AN EXACT SOLUTION OF THE RAYLEIGH —EE SANT EQUATION Wlly Sn lth Heat Transfer and Fluid Irechanics Laboratory Department of Nuelear DEngineering The University of Iichigan Oetober 1962

ABSTRACT.P A power series solution of the -RayleirlP-Beesa: t equation goverkning the collapse of a spherical cavt in an inviscil f-luid is presented~ IFuumerical values for the non-dri-ime sional solu<ion t = t(R) are obtainedWd wfhere R is ixorcIeossad ill ternas of +I-9t initial bub:)le radius Ro and t i:e is expressed in ofers of IRo V I/p A1IO ILEDQI i Tl'Financial support for the overal. researech investigations of which this study: s a part vas furnished by NASA and N3FE

-1 ~~n-roductiono In the earliest studies of the collapse or growth of a spherical eCru-`Cr within a continuous liquid medium Besant in 1859 and later iAylaeigh2in 1917,obtained by different methods a differential equat ion governing the motion of the bubble wall. This equation, based on the assumption ofan inviscid and incompressible fluid, is written. RR 2 R2 POO 2;here Pis the density of the liquid and p the pressure at a very large distance from the collapsing or groc-wing cavity, which is assumed to contain a vacuum, By integration in t, the above equation reduces to: 2P (1) of which no solutions -exist in the literature. An exact solution of (1) would be of considerable interest in connection with bubble dynanic studies, since it could be compared to the results obtained from more sophisticated analyses, such as those of Gilmore or Flynn, in which viscosity and compressibility are taken into account9 The function R = R(t) determines the radius of a collapsing bubble of initial radius Ro as a function of time, and will be therefore a single-valued function, In consequence, a solution of the forn t = t(R) will be equally useful and easier to obtains In fact, Rayleighl gave in his original paper a solution for the "time of complete collapse" required for the bubble to reduce its diameter to zero~ For all other tines, as pointed out by Lamb3, the solution is not so easily found0

2, Pover Series Solution, To express equation (1) in a more tractable form, let: ~K - of iE (2) and define a new dependent variable pas: (3) RO Thlus: w "1 dR dot Ro dt and equation (1) becomes: dt But for the collapsing bubble, dR/dt <O, and hence, also dP/dt < 0o Since E is a positive number, only the negative sign has physical meaning in the above expression, w.hich is then written: dt-K K (1 - f33)Yz The time required for the bubble to collapse from an initial radius Ro (i.e.e, = 1) to an arbitrary radius R is thus: For t= O, the time of complete collapse 2' is obtained: ar Kf.LA (5) As shown by Rayleigh2, this integral is easily solved in terms of the gamma function. Let:. s and then 3,32dS -ad.

The integral (5) becomes then: r= f K&'8(1.Lz) dz 3 loReSling now the wel known formula | x-(l- nol (m) r(n) 0 (110M) dx we have for this case, for m = 5/6 and n a 1/2: 3 r(4/3) After calculation of the gamm functions and replacing K by its value, (2), one finally obtains: | r.9l468 R0/. I (6) V'P as given by Rayleigh. Butt we are interested in calculating t wfhen the final value of 3 is different fron zero. For that case, equation (4) is rewritten as follows: t = i d _ K. 1 K O 3(Y&e3) (1 cl 3 ) or also, con idering equation (5): t =r (7) r The integral (') d will eist for all 0 (I31, since the integrand is defined wi-thin this interval, This integrand is then expressed in terms of the (*) See Reference 4, page 383, for example.

(X+ZX)'= 1 + 3!x + "2~ x;+:..- - v. e valid for any real numr m and for \,<1 For our case: m = 1/2 and therefare: $ (~ ~ (2n4l!1 B 1 (9) (2.'.f't" nl__ (2n) i 3 valid for i.e...within the interval of interest 04 Je l, and there: (2n-1) 1 I.3.:5...... (2n-3)2{-) and: (2n) 1!' 24,06......(2n4) (2n-2)2n The radius of convergence r* of the poer series (9) cal be found b using the formua s+) For this case: (2_1) _ Cn' (2n)iI (2l+1)1 Cn+l (2n+2)11 and therefore: (2a3) z (2n+2) 1 I. 2n+2 r* -I J 1 n, nIII 2n-"~ o0.1l <

and converges uniformly for: o0 P<rl 1 Since a power series can be integrated term by term within the interval of convergence, by substitution of (9) inb (8),re obtain I I= d0 (+ ) M2n-1)1! Inl (2n)!J 3n or: I tA( (3+ (2n)!I (10) For sl1, the intcgation term by term is not valid, since the power series (9) diverges at that point. For our purposes, this is not iuport-nt, since for - 1 the value of I is, from equation (5): I and in (7)} t - O K By substitution of (10) into (7), then: (2n)! 3n (+i2 Replacing r and K by their respective values (6) and-(2), the desired solution of the Rayleigh-.:esant equation is expressed: for Op c3l, and: for 1. This i also evident fron piysical oonsderations: (*p) Reference 4, pase 352, Theorem 37.

).4twaaerical Solution Practical applications of equation (12) require the calculaK;on of numerical values that are independent of the paL:tictlae.r')-1oblen under consideration. Clearly, then, the first step ie to express (12) in a dimensionless form. For that, one observes that -heo quantity: Ro ri lhas the dimensions of a time, Hence, if t is expressed in terms of this quantity, equation (12) becomes: t o.9l4.68 +~ (2nml)!I ( I (14) | s 0l2 Ai5 nml (2n)I! 3n +. valid for O< <1,, where p R/Ro. This non-dimensional equation Can also be written, in a more compact form: t o.91468 w S( ) (15) hieroe S( i) is defined as: o S(lB) 1 224745A4 (nt ),3n+20j1 (1) The next problem, then, is to dtermine how many terms one must calculate to obtain S(P) with an error less than a given walue G. ITo do that, let: n2l....n I nul 3n + n-1 anmd calculate: ____+1 ) 53n3 3n + 3

Then, the remainder of the series is given by the expression: I (Un17) 1 |.3 (2n+2)l 1(3n +3 (37) This formula gives only an upper bounld of the error. One notices that for. P-l, (1- p3). 0', and therefore the values of R. wll increase as E. approaches 1. If 1I is the maximum value of P in the intorval of interest, the actul error for 3 will be less than the value given by (17) for (3 =(2l. In consequence, ession (17) can be used to obtain an estimate of the error that twill result / computing only n terms of the sum, for different values of p. The folltrdng values 8were obtained: 0o90o lo.0*00054 20 0.0334 0.999 10 1.5285 20 0.5852 Using a 7090 IItli digital computer, values of t as given by (14) were calculated for different values of n and d3. A preliminary calculation show-ed that the actual error, for given n and (,is as _ n -. oe0 < p07 10 10'4 (.) See Reference 4,page 32o_

confirring that the values of E are ruch less than the values Rn predicted by using expression (17). Using then these values of n a fTnal machine calculation was performed, incrementing, by 0.O1 betw-een 0 and 0.99. The results are listed in Table I, while the corrsponding plot of t vs. 3 is presented in Figure 1. One must keep in mind that P represents a non-dimensional radius, defined R and that t is a non-dimensional time, related to the actual tire ta by the expression: t = *mu.nr Ro T It is interesting to note from the above that the usual considerations of dynaice Is ilarity apply to this case in that tirnes of collapse for different fluids would be equal as long as the available ":head drop", ie, p, vwere the same. w::l/F62

ts I t t t 0.99 0.016145 0,64t 0.733436 0029 0892245 0.98 0.079522 0.63 0.741436 0.28 0.894153 0,97 o.130400 0.62 0.*749154 0.27 0.895956 o.96 0.174063 0.61 0.756599 0.26 0.897658 0.95 0.212764 0,60 06763782 0,25 0.899262 0.94 0.247733 3 059 0.770712 0o24 0'900769 0.93 0,279736 0.58 0.7773981 0.23 0.902182 0.92 0.309297 0.57 0.783847 0.22 0,903505 0.91 0.336793 0.56 0.790068 0,21 0o904738 0.90 0.362507 0.55 0.796068 020 0.905885 0,89 0,.386662 054 0*801854 019 0906947 Oo88 0*.409433 0.53 0.807433 0,18 e0907928 0.87 0.430965 0 o52 0.12810 0.17. 0908829 0.06 0/0451377 0.51 0*817993 0.16 0,909654 0~.85 0.470770 0.50 0.2298| 0.15 0,.910404 0,o4 0.489229 0.49 00.827798 Oo.4 0.911083 0.83 0*506830 0.48 00832/31 0,13 0.911692 0.02 0.523635 0.47 0.-36o90 o,12 0*912234 0.01 0539701.0.46 0*841101 0.11 0.912713 0,80 0,555078 0.45 0.845308 0o10 0O913130 0,79 0o.569810 0.4 0 049277 0,09 O. 913409 0.78 0.583937 O.e43 0.853090 0.08 0O913793 0,77 0.597495 0.42 0.856752 0,07 0o914045 0.76 0o610515 0.41 0.860268 0O06 0016 o948 0.75 00623027 0*40 0.063640 005 o. 9144o6 0.74 0,635059 0.39 0o866872 o0o4 0o914523 0973 0,646633 0.38 0*869969 0.03 0,914604 0o72 0.657773 0o37 0 872933 0.02 0.914652 0.71 0o668498 0.36 0#.75768 0.01 Oo.914675 0,70 0 678630 0.35 0.878477 0,00 0 o91468 0,69 0,68784 0034 0.0881062 0*68 0*698377 0.33 0.883528 0,67 0,707625 0.32 0.885876 0,66 0.716542 0031 0.888110 0.65 0.725 14 0.30 0.890232.......1 - I f a! m in in I i A or, Error loss thanl 106 for O$0. O96 qAf.E I

I I o.6Fl | 0 0_ O. 0.2 0. 4 o.6 o08 1 Figol. Non-dimensional Tine vso Non-dimensional PLdius for Collapsing Spherical Void0

W;B3esant, "A Treatise on HrA1drodynar.cSi, Cnambridge University IPress,Ca7br1, 859o Lord Rayleigh On Vhe Pressure Dovoloped in a Liquid during the Co lapse of a _3 pherical Cavity', Phil Iag1. -A 94 (1917). HoLamb, "Iydrodynanics", Dover Publications, New York,.9945Q i, Uilfred Kaplan, "Advanced COleulus", Addison'.eeley Publishing Co., Reading, I.-s 1959. Charles D.Hodgma, "Hndbook of Physics and Cherstry" 33rd edition, Chei cal Rubber'Publishing Co, June 1 RoS.Burington, "HIndbook ofI.~themati a Tables", Handbook Publishers Inc., H-19a..o 7~ FoRoGilmore, " The Growt.h or Collapse of a Spherical Bubble in a Viscous Com-pressible Liquid", ieat Trars foer and Fluid Mlechanics Institute, 1952, Stanford University Press~ H*GoFlynn, "Collapse of a Transient Cavity in a Comnpressib1, LiquidV Part I, An Approximate Solution", TecholIlemo.Noo3g, UR".014-903, Office of Naval Research, ~1arch 1957o