e-

,.. .-. -. . Thr Bursri~~g of Snap Films. I!. Thcore~icul C o n s i d e r n l i o n 6 1

b~ S e n l q Rankd and Ram1 J. Mywllr 111 ,Y. Nwd, Lu A*d<#. Cai~-h 90Mb, and R-e Dwdmnu, n. I. Remrd . r&m Corn-. n ~ u l o n ~ o h . S ~ I A c d i i mm wahal k-O, ,~bn

.Cer~-cml I W ~ J P ~ ~ A of 1110 bumlil~ olll~in liquirl films ql low a.iwircopity em oonsklnrd, rt&cctin~ air &lnnbc. I L i. xlmnw.n tltnt ;m)- aimificnn1 bum~lc l~rccslin~ the rim or tile c?pandlnx holc ip rclnlell to la r~c chnnp of surf~tcc Lcnainn h* lhc Rlln shrink. nuld rhiekcns. SlclhOds lor cnicuiatii ffiwc cbnnp rmm alrmvnblc rla~trltil*. nm Ricm both lor the mncrnl cruc nnrl [or rclr+imilnr ln tn t i~~r in rvi~ich nll fmtuw or the numole cxpnd lincnrlf ail11 tilna. Thc latter rrprc3cnD a limiti~~a c m in .111ich tho rdantion of surfurn tcnsion is ~tr~lip~hlc rluil~~e LunrI~l& This limiting cast k dimwed in mom dctnil for ullirii~laiollo1 blmtin~ ond mcvcr:tl uxlar;nncrlnlly r.l~rrwl Ienturea 01 the buratinn d aunp films RE wmuntod lor quuiibli\-c$.

Aftcr a thin lirluid film such fir e aoap mm dmdow mlculnred for an undisturbed film. AII n-lc hns tha tinicvt holc, ~urlacc t n ~ n b n f o r c ~ nil1 CILUSC this bccn ah0 n o l d ky J+icbm,n, d of." holc to m w m l d Lhc film to disnpponr. n ~ i s i~ m nlpid Thc pmca t p p r is concerned with the lunhcr n pmcw thal lhn puldimbcd expcrimcntal o l u d i ~ .1IC dmiupmcnt of Lhc theon- of bunLill~inmuedi~turbod r d t c r akctch)-. A thvorcllcal cslimstc or the rulc :tt film am ran iinporL~l~t limiting nsc, and with the mlc of which rho holcgrbn~ 1r.s givcn by Duprdover nwlllury ngo.' Thia r~rimnLcwns t n d o ~ l l>y I(i1ylcig11: hut in -Mi" "SF, nL,b le,hNn b"nl Mr(insd hFhriun 1000, Culiek' a h f ~ m l lPn1 i l invdvml n bnvic Cmr chrrnld amkly. am ymndlco, a t i t . A,,,+[ lm. which c w ~ ~ ~ c r ; t l c d lho rnlc Ir n lnclnr af &. Tho IZ) A. Dtt~w@. Am". Chim. rhw., (4>11, IW !lmi); .m&t implc pmmir llvctl hnlh ;?up& ctlliel; thnt . Ll~ulquc do In Clnnlmr." G n ~ L h i i ~ - V 1 1 1 ~ . PWL 11115.

thcPrOCFsH ,,nn,"RoUJ lo conlrnElion of L,,CJUrfncC 0) l a r d Rnvldah. r'roc. RON. fn*.. 13. -1 IIROI): .Kdsm. 44.249 [IROI): "Sda~tiGc I'amr9;' Dmm, Sew Yark. N. 1'. Vd. 111,

of D puw liquid in that the rurfncc tension rcrnnills con- IW, p a s . sLqnt cvcry~rhcrc. Aa a comllary, thu film muat hc un- (0 F. E C. Crlirh I. Appl. IVr.. 31, 1 1 3 (1000).

dialurbd crccpt ,,t lhc dm of lllc llolc ~ 1 , ~ (6) A. 1. i Y c M a , Re. Tmr. Chi.. I'ls-Rru, *. 38s OW)): R u k am. TKhnsl,. Sl, 1149 (IDbB).

collapcd mnlcrii~l eollml~ in n tomidnl rim. This 1.a. la) w. E. J. Ad.pPhw,, 1m (lw, in ngrccmcnt a i th cxpcrimmtnl uutk nmilnl~lc on (i) L o%g*~qm d. ~i~,,lde.- vd. n. c.-vulur burrtine?-'a nlllooug)~ n lnrw hody of other aludios, ic- l'&, 187.7. p as: AlelL A d . Rw. I3dai.m. ra)r.3 clB1)-

clvdinp: Oibhs' cotuidcmniiona 01 sttrfnec clmticity," (8) G. van dm S l l m r ~ c , Ann. &c. SdmWw B.rall.l.11.1 ClW. showed thnl nolulioaz, nnd cspccidly lhih film olaolu- ciU C, Xtvvlnai wd P. s-,. Xlmdm

tionr, d o no1 hrrm it r n n s l ~ ~ n l e u r i * ~ rcn*ian during wn- ~ ~ I I B B ~ U . WW.. mr m a . rbirh mdd .d blar~d tractton. np~cinlly rnpitl contrnclion. nccent cmd- ~ ~ l ~ ~ ~ ~ ~ ~ ~ ~ ; = ~ ~ ~ T x g ~ y & ~ , ~ o ~ ~ l ~ mcnlrd work of XlcEntce nnd 3lyrrls. desvrib~l in ihc nr. K,,,nrsl,, 2L l,. (Iw,,

Aral pnpcr of thia ncrica,Ig nhou.s thrh ill fncl, film burst- (11) J. 1s. ~ i b b , . . ~ a ~ l = u l p-.- brim- G- + a. IW. in& oftcn involvea dia~urbnnea (the nllrml!$ clcnrly Nar Yak, 14. Y.. 1831. ~ 3 0 1 .

viaihlc on hi~h*@ photopmphs, which r o w mom IlPl W, n. h l c b ~ m md K. J. st- 3. Pk. L)mr. 75, %t# IIMP)s

rnpidly thnn he hole ilcr?lf, nnd nlw Lhnt urlrlcr sumo mdi l imr* - r h ~ - vclmlty of Ih@rlm_h&~ch_hrIow~thnt~t

--. . . > " r , c

h o u a forces ill this 1)rocw: i l devclol>s Lhc equntioak vclooity ofaoulld ill ir.:~lcr (1.4 n~m/rrcc) is mucl~ l o r ~ r for thc gcncrnl c w cr disturbed films end their solution ~111111 I , u rh l i~~~ V C I U C ~ L ~ C S I L I I ~ LIIC two S U ~ ~ I I O C J ol the film for another limiting ideal, thnt of "unidimc~isionnl" mmuin nt ntmosphcric prcxqurc .w that atby comprbasivc buntiug imm a linc rourcc in lllc nhvcncc of 6ipldficnht s t rcs in rolicvod bo~urcrillceting tho cxl>crimcnt.. wIuce rclnwtion. The ht tcr cnsc is pnrtioul~vly trnctnb1,lc nnrl cxperimentol cvidcnec indionteb thnt the Undinturbcd nndysis can m o u n t qunlitaLivcly lor nuilly of the T h i s limitirlg cam ma iriti~nlyac~l mmplcloly .lid thc ohsen-ed phcnomenn. results form the be& of the mmorling JkuQion of

Thc model rvllicl~ !re ml~all wn~idcr throughout is n moreusual bumtin~pmccscs. film having n uniform initial thickness, do, surface ten- A. Unir/hcneiortd Case. Co~r~/anl 1;clwily. Thc aion. m., and density, p. A burst may bc initiated st a simplmt cnsc conidera n b~imt. propnytir~g lrom a linc point nnd ihcn spread to produce a circular hole to give origin st n vclocity ~ s u m c d to bc colatant. I t yiclds iwodlmcnaio~~nl rudial bursting; nllcrnntivdy it thc s:mo rcloeit~ n~ tho rndinl cn.jc ~11icl1 lvill bc tun- originnte along an (infinite) lino and propugate uui- sidcred in thc fo l lor i~~p mction. Hcm u-c ded with a farmly ta hot11 .ridel a1 it to gin! unidimensinnd burst unit length of tllc rim u.hich move% naith n c o m b t ~ t ing. The dktanob r orz, rospcctivdy, from the origin is velocity, UK, inc- in mas b ~ ~ u ~ ~ d ~ p t l l in tho timc dl,

a the approprintc principd coordinate. 111 eithcr caw the and is nctod u p u by thc surfnu! tcrinion of thc hvo holc is limited by a rim of varying but large (wmpnrcd laws 01 the lilm, 12-0. C6nscrvntlor1 of momcntum to &) thickness wharespnd, mcvlurcd my nt the ccntcr then requirn ol m s s of its c m w c t i o n , is u ~ , the rnlc of capansion of the hole from its origin. Thc fllm beyond the rim may 2 4 = (u~~b.dl)ua (1.1)

he mentially undisl~rrbnl or i t may be dieltwbed in a Hcnee region, the ou~mk, which runs ahe?d of thc hole. It mny thcn have n thicknen, 6, and rrurfncc tension, e, a,, - (2m/b~)"3 (1.2)

varying with distnncc and with timc, I . If a is only n Thin ia tho rcsult ohrnincd by &lick.' nnd b u r c of function of 6 bccnusc other c h a n ~ w in rn ere slo* aom- its imprtnnec, rrc call it the Culick vclocity, uc, pnred to the spacd of bursting. wc dcal with thicknes- dclincd hy only dependent 6. Thia is nn important limiting e m arising, for example, when the nniount of adsorbed PC' = 2uJ(.p (1.3)

surfactant ren~nins cvlwtsnt a11d surface tcnsion is deter- It mny bc nolad that if mnsmrati6n or c n q y n.re mincd by the nrcn per adsorbcd molcmlc. If d m r p nssumcd, n factor of iivuld IX i n tduccd into (he tion occun, thcn w will relnx st e o ~ ~ t n n t 6. We shall right hmd sidc 01 eq 1.1 and the rim velocit.~ would Le show lhD~ lf h w s l i n ~ were to oecur nt conatnnt D, i . 8 , v i n c h by a. Thi was Dupe$: rcsvlt, whioh k = a , thorc would bc no visible nunole un lm thc visc03- clcnrly inwrrcct nnd implim Ihnt bursting is Com~dctClg ity of the liquid is mrtrcm:ig high. Air mi-'mce rvill frictionlm. be neglected thmIIghout. This lmt paint may he an Thc surlncc oncrR). or the lilm originnlly oenrp l in~ important limitation in applying thc theory to rcal the hdc of nrcn 1 in 2 A b Thc kinctic c n c m of thc nyotems. Spcdn io lnborntury coordinntw are indi- rim (%)(Ad~p)uc' - Awe is c~net ly hnll na hrp. eated by u and otllcr aymbola will hc inlroduwcl when Hence, undcr LIIW nwrnption$ cxnclly hnll tho rurrncc the wordinntc ~yotcm is changal. energy i s diulipntcd, N nlrcady notcd by Culick.

Certain factors which nrc n c g i a c ~ bcw~4c they do For wnp film eo 30, p r 1, nnd Q varies from not aFleet thc npplionhility of the theory lo mnp fi lm nbout lo-' tn nbout lo-', nll in CGS units. Hcncc, (as w n t w k d with tho aflcct of nir resistnnoo) nhould be Cnliek'a vclmity b of the onlcr or 8 X 10' In 8 X 10' mentioned explicitly. One is .slip betwmn the mono- cm/sec or nbout '&ZOO milcv pcr hour, lnycr nnd the immdintely subjncent liquid. This has B. Redid Cme. Conidcr a holc having n rndilu r 'never ban shown ta bc yiwiiicnnt for water. Anothm and a rim moving vlth u veloci~. u,i rrhich may vary is rho r e l n t i ~ V~ECOUS motion of thesurfnce with m p w t wilh r. \Vc w u m c thmt tho cm-ion of thc rim is to the central lnycr 01 thc film Thii would correspond ncglig;ble nnd thnt it incorpornta nll the mntnial of the t a n lagging ol thc center behind rhcaurlnce lnym ru the film ori~nnlly in the a m of the holc. Hcnce. thc total latter ~hr ink nnd nccclernte. The kinemstio vismnily msag of the rim u rr'bp 4 for n s d l wbr. d8, it U of wale? in about 0.01 cm2/oecor 1 u'/mec. Hcnce. the r'bpdO/2. Thc totnl momcntum is hdd in the rim rime for the trarrfer of momentum emer 1 r since the wt of the film h, by h ~ ~ ~ t h e s i s , undistnrbnl. Ihc dinlunw to tho ccntm of equitc thick f i l m - i ~ o l thc Tho momcntum within thc amdl amlor mnsidcrcd i. ordm of 1 pwc, i*., jnnt peter thnn ihe blur on r'Qm,1d0/2. Aim a wry short timc intcnrd, dl, tho n rnicmfl~h pho@rnph, and dcorennu with Lhe qUnm radim of Lhc hole hm i h c m d by dr - undl, the rim d the thicknes for Lhlnner Blma. Thc third nc~lcctal vclocity by duw oo Lhnt ila m.m tor the emnll =tar is racbr in tlic efTect of compmlbillly or wakr since the thon (r 4 dr)*hd8/2, n c g l c c t i n g ~ ~ c r - term in - ~p~ - - .- -~ -- - pp .

-7

in= to our ,mamptions. Cu~l~crvn t i~~n 01 lllomcnlum ing a mmplotclg :rbrut)t lnnsition hct~vcen undisturbed lilm nnd rim. Since viscou~ f o r m rcsiat thc c h n w of

2m(r .+ Zdrldedt - (7 + dr)%~(un -I- s h n p ul ;r lilm clcmcst,, tlucre mud bc n trn~~silion none of finite width in which cnclr clomont of film &=inks ;rnd

dur)tl8/2 - r'h.pund8/2 thicken? in tho course of flo~virtg into thc "klk" liquid or, noglccliw h i ~ h ~ ~ terms nlld rcp~sinp ulldf hy dv of thc rim. Although o w m i ~ h t be t@mptcd to look to

thiq vi.colur trarvitlon sonc for m c~plonntion of the dull = 2[(zsd@l - u~t*I(dr/a~t) s nureolc, nn odc r 01 mngnitudc oleuintion sufficm to

l(rk1 - u,,3 (dr/nrn) (1.5) ahow that the nurcolc of n sono film a n n o t hc .xcounrcd for in this wq'. Hem, tho velocity of thc rim mmnin? u ~ ~ c h a n ~ d Thc shrinking xnd t h i c h i n g of n film clcmont in tho

rllen it i* ulmtl ta Culick'svcloeity, and if itdiflera rmm tmnsitiorl zanc invo,voq shc3ring of its liquid cvntcnt this vnluc, idrvi~y3 tends towanl it. Thun, if the rim nt a rntn u, which is its fmctionnl c h m s of arm or vclmitp is conata~~t, i t is Culick'n velocity, nnd il d i t h i c k ~ ~ m pcr unit time. By wmon of the virmhsity of

turhed, it tends to return Lo it. Mort s ~ ~ i f i ~ l l y , if the ~e liquid, onera is disiplltcd in it rrt r;ltc of ,,-' per disturbed velocity := un, nL d i m r', in-tion of thc unit valume. hlgt cquntion Gvcn A chaetcristic time w u e i n t d with the p i r a w of

thc liquid through this trnn~ition zone mill be I = ffl-l- Tho dis~ipslcd encrm will then be of thc order of w'br per unit arcn. As we hnvc sccn above, hd f the surincc

For smdl rolatiw doviatiom, this simplifies to energy of tlrc film is dlsiwted durinc burst in~and if a c nssumc thnt nll of this is diwiwtcd bl- viqrnuq friction,

(I.') we obtain

This menns that any di l lc~n* from Culick's vflotity d m w m nrpidly a? n Iu~xction of tho r&tI~@ i m w or i-. tho dim or tile hole. An imp~rtnnnt wnqueneo is h t any cffoct of carly diiturbmnccs duc to thz first puncturing of the holc will he pnr~iculnrly rnpidly

For map nalutionrj in snter, the dcnomimtnr (odd is ~ h c Rflcdof Rim R a d i ~ . In tho nboadhcus ionw ,,bout 3 x ]p cm/me or 01 thc anmc order a3 CulicW8

hnve ~ g l c c t & tho dimcnqlonr 01 tho rim i k l r nrul t k tT~city of $he dm, yienec, dishnrn cod by ihc ncglect ahoulrl be jwtifia~l. The rim inulrpornks undcr sholmd clcmcnt in chnnetcri?ltic timc is eom- pwcn t wttmptions d l tho mskrinl of lhc film within parnhlo ,,.ithucr r & T]~,,~, filmclcmcntmll thickrm thc original ho~c n d mwt hnw ;m n~pmximnhly nRo~nl~vircom m i a ~ n c c s v,hilc rnnving n diaiancc circuvr cmwcctiol~. If thi* wcro U . T ! ~ circulllr, the palRbjC it. thickncw, ,vhiE)l is or the mielom dismctcr of the rim c r m - ~ c t i n n would be v u d to 1, ,,,,d thmc~urc bclo,v *c limit or p,...nt cwri- (rh!~)"', which for the ununl WnP film '"auld be 'm m,rltv. Even il n gcncrouj rnultil,lc or6 is xllmcd fot than n tmth to n hu~~drcdih 01 n miiiimchr t:hcn Lhc tho ,vidlhof lrsnrition zunc, il.atill rvould ncpligi- holc d i m rcttchcr 1 mn. Th*. evcn il Lhc rim CKV- hlC cnmnnmd to the obsrvml numlc. wl,i,+lt nrc W r mtion d c v i n t ~ somcwhnt from eirculnr, i h pmjcction

B,ld in millimctcrs, on thc plnne of ihc film, Zm, is v w emall in cumprison The nhaw n ~ r n m t ia bwd on the awmption that r i th rand cnn bc neulectcd in compurinan with it. Wlc vi3coaity of the in t rdmrdld liquid iq of thc @%nc

If the calOulnLion lending to eq ' *wd1 orderru tllnt of antar. This ia mnristcnt with -1Cs of '

takina into sefount the finitevnlucol the rim projection ,primcllts on Lhc on %hc formntion1l of 2a r.ld m e ~ u r i n g r to the center of this pmicction, the films, but cnn bedebated.j& ~ovcvcr, fnct thst m-ia of the rim h e m m (r -I- a)*& and iln momentum '. m p n d i n g l y incrcnncd. In order to be mre thnt 1 1 0 X. 1. %f,r*. K. B~MB, mas. hnkd."*rUma S-

alThdrThlnnl.rnnd I UibUomhy."-m h X-Yvtk. the film i# undinlurbcd, Lhc l v m ecting on the rim mny N. Y,. IUD, Ch*Pk IP. v h w d nt r + 0 + 2dr. O t h c m k the nnmo (16) J. ~ ~ r k ~ , . I.. C. BCWUR ."a K J. J. m r rrn, m n i n g mny ha followed nnd rlnec a cntcm nll ex- LWO It6 rlmr.

- -

. -.-,.. dime~~xiun, uul rhc uureolc &c 1101 stmng1~dcpendini;;n film t11ick1ic.q wuuld imply IL rcm:~~knbIc con~pc~imtinn ' hctaccll Ll~icl;l~cs~ n110 viucority in t l ~ c lllm if it !vcrr! n gc:~Wy i~~cmluud viscosity thnl h rwponsiblc for tllc ~rurmlc. Ilinnlly, buntinp experiment-r UII films lrOm $ glywinc-rich mlulion~" hnvc ~hown thnt Lho Ividlh or thc nurwle docs l ~ u t increw with thc vi~cositg of thc solution. x X

An indcpcndcnt urgumct~t that tho oboervcd nureolm m nut due to viaous lorws is thnt thcrvidthol a trnnsi-

1 t.iouiorle dctonninatl by vivcaus rctiistnncc to thickaning L1tm.n

I uould bc con~(f7nt rntlrcr thrul irtcrclLsing ill pmpxtio~r 1 to the timc dlnec thc beginning of thc bursting, as thc

I observed nnre:>lw do (nl lcnst approximnrcly). * . k /

I Nc mnclirde. therefore, that vircouq f n r m mnkc a /,< *;,:,d

nc~ligihle wntrihulion to the ob$crvCa width or thc t-o aunnio m d . w k rm mplnnatiot~ l ~ z w l on chmgm in x sur~acctc~rsio~r ~ccompnnyingvcry quick changcq ia film thiek~tcs nnd thelrforc of i b arch. Flrurc 1. Sch.maL(o p-nm~i~n of the irumtingcf L blm in

term8 of inborntory rmrdinrla i 4 1. mrtd of tha ofi&iIIbl

m. The Inl~~pte lDU~n of Amolea in poniibn or n flrn elcmenlin thc ~~nrlistur(nd am. X.

Tcrma ol Surface Teasidn Chrnges

Aauming th:tt the nurcolc in due to chsngol in aurlncc tcnsion, one m q risk rrhcthcr it is ilnsjitrla to detcrmim from n study ot tho nurwie sk-t thcsc chnngcs arc. Oncc such ir~lmmntinn is nvnilnblc, it shoulrl bc pw~ihlc to intcrprd it lurthor in molcculnr t m s nn3 to obtain information about nonci~uilibrium ntrfncc hchnviornnd nbout dcsurptio~i kincticn 011 a mierosceor~d scale. \VC will. thcrcfore, examinc flmt the pncral pmblcm of I@- lating obrcrvuble quar~litiea to chnngc. in rurlacc tension, rlthoughit doarrot lend tonnnlyticn'wluliom. Wo will thcrl prucccd to rr aimpier l~mltlcm which in x X+dx wlublc and pmvidrs yome inright intrr t l ~ c nmcllnnism involrd. F I ~ I R 2. Bchurutic c w c l i m d r htntinr film i d l d n u

A. ~,lidimmrimlal Rur~ing. Since, nl ymcnt, 'he rim) Jl 190 m l e c a d v . inahfa. Tb Jnlr md Hnm indiedm Ula -tiom of Uzc "nmc film .lcmmt which h*r moved In lhe individual fllm clcmcnb n r ~ n o l bo lag@ nnd lullowcd nRhl rind lhickncd d"rinr pmeoa~

vinudly, wc nrc rcwlrietd ta the ohmrr.~tinn of liner of oqunl thicktucm, 6 trhicl~ cormpond ta color fr lnps (or

.. . linm ol m w h c intenuity of mflcctd monochruml~(ic tho budinpp-; in them yet u~ldiwturbnl lilm, s - light). JII prirtciplr, nt [ k t , thc podtlnn ol mnny mch X. Figure 1. riho\*-r-r Name rclntlan~hips bclnoen thc t, lined cnn be dctcrmi~tcd nt, aacecmivc timc intewnl* to X, nnd 1 cwdinnta . i v c thc complete history ol lhc nltrcolc f.(x,l). I t is. Thc velocity of n lilm clcmcnt is ' thcreforc, 490 pmlhlc to obtnin owprimcnlnlly tho rnie of ndvence a l Lho pd t ion nt which n rnrlievlnr

u = (&/do, (3.2)

. thidtnerr occum, or, hricfly, the velocity of n given C0n3idm n film cmwxction at tao su&vc iwtanta thicknew lu shown in Figure 2. C~mcwrt ion 01 mm q u i m

thnt tho tom1 men gcr unit uidth bcttrcrn nome l i ~ ~ c in ua ,= (&//dl)r 0'') the undiaturhd film nnd n kivcn film ckmcnt mnain

Bin- film think- c h m ~ , thmc constnnt-thicknba mmlant during bunting. Xcglcctiw any dcnrity

/ linm movc with mqwet (n thc film matcrinl nnd r, iq ehnnm, thc nnmc npplin In vnlomc, nnd in the orom-

I gcncrnlly dlllcmlt from thc vclocily of a Rim clwnm(, mtion, b i~ nmn. Thu* I at, nt tltc inmtnnt whcn thoir puaitionr cuineide I To irlnfi!ify film olcmcntn It in convenicnl la intmducc t the uri~inhl pmilbn ul n film clcmcnt, iu diatnnm, X, (3.3)

fmm the orillin, the lihc nlmlg which tho film i q nevcml. 1 ' Thus. X k lnvlrrinnt for n pnrrioulnr film elomnl dvri n l in thc andMurbEd Rlm and ml the

-- - -~ - -~ ~ - - ~ P c T.T--ap

7

4 - .. .~ ~

eoordi~lnte of n lilnl cl~~mc~rt . I 'crlumi~~g tlrc dil- icrcl~tiuLio~l uivcs

(3.4)

and, lhercforc, rhc film c l ~ l n c r ~ t vclociby h ~ ~ ' C I I b ~ "

(3.5)

Tl~is yuntion pcnnits tho culculalion of Lke film elemcnl \-e.cLrcit,). a t nll pnilltx nnrl limc* .so tbn1 ld(z,l) in ohlsinctl. Diller@~~ti:diu~~ (rhich wc nwmc tor lhc nlv~llrtlL i n be pnssible) nim the ; d c m t i o n bu/N nt

x 3 c Figtnn 3. The qrnl~l!lia dr*rrilving r rho* w v e :ow poiat. This accclcmtion is chlc Le lhc pddierll uf in b,lrslina61m. .

s u t r ~ r r tcmioe forcn?Zw/ax nrd ir relilled to it by

w a r - ( 2 / f i ) @ ~ / b z ) . ,

(3.6) This mpmcnion is particularly useful in ahmining sl As s(z.0 in itlrcady 8smmcd u, bc known, thiq cqunlion ahnck nt the outer Ed@ af the numlc a-hi& rum into mrl bObolvcd at an]- p i n t and rimc to give (b/tlx)(z,.l). undi?Lwbcd film nnd mny bo mUcd the Jmnful +hock. Since the sudncc tenGu11 of the oldivturbed film is I n lhnt ease 61 e g, OI = and ui = 0 so that '2q 3.11 linum, it is Imwibl~ tn obhin thc v:tluc of the a u d m ~ U C C S to tcnriu~~ nt nny lloist nrrd timc by inlcvntin~r p ~ ~ ~ d u . ' ( k - 60) - 2(v0 - 04 (3.12)

.(z,o - v8 + (J:mo;h)dr) (3.71 ~cnce, r n c m w m c ~ ~ t of thcsped of the o n b eds m d ut t k thi&llobs immdiuWy bcforc nnd behind i t ik

.Qr,o& 1Yores. The nbovc trcutmetll is mlid lor aliicient to give thc surf~lcc tcnsion drop oocumnz cor~tinuou~lp vnr)'mg thicknwin. Whcn n shock wave across it. is prcsellt, 11% h.w h n imIucntly ohcrvcd 5y 3lcX11tcc llctunling now to cq 3.7 n o can wmbinc it with eq rnd >Iyrls," i.c., whcu Lhc thickncal jump lrbruptly 3.11 obtaincd gleviously to solve, in principle, Ule from ~ I I C vuluc sl to nr~othcr & (Lpiprc 3) with n carre prublm 01 obtaining tho ludaeo tonsiotl of the b ~ d n g rpndinp jump ill vclucity fmm u, lo u,, ~q 3.4-3.7 film from a kttowlcdgc of d ~ c sh7p or the aurrolc b r n o dubieun. To find the nquirctl C I I U I I ~ in sub tvhcthcr iLmrrlili~rti shock 3vnvcs or I I O ~ .

law tcnrlon (from VI tu a,), we cor~vidcr the change of Time Dsp~~dcnix. It iu likcl? thnt the nrcn chnngcs momci~tunl ncmw ihc ~houk lil~c in it coudinute i y ~ t c m and, U~mcfme, thidincm chilnpcs ~vlll be m~ impomnt nttachcd to L11i.l line which is mnvinq with n vclacity u.. vnrinblc dctermillieg C ~ ~ I I ~ C ~ in mdnm wn~ion. The maw nl ihc t,hinncr film c n t c n c ~ l l ~ c alrnck Iino per Ilcnm, a(6,l) ivill be a more sig~rificnnt function than unit width end lime ia JI P (u. - u,)p&. IIIIJ of thicker v(x,l). Since 8(?,L), thcnhege of thc numolc, is usumcd film Ienving it d l = (u, - *),A+ Sir100 L ~ C J C two to bc known, thc r q u i d Edunxfommtlon is, in princi- m ; ~ e r arc q u n l ns i ~ t d i c ~ ~ t d plc, trivial.

11 must bcnolcd lhnt this pro&rc requimonly the , u, = (4% - SII~I)/(E, -- 81) k n o ~ v l c d ~ ut Lhc mhnpc of tho x!~rcalo 4(r,l), but in\vllve.

In pn.uilrg rllmuRh slwc~ lillc, mw ulldcnon LWO d i l l c rcn t i~~ t iu~~~ nnd 1u.o inlcpl?lliona of this illlo* I

chanR. in ycloeiry ,,f ,,, - ,,, ulldcr lllc irlnunlco 01 mntioll, ~ l h n t quite l~curnto dnrn mas b~ be~li=d 1 nur(noc tcnaion ~ i f l ~ ~ , , ~ - eu ~~~~~~~i~~ d Bivc muni~lglul rcr;ults. Furthcrmom, in ~cncml, the 1 mommtum rerluircr ihnt nvfm tcwion mny bc m e t d to d c m d not onis on 1

thc momanlary rrtntc of thc hlm, bnrnclr its ihidinw, ! M(w - UJ - -2(6 - .I) (3.9) but n l s on 117 past hislow, which dctcrminu, rhoextent I

Eliminntlng u, bstwccn t b quatitio~u @vw to which it hns ~ppmnchcd equilibrium. Thli histots r i l l dcpcnd on z or 1 in n dcfinitc n y for Rny givcn

q - 4 - -dtb(us - u~)'/a(h - 8,) (8.10) Glm, but will vnn. wilh Lhc rak of bursting. and, t h m e

Allcrnmti~cly, ilu. la thc obsMvctl c~tmnliiy, olrc of (,,, A,,,,umn,hn~rhl.h.onsYno.ma,,-Mli

rhc other q~~nn~i t im, for cxnmple, u. mny be climinnbd b10 stnhmwly $ O M A ~ ra ~,hpli~. h t tho ~ I I U W 1 . t o & ImW al &ha .ham. Thb ,+I& In lb. aaidimmwk4W CM

&dl j i d 1 ( ( - a ) - I ) - ( a - r I ) ' - "d dd - -. 1 w

-*)li- - - - ----. .. . . - , . ...

-. , . - . ,

lore, nilh tho Lhicl;acss, cvcn lor films ol 11 +.II corn- cxtcl~t thut thc rcnl film d c p : ~ h from Illin l imi t i r~~ positiotl. ncc~usc ul rl~iti dur~blc tlcpcndcnw it rihnrild itehavhr, w nrcdcnlir~g 114th rclnxalior~ p h c ~ ~ o m c ~ ~ a u( bc pomiblc to xtudy (Ire cltcet ul this lursl IJsbv uvcr v timc acnlu conipnr~tl~lc 1viLI1 I~emlir~g Limcx dur, c.0.. (11

considcrnblc m n ~ c or cllccl?;. tho ~oluhiliiy or tltc rnunol:rycr rind thcrclnrc ail11 llic Altcr pcncrdisinp, Lllnc ~CSIIII)I to thc j~~.nati~lllly liinclics a1 il(i tlciurplion or with frictbtlsl dlcc& 11, t l ~ c

importnnt ntdiul ens@, tvc shrill tonrirlcr tho ximplc wlrc ~urroet~dir~g medium. in ahioh awlace Lcnvios i~ dcp~r l r lc~~l onlj. UII 111~ Thc trcutmcnl is bu~cd oa thc mctl~od 01 cl~nr:~clc~ir-

: thicknom of lhc film und is not utl~cn~-iac dcpcl~dcal an tic5 wlrich is d i s z n d in some 1cxtIxMk.s r ~ r gar cly~~.lnm-

i its p s t history. ~ c n . ' ~ B. Radial Cue. Wc nssumc thnt thc bunting pro- Thc cnmprcskio~r nf the mor~olnyer is in our c m duc

cms ia ~ymmctricnl nbo11P thc nn'girl. Thc hvscr:d nn- to thc d~rit~h-oru a l tllc Iilm nurfitcc na :I film clcrncnt ~oning of 1116 unidimcr~!iior~nl m cnn 111~11 he npl>liccl lhickcnn nnd thin fihrinlis~o CIIII IIC cnnvcr~ic~ltly CX-

I dircctlj-, rcplncing I by ~ . n r ~ d X by I?. with 1I1c mudi1ic:r- pmscd br tcrma ol n pnnunctu, e, \vhlch ~ ; U I hc clclined ! tior~ thnt tho volume of thc hlm lrol\r.ccn r a l ~ d r + dr is by any of the fnllnnirr~ rntios: of the p m r t nrcu, A,

pmportionnl not only to its thick~~om, 6, but ulrn to its n: a film elcmcz~t to its o r i ~ n n l m n , AD; uf its initial distencc lmm tho origin, r. If \re dcr10tC tllc origi~l:~i thickncq, &, to its prcscnt onc. 6; of its prcrerll 1%-idth, pcsi~ion of 11 film clcmcnt I I ~ It, eo~rscrv;riion 01 whtn~u dz, &I i& original VIE, ddX, tvhcrc X ia s ~ n i l ~ the orjginul givw aoordi~~ato of thc lilrn c l n n c ~ ~ t inr n unidirne~~slo~~d

burst - n/no = sda - (wax), (:$.I:$) (4.1)

Our first g ~ ~ m o s i:, ta dc.rerihe thc h m t i r y p m e m nnd the vel~eity of n film c l m n ~ t is given h? n w m i s r that tho funeliorr

s - *(-I (4.2)

u - (tI?/&)# = n,141 dcqcribing thc vmrin(ion ol m ~ r f l l ~ ~ tcnciol~ 11<1h shrink- 6r L ~ C is kr~o~\'n. filt~ee a vnrica bclrvnn 0 iuld I, this

in drict annlow with cq 3.3 and 3.5.'' l~~nction may be r c p m ~ t e d n8 n line or thc kind shown

T,,~ rrmxini,,p, trcntmcr,t uf the ul,idimmionnl in Eyrc4. It 11s tho vnluc u. nt n = 1 nr~d itsdcrivw

mn ba ndeplcd directly ,,, $kc rd&l since tho rn$inl tivo nt Lhnt point is positive. The dcrivnti\.e a t that

chnmct2r of hum- not rcnnnillK. I,, point cnnn~ ' ~c lc$n thnn h d l of Cihh'a film elr*icity

pnrf~culnr, nlthougl~ thc shock line hcmrncn ;L circle, n.c m*dulu'E cnn rntrict LIIC conaidcrntion of thc thicknmm nnd a! - 2dr'/[(dA)/A].,1 P Z(dv'/du).., (4.3) f m 3 invnlvd to vuch ;r n:vmw bud ml both side* of it %hnt the chnnpa ,,.it11 becrlmo ncC;lifih~c. q-lle T h t prime is u . d to emphneim that Gibbs' dcfinitial~

orRumcnt is annloRoua to ,hilt l u ~ ~ ill deriving (nelinl, nzlntrz to n tlvc quilibrium atntc for m c l ~ ihicknm or

I&) Culick's vcloci~y fur t l ~ e undislurlml nalirll awe. the whcrctm illc slntcs m ~ m m n t d h1 fii?um mnY

wc .hall rrturll 10 tho [ntq~rclntiun tll mpcrimcl~tnl h@ only lun~f-eclnilibri~m nnCT. From Our b o w l d t 3 drtD nftcr n mnro pncrnl Jiacumion of the dyrlnmics of 01 mnnoln~en tvc cnn W ~ c t l h a t thc rmrfnee tcnsion burding whcn u inn funclion of 6 only. C O I I ~ ~ I I U W to dnrcnw IL~ thc film nrcndmc3a~ll. Othcr-

\vise, t h ~ r c sccm to k no a primi limitalions on the 1V. Unidlmendond Rlrn Bmllng W b l Sarfna shnpe ol thc carvc. Tendon Is ThleknrukOnlr Depcndml Thc prcsnl porition. z, of !I film olcmcnt is n t~~nction

We now canltidcr the mcohnninm of Rlm burstill for of ils origil~nl podtion, S, nnd 01 limn. I the idenl limiting o m ol unidimcs~ionnl burn~i~lg wilh

I .

; mrfnce tension dependc~~t only or1 thc thkkcningol ihc a $(X.,l) (4.4) film wit ahrinka, nnd not on tho & hiobrv or UII tho Thc pnrtinl dcrivmtivtivr or 6 am, mpre(i<-rlp, the rut. of thickening. These airnplilyina mrnptinna pep hhrinknw nnd lhc vclocity ' mit much morc explicit wlutions of the bnsic cquatio:~~

; nnd thus morc insight into the mcuhnniam involml. &/ax - cl (4.5)

Ti10 mrnpl ion lhnt mrfnco tcnion rlcpcnd!~ only on &/bl - u Lhc thicknm of thc film nl~d L11crc1ura o ~ ~ l y on itd ahrinknge b aguivnient la nnying l h n t tho 61m outf~~ce C d i R o ~ ~ n t . i n ~ i o n nhom thnt

behaves during bunting in tlrc wny nn innoluhlc mono- aa/bc - au/m (4.0) layeriaavppgwd (o llchnve ill n Langrnuir t rou~h, The innaluble monolnycr i* ruuvmcd tn in cquilihtiem. (lm Ed.. v.,h ""sl,, U**,, *I#8 our shrinking fllm b in n qu&qeillhrium. To tho n o d c... IW.. N" ~ n k . N. Y.. IOBP.

~- ---- -- ... . .

Du bv b*r ba -- - -A -; .- -.I - ax (4.12) a t a1 ax

111 vies. of mq 4.0 nnd 4.7, ;orpcctivcly, we cxn .nplncc dim two equations by

I au a ~ . as - I -., - -- = - A - I 01 ax1 a1

(4.13) ax I

I in uhith o and Y uppear compldely wmmctrionlly. BY aubt~deti~lg tho mcond lrom the Rrst o m and rn nrmangii~g-, we obtnin

Q-, i& ) (u -d -o (4.14)

This mlntion mcnnc thnt n corlnin combination of dmivntives of tho dilimcncc botmcn wand u is acm, i.e.,

ci I that this diRcrcncc is unchnndn~ don& mrtain paths. This combinntion of dcrivntivcs daRnca n path in the

Rn~n 4. ?he ralntion 01 auui-n)a(lih.itlm v.ri*llon of * X,I p1811e who% d o p , dX/dl - -A, so thnt thc path is face Lcruian. with ubr inkx cibh.' fihn el.aSdb atending fmm high ,.dues X, u n d b th. pmmmkr d - LC turbed dim, into thc bursting port ar~d to the rim.

Howcvcr, accordina to i t . definition, u is zero in the un- The nccdcrntios 01 Alm clcmcnt im givcn by disturbed film and so in w necordin~ crj 4.11. Henco

these two q u ~ t i l l c a are cqunl thcrc, nnd by cq 4.14 __--------- am D" ' ad dv - y - (4.7) stay equal dong nny aueh lino. Sinco the line has nn at pa h p 6 0 b ~ 6 de ax ax nrbitrnry origin, it cnn bc mnde t~ n?ss through any

Hem, A is an important pmunetcl defincd by point oftltcX,fs1)nrrn1td weohtdh the result

I A - [(2/&)de/dml"' (48) u - ufotdIXnndt I

(4.15)

rnd dcFndinp, under our naumptiom, only on the Sinw w ia by definition a function of only the shrinkns, ahrinknEc, m. Ithna lhc dirnmsions of a v~locity in the a, this mlntion mcnnrr that u, thc vclooity d n film x , ~ ~ ~ t m , i.~., lcnfih of film ,,q ,unit time. element, Is ci-:. aincd by its h in l iagc only and c m be Tho physical mc:ming of n r~locity in thb system is the obtained f ~ m n knowledge of the *(a) function by eq

I rate nt which thc original film is undcr.qoing n p m w . 4.8 and4.9 which e v e I Thuy E.V., UI, ia thc velocity IJ which oridnnl film la I conaumcd hy ihc nrlvnnoing rim md U. tho vchcity at r - 1 Ma = ~ l ~ / r ~ l r / d t r l ' " dm (&I. I which thc origirtnl film p m u through the a h t o ol hnv-

! ing ghmnk ta m. We will yhnrr.nhortly that thcvnlue or To o b t n i ~ ~ tho menhin~ of A, rro now add rhc tuu U. is A. Wc may notc thnt A in mlated to ir in a eimple qua t iow (4.16) and mnnnflgc to obtain any sine@ iM Mlunre in, CXccl)t ror thc conntnnt foctor 2 / h , tho u l ~ p of tho r oa. e 1110t M shown in Figure 4. (4.17) Wc now inucducc u, r~nnthcr function of a and nho

dimcn.+'iohnlly a velociiy (hut in the z,L system) and ,gincc vc klrow ~h,,~ cqunla ,, cvvhm, wis Nl,,iion dehhed hy showa that lhcrc is b pnth in thc X,r spnco dong which

rnw s.'Ad= both w ~d u am constant. Since they have equal

(*'O) valum bnd u is a function of a only. this must Lr, thcrc lore, n parh of wnatnnt hdnkngc. This pnth n a line

kcordinp to thir duiinition whaae slopc, dX/dl = A, in thcretom u con$tnnt. Thug A is indeed qun l to the velocity U. of a linc of w t a n t

+!=-A d e

(4.10) ehrinkngc, in t e r n of orig;nnl film position, X, .u m have nnlieipatod cnrlii.

nhd Hcnee, in tho nurcalc or this type of qnerm. cnch

u - O r o r a - 1 ( 4 ~ 1 ) thicknus h w a cvnatant velocity nnd Lhe auroole rrx- pundm linmly with time. Tho burning myr be died

Furthermarc, sineo, neeording ta ~g 4.6 nnd 4.0, A and Wqimilor." w - dcpend - only on a Gsuotion 4.14 ahom thnl n cmnh qnmtity propn-

T p , , -- ~~ ~~ ~

3 - - T '~ 7

: I : I ( I I

?

4

I / t 3 I : P, r

1 : lJc F 1 ; - -- ..-...

I K

Figure 8. (a) Follr sidple hypothcllml .+sp=n ror lhc r- NNF. (b) COmponding of ~,~[(2/&ldr/d=l'~~. Ic) Cornagrmdl~g ah.* 01 11. (cq 4-18). Id) Detd of (0).

Flwn 6, The dU1lib11lBn of thl&ns€s md 01 vela~iti6 in u ~ d . s commpmding to tin @cm- or Eplm 5.

p t w to thc left in X,I nith .sped A. Quntlon 4.17 dmis the r igh txvnrd pmpngntion ofnnothcrqunntiW nnts. 1, mny he neon Lhnt nt Uc outer nip of tba nt tho anme w e d . n u s , A plnys n role simil*r to that nuroolc ,,,Illre - 1. thcmnncl tcrmr.anichcjnrtdu. - of the s p c d of mutld (depcndill~ on dmsity) in QWJ- u.; wllcrcsr, a ac ~hnl l scc in more dctnil lntn; %t the dynamics." rim whcrc u tcnda to zero this first tcrnl vanishen, r. =

In finrphienl t e r m have @ll*ndY fint u*' U, llnd the film clcmcntr llnvc the riri~ ve1.elocit.y which is vuicd with the slopc of the - m. a. plot W shorn in gfvcn by ihc i n t w l nvcrihcwhnlr r n r g ~ of ds. Epure 4. X m , if the vf~lvm of u. thus ohtnincd ma Wo cnn now eondnlet n pmfilcnf tho rumlo 6nef eel p b l t d n&n5t a, the m l m i b ~ of film clement, 4.18 givm n relnt,ion hctwcn cr nnd u. nnd thc lalter is W, is Riven necn~llng to the definition n fu (ccl4.9) hy pmmrilo,lnl the nf the film clement which area under thia curvc crtc~trlingfmm the right nidc limit I,, ,,nderaonc ,, thicknbillg, 0 - I/&. Thin b to tho m corresponding to the r l l i c k n ~ 01 this film ahown i n i;iprn 6 nlungn.ith the dlrtribulion olwloci- element. Thew mlnlivns nre i l lu l rn ld *or orcvernl tics t.hmnRh the numle lor the 8plcms of fi@rc 5. simplo EM^. in l:igurc 5. The coordiltntc~ in thin figure , y ~ , ~ k 1vWI, Since U. vnrics *ith thc slop@ of the o am mnde dimctnlrionlcas b~ blpminR mrfncc tension in ,, iillc, lhcrc mny bc region. in which it mny be i c a term# of thnt of the undiaturbd lilms ub m d for tlrinner films thnn for thickcr mrn 04 m,?v be noted velocitica in t e rn of Culick'a vel~city, uc. In Figure 4. Thin hnppens n h n the lint is ConUwW

A m d U. arc thus rcldcd to U nnd 10 s lo~c ofthe downwm.d~ rind mcons thnt the thickcr fllrnxould over- t~ w. a p l ~ t , but (I,$ not anohaervnhle l u a n i i t ~ since it tmkc the thinner oncm thnt thelnttcr d i ~ n p ~ m r s a n d an i~ in the XJ ly~tem. I t can be minted. h'"J'''rr nt,mnt chnngc 01 thiekneas, n "sllwk vavc,'' i s ucntcd. obsermhle u., which memured ill lhc z,( Inhorntow Undcr thw condilionn the o b v c mnlysis b d i ~ down coordinnt~ hy nnd vrc mud aupplcrncni it hy n drtcnninntbn or .rhock

velocity nnd nhock cxtcnt. (LC, its rhnngc or thickneeq).

(Ibl TI. 8. %2Int'rb.lrh nnl Tm, P. Rdm. "&+n nl Bh& W m The first tcm in thmc aprcamions m p m b the mn- g:,:t#gyvT$;. :-e r-l-'l. A-

tmation irom cU[ t a dz nnd tho mend t & ~ l l lb lm 11, h m b , "llyti-unW D*. ~ m k N. T. IDM. mrn~nt the movement of the film in lnbbrnlary mrd i - Chnprer X. ~- v . -- - ~- - ... .* . ~

t . - ~- .-?;;. ~. t~u~pnnt to the main portion which p w c s through thc (#,,I) coordinate8 of tho ori~inzl film repmpcnta a frontal shock pmcccding Lhc aureole, *, ia then =o and mr - 1.

hchind tbc rim, ie., wvithfn thc holc, tho filmdom not exist M that iit9 "aurInce tc~uiun" mny be rwudcd ns zero. Sincc no mntcdd m c s out 01 the rim, thm ia no outgoing momentum and the rim behnvcs like a shock wnvr leadinc to an infinitely !,hick film, is., ar 0. Renee, tho rim mny he repmentad ns a hock dcfrncd by nlinc pnssing through thc otigin nnd tnngcnt to tho c u m .

U I Q Thw, the bchnvior of thc film rnny bc doacrikd by the original considcmtions pmvidcd that all porlwns of the a vs. a curve which aan be suhtcndcd by tangents et two points, or tangents pming through the origin or through tho modinntea of the undisturbed film, be mpluccd by thcsc lincs. Wc may call such a cuwe n mvet a us, a plot (Note that i t is oonvcx downwnrd). It is this plot which must be used in cq 4.6 nnd 4.9 to obtain vclwitios for this idcnl onsc a. shvrn in Figure 5. If the e uc. ol curva licr enti?+ shnve the ding~nnl

lint oanncctlng tkc cwrdinntes 01 Chc ~~ndlsturbcd film td the origin (Figurc 7(d)), the convex plot becomes Lllis strnight line nntl only one tlhoek wnvc, nnmdy the rim itself, w pnscnt. Thc rim shock movcs into undis-

I 0: turhcd fllm nnd wc nre denling with the situntinn dir- a cussed in gcclion I. The vclocity of this rhnck -vat\.c is

mlplm 7. n~hc WML -plot.. mid r i & m the mvm the sJmc in tcrms of the originnl Rim lm i71 lahrato'iy plnr srul dn-hcd lim any part o l lhc nridnbl r m plot which wordinatca so that b ken npiaeed by slrnkbl lina c o m n d i n g ro aha* wavm. Cf, - (Z4-h)" - u, - uc (421)

nnd, nn would be expected, we And that the rim has Let U. be the shock velocity in the X,L coordinntos. ~ ~ l l ~ k t ~ vclocily-

Thc meap or film over which i t pugfa (per unit width me of n"mlc ,,,- that the m. and :irnel ia Ma- If we dcaignntc t h ~ qunntiticr Nrvedips belor the diwnal line mnntlnEj,cvbluea immodiatcly in fmnt 01 the shock by 1 and tho= for the film the origin. ~i~ - ] d r y immedintcly behind i t by 2, thcchr~ngc it1 vclocity nt thc i, then enording Eq 4.16 by #hock iS

uj - u, - U.q - U.m U.(a, - 4) (4.19) ur - l l ~ e (*.a) - - - . . - .-

Conncrvntion of momentum dvcu with U. takcn nlbng the convex plot of the s w. a euWc. U.' - (2/h)(ur - cd/(aa - aJ (4.20) Thus, ua is tho avcmge of U. ovcr nII a vnlw.

It mn hc nhoun thnt u= is ncvar larger than Cdick'a which differn from the definition or A = U. (eel 4 8 ) by vcloeitg lor the ideal vsm under diauJsion. Far having difimncc quotiant i m m of a de*ntive. this let ue call fmtmnnl devinrion of Thua, thonhock velocity is rclnlcd to r c h ~ n l cunnecling ua lmm ile nvemge vnl"c r, thnt two polnb on the s ue. a linc in tho nnmcvlly m U,i$ to the t a n ~ m ~ nt n point. I t an be men Pgurc 7) that U. - un(I + I(&)) the nctunl shock eormmnds to the chord aubtending

(4.W

the inrgcst pomiblc p ~ r t 01 the CUW. because olhcrwlm Let Us aIso nok that aeemdin~ tn eq 43, ps(U.'D k it would eiLhcr bc overrunning slower moving film ihC8loFC of thc coltvex s w. a plot. I n ~ ~ n t i o n dong thleknmm nhmd or being overrun by thnt bchind it. thh line bv@J The number of ahoeks achldly p r m t dcpehdr, of ,mum, an ;he shop of the s w. a C U W ~ . - ( & ~ I ~ v . ~ ~ o (4.w

Of q w i a l interest are condltiona ncnr a - 1 nnd a - 0, If the cuwb hmiho_ahnpo a h m ~ ~. in Fiyre - Vb), h a Cambinlrlg thw L a o quatiom, we obtain .. .. --- ."" ~~ .- m

L ... ~ ~ ~ . , . . , . : , . .v. . .~ . .. - 2 d & - 1'uIIt(1 + j(e))lda - , f i d r - fu&S,"~(r)dF = fud.I(p) [5.:3)

uI,z[[~doil. + lJtf(a)aoil. + [F(a)dQ] ((1.25) w h m ~ ( t ) a ~ ~ l d ~ r 0 1 h ddinite i11te~m1 010 tmm p = o 0 (or, whet is aquhicnt, from the rim) to p. For hrgc

Tho first of t h m iatcpmk is cqunl to unity; the m n d P, I " P. Exprrimc~~t~Ily, I may be obkinnble morr vnni-he., since it is pmporlionnl to the total devillMon ~ & I Y by inteerating imm p to n point g. in the ondig

lmm the nrrrngc; N I ~ the third is a c c d l y >0. Wc tu'bcd obtal~r, thmlom

1 - PO - I @ ( ~ ) d p (6.4) un' = (?odpbo3/(1 + l ~ ( . i d Q ) (4.26)

Tha speed, u, of a fllm clcmcnt is the rnto or c h m p ivhich how% that thc mnrimum vnluc or ue is o b k i n d of x, kcepin~ cxrhWion S.3 Conqtant Hcncu uhctl the devi&tiona vunish. I t is OICII cqunl to uc; furthermom, u, E un; uvd the w n m s N. Q plot is n

+ fuk(cUid~)(-x/L%) + stmight line nr~d nr dcnl s i t h bumling illto undislurlxd u lu&(dT/dp) ( l i - 0 (5.5) film. It muy be n o t d thttt irucliunrl dcvintion~ rmm or this mnaimum rim wlooity dcpcnd an the nvcmgc squnrc of dcvintions of u. fmm ue, hcncc, ~vhcn thcm u - (-4 + B./~)IB = ruo - d / ~ (5.6) latter dcvintions nrc srnbll, thc dlKcmncc bct%s.ccn the F~~ inre p, bma e .M. rim apccrl and Culick'sq~~cd isww &I. Thc nccclmlion of a film dcmmt can be exprrnrd V. The hlerpret.lion of SeU-Siiar Result# lu lollo~*i

In tho prcvious miion we have U ~ O F V I I ihnt When P dtf f tcnsion is "thickncs+only dcpcndcnt," Iinm of & 9 - (-: + 3 s; - equal thiekncsr pmpngato at constant spcctls ahich de- c m o from thc outer edge of Lhc surmlc to the rim of F(u - m) $ - -p$) $ (5.7) the hole. Thus, ns thc sureale expands 4 t h time, it x remains =If-similar, all af it? dimensions inemssin~ d e m t i o n must be q,ul proportionntely to lhc time since thc hole originated. This similnrity of wrly and I ~ ~ t c e t a ~ e a ol the aureole re- (2/ps)(bu/az) = WM(~U/~P)CJ;X; (3) wlts fmm thc lnck of n timc conntxnt lor any relaxation tb tm wple9jionb gins proees? in the thmry .a, that them ia no criterion for definin~ thc monning of "cnrly" or "late." For thc - W ( d ~ / d p ) / 2 mmc muson rndinl hunting will he dl-nimilnr whcn thc - ( f id /2 ) ( r t r /P ) (dB/dp) (r 7: surfncn tcnsion in "thicknc.-wnly dopendcnt." We shnll now consider the prnhlcrn o l obtaining thc rclstion that

I between surfncc inlainn nnd thicknern lor auch adr- duldp - (&~'12)(I/B)'(liB/dp) (5.10) similnr bards, both unidimcn.io~~nl nnd rnciid.

Unirlimrn#im~l ASCII-Timilat Runlin~. A convenient Thin cxpro-1~ mn be i n w l d numrricnlly Lo p wr<linntc ir the vcloeity rclnt.ivc in some mkmnec *id@ 8 M a funclion of P. Binm 6 is k n m l 8

vclocily u. fur~clion 4 p, thia g i m thcn u M E Tunqion or 8 or m. The above icmlt cnn bo pncmlisad to the d i n 1 burst

P s z/lul ('") M &ow11 below, but lor the unidimendond nyntem The choim of u, in nthilnuy ntrd dspcnde on mtprti- furtl~er inlerpretatiano m pmiblc. Thue, i t is uacd mer~hl or computnl io~l convmiencc. It mny, lor M the ind~.pendcnt ve.rinblc, eq 5.10 cnn be w r i t t e n

krample, be set cqunl to uc or uu depending on cimm- stance. Thc origin corrcrponds to p - 4 and at do rn (*'/2)(1/01'@ (5.11)

sufficiehtly lnqc p thc film is undinturbed. Thc flm which can be numcrienlly i n t q m k d ta &I directly m thicknas a n bc apross~d in temu of the thkkmlng, p, m a lunction olB and thcrclorc or a or 5. which ia tho~cipmcmlol thc hmployed previously The value of the importnht pnmmemr A I li. am

d e3 486~) (62) be introduced uait~g cq 4.18 to pie

8 Thw, B L unity in undiaturbd him UIKI inemmen ma the A = (% - u = C J - u @lZ)

rim is nppmnchcd. For p values within the hole, wn I n vim of q 6.6, this cnn bc simplified to

1 will fe@ 8 ira hnving thcvduc ccm. The volumc ol liquid pcr unit wldth (a the Idt of r A - u J ( p ) (6.1a)

point, x , In ablllined hy lntcgm~tion of tk Rlm thicknsr Radhl Prwr*onol Bursing. The ntlnlyria folbwa v -

- ~- .-.. . * - -.:_.~ - --

P - ~/[UQ (5.14) which can ngsin IIC inhgmt*l sumricnlly hut is wn-

and 8 rctd~ra iln dchnilion. Tho r.olumc of lirluid mn). aidcrnbly more complicatml e ld drm no1 lend ilaell lu be wtor of QnC rndibll (to el;minah dm~llificntion or fu~hhcr nlollc: the lirl- am-

ploynl in tho unidimensionnl cmc. numcncnl inctom) b y

I is non. thc intcgrnl alp# from the origin t o p andean bc cvnluntcd n l s ~ by

Thc velocity 01 IL film elcmcnt is amin ohtoincd by ?=king the time dcvivat,ivr with t.lm vnlume nf exprcc sbn 5.15 unchnnmd so t b t

2r + r-(dT;dk?)(-r/l=*) f

qt(lW/dp)(l/WJ = 0 (5.17)

Hcncc

u - [(p%r/llb) - ~~1/(81r/u.) =

PO - ~~~r~lcolpe) ($.is)

The mcleration of a film elomcnt is then

Condudon The n h v o discusion ahom thnt the cxiatc~lce of+

w r m l m p n ? ~ d i ~ t h f n d v f i ~ ~ c i n g rim d n lrurnting film paints to laryc c h n n p of mrf:~m th~aion :IS the film ahtinkn. Thesc chsngm cnn hc obbincd imm crpcri- mentnlly ncccsible qtrnntitics by pmeedul.~; described in sections 111 end V. Thc treatment or the limilinp cncc 01 unidimcnsionnl bursting with the ncdcet 01 m y ~llrmtion6 or surfnco tmfion as ~ v e n in scotion N cannot be mdi ly gcnenlizcd to rndinl buMingnor does i t t&c into mnsidcmtior~ the ctrccts of uir rcri%tance. Nevcrtholen, it cxnlnina n nufihcr of mpcrimcntnl ob- aervntiona nsd pcrmits tha drns.int of mmc qclnlitativc cot~clusionn. Thua it *horn that:

(A) V e y lnw ch~ngm of surfnw tcnaion with shrinkqc must bc p m n t s i n a a i l ~ ~ c n r dccrensc, orane that ia l a rapid lor initin1 than lor f ind rhm, would land on1y.h a rim oinc&iWc width and an undisturbed film nhend ol it (Figutc 7(d)).

@) Only ccrbin parts of tho relntion bcturen Sum f a a tension and shrinliqc a n bc a d h l c ape&- mentnlly, whereas others-tho conerrve O n e N C un- realized, b6ng "hidden" in shock TU8W (JGgUw 7)- This cxplrinn the cxintmrcc of such shock rnves and the conshmt speed of any thnt fmnb the aureole.

(C) The rim specd is rclntiwly inansi t in to the a iacncc of, and chnngcs in, the aunolc unlcs tho lnltcr is hirhly dcvcloped, bccau* or Bet-order compensihting cflc$ if dovintiana (cq 4.20 and Pipm 5). 00n the

'p/w)(dm'dp' (5'm) olhcr hnnd, nn nureolo rhihicll runs far n h d of thc hole Q u n t i n ~ them two arpfimio~r rrivra' cnn be m i n t e d with n eon~idcrnhly m l u d wd.

It nhould 11c notcd, honwcr, that the dcmo~~strntian of -- - - dr

m--{&d~)(d~/dp) - - (psdud~) x the miatcncc of n mnxirnc~m rim velocity (a 4.20) h n ~ -. a?, not h n gcnernlircd to the rndinl CMC.

[-w + (2*w~'fl + @'U@')(@/'p)] ('") A~kho~clsllpntettl. We nrc indchtd Lo Dr. C. k n a m for hclplul wmmcnls. -