THE UNIVERSITY OF MICHIGAN College of Engineering Department of Mechanical Engineering Cavitation and Multiphase Flow Laboratory Report No. 01357-14-T 08466-5-T THERMALLY CONTROLLED SPHERICALLY SYMMETRIC BUBBLE GROWTH; UNIQUENESS OF THE SIMILARITY SOLUTION TC. C. Scott B. R. Hao F, G. Hammitt Financial Support Provided by: National Science Foundation Grant No. GK-1889 and Army Research Office Contract No. DAHC04 67 0007 September 1969

ABSTRACT The uniqueness of the similar solution of the problem of spherically symmetric thermally controlled bubble growth is demonstrated using a generalized group-theoretic analysis. The inability of similar solution methods to account for viscous dissipation is also shown. 1

THERMALLY CONTROLLED SPHERICALLY SYMMETRIC BUBBLE GROWTH; UNIQUENESS OF THE SIMILARITY SOLUTION INTRODUCTION The problem of thermally controlled spherically symmetric bubble growth in an infinite liquid has been extensively treated by Scriven [2] and further examined by many others. By neglecting viscous dissipation, the solution to the above described problem follows from the introduction of the transformation 9 T by which the energy equation is transformed to an ordinary differential equation. Under this transformation, the bubble radius is required to vary as R- Ha (z) f being a suitable constant In problems such as this, the selection of the proper transformation is often arbitrary. That is; the one that works becomes the correct one. Any given transformation such as equation 1 must satisfy both the governing equation and the boundary conditions. *Numbers in brackets [] refer to references at the end of this paper 2

From physical reasoning, there should be only one way in which the thermally controlled bubble can grow. It is possible to derive the transformation of equation 1 from purely mathematical reasoning and to show its uniqueness. The particular method employed follows from Lie group theory and has been developed by Na [1] and applied to problems such as the boundary layer equations which also yield to similarity transformations. For thermally controlled bubble growth, the governing equations are continuity and energy. Following Scriven, they may be written as av (dRJ (3) r r ^. VI Lo J <'; CL For completeness, viscous dissipation has been retained in the energy equation. A general solution to the above equations is now desired. In particular, the question of solution by similarity methods and the number of transformation groups possible is to be investigated. SOLUTION By substituting equation 3 into equation 4 and introducing the dimensionless parameters H^. i L- t; ^ aF,! R 5

the governing equation becomes: at ~~~f R C9 2 it dE The search for all possible similarity transformations now proceeds following the methods outlined by Na 11]. By making the following definitions 4J, j2 f ) B;'! equation 5 may be written as: am, ig ^ P~~aO P6 ) The requirement that equation 6 be invariant under a given transformation is )6 )4 +/)G )ci )8 )t* )G 44~ ~ )P Id~ ~I + + )B Cr where L~ aw zz bz /Z )' U r~f- t ~p 10~ P a if rT r ^t J */'jr *, I *^ )8, L 3F ) ^ ~ ~ al [tr ^ i'lV ) 2 Lf10 +P~ ) ~ ^ te+ l,~f Z,P, )r f Z4I

and W(t,r,Q,P,q) is the characteristic function sought. With W known, the similarity transformations follow from the absolute invariants which are given by the solution of d-r C dF d ) Taking the indicated derivatives of equation 6 and substituting them into equation 7 results in: ~ +#1p #e + otx -PI where t b. u} f 4P gi t @ I dew "~' J& P Jt -2l ) vansv )1 7 } ~ and Since W is independent of P12, the coefficients of equation 9 are set equal to zero. Equation 12 then indicates that W is linear in P so that W(tJ)6It t) c (t e ) +' 1(,^ ) (I/)

Substituting this relation into equation 11 set equal to zero along with from equation 6 results in: ~ —i t i rt I* — - b T~ Since W2 is independent of P, setting.oefficients of like powers of P equal to zero indicates that W2 is a function of t alone. Thus:,w -, ( w, C eC, ) - pvC) (17) Substituting this relation into equation 10 along with equation 15 and setting the result equal to zero yields: 8. +B P Pp p o Cr)() where _ - ^ ~l ^ Z T - F7I ^jF -/F L( +L L Lr q (2/) 6

Since W1 and W2 are independent of P, each of these coefficients may be set equal to zero. Equation 21 indicates that W1 is linear in q tI o 8 ) - ^ (t; ) + I - (t( J ) (Zz) Substituting equation 22 into equation 20 and equating it with zero gives J-2 - - - = o (z ~3) from which it follows that W12 is independent of 0 and linear in r. - t L ) - SIz ( ) t/ L 9F0 (+ ) 6 y) At this point, the characteristic function is given by w = l ^ t ) + 9 iU (8) f 4u Y) o it,( ) lpt (I) (Z ) Substituting equation 24 into equation 22 and putting the result into equation 19 set equal to zero gives P. 1 + P+ s^ = o, +6) where n~ 5 ZF p -6fw t PZ Y F M JN C Le - FJ,F a W t, tt ( z J 1. - 7

And, since W, Wil W and W are not functions of q, these coefficients may be set equal to zero. Equation 29 then gives ^..~~~ (^(411 A-,^D*- i^/p ^ which, when substituted into equation 28 set equal to zero, results in 32 /~ -1~ -~L yI~ _ 5t II C ZL Since all functions on the right side of equation 31 are functions of T alone, W1 is a quadratic in r. 112 ^uz^P)~I' I ('C ) t(-P1 ^ c Substituting this relation back into equation 31 gives: 2. 2 fd -s ~ ~.. -0^, NW -ffo% C1 u>I,- j R/C) W ~'I~z'2 ~tC ^ t ZWit I^ Tli -$ Turning to the remaining coefficient of q (equation 27), and setting it equal to zero results in Fa rf^=o (3v) where E^. tP 6W,-6F I,,-^-^ f^ F"- I~>T - / IiF 6V' L -^$ WW3 (35*)y Pi,.llct-Z,]-2 wL J P 82

All of the functions in equations 35 and 36 are independent of Q so that they may be set equal to zero. Also, since all of the functions in equation 36 are independent of r, setting coefficients of like powers of r equal to zero yields: 029 ~ WI % I n (37) WI2 Z.E 0 L z) -zW,,at - t - 0o (3') a Y, + _= o = The same situation applies to equation 33 resulting in: S//W - =6 (Y) - 2W. 2i - W it 23 - ~ J.e t:(r dery3i (Yu) df The characteristic function may now be found by solving equations 37 - 45 and equation 35 set equal to zero. Thus, from equations 39, 40, 43, 44, and 45 Wlr = z (Yce c (rt) 8W,~ 5= yc3 Co (-v) 9

lu,,( = Z 4 + CS (ro) And, equations 37, 38, 41, and 42 also require Y3 [-z ct - 2Z -o l, (s) I C, - O (t3) C2 = c3 (Y) while substitution of these relations into equation 35 equated to zero gives: L ZF /uk-wi [L l C -yF ifz ti CJ - r' tr T 6C3-fCi C. 3 tt',J = From equations 51 and 52, C = C5 = (i.e. W1 = 0). Thus, equation 52 gives 1 ry _ h >>) (S6) so that W2 must be linear in t. W?,,.t + c, and cy which integrates to bs t t t cstt tsc where b is the integration constant. Furthermore, since ^ * P Jt equation 57 yields the restriction on R. Cv t-C 3 C<< 10

Or, = C. ct c, C + c (S-) From the restrictions on 0 and W2, equation 55 becomes lt Z~< af t C't C 1wil cc Cf +f f b7 CcZ l () ~~;-7 F~ WtN O -- - - (=0 The characteristic function is thus given by w = U,, (tcF) +ef c(,-+~f.-c t +.G2 4FCt t f] f-P fc t+cJ I(o) From which the absolute invariants are given by Substituting equation 60 into the relations given for, f, and d yields: d - Jr & t PARTICULAR SOLUTIONS Consider the case when the bubble growth rate is finite, ( $ / 0 ). Then, equation 54 requires that C3 = 0 and equation 58 requires that C6 be finite. In this case, equation 62 becomes C - _ct C - w ill The first absolute invariant follows from the solution of cylt*t <^ W c, c -1 1 1Z

Or, t o= Cc rzT r- = r,; (Y SL) (bY) There are now three choices for W11; either WllI is a constant or zero or it is a function of r and/Oh t By examining equation 59, one sees that the first two choices are impossible if this equation is to be satisfied. However, if viscous dissipation is ignored ( F = 0 ), the general case of Will = -C8 is valid. In this case, the second invariant is 111 8 found from tC? -+Ca T where C8 = -W ll Solution of equation 65 gives:, _-~r = <o"s r _ 0T byf)) a - T~ ( Ce ^<<-F + C.? C,~ With no viscous dissipation, the governing equation becomes, t.F ~ } -, AF 3 (f7) Substitution of the two similarity variables defined above along with 6 from equation 57 results in: MOM q ~ F' _O (,s) _.__-_ - _ _ _ -f 0 f.* I[ ^ 3 f< r^Jg'+ 7; J, _ fC7 from which Y = 1/2 is an obvious requirement so that + tt<+, rf+ a- j -~f,'f.. (691 12

Equation 69 is the most general form of the transformed equation. The only transformation possible being the linear group. Therefore, without examining the particular boundary conditions, a similar solution requires that the bubble radius vary according to -3 = SE +cC3 c+ (?o) A consideration of the boundary conditions of the problem itself, namely e (C. o) -= C,.f)=,(7/) < (ft ) = srw/ (it) shows that the dependent variable can be none other than This requires that Will = C1 = 0 and thus = 0. Also, since Will = 0 is incompatable with equation 59 unless F = 0, similar solutions are not possible unless viscous dissipation is neglected. Equation 71 also requires C7 = 0 in order that it may transform to Finally, the heat balance at the bubble wall may be transformed oIly if c = 0 in equation 70. The final form thus coincides with the results of Scriven. 13

REFERENCES 1. Na, T. Y., Abbott, D. E., and Hansen, A. G., "Similarity Analysis of Partial Differential Equations", Technical Report, NASA Contract NAS 8-20065, The University of Michigan, Dearborn Campus, Dearborn, Michigan, March, 1967 2. Scriven, L. E., "On the Dynamics of Phase Growth", Chem. Eng. Sci., Vol. 10, 1959. pp. 1-14 3. Forster, H. K., and Zuber, N., "Growth of a Vapor Bubble in a Superheated Liquid", Journal of Applied Physics, Vol. 25, No. 4, pp. 474-478 14

NOMENCLATURE a,b,c,C...C9 = constants C = specific heat hfg = latent heat r = radial co-ordinate R = bubble radius R = reference bubble radius o t = time T = temperature TX = temperature far from the bubble sat = saturation temperature u = radial velocity f = dependent similarity variable 0C = termal diffusivity 6 = Bl-ft]/t?Z9 ~= independent similarity variable A = thermal conductivity /4 = viscosity /^, ~e= density of liquid and vapor t/e