On the force and torque on systems of rigid bodies: A remark on an integral formula due to Howe

Transcription

1 PHYSICS OF FLUIDS 19, On the foce and toque on systems of igid bodies: A emak on an integal fomula due to Howe Clodoaldo Gotta Ragazzo a Instituto de Matemática e Estatística, Univesidade de São Paulo, São Paulo, SP, Bazil Esteban Tabak Couant Institute of Mathematical Sciences, ew Yok Univesity, ew Yok, ew Yok Received 1 ovembe 006; accepted 1 Febuay 007; published online 9 May 007 M. S. Howe J. Fluid Mech. 06, pesented integal fomulas fo the foce and toque on a igid body that pemit the identification of the sepaate influences of added mass, nomal stesses induced by fee voticity, and viscous skin fiction. Hee a simple extension of Howe s fomulas is done fo systems of seveal igid bodies. The goal is to sepaate these effects on each igid body. These new fomulas may help in undestanding the hydodynamic inteaction between bodies. As an example, the inteaction foces on a pai of bubbles ising at high Reynolds numbe side by side ae computed. 007 Ameican Institute of Physics. DOI: / I. ITRODUCTIO The poblem of finding the foce on a igid body moving inside a fluid is cental in many banches of engineeing, so thee is an enomous liteatue dedicated to the subject. In pinciple, if the motion of the fluid is known, then the computation of foces educes to the integation of the fluid stesses on the body suface. The point is that, except fo bodies moving at vey low speeds, thee is no ealistic poblem fo which the fluid velocity can be detemined. Thus, many foce and toque integal expessions altenative to integating stesses on the body suface wee poposed. These altenative expessions may be moe suitable fo expeimental measuements, fo numeical evaluation, o just fo theoetical insight. The eade can find a patial list of the liteatue on the subject in Refs This pape is concened with a paticula fom of foce and toque expessions poposed by Howe. 1 These ae based on fomulas used in the theoy of aeodynamic sound peviously obtained by the same autho. 5 7 Expessions of the same type wee obtained independently by Quatapelle and apolitano 8 and Chang. 9 The advantage of the foce fomulas poposed by these authos, especially those in Refs. 1 and 9, is that they allow a patial identification of the sepaated influence of added mass, skin fiction, and nomal stesses induced by voticity see also Ref. 10 about this sepaation of effects. Hee we genealize the foce and toque expessions given in Howe 1 fo a single moving body, to the case of a system of many moving bodies, in such a way that the above-mentioned sepaation of effects can be, at least patially, pefomed independently on each body. Howe himself indicated implicitly this genealization in Ref. 5; it is also natual in the context of the wok of Quatapelle and apolitano 8 and Chang. 9 The paticula case of the inteaction between a plane wall and an aifoil fixed to it was explicitly teated by Howe 11 using the same idea as that used a Electonic mail: agazzo@ime.usp.b hee. So, ou main contibution in this pape is to pusue the details of this natual genealization of the wok by Howe to any system of many moving bodies and to show that it may be useful in the study of the poblem of hydodynamic inteaction between bodies. Ou wok is divided as follows. In Sec. II, we pesent a genealization of Howe s foce expession to the case of many bodies. The way we obtain this genealization is the same one that Chang 9 used to get his fomulas, which is a little diffeent fom the pocedue of Howe s. 1 Section II has a subsection about the paticula case of no-slip bounday conditions. In Sec. III, we genealize the toque fomula of Howe to the case of many bodies. The pocedue is vey simila to that used fo the foce fomula. In Sec. IV, we compute the dag and the lift on a pai of bubbles ising side by side with the same constant velocity and at high Reynolds numbe. Fo this, we wee guided by the detailed numeical wok of Legende et al. 1 The idea was to apply the foce fomula obtained in Sec. II to a system in which the velocity field could be estimated analytically. The poblem and its difficulties ae discussed in Sec. IV. II. FORCE FORMULAS The hypotheses and notation used in this pape ae as follows. Conside a system of igid bodies whose boundaies ae smooth i.e., continuously diffeentiable sufaces. In each body, labeled 1,,...,, fix a efeence point and an othogonal efeence fame K centeed on it. Let K =e 1,e,e 3 denote a efeence fame fixed in space. The configuation of the system is detemined by position vectos R and othogonal tansfomations T, whee R is the position of the efeence point of solid with espect to K, and T :K K descibes the oientation of body with espect to K. T is the attitude matix of body. The velocity and angula velocity of body with espect to K will be denoted by Ṙ and, espectively. The set of points of body will be denoted by B, and its bounday by B. The /007/195/057108/8/$ , Ameican Institute of Physics

2 C. G. Ragazzo and E. Tabak Phys. Fluids 19, bodies move in a pescibed way, without collision, though an incompessible liquid of density and dynamic viscosity. The egion outside the bodies and occupied by the fluid will be denoted by Fl. The egion Fl can be eithe bounded o unbounded. If Fl is bounded, then its extenal bounday will be denoted by B. In this case, B can be undestood as the bounday of a distinguished body B which contains the liquid and all othe bodies in its inteio. If B is at est, then Ṙ and ae null fo all time. The case of patially bounded domains can also be teated but it will not be explicitly consideed hee. The velocity field of the fluid will be denoted by v and its voticity field by =v. IfFl is unbounded, it will be assumed that the velocity and voticity fields decay sufficiently fast at infinity, such that the seveal suface integals at infinity that appea below can be neglected. It suffices that v=o1/x and =O1/x 3. otice that these suface integals at infinity may also be zeo unde milde hypotheses as in the case in which the velocity and voticity decay vaies with the diection with slowe decay within naow wake egions. Howeve, we will not ty to pesent the most geneal hypotheses unde which ou esults hold. Decay estimates fo the velocity and voticity can be found in seveal places such as, fo instance, the appendix of Ref. 1 o lemma.1 of Ref. 3 The pessue p is assumed bounded at infinity. In Howe s fomulas, a pominent ole is played by cetain velocity potentials. Hee, fo given body and diection e i, the analogue potential is povided by the solution of the following poblem: i x,t = 0 fo x Fl, i x,t nx,t = e i nx,t fo x B, i x,t nx,t = 0 i x,t = O1/x 3 0 fo x B,, as x, whee the last condition applies to the case in which Fl is unbounded. The iotational velocity field i is that of an ideal fluid initially at est acted upon by the motion of body B with velocity e i while all othe bodies emain at est. Thoughout this pape, nx,t denotes the unit nomal vecto at bounday points xb at time t, which is diected towad the inside of B. Equation 1 depends on t though the position and oientation of bodies B, =1,...,, which change as the bodies move. So, i also depends on t. In the usual case of a single body, it is enough to solve Eq. 1 at t=0 fo i=1,,3. The solutions to poblems 1 fo all othe positions and oientations can be obtained though convenient tanslations and otations of those at t=0. In the case of many bodies, this is not tue anymoe. evetheless, if the paiwise distance between bodies is sufficiently lage, useful appoximated solutions fo poblem 1 in tems of one-body solutions can still be obtained see, fo instance, Refs This will be done fo the poblem teated in Sec. IV. The following deivation of foce expessions is based on that of Howe, 1 although we stat diffeently witing the i 1 component of the foce, F i, which acts on body B as the integal of the stess tenso ove B, F i =B pe i n B xj v i + xi v j n j, whee the sum convention ove epeated indices is assumed. Using the identity 0=B v e i n =B xi v j n j, the viscous pat of can be ewitten as B xj v i + xi v j n j = B xj v i xi v j n j = B e i n. To handle the pessue tem, we multiply the avie-stokes equation t v + p + v = v by i, and integate ove the egion Fl occupied by the fluid, v i t v i p + Fl +Fl = Fl i v i. Fl Each tem in this sum will be analyzed sepaately. Fist, Fl i p divp i pe i n, =Fl =B whee we have used the facts that i n equals e i n if n B and vanishes if nb, with, and that the suface integal at infinity of p i n is zeo. Fo simplicity, we will always efe to the case of an unbounded fluid domain. otice that the expession on the ight-hand side of the above equation is exactly the pessue foce tem that appeas in Eq.. In a simila way, we get and Fl i v = divv i Fl = B v e i n 3 4 5

3 On the foce and toque on systems of igid bodies Phys. Fluids 19, v i = div i Fl Fl B e i n div i v =Fl e = =1 B i n. =Fl v e i Finally, v e i v v e i =Fl +Fl Fl i t v t i v v t i =Fl Fl =Fl v x i +B v i v n. 7 = i v v v i tfl Fl Fl v t i Using this identity in Eq. 6, one easily obtains Howe s expession, F i = tb i v n B t X i + v X i v n = div i v tfl Fl divv i v + Fl X i v + B X i n. Fl = t divv t i n i v =1B t i + v i v n, B =1 whee we have again used the fact that the suface integals at infinity vanish, due to the decay hypotheses. Collecting all the above tems, we obtain a foce expession that is an analogue to Howe s equation.11 in Ref. 1, n F i = t i v B + =1 =1 B t i + v i v n v e i n + i v B Fl + =1 B i n B e i n. Remak: Since each B is a igid body, the facto v n that appeas in all suface integals above is given by v n = Ṙ + x R n, whee xb. Remak: In the case of a single body, denoted just by, Eq. 6 is not exactly equation.11 in Ref. 1 obtained by Howe. Indeed, Howe s expession is moe compact, and it is witten not in tems of the function i, but of X i =x i R i i. Fo a single body in R 3, the following identity holds: 6 If thee is moe than one body, howeve, it is not possible to eliminate the integals v e i n fom the sufaces of all bodies. So, although Eq. 6 can be ewitten in othe foms using functions like X i, the new expessions do not look simple than Eq. 6. Remak (Ideal flows): Howe 1 makes an inteesting analysis of foces in inviscid iotational and otational flows fo the case of a single body. The analysis becomes much moe complicated in the pesence of moe bodies. Let us biefly discuss the case of iotational flows. In this case, only the fist two lines of Eq. 6 ae non-null. Fo a single body in R 3, denoted by B, it is possible to show that the second line is also null. Indeed, in this case v= and Eq. 7 becomes B e i n =B xi v n. Then the second line of 6 becomes B t i + i v n B xi v n =0. This last identity is poven in the appendix of Ref. 5. So, in the case of one body. the only emaining tem in 6 is the one in the fist line. It coesponds to the well-known addedmass foces of potential flows see Ref. 1, Sec... With moe than one body, the second line of 6 does not cancel out anymoe. In Sec. IV, fo instance, the potential flow of a pai of sphees moving paallel to each othe with unifom velocity will be analyzed. Fo this system, the tem in the fist line of 6 vanishes, but the one in the second line does not, yielding instead an attactive foce between the two sphees. Fo abitay motion and body geometies, even in the context of iotational flows, the fist two lines in Eq. 6 can give ise to vey complex inteaction foces see, fo instance, Ref. 15 fo a teatment of this question using Lagangian fomalism. It is woth mentioning that the most complete analysis of the foces and toques on a defomable single body moving unsteadily in a weakly nonunifom, non- 8

4 C. G. Ragazzo and E. Tabak Phys. Fluids 19, stationay, iotational flow field was given by Galpe and Miloh. 17 A. o-slip bounday conditions Some simplifications of expession 6 occu when a noslip bounday condition is imposed. In this case, the velocity at the suface of each body is given by v = Ṙ + x R, fo x B. 9 This implies that B v e i n = VolB Ṙ e i whee VolB is the volume of B and =B x R + e i, is the un-nomalized cente of volume of B, measued fom its efeence point. In paticula, if R coincides with the cente of volume of B, then is null. Moeove, if i x,t is known, then the othe tems in lines one and two of Eq. 6 ae also known. Remak (Skin fiction): The no-slip bounday condition hypothesis implies that the stesses on the suface of each body can be witten as pn+n. So, the last tem in Eq. 6 is an integal of tangential viscous stesses on body B, epesenting the so called skin fiction. The othe tem in the last line of Eq. 6 aises fom the integal of the pessue on the suface of B. Howe gave this intepetation in Ref. 18 Sec..3 and Ref. 19 Sec and illustated the diffeence computing these tems fo the Stokes dag on a sphee. The distinction between both tems is moe evident hee than in the case of a single body, since the integal of the fist tem in the last line of Eq. 6 is ove the suface of all bodies, not only B, and this is not consistent with the concept of skin fiction. This point is pusued futhe in Ref. 10. Remak: It is woth pointing out anothe big diffeence between the case of a single body and that of moe than one body. In the case of a single body in R 3, it is possible to obtain the thee functions 1,, and 3 fo any tr we will omit the index to simplify the notation since they ae known at t=0. This is a consequence of the invaiance of the Laplacian unde tanslations and otations. The constuction is as follows. Let Tt be a otation matix that descibes the oientation of the body and which is given in the following way: At time t=0, choose an abitay point x 0 in the body distinct fom its efeence point. If x is the position at time t of the chosen point, then Tt is such that x=rt+ttx 0 R0. To simplify the notation, let us suppose that R0 =0, T0=identity, and let us denote the function x 0 Rt+Ttx 0 by x t x 0. otice that t is the flow of the igid body motion velocity field Ṙ+x R. If jx, j=1,,3, denote the functions j x,0, j=1,,3, at t=0, then it can be shown that fo any t, i x,t = T ij t j t 1 x. Using this, we conclude that on the suface of the body, whee the velocity field is the igid-body one due to the no-slip bounday condition, the following holds: t i x,t + v i x,t = d dt i t x,t t=0 = d dt T ijt jx t=0. Finally, fom this identity, we obtain t i + v i v n = ijk k x,tv n. 1 B jb This last equation is essentially expession.15 of Howe, 1 which can be easily ewitten in tems of added-mass coefficients. Fo the case of moe than one body, howeve, this vey explicit constuction does not wok. Something simila can be done when appoximated explicit functions i ae used, as in Refs. 15 and 16. Yet the analogue to Eq. 1 becomes much moe complicated. III. TORQUE FORMULAS The following deivation of toque expessions is again based on that of Howe. 1 evetheless, hee we assume fom the beginning that, if is diffeent fom zeo, then no-slip bounday conditions hold. So, if 0, the foce pe unit aea on the suface of body is pn + n, 13 while if =0, this foce educes to pn. Remak: The eason fo assuming no-slip bounday conditions fom the beginning is that expession 13 is not tue unde moe geneal bounday conditions if 0. Conside, fo instance, the velocity field v= x 1, x,x 3 and the suface =x 3 =0. This suface is igid, in the sense that v n=v e 3 =0. The deviatoic stess tenso d ij = xi v j + xj v i is not tangential at. Indeed, the deviatoic foce pe unit aea on is e i xi v j + xj v i j3 =4e 3, which does not agee with 13, since in this case =0 and =0. Howe in Ref. 1 also stats fom expession 13, but does not discuss this point. Thee ae poblems, like the bubble one in the next section, whee the flow is essentially iotational, viscosity is not null, and yet the bounday conditions ae not the no-slip ones. Fo this sot of poblem, it is not clea that the toque expessions of this section apply. The i component of the toque about the efeence point R of body B is M i =B pe i x R n + B e i x R n. 14 To find expessions fo the toque, we do not use the auxiliay potential function i of the pevious section, but in-

5 On the foce and toque on systems of igid bodies Phys. Fluids 19, stead the potential i, which is the solution to the following poblem: i x,t = 0 fo x Fl, i x,t nx,t = e i x R nx,t fo x B, i x,t nx,t = 0 fo x B,, 15 FIG. 1. Diagam showing the coodinates used to descibe the two-bubble system. i x,t = O1/x 3 0 as x whee the last condition applies to the case in which Fl is unbounded. To handle the pessue tem in Eq. 14, we poceed as in the pevious section, multiplying the avie- Stokes equation 4 by i and integate v ove Fl, i t v i p + Fl +Fl = i v i. 16 Fl Fl Each tem in this equation can be handled in a way simila to that in the pevious section. Fo instance, the pessue pat of 14 can be witten as Fl i p divp i =Fl =B pe i x R n. Hence, essentially epeating the same steps that led to expession 6, we obtain n M i = t i v B + =1 =1 B t i + v i v n B v e i x R n + Fl i v + =1 B i n + B e i x R n, 17 whee the last tem is the one popotional to in Eq. 14, witten in a diffeent way. Remak: Using the no-slip bounday condition, we can wite B v e i x R n = e i Ṙ + e i I, 18 whee is defined in 11 and I is the moment of inetia opeato given by I =B x R x R. Remak: In the case of a single body, the following identity holds it is the analogue to Eq. 1: B t i + v i v n = ijk jb k x,tv n, an integal that again can be ewitten in tems of the body s added-mass coefficients. IV. A PAIR OF BUBBLES RISIG SIDE BY SIDE AT HIGH REYOLDS UMBER Ou goal in this section is to illustate the use of the foce fomula obtained in Sec. II. The system that we will conside consists of two spheical bubbles ising with unifom velocity side by side. This poblem was ecently computationally studied by Legende et al. 1 This detailed and athe complete pape guided us on the subject. The diffeence between this system and that of two solid sphees lies in the bounday conditions, the no-slip fo the igid sphees, and the no-tangential-stess fo the bubbles. This second bounday condition is much milde in the sense of geneating voticity. The geomety of the poblem is the following. Conside a pai of spheical bubbles of adius a centeed, espectively, at y = l/ and l/ with espect to a Catesian efeence fame e x,e y,e z, see Fig. 1. We will denote the bubble at y = l/ by B and that at y=l/ by B. The bubbles move with constant and equal velocity Ue x. The foce on each bubble has two components: a dag in the diection of the motion, e x, and a lift in the diection tansvesal to the motion, e y. These foces ae usually given in dimensionless fom as dag and lift coefficients, C D and C L, espectively, obtained dividing the foces by a U /. Hee we will be inteested in the limit of high Reynolds numbe, Re =au/, and fa apat bubbles, namely S=l/a lage. In these limits, the leading-ode tems of C D and C L on bubble B ae, as given in Ref. 1, C D = Re S 3, 19 C L = 6 40 S41 0 Re. The tem in C D, which is independent of the distance, was fist obtained by Levich, 0 and that popotional to S 3 was

6 C. G. Ragazzo and E. Tabak Phys. Fluids 19, obtained by Kok. 1 Both Levich and Kok obtained thei esults equating the viscous powe dissipated by the iotational flow to the wok done by the dag. Hee both tems in 19 will be obtained fom the suface integals in the last line of Eq. 6. The tem in C L that is independent of the Reynolds numbe, an attactive foce, is due to the iotational velocity field of the two sphees. This tem can be found, fo instance, in Ref.. Hee it is obtained fom the integals on the second line of 6, when only the potential velocity field is taken into account. The tem popotional to Re 1 in C L, as fa as we know, was not analytically deived yet. In Ref. 1, Legende et al. estimated this tem numeically, afte aguing that it should be of this fom based on an ode of magnitude analysis like that in Mooe. 3 In pinciple, it is possible to obtain this pat of C L fom Eq. 6. It comes fom suface integals in the second line of 6 and fom the volume integal in the thid line of 6. The computations ae toublesome and it will not be pusued hee. We point out that with the use of Eq. 6, we wee able to obtain in a unified way all known esults on this poblem, which wee peviously obtained using vey diffeent aguments. Moeove, along the computations in Sec. IV C, one can see that we ae able to sepaate the diffeent contibutions of the nomal viscous stess and the pessue stess to the dag. This cannot be done though powe dissipation methods, and it was peviously known only in the case of a single bubble. 4 This illustates one of the main advantages of Howe s fomula, namely the identification of diffeent effects on the oveall foces. A. Bounday conditions and velocity potentials The idea on how to study the flow aound bubbles at high Reynolds numbe 0 Sec Refs. 3 6 can be biefly descibed as follows. At high Reynolds numbe, the flow outside a system of bubbles is essentially iotational except fo a bounday laye aound each bubble of thickness Re 1/ and a wake of beadth Re 1/4. If the velocity potential of the system is denoted by, the velocity field of the fluid can be witten as v=+u, whee the velocity coection u is essentially diffeent fom zeo only inside the votical egion. At the bounday of each bubble, a feetangential-stess bounday condition must be satisfied. If only the potential pat of the flow is taken into account, this condition is violated. In fact, in ode to satisfy the feetangential-stess bounday condition, it is necessay that =u satisfies =n n 1 on the suface of each bubble. In ode to poceed, we need the velocity potential of a pai of sphees as shown in Fig. 1, as well as the potentials defined by Eq. 1. Velocity potentials fo a pai of sphees moving at abitay angle with espect to thei line of cente can be found in Refs. 7 and 8, and appoximations, such as that pesented below, can be found in Refs. 13 and 14. Upto ode 1/l 4 and in a neighbohood of the sphee B, the function is given by = UA 0 + A 3 + A 4, whee A 0 = 1 a 3 3 x = 1 a 3 cos, A 3 = 1 a 3 l a 3 a 3 A 4 = 1 a5 l43+ 5x y a 3 a 3 3x = 1 l cos, 3 = 1 a 3 l 43 + a5 3cos sin cos. Hee x,y,z ae Catesian coodinates with oigin at the cente of sphee B and axis paallel to those of system x,y,z shown in Fig.1, and,, ae spheical coodinates centeed at sphee B, with x = cos, y = sin cos, z = sin sin. The same is given in a neighbohood of sphee B by = UA 0 + A 3 A 4, 4 whee A 0, A 3, and A 4 ae the same functions given above but evaluated at x,y,z and,,, which ae Catesian and spheical coodinate systems, espectively, with the oigin at the cente of sphee. otice that e =Ue x e and e =Ue x e. The solution to poblem 1 fo 1 1,, and 3 stand fo the diections x,y,z, espectively in a neighbohood of B and up to ode 1/l 4 is given by 1 x,y,z = A 0 and nea B by 1 x,y,z = A 3 A 4. ea B, 1 is given by 1 x,y,z = A 0, and nea B, it is given by 1 x,y,z = A 3 + A ea B, the solution of 1 fo up to ode 1/l 4 is given by x,y,z = E 0 and, nea B,by x,y,z = E 3 + E 4, whee E 0 = 1 a 3 3 y = 1 a 3 sin cos, 9 30 E 3 = 1 a 3 a3 l3+ 3y = 1 a 3 l 3 + a3 sin cos, 31

7 On the foce and toque on systems of igid bodies Phys. Fluids 19, E 4 = 1 a 3 3 l + a5 53y 4 = 1 a 3 3 l + a5 33 sin cos 1, and again, nea B, we must eplace x,y,z by x,y,z, etc. Fo, simila expessions hold. B. The pue iotational pat of the foces We will just compute the foces on bubble B those on B then follow fom the symmety of the poblem. Ifthe flow is iotational, only the fist two lines of Eq.6 contibute to the foce. In fact, since the velocity of the bubbles is constant, only the second line of Eq. 6 is nonzeo. The iotational contibution to the foce in the diection of the motion e x is null. This is D Alambet s paadox. In the context of Eq.6 with i=1, the best way to see this is to conside both bubbles as a single igid body. By symmety, the foce on the pai is twice the foce in each one. But fo a single igid body, the second line of Eq. 6 is zeo due to Eq.8. We tun to the computation of F. Substituting, 4, 9, and 30 in the second line of Eq.6, neglecting all tems of ode lage than 1/l 4, emembeing that n= e on B and n= e on B, and that, since the bubbles ae tanslating with velocity Ue x, we have =x Ute x, which implies that t = U e x the same is tue fo t, etc., we obtain that the iotational pat of C L is 6/S 4, which is the dominant tem in the expession 0 fo the lift. C. The dag coefficient C D ow we compute the integals in the last line of Eq. 6. Again we conside just the foces on bubble B. Fom the bounday conditions 1 with Ṙ =Ṙ =Ue x, we have on the suface of B that = e + e, whee = a sin, =+ U a sin + a. 3 The same expessions hold on the suface of B, eplacing,, by,,. A simple symmety agument involving paity with espect to the eflection can be used to conclude that all functions that ae being integated in the last line of Eq. 6, fo i=, which coesponds to a lift foce, ae null. Theefoe, thee is no contibution of the last line of Eq. 6 to the lift. ext we compute the contibutions of the last line of Eq. 6 to the dag i=1, namely in the diection e 1 =e x. Fist conside the integal + B e 1 e. This tem comes fom the integal of the viscous stesses on B, Eq., though the modification in Eq. 3. Fo a bubble, the tangential stesses ae null. So, although it is not obvious fom the expession above, the esulting foce obtained fom the above integal is due to nomal viscous stesses acting on the suface of the bubble. Again, this shows that the stess expession given in Eq. 13 does not hold fo the no-tangential-stess bounday condition as aleady noted in the emak below Eq. 13. Afte substituting expessions 3 fo in the above integal, using some paity aguments, and pefoming some elementay computations, we ae led to the following esult: B e 1 e = 8Ua 4U a4 l 3 + O1/l6. The pat of this foce that is independent of l is a piece of Levich dag, the othe is due to the inteaction between bubbles. The othe integals in the last line of Eq. 6, B 1 e B 1 e, ae due to pessue stesses. Using as in Eq. 3, and the expession fo 1 in 5, we obtain, afte some computations, B 1 e = 4Ua U a4 l 3. The tem 4Ua is the last piece of Levich dag, which comes fom integating pessue stesses. Finally, using as in Eq. 3 and 1 as in 6, we obtain B 1 e = 6U a4 l 3. Adding all these contibutions and dividing by a U /, we obtain the expession 19 fo C D. otice that the dag is completely detemined by the suface integals in the last line of Eq. 6, and that these integals depend only on the voticity on the suface of both bubbles. This esult is in ageement with those of Kang and Leal 4 and Stone. 6 In paticula, Stone showed that fo any bubble of any shape unde steady tanslation, the dag coefficient up to ode Re 1 depends only on the voticity on the bubble suface, and it does not depend on the voticity within the fluid this was demonstated using an identity that elates the enegy dissipated by the potential flow to the voticity on the suface of the bubble. The same applies hee, as shown in the following. Afte all integals aleady computed in Secs. IV B and IV C ae subtacted fom the foce expession in Eq. 6, the following tems ae left: B u i n + B i uv n + i uv n + i B Fl u B i n + u i. Fl

8 C. G. Ragazzo and E. Tabak Phys. Fluids 19, It is convenient to futhe tansfom these integals: the fist plus the second line and the thid line of the above expession become, afte some computations, the fist and the second integal, espectively, of the following: + u i u u i. 33 Fl Fl The ode of magnitude of each integal above with espect to Re 1 can be estimated using the analysis pesented by Mooe 3 of the flow aound a single bubble see also Ref. 5. The fist conclusion we get fom this analysis is that up to ode Re 1, the contibution of the second integal in 33 can be neglected in compaison to the fist one fo both components of the foce. This is vey natual since u goes to zeo as Re. So, we ae left with the fist integal of 33. Let us conside the fist integal in 33 with i=1, which coesponds to the dag diection e 1 =e x. In this case, it is convenient to wite =U 1 + 1, which holds due to the definition of 1 and 1 in 1. Then 1 = U 1 1 = U and a paity analysis with espect to the vaiable y we ecall that x,y,z ae Catesian coodinates with the oigin at the cente of volume of the two bubbles implies that the fist integal in 33 is zeo. Theefoe, at least up to the ode of magnitude consideed hee, thee is no contibution to the dag that comes fom integals in Eq. 6 that involve the voticity inside the fluid. We can also show that the opposite situation aises fo the lift: All integals in the last line of 6, which involve the voticity on the bubble suface, do not contibute to the viscous pat of the lift. The viscous pat of the lift depends only on the voticity within the fluid. ACKOWLEDGMETS This wok was patially suppoted by FAPESP São Paulo, Bazil, Gant o. 04/ C.G.R. is patially suppoted by CPq Bazil Gant o /96-0. The wok of E.T. was patially suppoted by the Division of Mathematical Sciences of the ational Science Foundation. 1 M. S. Howe, On the foce and moment on a body in an incompessible fluid, with application to igid bodies and bubbles at high and low Reynolds numbe, Q. J. Mech. Appl. Math. 48, F. oca, D. Shiels, and D. Jeon, A compaison of methods fo evaluating time-dependent dynamic foces on bodies, using only velocity fields and thei deivatives, J. Fluids Stuct. 13, R. B. Guenthe, R. T. Hudspeth, and E. A. Thomann, Hydodynamic foces on submeged bodies-steady flow, J. Math. Fluid Mech. 4, T. Sapkaya, A citical eview of the intinsic natue of votex-induced vibations, J. Fluids Stuct. 19, M. S. Howe, On the estimation of sound poduced by complex fluidstuctue inteactions, with application to a votex inteacting with a shouded oto, Poc. R. Soc. London, Se. A 433, M. S. Howe, On unsteady suface foces, and sound poduced by the nomal chopping of a ectilinea votex, J. Fluid Mech. 06, M. S. Howe, Contibutions to the theoy of aeodynamic sound, with applications to excess jet noise and theoy of flute, J. Fluid Mech. 71, L. Quatapelle and M. apolitano, Foce and moment in incompessible flow, AIAA J. 1, C. C. Chang, Potential flow and foces in incompessible viscous flow, Poc. R. Soc. London, Se. A 437, J. C. Wells, A geometical intepetation of foce on a tanslating body in iotational flow, Phys. Fluids 8, M. S. Howe, Unsteady lift and sound poduced by an aifoil in a tubulent bounday laye, J. Fluids Stuct. 15, D. Legende, J. Magnaudet, and G. Mougin, Hydodynamic inteactions between two spheical bubbles ising side by side in a viscous liquid, J. Fluid Mech. 497, A. B. Basset, A Teatise on Hydodynamics Vol. 1 Dove, ew Yok, 1888, epinted in H. Lamb, Hydodynamics, 6th ed. Dove, ew Yok, C. Gotta Ragazzo, On the motion of solids though an ideal liquid: Appoximated equations fo many body systems, SIAM J. Appl. Math. 63, Wang and P. Smeeka, Effective equations fo sound and void wave popagation in bubbly fluids, SIAM J. Appl. Math. 63, A. Galpe and T. Miloh, Genealized Kichhoff equations of motion fo a defomable body moving in a weakly non-unifom flow field, Poc. R. Soc. London, Se. A 446, M. S. Howe, Voticity and the theoy of aeodynamic sound, J. Eng. Math. 41, M. S. Howe, Theoy of Votex Sound Cambidge Univesity Pess, Cambidge, UK, G. K. Batchelo, An Intoduction to Fluid Dynamics Cambidge Univesity Pess, Cambidge, UK, J. B. W. Kok, Dynamics of a pai of gas bubbles moving though liquid. Pat I: Theoy, Eu. J. Mech. B/Fluids 1, L. van Wijngaaden, Hydodynamic inteaction between gas bubbles in liquid, J. Fluid Mech. 77, D. W. Mooe, The bounday laye on a spheical gas bubble, J. Fluid Mech. 16, I. S. Kang and L. G. Leal, The dag coefficient fo a spheical bubble in a unifom steaming flow, Phys. Fluids 31, J. F. Hape, The motion of bubbles and dops though liquids, Adv. Appl. Mech. 1, H. A. Stone, An intepetation of the tanslation of dops and bubbles at high Reynolds numbe in tems of the voticity field, Phys. Fluids A 5, T. Miloh, Hydodynamics of defomable contiguous spheical shapes in an incompessible inviscid fluid, J. Eng. Math. 11, A. Bisheuvel and L. van Wijngaaden, The motion of pais of gas bubbles in a pefect liquid, J. Eng. Math. 16,