. The following b o u n d a r y conditions (a ~-~ d) and the initial condition (e) will be used. (a) The condition at infinite distance q~=0
for
r-->oo.
(2)
0 =--.
(~)
(b) The condition at the solid wall r
c~O
-- 0
for
2
(c) The condition of the contact angle Ro
_ _
lf~ + sg where
R o =
+ cos ~ = 0
for
0 =--,
(4)
2
OR ~-.
8=0 ~ Liquid
/~ ~/
8=90 ~
0=90 ~ Solid
Wall
I
Fig. 1. The spherical co-ordinate system
A. S h i m a a n d Y. S a t o : T h e C o l l a p s e of a B u b b l e A t t a c h e d t o a S o l i d W a l I
87
(d) The conditions at the bubble wall
Or
R 2 O0 O0
for
at
r :
R(O, t)
(5)
and 0~- - } - 2 - \ O r ] @ ~ \ O O ]
~---7 Poo -- fig -- Pv @ cr ~ @ ~
for
r = R(O,t),
(6)
where Poo is the pressure in liquid at infinity, pg the gas pressure inside the bubble, p~ the vapor pressure inside the bubble, ~ the surface tension of liquid, ~o the liquid density, t the time, and R~ and R e the radii of principal curvature at the bubble wall, respectively. The mean curvature is given by E4] 2
~
- ~
= 2
R ( ~ + Rg)li" q-
(R~
~)572- j '
(7)
where OR
8ZR
Ro -- ~ ,
Roo -- oO~
(e) The initial conditions are OR
R o=R(O,O),
3 3.2
0~-=0
at
~=0.
(8)
Analysis by Variational Method Functional
The problem governed by (1)~-~(6) can be shown to be equivalent to a variational problem: The solution of the above boundary-value problem is given by an extremum of the functional
J=
dt I f
2~ tl
sinOdO
~ qbt + ~~~ r a qS~2 + -2r
r~ d r _
L0
pg @ )0v -- jgO~ R 3 --~ f i R FIR 2 --~ R
3
@
2 ~ COS ~ "
R2]o=.~/,~
(9)
where q5 = 0 at r -~ co. The functional J is regarded as the energy in the whole system under the assumptions (i)~(iv). Therefore, the motion of the bubble will progress to minimize J. The first variation of (9) gives
dJ :
2=
dt|f tI
sin0d0
tO
~ d~bt+ qoc~c#r + ~ q ) o e g q ) o r 2drqI~
=/2 +
sin0d0
--0
q~t+2
~" + 2 ~-~ R
--
0
(2R~ + R~I ~R + zero ~RoI
]
--
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Ingenieur-Archiv 48 (1979)
After integration by parts, we have
2~z
dJ
,.[ f
a/2
tt
o
co
sin
dt
0 dO
r ~ dr 0
- ~ (r~(I)r)r @ r e sin~-O
R
~1~ + 0
~/2
; s i n 0 d 0 0 R ~ {~
,2,
Pg -{- Pv -- poe
-+- ~ c ~ ' ~ +
+ 2 r
0
( R
-
Ro
oo~
0
R~ + ~R~
RR00~I
-
(1o) s~(=l'~, t)
Then (t) as Euler's equation, and (5) and (6) for r = R(O, t), (3) and (4) for 0 = re/2 as natural boundary conditions were obtained from the stationary condition dJ = 0 of functional J.
3.2
Approximate Solution
Considering the boundary conditions (a) and (b) and the condition of contact angle (c), the trail functions of q~ and R will be expressed as follows: ~b(r, 0, t) = %(t) x ~92,(t) 20% (cos 0) 7 7 +,,~1
(tt)
and N
R(O, t) = Ro(t) So(O) + X R~(t) &,,(O)
(~2)
where 9o(t), 92~(t), Ro(t), and Rz~(t) are unknown coefficients, which depend on time; P2,,(cos 0) is Legendre's polynomial; Io(0 ) and I2~(0) are functions of angle 0 only. Thus the variations of ~0 and R are written as follows"
(13 a) and N
(~ 3 b)
OR = ORo Io(0) + X dP2k &k(O) Substituting ( 1 t ) ~ ( t 3 ) i n t o (10), we obtain
2~
k=o ~,t
R2k+lj &P2k -LO
N
-
-
~
1
2
1
sin 0d0o~R ~ q~t + 2 ~ + ~
e ~ +~
I2k ~R~k 9
2
~b~ +
Pg + Pv -- poo
e (r
A. Shima and Y. Sato: The Collapse of a Bubble Attached to a Solid Wall
89
Since d~%k and dR,% are a r b i t r a r y , we obtain the following equations as E u l e r ' s equations: a/2 Rt -- go, + - ~ T - ]
~
sin 0 dO :
(~5)
0,
0
.~/2
got + 2 fro, -[- 7R 2 goo +
q
-
R2Ie~ sin 0 dO = O,
+
(16)
0
where k = 0, t .... ,-h r 9 In vector n o t a t i o n (t5) and (t6) can be a r r a n g e d as follows:
(~7) (~8)
M,P + F [ ~ = O , Fr4~ + S = o , where the column vectors are defined as
q~=
[~0
... ~xTT,
(t9)
R=
[R o Re ... R2A~]T ,
(20)
~
S=[S0
S, ... Sx] r ,
(2ta)
~/2
dO ] 0
,{
+ a R~ @
o Poo-Pg-P~
R e I2ksin0d0
(21b)
2/)
a n d the m a t r i c e s are given b y =/2
M =
IMp,,]
,
R2,,+2k+ 1 (2n -}- t) P2,a @ ~-" dO J s i n O d O ,
Mkn =
k, n = O, t . . . . . N ,
(22)
0
~/2
F =
[-~k.],
~Tkn =
f -l::'Zk I2n -sin R2k_ 1
0 dO,
k, n = O, t . . . . . AT.
(2:})
2t/-1FR
(24)
0
F r o m (t 7) =
--
.
S u b s t i t u t i n g (24) into (48), we obtain the e q u a t i o n of the m o t i o n of the b u b b l e as follows:
= F-~E(aiM-1F -- F) R + M(r-~) ~ S~
(25)
where
s
= -
R2,,+2~+~ (2~ + 1) (2n + 2k + (26)
0
RRO dP2n. - dO --@ (2n -l- 2k @ 2) R
Ro dO ] s i n 0 d 0 '
k,n=O,l,...,N,
J
•/2
P
=
[PkJ,
P ~ . = - ( 2 k - t) f ~P2k&~ f~ sin 0 dO 0
k,n=O,t,...,N.
(27)
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Ingenieur-Archiv 48 (t979)
T h e m o t i o n of the bubble m a y be d e t e r m i n e d b y solving t h e simultaneous o r d i n a r y differential equation (25). Here the gas inside the bubble is assumed to obey a polytropic change. I n this case, pg is given as follows: G = Peo
[ co/1' Ly-~- j
(28)
where Pgo is the initial gas pressure inside the bubble, y the polytropie index, and V(t) is the bubble v o l u m e at time t, which is given as follows: a/2
V(t) = v ~r f Ra(O, t) sin 0 dO.
(29)
o
3.3
Non-Dimensional Expression of the Basic Equations
For the purpose of non-dimensional expression of the basic equations, the following quantities are i n t r o d u c e d : R -- Ro'
Roo'
poo -- p v '
(30)
where R o is the radius of c u r v a t u r e of the bubble surface at time t = 0. D r o p p i n g bars in the o b t a i n e d equations, the e q u a t i o n of the bubble m o t i o n is rewritten b y the same expression with the exception of the following c o m p o n e n t :
0 --
I --pg+~--
~-t-E
R 2 Izksin0d0,
(21b')
where W --
4
Results
of Numerical
Ro(P~ - Pv) 2~
(Weber number) .
Calculation
We calculated each case of the c o n t a c t angle e = t00 ~ 90 ~ and 60 ~ assuming t h a t the radius of c u r v a t u r e of the bubble surface is c o n s t a n t and equals to R o at time t = 0. The following e q u a t i o n was used as a trial function of the bubble radius R : N
R = RJo(O) + X R2,,{P2~(cos 0) -- P2~(o)}
(31)
where Io(0 ) is the function expressing the initial bubble shape which is given as follows:
Io(0 ) = cos ~ cos 0 + V't - (cos 0r sin 0) 2 .
(32)
The calculations were carried out keeping 8 terms (up to N = 7) in (3t).
4.1
The Relation between the Contact Angle o~ and the Variation with Time of Bubble Shape and Bubble Wall Velocity
Figures 2 a~-~c show the variation with time of the bubble shape for each case of c~ = a00 ~ 90 ~ and 60 ~ respectively. These figures are for the case of W = t00, Pgo = 0.1, Y -- 1.4, and z represents non-dimensional time.
A. S h i m a a n d Y. Sato: T h e Collapse of a B u b b l e A t t a c h e d to a Solid W a l l
r
=
W=IOO Pgo=O.J Y= 1.4
8=0~ ~
O
9t
&
W=IO0 8 = 0~
7
-
=
5go= O. I
~
7"= 1.4
o. oo
8=90 ~
~=9o
o
b 8 =0 ~
W = IO0
• ~ = 6o ~
8:90 o~Th2//////////////x///w////)///22y//~
Fig. 2a--c.
8:90 ~
V a r i a t i o n of b u b b l e s h a p e wiLh t i m e for different contac~ angles cr a t 0 0 ~ b 90 ~ c 60 ~
W=lO0 p~o=o.I
8=(Y
~
y = 1,4
5' = 0 ~
cz =6o ~
[
/'
', r=l.O
/! W//////////~,
R(O,r):VELOQTY VECTOR
/
8=90~
~, 7-=1.085 \
/
x
'
~ 8=90~
F i g . 3. D i s t r i b u t i o n of v e l o c i t y / ~ ( 0 , •) on b u b b l e s u r f a c e 7*
Ingenieur-Archiv 48 (t979)
92 0
.ec -I.5
93 - 6
-2.0 -25 "
0.4
0.8
I. 2
0
T ~ a~
a=too ~
0.4
0.8
1.2
-I0 0
T
b
a= 9 o ~
e
0.4
0.8
1.2
o~= 6 0 ~
Fig. 4 a - c. Relation between contact angle ~ and variation with time of /~(0, T); W = 100, Pc0 = 0.1, y = t.4
For the case of c~ = 60 ~ the projecting part of liquid was formed on the bubble surface against the solid wall in the final process of the collapse. Figure 3 shows the distribution of velocity on the bubble surface in this case. The velocity at the bubble surface increased considerably in the neighborhood of the projecting part. It can be considered that the jet of liquid would be formed as the collapse proceeded further. The shape of this projecting part was quite different from that of initially prolate bubbles with the contact angle c~ = 90 ~ treated by Chapman and Plesset [3] and Shima and Nikajima [4], On the other hand, for the cases of e = t00 ~ and 90 ~ the remarkable changes of the bubble shape did not appear within the range of calculations. Thus, the jet formation could not be predicted. The behavior of the bubble with the contact angle of e = 90 ~ coincided with the case of spherical collapse. Figure 4 shows the variation with time of the bubble wall velocity in this case. In the figure the axis of ordinates shows the non-dimensional velocity. For the case of ~ ~- 100 ~ the velocity along the solid wall (0 = 90 ~ is larger than the velocity along the symmetric axis (0 = 0~ For e -~ 90 ~ the bubble wall velocity is independent of 0. In either case the rapid increase of the bubble wall velocity cannot be found within the range of calculations. On the other hand, for the case of c~ = 60 ~ the velocity along the symmetric axis is large compared with t h a t along a solid wall. The difference between these velocities increases rapidly as the collapse proceeds. For example, the velocity at T = t.085 is about t0 m]s under the pressure 1 atm for 0 = 90 ~ while about 80 m/s for 0 = 0 ~ Therefore, when the collapse proceeds further and the jet strikes against the solid wall, it can be predicted that the jet will exert the destructive action against the solid wall. We find from these results that the contact angle plays an important part on the process of the bubble collapse.
4.2 Effects of W, pg, and V Let us next examine the effects of W, pg and y on the variations with time of the bubble shape and the bubble wall velocity in the case where ~ =- 60 ~ (i) Effect of W. Figures 5 a and b show the calculated results for W = t0. In Figure 5 b the results are also compared with the case of W = tO0. I t is seen from these figures that the rate of the bubble collapse in the case where W - = tO is accelerated as compared with that of the case of W = tOO. Also in the case of W ~- t0, the bubble has a roundish shape and the projecting part is formed sharply at the end of collapse. Figure 5 b shows the variation with time of the bubble wall velocity. In the figure, a solid line shows the case of W =- 10 and a dotted line the case of W = t00. The bubble wall velocity in the case of W = 10 is larger than that of W = 100. The velocity along the solid wall (0 = 90 ~ hardly depend on the value of W, but the velocity along the symmetric axis (0 = 0 ~ differs remarkably depending on the value of W.
A. S h i m a a n d Y. Sato: The Collapse of a B u b b l e A t t a c h e d to a Solid Wall
93
~0
I
~'~o=0. I Y=I,4 ,2 = 60 ~
8 .....
w=lo
,4~
W=100
i
T 8=0
T =0 /
~
~
~
i [0=021 r /
W=IO 7" = 1.4 ,2 = 6 0 ~
!
~
9C 0.4 &
b
0.8
1.2
T
F i g . 5. a Variation of bubble shape w i t h time, b Effect of W on ~(0, T)
(ii) Effect of pg. Figures 6 a and b show the calculated results for pg = 0. In Figure 6b the results are also compared with the case of pgo = 0.t. The variation of the bubble shape in the case where pg = 0 is almost similar to that of the case of pg ---- 0.1. However, the rate of collapse becomes large as compared with that of the case where Pg0 = 0.1. Figure 6b shows the variation with time of the bubble wall velocity. Both of the velocities along the solid wall (0 = 90 ~ and along the symmetric (0---- 0 ~ is larger than that of the case of Pgo-= 0.I.
I0
I w = Ioo 7=1.4 Q = 60 ~
I 6 8=0 ~
W= IO0
rJ
II
II
i
i
I--,
8=0~
1 / ///I
4
2
jjjjjjjjjjjjjj ,,, jjjjjjjjjjjjj!fj9o~ e 0
&
b
Lz" 1
Q4
Q8 T
Fig. 6. a Variation of bubble shape with time, b ]Effect of pg on/~(0, ~)
0~
1.2
94
I n g e n i e u r . A r c h i v 48 (t979) IO
b
W = I00 Poo=O.t o~ = 60 ~ ~'=t,O -
-
. . . .
i
? ' = 1 . 4
r
6 8=0 ~
W=IO0
e=o~~!
I-,
}/H
4 - -
2
0 &
0.4
F i g . 7. a V a r i a t i o n of b u b b l e s h a p e w i t h t i m e , b E f f e c t of 7 Oll
0.8
1.2
2~(0, T)
(iii) Effect of 7. Figures 7a and b show the calculated results for 7 = t.0. A comparison with the case of 7 = t.4 is shown in Fig. 7b. The variations of the bubble shape and the bubble wall velocity with time are similar to those of the case where ~ = t.4.
4.3
Variation with Time of Coefficients R~(~) as a Function of Time
Figure 8 shows as example of the variation with time of coefficients R~(z) as the function of time. The amplitude of the coefficients R . increases with the collapse of the bubble. The astringency of R . tends to fall at the end of collapse. This is a reason why the calculation was terminated on the way.
0.2
- -
,7" t-,
at ,-r -0.2
-0.4
- -
--
0
W = I00 ~go= 0 . I y=l.4 a =60 ~ 0.8
0.4
1.2
7-
F i g . 8. V a r i a t i o n of coefficients
R2n w i t h
time
A. Shims and Y. Sato: The Collapse of a Bubble Attached to a Solid Wall
5
95
Conclusion
We analysed the behavior of a bubble attached to a solid wall by considering the effect of contact angle ~. In this analysis the effects of the compressibility and the viscosity of liquid, and of gravity were ignored. The contact angle ~ assumed to be constant through the whole process of the bubble collapse. We also assumed that the radius of curvature at the bubble wall was constant and equaled to R 0 at time t = 0. The calculations were performed for each case of the contract angle ~ = 60 ~ 90 ~ and t00 ~ As a result, the following matters were found: (t) The deformation of the bubble shape in the process of collapse is remarkably different depending on the size of contact angle. In particular, for the contact angle e < 90 ~ the deformation at the end of collapse is remarkable. (2) The smaller the Weber number, the larger becomes the rate of the bubble collapse and the more remarkable becomes the deformation at the end of the collapse. The existence of gas gives a strong effect of the rate of the bubble collapse. (3) For ~ ~ 90 ~ the projecting part appears at the bubble wall in the process of the collapse, and the velocity of the projecting part increases remarkably with collapsing of the bubble. Therefore, it m a y be conjectured that the jet produced b y the projecting part strikes against the solid wall as the collapse proceeds further and it will cause the destructive action on the solid wall.
References
1. Naud6, C. F. ; Ellis, A. T. : On the Mechanism of Cavitation Damage by Nonhemispherical Cavities Collapsing in Contact with a Solid Boundary. ASME Transact. J. Basic Engng., Set. D, 83 (t96t) 648--656 2. Plesset, M. S. ; Mitchell, T. P. : On the Stability of the Spherical Shape of a Vapor Cavity in a Liquid. Quart. Appl. Math. 13 (1955) 4t9--430 3. Chapman, R. B.; P1esset, M. S. : Nonlinear Effect in the Collapse of a Nearly Spherical Cavity in a Liquid. ASME Transact. J. Basic Engng., Ser. D, 94 (1972) 142--146 4. Shims, A.; Nakajima, K. : The Collapse of a Non-Hemispherical Bubble Attached to a Solid Wall. J. Fluid Mech. 80 (1977) 369--39t 5- Hsieh, D . Y . : On the Dynamics of Nonspherical Bubbles. ASME Transact. J. Basic Engng., Ser. D, 94 (t972) 655--665
Received Augusl 26, 2978
Prof. Dr. A. Shima Institute of High Speed Mechanics Tohoku University Sendai, Japan M. Eng. Y. Sato Mitsubishi Heavy Industries Ltd. Chemical Plant Engineering Center Tokyo, Japan
