Nonequilibrium effect on vapor bubble growth determined in study of boundary conditions at liquid-vapor interfac

Bornhorst, W. J.

Hatsopoulos, G. N.

English

NASA Technical Reports Server

. 
4 
Study of t he  Boundary Conditions at a Liquid-Vapor Interface 
through Irreversible  Thermodynamics 
bY 
Walter J. ~ornhor s t  
and 
George I?. Hatsapoulos 
Quarterly Progress Report 
Mach - May, 1966 
f o r  
National Aeronautics and Space Administratian 
George C. Mzrshall Space Fl ight  Center 
GPO PRICE 9- Huntsville, 
At*: PR-EC 
Department of Mechanical Engineering 
Massachusetts I n s t i t u t e  of Technology 
https://ntrs.nasa.gov/search.jsp?R=19660030652 2020-03-16T17:50:23+00:00Z
Assistznt Professor of Xec; t icel  Engineering 
Xassachwetts Ins-ti%zte of 'i!ec3?ology, Cadmidge, Mass. 
AZSliRCICT 
Tie object of t h i s  theoretical investigation is t o  d e t e m n e  the 
I-ortmce of the n o n e q ~ E k i m  region, v*hic.fl exists at the bubble mil, 
on t h e  grcrrth of a m ~ o z  bujble. A l l  previous investigators concerned 
~5th bubble grovth ksve c q l e t e l y  neglected Chis nonecpilibrium effect.  
I n  the pre-dous paper B o n h o r s t  and Hztsopo~los investigated the phase 
change problem, an& the resul ts  xhich were o3tained are applied here t o  
solve the special problez of vzyor bubble E;rc&h. 
9-e results ere preserted in t e r n  of z txo-parmeter set of curves. 
These resc l t s  shov t3zt for  pressmes >&ere eq?erimexttzl resul ts  are 
~-~-cCkble,  the no~eqailUx5-m effects are v2thin e q ? r i m n t z l  error; hw- 
ever, for  pressures ir5ich are smmkat lcT;Zr, the nonequilibriun effects 
becme very importat .  
1. nm.onmox 
\*?hen a vzpor bubble &ro~S,  nzter ia l  pzsses from one phase t o  moYner 
E z t  a net finite r a t e  ne= the  bubble wall . .  
eqKLibrim r e d o 9  mst e d s t  zt %he interfzce t o  a l l o w  for  t h i s  phase 
A very th in  but finite non- 
7 
This work vas sponsored by EASA contract x~s8-20013. 
-2- 
trcnsition. 
sv.ch as temperature a d  che3ica.l potential, lose t he i r  meaning, as 
also does tbe Fouirier hzat cmductim l a i r  alcng with other c o n t i m a  
e q m t i o s .  
pz3s thro@ this  noneqJili3riun regicn x l t h m t  2 corresponding drop 
in t eqe rz tu re  and cfieriicel 2otmtizL. Subsegyently, ye sBe2.1 speak 
of -L?-,ece &zxgec as ~A~rontinuLties,  realizing, hcxewr, thae they 
realll-y resul t  ~ f r c x n  a gap i n  %e t eqe ra tu re  a d  chenical potential  
3rofiles. 
In this nonequilibrim region t h e r n m d c  properties, 
We should not e-ect a f inite r z t e  of energy or w s  t o  
I3 order t o  connect t h e  bulk collsemt5n e p t i o n s  8cross the 
ronequi l ibr im region, it i s  necessary t o  hime z set of equztions vbicb 
r e h t c  the Znterfzcial d i scc~ t inu i t i e s  t o  t're flux of nass end ecerzy 
across the interface. 
vzs described in the pre-cims 322er by Bomhorst and Iiz-tsapvdcs 1x1. 
Tcis set of eqcatiozs, v-hich K i l l  be used here, 
In previous bub3le =crith m-dy-ses, t h e  effects of this non- 
eqailibri-aia region a t  the brrbble xiall have been cozpletely neglected. 
In other -$or&, it has elvays been sssmed Y z t  thz tenpemtcre and 
ckzaiczl potentis1 are  eqmL in the limid 2nd -i'qcr p'mses st thz 
?-ker%ce. 
md  t eqe rz tu re  at t c e  ia%erfzce sre r e h t e d  bg sattzl-ztion conditims. 
Su.xeqtlently, ve shall refer t o  t 3 i s  zss.~r$Azx zs the equi l ibr ium 
aszurrrptim. 
?pxen  rates  withaut -&ng this zssxzztion. 
s h a ~  tkat the equilibrilun zssc&5m muses bxge errors f o r  bubble 
grs&h rates zt loiq pressczes . 
%5s zssmiziticz is e p i v z l e n t  t o  assmixg t'?.t the pressme 
The pu-ose of %his pzper i s  t o  o5tFzia ecuations foz 3u551e 
!RE- res1.LL.s thus obtaimd 
! .  - 
-3- 
Tae first solution fo r  the problem of bubble growth was reported 
by F2,yleigh. [2] I n  his emlysis he considered mly dynamic effects 
. z ~ d  c o q l e t e l y  neglected t5e cooliw effect of evcporation. 
clxsion vzs that bu3bles grm ct a constarit velocity. 
ma Zick[f! ma othersrb, 51 obtained a s p q t o t i c  solutions 
H i s  con- 
U t e r  Plesset 
c 
for SufiYCiently large valizes of the mdius R) -id1 coq le t e ly  neglected 
uOiiEq&X.?ij52= effcets b:t i~cL-&eC the coolins effect of evaporation. 
T k L r  conclusion was tkz t  the cooling effect  is very importgnt 
t h e  bubble - i  velocitjj i s  pro2ortionai t o  the inverse square root of 
drx. This conclusion is iz WesneEt wit'n eqerimental bubble gcr,-th 
roesmezents =de by Eergsr&edim. [6] These r,.-esmments do not justie- 
%Le ecgiEorim assu!-l loi? Tor the geneml czse of bubale gro-wth because 
the expe r ina t a l  conditions of DergzraSedian's e-erinents were such thzt 
%he nonequilibriun effects %-ere miqor tznt  for h2s specific tests. 
that 
L 
2. " V . . I C N  OF BU33IZ PCiEEI EWLnIOX 
The goal of t3e present m2lysis is  to reduce the zppropriate con- 
sentzticm equations t o  z single a f f e r e n t i z l  eqwtion far the strbole 
rill velocity R. 
Tcrr both ezrly and I z t e  3iis-Dle growth. 
2.1 Rayleigh EquGtion 
This sei-ation should be st lcast azroximtely valid 
The nomntum equztion f o r  a sgherically symietric flak- of gn incczi- 
pressible 3ewto~zn fluid -*5th no body forces i s  
I .  -4- 
ur2 = qt) y 
:&ere 
ated zt the bubble wall by cozsidering as o v e r d l  a s s  bzhnce for the 
( t )  re2resents ul zrbitraqy fumtion of ti% vEch cesl be emlu- ’1 
- R2 U = R T  y 
r 
%*here we have assumed t’mt 
md clso thzt 
ilssuzption (L) is certainly very good es_cecizUy at low pressures, 
aid  t%e second assumption czn be physicalljr jtxstified since the bubbiP 
i...J2us chages orders of o - m k d c  while th? vaFor density vn8iz.tions 
~ Z F ?  relatively smll.f3] 
sti*Jtei! in to  (7), sad s i x e  tke ~ s c o u s  t e z  is identically equal t o  
zero, the resulting eqGat5.m c a  be integrated f r o n  the bubble v&I. 0x5 
50 ialinity. scross the interfzce 
C-uc t o  surfzce tensicn: %-e o3tcin the Emilizr 3ayleiglI eqwtion 
-- 
T?e eqression for u Given by (3) czn be sub- 
If Tie a l so  irxl-aie the pressure 
-5- J 
2.2 Er-ergy Equation 
The exrw equaticn a p r c p d a t e  f o r  the lip53 phase is 
xhere we w6-e use of equation (3) t o  express tbz convection velccity 
u 5a terms of R. Tne 
by q2155ng the first 
b2.3bLe 
3 w m & ~ r y  conditio2 zt t%e bubble xall is obtained 
l m  t o  a control voLme ccnsisting of the x q o r  
s m l l  t e r n  due t o  c m r e s s i b i l i t y  effects a d  
kke texperahre slope i n  the l iquid atst be srrch t o  ycupply the r ewi red  
e z z ~ g y  t o  enporate  tne liquid. 
To solve Equatim (7) i n  conjunctio3 x i th  (8 ) ,  ve ewloy En agroxi- 
-rL L 4 s v ~  i n t e p - i  technlqce devdoped bg- W8cck.  [7] This met??& ccilsists 
of sssuning a second-orcier t enpra tu rc  distribution in t5e l iquici  therm1 
bcmckry 7kyer. 
fke integrated distrilxtim. t o  szttisfy the enerm equation. 
The bm&ry l cp r  thic'mess is detemiccd by recpiring 
*-e &tai,ls 
of this a n z l p i s  along vith e. discussion comer ing  the val idi ty  o f  the 
integrzl zpproxizztitian i s  glven i n  Ap2endix A. Tne resul t  xhich is shoim 
to be valid fo r  low press ins  is 
-6- 
-&ere T 
i: fwct ion of fluid p r q e r t i e s  definel: by 
is the teuper&re 02 the liquid 
f i  
2.3 Irre\-ersible-Flm Ecpatims 
1x1 order t o  crniibixe scpations ( 6 )  and 
zt the bubble wall and % i s  
(g), ;:e need a relation betveen 
T .?,ad P If we were t o  admit tbermoZynaric equilibrium a t  the buB1e 
~ m l l ,  s;x& a relation vmld be given by the Clzpeyron equatioa; hmever, 
here ve sEaU avoid Yne ecpilibrium assmption by employing Equations 
{23) ~ .n& ( U )  f'rm the pre.r;,ous pa9er by 3ornhorst m d  Hatsqoulos. El] 
rr lilese equations ozy be -,=itten es 
&i g ' 
::kre 3 z~n.5 J desote thz e n e r g  znd mss f l u ,  respectively, K Lk, 
U i ? 
s7_d I. represent tramsport coefficients, 8xE %fie &efining eqcatioz f o r  ii 
s+? Pf - Pfs 
-7- 
-,.:here L~~ kxs replzced by mother transport confficient C, defined by 
tion, i i e  ca i  o5tzin 
-8- 
?&ere ATs is the superkezt of the l iqaid a t  infinity. It should be 
rotea t b t  t he  use of t:?e Clapeyron relation here iwlies nothing s;ba;zt 
t3e actuel eqi l ibr inm of the liquid-vzpor systen. 
(18) c m  be cozibined t o  y5eld 
Eqv,atlos (16) and 
2.4 Piral Differential Eqdztiox for Eubble Grmih 
Tnc f i n a l  dLffei'ectia1 e p z t i o n  is  o5tztaed fro= (19) by substikki_i?g 
. for (Pe - Fs) fro= E q a t i o a  ( 6 )  and for  (Tp - Tfi) frm Eqcation (9). 
.- Llie zesult is 
Li@d 3 n f a c e  Noneq=riEbriu=1 
Teas ion EfTect Ine r t i a  
Tie physical inteT-Jretz%iozl of Equzklo;? (23 )  i s  that the potentie 
. .  - . .  
-9- 
inezzia effects were iden%icr,lly zero. 
velocity R . 
diaenslcnless quantities 3 zn? R are 
I?e shall CZU. this bazbble wall 
The defining ec;ll?Aions for Ro, 30, zcnd the correspooang 
A +  
0 
-,-here 0 aaa M w e  functims, of the f luid TrcFer"-Lies aaa the initisl 
con5itions. m e  Cerining equzhions f o r  ,C e ~ , d  K are 
! 
(2b)  
It is rezdily seen t3zt 
vi i le  M i s  8 nezsure of iffie l i f p i d  i ze r t i a  effect .  
is a zecsure of the noneq~libr ium effect 
1 
-10- 
Gn eac3 figure a table is gim which re lates  the pastiailax values 
of 0 and M t o  a specific qera t iag  condition for  a specific liquid. 
Since K is  not a flmction of the  condensation, coefficient vc, we may 
tag a value of rc t o  ezch a u e  of @ for a given werating condition. 
The l inear  requirezent xhich can be obtzined from Equation (16), 
ezd the definition for  R E L E ~  ,@is that  
m e  mxd!im.m value of JfP/? for the cases considered i n  Figs. 3, 4, and 
5 is l i s t ed  on each figne. 
l i n e a  except for  nitrogen There d$?/P mzy be is large as 0.6. 
ir-g the superheat ATs a d  inceasing the external pressure P, , we c a  
get cases fo r  nitrogen yhicb ere l i ne=  aod still exhibit i q o r t m k  non- 
equilibrium effects. 
3.2 Significance of Xesulks 
A l l  the cases are  seen t o  be approxLmztely 
By redac- 
The resul ts  ob ta ixd  clezrly indicate the iqortance of the nonew-  
librim effects by ths spread of the m s  on Figs. 3 through 5. 
should be noted that the noDecpilibrium and ?mzentum effects decay fcster  
k i th  bcbale radius for the smlller values of p and M. 
30th ,c3 md M is t o  g r e s t l j  increase the extent of what has been called 
the  initial bubble grcxlth, tbt is, the gro?&lI up t o  the point vhere the 
It’ 
The effect of 
asymptotic solution becmes reasonably accur2te. 
In order t o  i U - s t r a t e  tfie points mentimed above, ue shall con- 
sider the e-le of a ~ p o r  bubble grcmring i n  vzter at an external pres- 
m I?, e& t o  0.178 pi. 
‘ 
Tne correspmfing saturaeion teqera ture  
-12- 
T is found to be 5lOo3.  For this value of To md a value of the con- 
imszt ion coefficient 5- eqml t o  unity, we find kom Fig. 1 a vake 
of 
m t e l y  equal t o  
3 obtzitl a vzlue for P of 1.0 a d  a Value for M of 10 . 
corresponds t o  N equal. t o  10 , t h e  m e s  on this figure represent the 
czsc zt “ZiiZ. Tzz 52,ffermt. V Z ~ P S  of 6 correspond t o  various valces 
of %be cmaensation coefficient Gc. 
S ~ I - P T S  the  real bubble velocity (ft/sec) versus the  real bubble rzdius 
(f%) fo r  vl,xiw values of Gc. For this ccso we conch3e that the nm- 
eqa2KDrj.ua effects a r e  v e r j  imgortznt for a lzrge portioa of the ImYble’s 
0 
C 
2 5 /A T~ eqpl  t o  0.01 m-6 fro3 Fig. 2 a velce of M/( A T ~ )  approxi- 
If ve choose t o  t&e ATs equal t o  10°F, ve t h s  
Since Zig. 4 
3 
I 
For the e:-ss@e considered, Fig. 6 
p x t h .  
3.3 Suggested Experittents 
A l l  existing experixe?trl data an bubble grawth rates correspond 
t o  such low values of /3 and 14 thzt the nonequilibrium ef&cts  we mix- 
p0-rtm-t i n  tl?ese ex-perizmts. 
3ubbl.e grovth measurezxdx zre needed at low pressures with tbe corres- 
ponang high values of ,G zn?i N. 
I n  order t o  &heck the ?resent theory? 
Since bub‘ole grovth r&es have bem folltld t o  be such a strong f‘mc- 
t i on  of ,6 , end therefore clso of 
measured frm bubble S m A h  emeriments perfoned at low pmss?mes. 
ojvious ad-mntsge of this type of experiment is t3e elinination of the 
contzxinatioa problencs ~,Mch seriously pbg-red most p rev iw zttempts 
G& it is mggested that Gc be 
One 
t o  raezs?.re rc, especlzlly those for tbe ncn-?7,eeCals uch 8s water. 
1. 
2. 
3. 
4. 
5. 
6. 
7.  
Bclmhorst, W. J. and Hztsopoulos, G. I?., J. Appl. Me&. (To be published). 
RaJrleighy Phil. M2g. sy 94 (1917). 
Plesset, M. S .  and ZKick, S. A,, J. A-1. Phys. 9, 493 (19511.). 
Forster, H. K. and Zujer, N., J. Appl. Phys. 3, 475 (1954). 
Scriven, L. E., me=. ET& Sci., 2, 1, (1959). 
Dcrgarz%eciia.n, P., j. of Exis X X ~ . ,  9, 39 (I?&]. 
14urdock, J. , Internal Regort i n  Xecharnical Ecgineerir! Departmnt , 
3hss. Inst. of Tech., 1965. 
EFZTDIX A 
E x r s  IDtegrzl Technique 
Most of the concepts i n  this appendix vere taken from a report by 
!kYrlo&. 177 
Tne object is to solve 
?.;here the n&ztion is the Sam es used i n  the b d y  of this p q e r .  
The other bcmndzrf cocdition far from the bubble ~-211 is 
(A-1) 
-14- 
a d  
T(r, 0) = Tp . (A-3) 
A t h e m 1  baun&ry laTer of thiclrness 6 is sssmed to exis t  i n  the 
!The tenperature prof i le  assmed i n  Vne qiq?rid oatsi.de the vapor bzb32.e. 
- .I_--- 
U W U I ~ ~ J  h:-er ic 
Tryere Tfi is the temperature of the liquid at the bubble w a l l .  
s i q l y  3 second-order cmve -vhich sz t i s f i e s  
‘This is 
T(R + d  , t) = T, 
%re it is conuenieat for us t o  define 8 IEW variable 
Cof9iniag (A-1) and (A-k) 15th (A-6) and theo iategrating from R t o  R +J , 
ve cbtzin 
-15- 
This different ia l  eqwtion czn be greatly s i q l i f i e d  by assuming thzt 
(A-8) 5c-Y 
which is j m t i f i e d  later in this appendix. 
Z:aua%ion (A-7) becomes 
135th t h i s  si?qlificetioa 
TcLs equztim can be i3tegrated t o  yield 
viiere t = 0 corresDonds t o  the tine when 
(A-9) 
(8-10) 
(A-12) 
If we coribine t h i s  775th the bomdary con2ition given by Equztion 
{A-2), ve find t h a t  
-16- 
Replacing in (A-13) from (A-lo), we get 
%-.-is c m  be integrated by noting the variable chmge 
ad, t3erefore, 
Com5inLng this resul t  with (A-13) we find t’ne desired result t o  be 
(A-14) 
( A-16) 
(A-17) 
(A-18) 
In order for this resul t  to ,e nemJ1-&‘ul, it must be trde that 
&/R << 1 (A-19) 
-am (A-16) ire cilil m i t e  
* 
I -  
i -  -17- 
(A-19) can be reslzczd by the condition thzt 
(A- 21) 
9 1 s  confition is usuallly satisfied, especially zt l m  pressures. 
Instance, kxter at 50'~ &-res a vakae of spproxiinzteu 10 
For 
-3 . 
SztisPying condition (A-21) does not coq le t e ly  justif'y the use of 
t h e  in5egml technique since s;e b v e  only szt isf ied the energy eqwtiozl 
f o r  t h e  integrated d5stiib;ztion and. not at ezch Doint i n  t??e boun7&z33 
L3yer. The integral  tecks5q.x i s  believed t o  gi-Te a god. a n r o x i m % i o n  
for t'ne following tvo rezsona: 
1. I n  the l i n i t  of iarge R (asymptotic solution), this technique gives 
results xhich are ritlzin 2$ of the r e s u Z t s  ob'ainea by Plesset and 
Z-*%ck[3] and Scrivez[5] f o r  the z s y q t c t i c  region of 'rabble S O i C l .  
If a l i n e s  teqerata-e profi.19 is used instezcl of tine parabolic 
onet Yle results chmg? by only 1%. 
t 3 e  tmperatxre zzofile to be r e k t i v e l y  mimportznt . 
2. 
Zais izGcctes the shz2e of 
A33XXCX 13 
S i q l i f i c a t i o n  of Mzss Flvx Equztim 
I Tke ime-rersible EZSS flux (Equation (15)) c m  be m i t t e n  as 
-18- 
where the notztion is tbe s m e  es i n  the body of $his paper. The pur- 
3ose here is t o  shov tht tfie first tern i n  the brzckets is very swll  
ccr;Dz;red t o  the  second. 
This is accorqlished by noting that tke slope of the =par t e q e r a -  
b e  profi le  at  the bubble vall starts out as negztive a d  rerains negz- 
hive tI??cu&mt bubble grGth. 
the t eqe ra tu re  at the k~b31e trzl l  decreases mchgfaster due to 13;teEt 
beat requiremnts t k n  does the bilk teGeratcr-e of the v q o r  Cue t o  
A. The profile S&&L%S out negative becmse 
tkc vork of the pressure forces. To shov this, ve calculzte the order 
of =@tu& of the ml;iri te-Terature decrezse inside the bubble when 
t h e  buSble volume doubles froia its initial unst5ble equilibrim sA&te. 
For tPLs chvlge the vork is  
(3-2) 
vh2re rj, is the ioitiel m s s  of vqor i n  t'ae bubble. 
ci-mge in the average tezipera-kure T 
The correqon&ing 
of tine vapor mst be m 
(3-3) 
1.le fom a r a t i o  by dividing 3y ATs and use t k e  aerfect gas l2v t o  a b t s h  
(3-9) 
0 This  r z t i o  is extremely SEAI.. 
of about 
For emple ,  TEter at 50 F gives a =he 
-19- 
Tne corresponding Cecresse in  the tTau temperawe c a  be sham t o  
mch Larger thzn 6 Tm. I n  AEendir. A it was fa;u?d t h a t  
. 2 < R 2  R =  3 . (B-5) 
R3 - Bo 
iyov ve mint t o  find the value for 8, when the bubble volume has doubled. 
or us-h~le the definition of the dimensior?less -,-elocity, we find t'nat 
then procoed t o  shmr that c,r T- is s d l .  
-20- 
The e n e r a  flux into %be bubble at the bubble wall. can be eqressed 
ory 3y using the irreversible 
Y 
energy flux, 
~ c ; l ; ~ i ~ i n g  (B-8) an2 (B-9) ve get 
as 
(B-9) 
(B-10) 
3 m  (B-10) ?re see thct J T x3.l kzve i ts  largst absolute value ?&en 
kg( 2 T/ 2 r)r=R, has its Largest algebraic value, which IVES shmn cbx-e 
50 be zero. I?nzs it- f o l l o ~ s  tFa% 
(3-12) 
I c 
-23.- 
e 
The c p n t i t y  1 hzs been defined so t hz t  it i s  the ra t io  of the 
In  ather trords %-ie need t o  shov 5 t e x s  viich d d  i n  Equation (B-1). 
t o  be mall. Anticipating 2 t o  be a smaU. xrde r ,  we can simplify - 
(3-12) as 
(3-16) 
=e r i&t-hnd side of (B-16) is znroxia te ly  10-I indicating that  %ne 
t e n  Tie neglected is 2% lest one order of ZgLtude smaller than the 
otAer tern i n  Eqpztio2 (B-1). 
Critical 
I ernperature TC Fluid - 
tiydroqen 60" R 
a 
N i Erog e n 228" R 
O x y g e n  278 O R 
I i 6 5 " R  
**?n-.-%***,, $$ lGi  CIUl J 31 ! 9 0 R  
Potassium 3555" R 
Reduced Temperature TR = To& 
.XERO'i 
C O P Y !  
i XERO 
. C O P Y  
. 
r- 
tx, 
I I I I 
. io .20 .30 40 .50 .60 . 70 .e0 
Reduced Temperature TR = To/Tc 
Fig. 2 Dimensionless Fluid Parameter M Versus Reduced 
Dimension ~ S S  Bubble Wall Velocity R x I C ~  
N 
0 0 
0 .,. 
z 
0 
I f  
U 
v, 
0 
Ic 
0 
3 
-9 -. 
T r  
i! -- 
I ,  
. . ,  
I I 
Fig. 3 Dimensionless Bubble Wall Velocity Versus Dimensionless 
I 
--. . 
. 
Q 
. Dimensionless Bubble Wall Velocity R x IO2 
0 '  Iv -P 
w 
1 I I I n 
Fig. 4 Dimensionless Bubble Wall Veloci y Versus Dimensionless 
B&&le Radius with M Equal to loYERO 3 
I--- $ h' - .T -- .-_ -. - . ---- .---. -_- rlcopI,- - 
'- , , ,. . , 
CI1 i n  . 
c 
e 
Dimensionless Bubble Viol1 Velocity R x IO2 
0 
0 - p p -b 01 
Fig. 5 Dimensionless Bubble Wall Velocigy Versus Mmensionzess 
Bubble Radius with M Equal to 10 
COPT I 
. -. )I___ 4 XERO 
? C O P Y  ' 
XERO' 
0 
.. 
_.  . .... . 
, \  
Bubble Wall Velocity R (ft/sec) 
0 
P 
Fig.  6 Bubble Wall Velocity Versus Radius fo r  Water with To 
Equal t o  50 0 F and A T, Equal  to 10 OF 
"' ,:o 
n, < - XERO _ -  , I .< '  . - -_  - .  . XERO Y- - , C O P Y ~  " ,COP";- -- , t ,  


Bubble formation and growth - Study of the boundary conditions at a liquid-vapor interface through irreversible thermodynamics Quarterly progress report, Mar. - May 1966

Bubble formation and growth - Study of the boundary conditions at a liquid-vapor interface through irreversible thermodynamics Quarterly progress report, Mar. - May 1966

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