Cavitation Bubble Dynamics in Non-Newtonian Fluids

Transcription

1 Cavitation Bubble Dynamics in Non-Newtonian Fluids Emil-Alexandu Bujan Depatment of Hydaulics, Univesity Politehnica Buchaest, Buchaest, omania Cavitation phenomena play impotant oles in many aeas of science and engineeing. The most inteesting effect of the non-newtonian popeties of the liquid is the eduction of cavitation damage and noise. This aticle eviews expeimental and theoetical effots to undestand such phenomena. Two majo aeas of eseach ae descibed, namely the dynamics of cavitation bubbles oscillating in a liquid of infinite extent, and the behavio of bubbles collapsing nea igid walls. POLYM. ENG. SCI., 49: , ª 2008 Society of Plastics Enginees Coespondence to: Emil-Alexandu Bujan; eabujan@yahoo. com DOI /pen Published online in Wiley InteScience ( VC 2008 Society of Plastics Enginees INTODUCTION Cavitation phenomena play impotant oles in many aeas of science and engineeing, including acoustics, biomedicine, botany, sonochemisty, and hydaulics. They occu in numeous industial pocesses such as cleaning, lubication, pinting, and coating. Although much of the eseach effot into cavitation has been stimulated by its occuence in pumps and othe fluid mechanical machiney, cavitation is also an impotant facto in the life of plants and animals [1 4]. The most intensively studied consequence of cavitation is the occuence of eosion o cavitation damage to solid sufaces in the nea vicinity of collapsing cavities, but othe widely known effects include educed hydaulic pefomance and the geneation of excessive vibation and noise. Seveal books that seve as a valuable esouce fo the field of bubble dynamics and cavitation ae available [1 3], togethe with the extensive eviews by Plesset and Pospeetti [5] and Blake and Gibson [6]. The latte addess geneal aspects of bubble dynamics and cavitation in Newtonian fluids. In this eview, we focus on bubble dynamics in non-newtonian fluids. Such fluids occu widely in pocess engineeing, and it is essential to undestand that the effects of non-newtonian popeties on bubble dynamics and cavitation ae fundamentally diffeent fom those of Newtonian fluids. Aguably, the most significant non-newtonian effect in the context of bubble dynamics and cavitation aises fom the damatic incease in viscosity of polyme solutions in an extensional flow [7], such as that geneated about a spheical bubble duing its gowth o collapse phase [8, 9]. Specifically, polymes, which ae andomly oiented coils in the absence of an imposed flow-field, ae pulled apat and may incease thei length by thee odes of magnitude in the diection of extension [10]. As a esult, the solution can sustain much geate stesses, and pinching is stopped in egions whee polymes ae stetched. This extensional thickening leads to the chaacteistic beds-on-a-sting pofile of polymeic jets, as seen by Yain [11]. Despite the inceasing use of non-newtonian liquids in industial applications, a compehensive pesentation of the fundamental pocesses involved in bubble dynamics and cavitation in non-newtonian liquids has not appeaed in the scientific liteatue. This is not supising, as the elements equied fo an undestanding of the elevant pocesses ae wide-anging. Consequently, eseaches who investigate cavitation phenomenon in non-newtonian liquids oiginate fom seveal disciplines. Moeove, the esulting scientific epots ae often naow in scope and scatteed in jounals whose foci ange fom the physical sciences and engineeing to medical sciences. The pupose of this eview is to collect the infomation to be gleaned fom these studies and oganize it into a logical stuctue that povides an impoved mechanistic undestanding of bubble dynamics and cavitation in non-newtonian liquids. One main message is that the intoduction of ideas fom theoetical studies of nonlinea acoustics and moden optical techniques has led to some majo evisions in ou undestanding of the dynamics of cavitation bubbles in non-newtonian fluids. SPHEICAL BUBBLE DYNAMICS The investigation of the dynamics of spheical cavitation bubbles is of no diect inteest fo the explanation of cavitation eosion, because bubbles close enough to a bounday to cause damage will always collapse aspheically. Nevetheless, it povides the basis fo the intepetation of data obtained fo the asymmetical collapse of bubbles in non-newtonian fluids and is to date the only means of compaing expeimental esults with theoy. POLYME ENGINEEING AND SCIENCE -2009

2 Geneal Equations of Bubble Dynamics Conside a spheical bubble of initial adius 0 situated in a compessible viscoelastic liquid. Until the efeence time, t ¼ 0, the pessue is unifom at p! and the liquid is at est. At t ¼ 0, the pessue inside the bubble is deceased instantaneously to p 0 and the bubble begins to collapse because of the pessue diffeence between the inside and outside of the bubble. The bubble keeps its spheical shape thoughout the motion and the cente of the bubble emains fixed and is the cente of a spheically symmetic coodinate system. In pinciple, the quantities associated with the bubble collapse, such as velocity and pessue, can be detemined fom the solution of the consevation equations of continuum mechanics inside and outside of the bubble joined togethe by suitable bounday conditions at the bubble inteface. Neglecting the effects of gavity, gas diffusion and heat conduction though the bubble wall, the govening equations may be expessed as follows: continuity: qp qt þ qðv Þ þ 2 v ¼ 0; (1) q momentum: qv qt þ v qv q ¼ 1 qp q 1 ð tþ ; (2) whee v is the adial component of the velocity field,, the liquid density, p(,t) is the pessue in the liquid, and t is the stess tenso. Equation of state fo the liquid: A widely used equation of state fo liquids is the Tait fom: p þ B p / þ B ¼ n ; (3) whee the subscipt! efes to the values at infinity, and B and n ae constants having, fo wate, the values n ¼ 7.15, B ¼ atm. Equation of state fo the gas inside the bubble: 3k 0 p i ¼ p 0 ; (4) whee k is the polytopic index. Bounday conditions at the bubble wall ( ¼ (t)): kinematic bounday condition: dynamic bounday condition: / v ðtþ ¼ d dt ¼ : ; (5) p B ðtþ ¼p i ðtþ 2s ð t Þ ¼ ; (6) whee p B is the pessue on the liquid at the bubble wall and s is the suface tension. Seveal comments elevant to bubble dynamics in non- Newtonian liquids ae appopiate hee. In a compessible liquid, the stess tenso consists of two pats. The fist pat is the shea stess tenso s s that depends on the ateof-stain tenso. Fo a puely viscous liquid, this tenso has the fom t s ¼ 2Z ġ tðġþi ; (7) 3 whee Z is the shea viscosity of the liquid, I is the unit tenso, and c is the shea ate. The second pat is the isotopic tenso s i ¼ f 0 Iwithf 0 being a function of invaiants of the ate-of-stain n tenso, i.e., o f 0 ¼ f 0 ði 1 ; I 2 ; I 3 Þ whee, I 1 ¼ tð_cþ, I 2 ¼ ½t ðþ _c Š 2 t ð_c 2 Þ, and I 3 ¼ Det ðþ. _c Fo Newtonian and linea viscoelastic liquids, s i has the fom t i ¼ l v tðġþi; (8) whee l v is the second coefficient of viscosity. Fo nonlinea viscoelastic liquids, whee the shea stess tenso has a finite tace, t(t) = 0, thee is an additional contibution to the mean pessue p ¼ t½ pi þ sš that esults in its vaiation fom the pessue p in the liquid suounding the bubble. We futhe note that Eq. 3 applies only to isentopic changes, but can be applied with easonable accuacy in geneal as n is independent of entopy and B and p! ae only slowly vaying functions of entopy. Finally, Eq. 6 assumes that the gas liquid inteface is clean, i.e., the only molecules pesent ae those of the gas and the suounding liquid. Howeve, whee sufactants ae adsobed onto the bubble suface, a suface stess tem needs to be added to Eq. 6, which includes the effects of suface viscosity and suface tension gadients. The latte occus when the concentation of sufactant molecules on bubble suface is not constant esulting in an additional adial foce that aise fom the vaiation in the concentation of suface active molecules. A futhe appoximation that was intoduced in Eq. 6 is the neglect of the suface viscous tem which, in the case of a spheical symmetic motion, is defined as s ;s ¼ 4a : s = 2, whee a s is the suface dilatational viscosity [12]. Although this pocedue is justified fo dilute sufactant solutions, it may be noted hee that the pedictions of a pue inteface model ae of inteest in themselves in view of the fequent use of such a model in the study of bubble dynamics in non-newtonian liquids. Equations of Motion fo the Bubble adius Hee, we shall estict ouselves only to the case of linea viscoelastic liquids fo which the stess tenso is taceless, i.e., the sum of the nomal stess components is zeo. It should be emphasized hee that these models ae not 420 POLYME ENGINEEING AND SCIENCE DOI /pen

3 entiely satisfactoy fo the desciption of viscoelastic flow behavio. Howeve, studies of idealized models may povide a qualitative insight fo moe ealistic systems and also quantitative esults about thei intemediate asymptotic behavio. Moeove, these models have the main advantage of being tactable and, thus, they allow us to obtain an elegant solution by educing the poblem to a nonlinea diffeential equation. The nea field is a egion suounding the bubble with typical dimension, the bubble adius, the fa field scales with a typical length c! T, whee c! is the speed of sound in the liquid and T a chaacteistic time, such as the collapse time. If one assumes that is of the ode : T, with : a typical adial velocity of the bubble wall, the atio of length scales is just the Mach numbe of the bubble wall motion. Once cast in these tems, it is clea that, to lowest ode, the nea-field dynamics ae essentially incompessible while the fa field is govened by linea acoustics. Howeve, the pictue becomes consideably moe inticate fo a nonlinea viscoelastic liquid [13]. The analysis leads unambiguously to the following equation fo the adius of a spheical bubble situated in a linea viscoelastic liquid [14 16]: :: þ 3 2 : ::: : :: : þ6 þ2 3 c 1 ¼ H 1 1 Z / qt q þ 3t d; whee H is the liquid enthalpy at the bubble wall H ¼ nðp 1 þ BÞ ðn 1Þ 1 " # p B þ B ðn 1Þ=n 1 p 1 þ B ð9þ : (10) The stiking featue of Eq. 3 is the appeaance of the thid-ode deivative of the bubble adius with espect to time. This is just a consequence of using Taylo seies expansions to expess etaded-time quantities, e.g., :: ðt =c 1 Þ :: ðtþ ð=c 1 Þ :::. A simila tem aises in Loentz s theoy of electons. Loentz was consideing peiodic displacements x at fequency x and thus set ::: x x 2 x : and identified this tem with adiation damping. Late eseaches, howeve, wee deeply puzzled by this thid deivative although thee is nothing mysteious about it [16]. Fo c 1!1, the incompessible fomulation is ecoveed, namely: :: þ 3 2 : 2 ¼ H 1 1 Z 1 qt q þ 3t d: (11) which is known as the ayleigh Plesset fomulation. Futhemoe, if one wites 2 : 00¼ a 2 : 00þ ð1 aþ 2 : 00 and uses the incompessible fomulation in the fom 2 : : 2 4 ¼ H 1 1 Z 1 q q þ 3t d (12) to evaluate the fist tem and Eq. 11 to expess the thid deivative of the adius, which appeas on expanding the second tem, one finds :: 1 a þ 1 : þ 3 c 1 2 : 2 1 3a þ 1 : 3c 1 ¼ H 1 þ 1 a : þ H : 1 1 þ 1 a : c 1 c 1 1 c 1 Z 1 qt q þ 3t d 1 Z d 1 qt 1 c 1 dt q þ 3t d ð13þ which epesents an extension of the geneal Kelle Heing equation to the case of a bubble in a linea viscoelastic liquid. Fo a Newtonian liquid, by taking a ¼ 0, Eq. 13 becomes identical to the equation poposed by Kelle and Kolodne [17], wheeas the value a ¼ 1 bings it into the fom suggested by Heing [18]. It will be noted that, by dopping tems in c 1!, Eq. 13 educes to Eq. 11, which is theefoe seen to have an eo of the ode c 1!. The abitay paamete a (which does not seem to have any physical meaning) must, of couse, be of ode 1 so as not to destoy the ode of accuacy of the appoximate Eq. 13. Because of the pesence of the thid time deivative of the adius, the fom Eq. 9 of the adial equation is hadly moe attactive than Eq. 13, if fo nothing else than fo the need to pescibe an initial condition fo. Actually, this is a mino difficulty because, to the same ode of accuacy in the bubble wall Mach numbe, an initial condition fo :: can be obtained by substituting the given initial conditions fo and : in the incompessible fomulation Eq. 6. Howeve, in view of its uniqueness [14], it is pope to conside Eq. 9 the fundamental fom of the motion equation of a spheical bubble in a compessible linea viscoelastic liquid. With efeence to Eq. 13, it should be noted that a elated equation is that due to Gilmoe (see, fo example, [19]): ::! 1 : þ 3 C 2 : 2 1 1! 1 : ¼ H C 1 þ :! Z 1 C 1 d 1 C dt Z 1! 1 þ : þ C C qt q þ 3t d qt q þ 3t d wheeby C is the speed of sound at the bubble wall. 1 : C! H : ð14þ DOI /pen POLYME ENGINEEING AND SCIENCE

4 C ¼ c 2 1=2; 1 þðn 1ÞH (15) and whose deivation elies on the Kikwood Bethe appoximation [20]. In this appoach, the speed of sound C is not constant, but depends on H. This allows one to model the incease of the speed of sound with inceasing pessue aound the bubble, which leads to significantly educed Mach numbes at bubble collapse. To close the mathematical fomulation, an equation fo the shea stess in tems of the ate-of-stain is necessay. Non-Newtonian Puely Viscous Fluid. A lage numbe of theoetical studies on the behavio of spheical bubbles in puely viscous liquids have been published. The powe-law model was adopted by Yang and Yeh [21] and Shima and Tsujino [22] in thei investigations. In addition, the Casson model [23], the Ellis model [24], Sisko model [25], the Caeau model [26], the Powell-Eying model [27], the Shima model [28], the Williamson model [29, 30], and the Bueche model [31] have been applied. Bujan [14] deived the equation of motion fo a spheical bubble and the pessue equation in a compessible puely viscous liquid by using the Williamson model, which well epesents the heological popeties of caboxymethylcellulose (CMC) and hydoxyethylcelullose (HEC) polyme aqueous solutions. It was demonstated that, fo values of the maximum bubble adius smalle than mm, the shea-thinning chaacteistic of liquid viscosity stongly influences the behavio of the bubble and the heological paamete with the stongest influence on the infinite-shea viscosity, g!. Fo lage bubbles, in spite of the consideable diffeences of the appaent viscosity of the liquid, g, the behavio of the bubble emains the same as that of an equivalent Newtonian fluid with a viscosity g!. The effect of polyme additives leads to a significant decease of the maximum values of the bubble wall velocity and pessue at the bubble wall and to a polongation of the fist collapse time of the bubble. On the othe hand, fo values of the initial bubble adius 0 [ mm, sound emission is the main damping mechanism in spheical bubble collapse. Linea Viscoelastic Fluid. The ealiest theoetical teatment of bubble collapse in incompessible linea viscoelastic liquids is that of Fogle and Goddad [32] who consideed the collapse of a spheical bubble in a liquid model including stess accumulation with fading memoy. Late, Tanasawa and Yang [33], Yang and Lawson [34], Ting [35, 36], McComb and Ayyash [37], Tsujino et al. [38], and Agawal [39] used an Oldoyd model, and Shima et al. [40] a Jeffeys model. Moe ecently, Ichihaa et al. [41] and Ichihaa [42] studied the bubble oscillation in the context of magma fagmentation using a linea Maxwell model. A theoetical teatment of spheical bubble dynamics in a compessible viscoelastic fluid was fomulated by Bujan [15]. In this study, the thee-paamete, linea Oldoyd model was employed to epesent the heological behavio of a viscoelastic liquid. The heological equation of this model is epesented as follows [43]: t ii þ l 1 Dt ii Dt De ii ¼ 2Z e ii þ l 2 Dt (16) whee D/Dt is the mateial time deivative, l 1 is a chaacteistic elaxation time (fo the stess), Z, the viscosity coefficient, l 2, a chaacteistic etadation time (i.e., elaxation time fo stain), and e ii ae the stain ate components. It should be noted that the assumption of a single elaxation time l 1 is ove simplistic, even if the polymes ae monodispesed. athe, one would expect a long chain to have a distibution of time scales, coesponding to vaious subchains that compose the polyme. In pinciple, thee is no poblem in incopoating such a distibution of time scales in the model, but it would violate the fundamental desideatum of simplicity. Usually, one chooses l 1 to be some aveage of those time scales, but pehaps it is moe easonable to assume that stong flows will be dominated by the longest elaxation time scale of the system. The intoduction of viscoelastic liquids into the bubble dynamics analysis ceates two independent sets of paametes: the eynolds numbe, defined as e ¼ max q! U/g, whee U ¼ (p! /q! ) 1/2, and the Deboah numbe which is defined as the atio of the chaacteistic time of the fluid and the chaacteistic time of the bubble collapse, De ¼ k 1 U/ max. The effect of Deboah numbe on the behavio of a spheical bubble is illustated in Fig. 1, which shows the maximum dimensionless velocity of the bubble wall plotted as a function of the minimum bubble adius, fo thee values of the eynolds numbe e ¼ 10, 10 2,10 3 and k 2 /k 1 ¼ Figue 2 shows the influence of the atio k 2 /k 1 on the maximum dimensionless velocity of the bubble wall and minimum adius of the bubble, fo thee values of the eynolds numbe e ¼ 10, 10 2, 10 3 and De ¼ 10. It can be seen that the liquid elasticity acceleates the bubble collapse, in ageement with the pedictions of Ting [35], Tsujino et al. [38], and Agawal [39], wheeas the effect of liquid viscosity and etadation time is to deceleate the bubble collapse. These esults futhe indicate that, unde conditions compaable to those existing duing cavitation, the effect of liquid heology on spheical bubble dynamics is negligible fo values of the eynolds numbe lage than 10 2 and the only significant influence is that of liquid compessibility. The noticeable effect of liquid heology was found only fo eynolds values smalle than In both situations, as in the case of a shea-thinning fluid, the 1/ law of pessue attenuation though the liquid is not affected by the viscoelastic popeties of the liquid. The esults pesented by Fogle and Goddad [32] show that fluid elasticity can have an impotant effect on bubble collapse. Howeve, fo conditions simila to cavi- 422 POLYME ENGINEEING AND SCIENCE DOI /pen

5 FIG. 1. The effect of Deboah numbe on the dimensionless maximum velocity attained duing the fist collapse, j 0 j max, and dimensionless minimum adius at the end of the collapse, min, fo v ¼ k 2 /k 1 ¼ The filled symbols indicate the esults obtained using the incompessible fomulation, the open ones using the compessible fomulation. Cicles: Newtonian liquid, diamonds: De ¼ 10 22, squaes: De ¼ 10 21, tiangles (D): De ¼ 1, tiangles (!): De ¼ 10, and hexagons: inviscid liquid. tation, one would not expect to be in a paamete ange whee diffeences fom Newtonian esponse ae appeciable. In fact, when the chaacteistic time fo bubble collapse is in the micosecond ange, as it is fo cavitation, ayleigh Plesset inetial solution appeas to be entiely satisfactoy. An analysis of suface-tension diven oscillations of a bubble was pefomed by Inge and Bak (1982), who also esticted the heology to linea viscoelasticity. They found that the effects of elasticity ae small and compaable to viscous effects. We close ou discussion of spheical bubble dynamics in quiescent viscoelastic liquids with the impotant theoetical contibution of yskin [44]. By incopoating the polyme-induced stess calculated using a yo-yo model which accounts fo the unaveling of the polyme molecules, yskin computed the gowth and collapse phase of a vapo bubble. He concluded that the gowth of the bubble is not affected by the polyme, but the final stage of collapse is. He showed that thee is a total aest of the collapse, with the bubble wall velocity educed to nealy zeo when the bubble adius becomes about 10% of the adius at the initiation of collapse. Nonlinea Viscoelastic Liquid. The numeical simulation of spheical bubble collapse in nonlinea viscoelastic liquid is complicated by the fact that the stess tenso has a finite tace. In contast to the case of a linea viscoelastic liquid, the stess tenso has two components instead of one and, theefoe, the poblem cannot be educed to a single diffeential equation. Kim [45] solved the continuity and momentum equations in a Lagangian fame fo the study of the fee oscillations of a spheical bubble in an Uppe-Convective Maxwell liquid. He implemented the Galekin-finite element method fo solving these equations and compaed some of his esults with those obtained by Fogle and Goddad [32]. The significant paametes of his study ae the eynolds and Deboah numbes. He noted that, fo values of the eynolds numbe smalle than 10, the fluid elasticity acceleates the collapse in the ealy stage of the collapse, wheeas in the late stages, it etads the collapse. The diffeences between a viscoelastic and an ideal liquid become smalle and smalle as the eynolds numbe o the Deboah numbe inceases. Simila tends wee epoted by Butyan and Kapivsky [46] who used an Oldoyd model, and Shulman and Levitskiy [47] and Jimenez and Cespo [48] who investigated the behavio of spheical bubbles in an Oldoyd-B liquid and Uppe Convected Maxwell liquid, espectively. Heat and Mass Tansfe Though the Bubble Wall Ting [36] employed an Oldoyd thee-constant model with chaacteistic elaxation and etadation times multiplying the covaiant convected time deivatives of the stess and stain ate, espectively. He allowed fo themal effects because of the phase change of wate being evapoated o condensed. The esulting integodiffeential equation was solved numeically fo the case of a 500 ppm solution of polyethylene oxide (PEO). He concluded that viscoelasticity has a vey limited etadation effect on bubble gowth and collapse, povided the mateial constants ae compatible with dilute polyme solutions popeties. It also appeas fom the wok of Ting that the effects of heat and mass tansfe ae not impotant unde cavitation conditions. Zana and Leal [49] numeically solved the consevation equations of mass and momentum along with a gas diffusion equation fo a single bubble collapse. A complicated constitutive equation incopoating seveal mateial paametes was employed and the FIG. 2. The effect of atio v ¼ k 2 /k 1 on the dimensionless maximum velocity attained duing the fist collapse, j 0 j max, and dimensionless minimum adius at the end of the collapse, min, fo De ¼ 10. The filled symbols indicate the esults obtained using the incompessible fomulation, the open ones using the compessible fomulation. Cicles: Newtonian liquid, diamonds: v ¼ 10 21, squaes: v ¼ 10 22, tiangles v ¼ 0, and hexagons: inviscid liquid. DOI /pen POLYME ENGINEEING AND SCIENCE

6 esults wee compaed with the coesponding Newtonian case. They found that viscoelastic effects coupled with gas diffusion had pofoundly impacted only the dissolution of gas bubbles. Fo a study of some othe situations whee diffusive effects ae impotant, the eade is efeed to the wok of Buman and Jameson [50], Yoo and Han [51], Shulman and Levitsky [52], and Veneus et al. [53]. Expeimental esults In all expeimental studies, no significant influence of the polyme additives on spheical bubbles was obseved. Ting and Ellis [54] used PEO and Gua Gum aqueous solutions in concentation as high as 1000 ppm, Chahine and Fuman [55] used distilled wate and a 250 ppm solution of PEO (Polyox WS 301) with a viscosity two times lage than that of wate, and Kezios and Schowalte [56] used diffeent polyme solutions whose viscosity was up to Pa s. They indicated that the time and amplitude of the fist and second ebounds wee unaffected by the polyme additive. It should be noted hee that the bubbles geneated in thei expeiments wee extemely lage, with a maximum adius max [ 1 mm. The negligible effect of polyme additives on gowth and collapse of spheical bubbles has also been noted by Haa [57]. Bujan et al. [58] epoted that fo bubbles whose maximum adius is lage than 0.5 mm, the polyme additives, even in the case of polyacylamide (PAM) fo which the aqueous solution display maked viscoelastic effects, did not affect the behavio of bubbles in any significant way. Howeve, fo bubbles whose maximum adius is smalle than 0.5 mm, a slight polongation of the oscillation time was obseved, which inceases with deceasing maximum bubble adius. Moe ecently, Bazilevskii et al. [59] have investigated the gowth and collapse of spheical bubbles with maximum adii of about 0.1 mm geneated in polyacyamide aqueous solutions in concentations of up to 0.6%. They noted that the gowth phase of the bubble is not affected by the polyme additive and, at high polyme concentation, they also obseved a slight incease of the collapse time of the bubble in compaison to the case of wate. It is woth noting hee that a diect compaison between expeiments and numeical esults is difficult owing to the limitations in the constitutive equations used and/o in the heological data pesented in all of the afoementioned studies. It is clea, howeve, fom the expeimental wok that even a stong shea-thinning component of fluid viscosity and a high degee of elasticity of the fluid suounding the bubble cannot influence the collapse of spheical bubbles damatically. The maximum adius of the bubbles geneated in these expeiments is lage than mm, and the viscosity of the polyme solutions used as testing liquids is smalle than Pa s, so that the eynolds numbe associated with the bubble motion is lage than Obviously, the collapse of such lage bubbles is dominated by inetia, iespective of any details of fluid heology. It should be noted hee that a significant eduction of the maximum bubble size can be obtained by using lase pulses of picosecond o femtosecond duation. Such a shot pulse offes the possibility to poduce bubbles with a maximum adius of the ode of mm. Using such small bubbles, it is possible to achieve small enough values of eynolds numbe to detect the influence of liquid heology even in the case of dilute polyme solutions. Numeical pedictions in spheical bubble dynamics is possible, but thee is a need fo expeimental esults using well-chaacteized liquids, which can be descibed by moe sophisticated constitutive models than those that have been used peviously. Bubbles in a Sound-Iadiated Liquid A spheical bubble in a liquid can be viewed as an oscillato that can be set into adial oscillations by a sound field. Fo vey small sound pessue amplitudes, these oscillations can be consideed as being linea about the equilibium adius of the bubble. The esponse then is that of a linea oscillato. Going up in the diving amplitude will bing out the effects of nonlineaity manifesting themselves in the occuence of seveal esonances. The behavio of a bubble in a sound field can be descibed by the theoetical models outlined in section. The theoetical desciption stats with bubble nuclei with adius 0. At time t ¼ 0, the pessue inside the bubble nuclei, p 0, is balanced by the static pessue in the suounding liquid, p 0, and the suface tension, : p 0 ¼ p 0 þ 2s : (17) Povided that the bubble in the viscoelastic liquid is subjected to a peiodically vaying pessue, the pessue p! fa fom the bubble can be expessed by p 1 ¼ p 0 ð1 þ A sin 2pftÞ; (18) whee A is the atio of the pulsating pessue amplitude to the static pessue and f is the fequency of the pulsating pessue. The behavio of a single spheical bubble situated in a sound field and in a puely viscous liquid was investigated by Shima et al. [60], Tsujino et al. [61], and Bujan [62]. On the othe hand, Shima et al. [63] studied the bubble oscillations using a linea viscoelastic elationship to descibe the liquid heology. Figue 3 shows an example of fequency esponse cuves of a bubble situated in a Williamson liquid, as pedicted by the incompessible fomulation of Bujan [62] at A ¼ 0.4, fo wate and PEO solutions in concentation of up to 1.5%. Hee, the nomalized maximum adius, ( max 2 0 )/ 0, duing one peiod of the diving fequency afte the solution has eached steady state is plot- 424 POLYME ENGINEEING AND SCIENCE DOI /pen

7 FIG. 3. Fequency esponse cuve of a spheical bubble oscillating in wate (dashed line), 0.5% caboxymethylcelullose (CMC) solution (solid line), 1% CMC solution (dotted line with one point), and 1.5% CMC solution (dotted line with two points). The initial bubble adius is 0 ¼ 0.1 mm and the amplitude of the oscillating pessue field is A ¼ 0.4. ted as a function of the atio between the fequency of the sound field f and the esonance fequency of the bubble f 0. The maximum esponse occus when f/f 0 is nealy equal to 1. Othe peaks ae seen at o nea f/f 0 ¼ 1/2, 1/3, 1/4. They ae known as the hamonics of the esonance esponse. These peaks have been labeled with an expession m/n, known as the ode of the esonance [62]. The case m ¼ 1, n ¼ 2, 3, :::, denotes the well-known hamonics, the esonances n ¼ 1 and m ¼ 2, 3, :::, ae called subhamonics of ode 1/2, 1/3, ::: The esonances n ¼ 2, 3, :::, and m ¼ 2, 3, :::, ae called ultahamonics. It is clea fom this figue that the esonances ae stongly damped o even suppessed with inceasing polyme concentation. Although the hamonic esonances of ode 2/1 and 3/1, espectively, ae found in wate and in all polyme solutions, the subhamonic esonance of ode 4/1 is not found in the 1% PEO solution, while the subhamonic esonance of ode 1/2 is found only in wate and in the 0.5% PEO solution. Fo f/f 0 ¼ 0.641, the ultahamonic esonance of ode 3/2 was found only in wate. The numeical calculations indicated that the heological paamete which is influential in this espect is the infinite shea viscosity g!. The lage the value of g!, the smalle ae the values of the nomalized bubble adius duing one peiod of bubble oscillation leading finally to the obseved damping of the esonances. We also note that the nonlineaity of the bubble oscillation has a softening effect. The values of f/f 0 at the point of pimay esonance move to the low fequency side and this value inceases with the polyme concentation. Fo example, the value of f/f 0 at the pimay esonance is 0.8 fo a 0.5% PEO solution, fo a 1% PEO solution, and fo a 1.5% PEO solution, espectively. It was also found that the incease of polyme concentation leads to a eduction of the maximum pessue inside the bubble. Simila obsevation has been made by Shima et al. [60] and Tsujino et al. [61] who consideed the bubble oscillations in a Powell-Eying liquid and in a Caeau-like liquid, espectively. Shima et al. [63] obtained the fequency esponse cuves of spheical bubbles using a thee-paamete linea Oldoyd model. They found that the hamonic and subhamonic esonances ae moe easily geneated in elastic liquids and the nomalized maximum adius, ( max 2 0 )/ 0, inceases with the elaxation time of the liquid k 1.On the othe hand, the incease of the etadation time k 2 leads to a decease of the nomalized bubble adius and to a stong damping of the esonances. In paticula, the subhamonic esonance of ode 1/2 and the hamonic esonances of ode 3/1 and 4/1 ae the most affected ones. Moe geneally, the authos noted that fo k 1 /k 2 [ 10 the values of ( max 2 0 )/ 0 ae lage than the coesponding values in a Newtonian liquid, wheeas fo k 1 /k 2 \ 1 ( max 2 0 )/ 0 is smalle. Simila tends have been obseved fo the pessue at the bubble wall. ecently, numeical investigations on the nonlinea bubble oscillations in viscoelastic liquids have been caied out by Allen and oy [64, 65], using the linea Jeffeys model, as well as the Uppe Convected Maxwell model, and Jimenez-Fenandez and Cespo [66] who used a diffeential constitutive equation with an intepolated time deivative which includes the Oldoyd-B model and the Uppe Convected Maxwell model as paticula cases. Thei esults confim the pevious tend quoted above on subhamonics enhancement in elastic liquids. It was also shown that the fluid elasticity poduces a significant gowth of the amplitude of bubble oscillations. Up to this point, all quantities have been given a single value fo each solution if, afte eaching steady state, the solutions have the same peiod as the diving pessue. But this not always the case, especially at high pessue amplitude whee nonlinea effects ae moe pominent. One of the most significant developments in bubble dynamics is the ealization that the bubble esponse to a time-peiodic pessue field can be chaotic, even when the bubble is assumed to emain spheical. An example of chaotic oscillations of a single spheical bubble situated in a viscoelastic liquid is given by Jimenez-Fenandez and Cespo [66]. They concluded that liquid elasticity may enhance the chaotic oscillations of bubbles even at modeate values of the diving pessue field. No influence of liquid elasticity on the numbe of collapses in a fixed amount of time was obseved. DOI /pen POLYME ENGINEEING AND SCIENCE

8 Aspheical Bubble Dynamics Although the events duing bubble geneation ae not influenced by the viscoelastic popeties of the suounding fluid, the subsequent bubble dynamics is pimaily influenced by the bounday conditions in the neighbohood of the bubble and the popeties of the fluid. A spheical bubble poduced in an unconfined liquid etains its spheical shape while oscillating and the bubble collapse takes place at the site of bubble fomation. When the bubble oscillates unde asymmetic bounday conditions, it is usually exposed to pessue gadients. This leads to a faste collapse of the bubble section exposed to a highe pessue and to the fomation of a liquid jet even fo an initially spheical bubble. When the bubble collapses in the vicinity of a igid bounday, the jet is diected towad the bounday (see, fo example, [67]). The pessue gadient causing the jet fomation is due to the lowpessue egion between bubble and igid wall developing duing bubble collapse. Duing the initial collapse phase, the bubble acquies the fom of a polate spheoid. This shape also contibutes to the fomation of the liquid jet. A bubble oscillating between two paallel flat igid walls is subjected to two opposite pessue-gadient foces and the collapse is chaacteized by the fomation of two liquid jets that ae diected towad each wall [68]. Bubbles Nea a igid Wall Of utmost inteest is the case of a bubble nea a igid bounday because bubbles ae the souce of cavitation eosion. The use of a nomalized distance c ¼ s/ max whee s is the distance of the bubble inception fom the bounday has poven advantageous to classify bubble dynamics nea a plane igid bounday. Bubbles with diffeent max but the same c-value exhibit simila dynamics, thus giving the chance to specify the degee of asymmety of bubble collapse: cavitation bubbles with a small value of c ae moe influenced by the bounday, thus collapsing with a moe ponounced shape vaiation than those with a lage value fo which collapse is moe sphee-like. This statement, howeve, does not apply to bubbles too close to the bounday, whee c 0 and the bubble adopts a hemispheical shape, i.e., appoaches a spheical symmety again. Figue 4 shows a seies of high-speed photogaphic ecods of bubble motion in wate, a 0.5% CMC solution with a weak elastic component, and a 0.5% PAM solution with a stong elastic component fo the case whee c ¼ 3.17 [58]. The liquid jet, which is developed on the uppe side of the bubble leading to the potusion of the lowe bubble wall, can be seen in the case of bubbles situated in wate (top sequence). A simila bubble shape is found in the CMC solution, but, in this case, the jet is not as stong as in the case of wate. The most inteesting behavio fo a bubble situated in the vicinity of a igid bounday was found fo the case of the PAM solution. The liquid jet is not obseved and a flat fom of the bubble shape is the dominant aspect of bubble motion afte the fist collapse. In the case of the PAM solution, the maximum jet velocity was found to be 88 m/s, a value which epesents about 78% of the coesponding velocity in wate (113 m/s). Fo the CMC solution, the jet velocity, 102 m/s, is almost the same as that fo the case of wate. Simila obsevations have been made when c was educed to 1.67 (see Fig. 5). At fist sight, the addition of polymes into wate has a less significant effect on the bubble collapse. The liquid jet inside the bubble was obseved fo wate and both polyme solutions. Howeve, a significant influence of the polyme additives was noted fo the velocity of the eentant jet. Although in the case of wate the maximum jet velocity is 104 m/s, only 63 m/ s was measued fo the PAM solution. In a pevious expeimental study, Chahine and Fuman [55] indicated that although bubble gowth is not sensitive to addition of 250 ppm of PEO to wate, the collapse sequence and the shape nea a igid bounday ae appeciably affected. In paticula, they also obseved that the polyme additive intoduces a etadation effect ove the initiation of the eenteing jet developed duing bubble collapse. Because it was substantiated that the viscoelastic popeties of the suounding liquid might affect the collapse of a cavitation bubble situated nea a igid bounday, futhe studies have investigated the dependence of the pessue amplitude of the acoustic tansients emitted duing bubble collapse with c [69]. In this study, two polyme solutions wee investigated, namely a PAM aqueous solution and a CMC aqueous solution, both in a concentation of 0.5%. The extensional popeties, in the fom of an appaent Touton atio (T ¼ g e =g), fo both polyme solutions wee measued in uniaxial extension using a heometics FX opposed-jet appaatus with 1 mm diamete nozzles. The geneal behavio of the PAM solution is that it is extension ate thickening, which is a geneal chaacteistic fo flexible polymes. The appaent Touton atio fo the PAM solution was initially at a value of T 4.5 at low extension ates and then it inceased to attain a maximum of T 70 at extension ates of _e 4000 s 21, indicating a stong elastic component. The appaent Touton atio fo the CMC solution was elatively constant at a value of about 5 fo all the extension ates investigated, indicating a elatively less elastic behavio of the polyme solution. Figue 6 shows the amplitude of the acoustic tansients emitted duing fist bubble collapse, p max, as a function of c in wate and both polyme solutions. It can be seen that the lagest values of the maximum amplitude of the acoustic tansients ae obtained in wate. Fo the elatively less elastic 0.5% CMC solution, the bubble dynamics do not diffe substantially fom that in wate and the maximum amplitude of the acoustic tansients emitted duing bubble collapse is almost simila to that in wate. Fo the elastic 0.5% PAM solution, howeve, a significant 426 POLYME ENGINEEING AND SCIENCE DOI /pen

9 FIG. 4. Pictue sequences of the behavio of a lase-induced bubble nea a igid wall in wate and polyme solutions fo c ¼ Top: wate, max ¼ 0.63 mm; Middle: 0.5% caboxymethylcelullose solution, max ¼ 0.47 mm; Bottom: 0.5% polyacylamide solution, max ¼ 0.63 mm. Fame inteval 4.8 ls, fame width 1.7 mm. eduction of p max was obseved. We futhe note that the most ponounced eduction of the shock pessue in the PAM solution was obseved fo c \ 0.6 and c [ 1.5. Figue 7 shows that the velocity of the liquid jet developed duing the final stage of bubble collapse ange fom about 10 up to 50 m/s. Futhemoe, the jet velocity shows a dependence on c simila to the pessue amplitude of the acoustic tansients emitted duing bubble collapse: Thee is a minimum fo values c 1 and the jet velocity deceases with inceasing the extensional viscosity of the liquid. The effect of the viscoelastic popeties of the liquid on the sound emission duing fist bubble collapse can be undestood in a heuistic manne. A spheical bubble geneated in a liquid of infinite extent etains its spheical shape while oscillating. When the bubble is fomed nea a igid bounday, the collapse is associated with the fomation of a high-speed liquid jet diected towad the bounday. Howeve, examination of the high-speed photogaphic sequences shows that the bubble emains nea spheical fo much of its collapse peiod (between 90 and 95% depending on c), only developing significant nonspheicity at the end of the pulsation. The flow is thus pedominantly uniaxial in extension duing most of the collapse and the viscosity of both polyme solutions is significantly lage than that of wate. Theefoe, a lage pat of the maximum potential enegy of the bubble is dissipated duing the collapse phase because of an inceased esistance to extensional flow, which is confeed upon the suounding liquid by the polyme additive. Consequently, less enegy is available fo bubble collapse, the bubble content becomes less compessed than in the case of wate, and the pessue amplitude of the shock wave is diminished. Fo lage c-values, the etading effect of the igid bounday on the fluid duing collapse is small. Theefoe, the bubble emains nealy spheical and the liquid jet develops only in a vey late stage of the collapse. Fo c \ 0.6, the bubble is nealy hemispheical and the flow is diected towad the bubble cente fo most pats of the bubble suface, as in the case DOI /pen POLYME ENGINEEING AND SCIENCE

10 FIG. 5. Pictue sequences of the behavio of a lase-induced bubble nea a igid wall in wate and polyme solutions fo c ¼ Top: wate, max ¼ 0.6 mm; Middle: 0.5% caboxymethylcelullose solution, max ¼ 0.45 mm; Bottom: 0.5% polyacylamide solution, max ¼ 0.6 mm. Fame inteval 4.8 ls, fame width 1.7. of a spheical collapse. In both cases, the bubble assumes spheical symmety fo most pat of the collapse, thus the fluid elements expeience a stong uniaxial extensional flow, and theefoe, the enegy dissipation duing bubble collapse is the lagest. The explanation fo the significant eduction of the jet velocity is simila as fo the acoustic tansients emitted duing bubble collapse. The pesence of the polyme additive confes on the solution an ability to sustain highe extensional stesses than its Newtonian countepat. This enhanced esistance to extensional de- FIG. 6. Pessue amplitude of the acoustic tansients emitted duing fist bubble collapse as a function of c. The pessue values ae measued at a distance of 10 mm fom the ultasound focus. FIG. 7. Maximum jet velocity as a function of the stand-off paamete c. The jet velocity is aveaged ove 3 ls. 428 POLYME ENGINEEING AND SCIENCE DOI /pen

11 fomation educes the intensity of the eentant liquid jet developed duing bubble collapse. Fo c \ 0.6 and c [ 1.5, whee the spheical symmety is peseved duing most pat of bubble collapse, the extensional flow becomes dominant and the eduction of the jet velocity is the lagest. Using a petubation appoach, Haa and Schowalte [70] investigated the effect of viscoelasticity on the dynamics of single nonspheical bubbles situated in a quiescent viscoelastic liquid. The constitutive equation they used is of the Maxwell type, simila to that used by Fogle and Goddad [32]. They showed that the effects of fluid heology on nonspheical bubble dynamics ae lage than on spheical bubbles. Nevetheless, gowth and collapse of bubbles in an initially unstessed liquid emain dominated by inetia. Thei method is, howeve, limited only to small oscillations of the bubble and cannot descibe the motion of the eentant liquid jet. It is well known that bounday integal methods ae paticulaly well suited to this class of poblems as they involve discetization of the boundaies only. Howeve, application of this method is possible only in the ceeping and potential flow limits. The estiction of the bounday integal methods to potential flow poblems pecludes an exact accounting of the ole viscoelastic effects play in the dynamics of cavitation bubbles nea boundaies. It is, howeve, possible to include weak viscoelastic effects in the bounday integal fomulation if it is assumed that these effects ae limited to a thin egion nea the inteface so that the bulk of the fluid emains iotational. Lundgen and Mansou [71] pefomed an analysis to include weak viscous effects in a bounday integal simulation fo an oscillating dop, wheeas Boulton-Stone [72] applied the bounday integal method to study the effect of sufactants on the behavio of busting gas bubbles. The development of compute codes that would pemit the calculation of bubble collapse in a viscoelastic fluid and nea a igid bounday has been slow. Owing to the difficulties involved in implementing both moving boundaies and viscoelasticity, esolution has not been possible anywhee nea the expeimentally attainable limit, even with pesent-day computes. Numeical simulations could contibute to a bette undestanding of the dynamics by poviding pessue contous and velocity vectos in the liquid suounding the bubble, which ae not easily accessible though expeiments. Bubbles Between Two igid Walls When a bubble is initiated between two paallel igid walls, an annula flow is developed duing bubble collapse. Fo a sufficiently small distance between the walls, the annula flow leads to bubble splitting and the fomation of two opposing liquid jets diected towad each wall. Chahine [68] and Chahine and Moine [73] conducted seveal tests using geomety with bubbles geneated in wate, and 125 and 250 ppm of PEO, espectively. They found that, although the gowth phase of the bubble is unaffected by the polyme additive, the lengthening effect on the oscillation peiod of the bubble is significantly educed in the case of polyme solutions, and the depatue fom spheicity of the bubbles is consideably delayed. No esults wee pesented by these authos with espect to the influence of polyme additive on the velocity of the liquid jets fomed afte bubble splitting. Bubbles in a Shea Flow Vitually, all of the pevious obsevations and analyses have focused on bubble collapse in a quiescent liquid, despite the fact that a numbe of expeimentes have commented on the defomation of cavitation bubbles by the flow (see, fo example, [74]). Some of the ealy obsevations of individual taveling cavitation bubbles by Knapp and Hollande [75] make mention of the defomation of the bubbles by the flow. A detailed investigation of the effect of a contolled shea flow on the defomation of lase-geneated bubbles was conducted by Kezios and Schowalte [56] using PAM and PEO solutions in concentations of up to 2000 ppm. The main pupose of thei wok was to undestand the ole played by a pe-existing stess field at the moment when cavitation bubbles ae geneated. They demonstated that the depatue fom spheicity is significantly educed in polyme solutions, in paticula in the highly elastic PAM solutions. They also noted that inceasing the concentation beyond a citical value eveses the esults and they speculated that this can be caused by the elative incease of the solution viscosity when compaed with its elasticity. Ligneul [76] also pefomed expeiments with spakgeneated bubbles in the shea laye developed by a otating cylinde. By compaing the behavio in wate and solutions of PEO with 50 and 250 ppm concentation, he concluded that the influence of the polyme additive is to maintain spheicity duing bubble collapse. The effect of viscoelasticity on cavitation chaacteistics in flow between eccentic cylindes in elative otation has been epoted by Ashafi et al. [77] who found that fo low speeds of otation, the liquid s fee suface depated pogessively fom the initial hoizontal (est) configuation. With futhe inceases in otational speed, a povocative fingeing mechanism appeaed, geneating a seies of cavities, the numbe of which inceased with otational speed and eccenticity. The elastic liquids wee found to geneate moe cells than thei Newtonian equivalents, the shape of the cavities exhibiting distinctive cusp-like extemities. In this study, fluid elasticity was found to pomote cavitation. Shock-Wave Bubble Inteaction The inteaction of a shock wave with a bubble in a liquid is of special inteest because of the shock-induced fo- DOI /pen POLYME ENGINEEING AND SCIENCE

12 mation of a high-speed liquid jet. When a shock wave eaches a esting bubble, it will be almost completely eflected because of the shap incease in acoustic impedance at the bubble wall. The esulting momentum tansfe acceleates the bubble wall and stats the collapse fom this side. Togethe with focusing effects duing the collapse stage, this situation finally leads to the fomation of a fast liquid jet in the diection of wave popagation. Shima et al. [78] examined the shock-induced collapse of bubbles situated in wate and PAM aqueous solutions. They used the steak technique to visualize the collapse phase of the bubble and esticted thei investigations only to collapse time. The bubble adius in this expeiment was vaied between 0.01 and 1 mm. They obseved that, fo bubbles smalle than 0.05 mm, the collapse time in PAM solutions with concentation of 0.05 and 0.1% is shote than that in wate. They explained this esult as a consequence of the elaxation effect of the polyme solutions. Outlook and Challenges In this eview, we have collected and oganized infomation on the dynamics of cavitation bubbles in non- Newtonian fluids that until now was widely scatteed in the scientific liteatue. We developed a famewok to identify elevant mechanisms govening the motion of cavitation bubbles situated in a liquid of infinite extent o nea igid boundaies. Nevetheless, much wok emains to futhe impove the mechanistic undestanding of bubble dynamics in non-newtonian fluids. On the expeimental side, futhe investigations ae needed to bette chaacteize the final stage of bubble collapse. Thee have been seveal obsevations of the poduction of light fom lase-induced cavitation bubbles collapsing in wate [79 82]. An estimate of the inteio bubble tempeatue at the moment of light emission can be obtained by fitting the specta to a blackbody fom, and in geneal this yields esults of about 8000 K. An extension of these studies to investigate the effect of viscoelastic popeties of the liquid suounding the bubble on the chaacteistics of the luminescence is highly desiable fo a bette undestanding of the cavitation phenomenon in non-newtonian fluids and associated damage to neaby boundaies. On the modeling and computational side, wok is necessay to integate the pesent knowledge on the hydodynamics of bubble gowth and collapse to build a model of the entie pocess dynamics. The challenge hee is to devise models that include both moving boundaies and viscoelasticity. It is likely that the constuction of a model that accommodates this consideation necessitates a computational athe than an analytical appoach. The validity of esults obtained though such an appoach depends citically on the accuacy with which the computational model epesents liquid heology. Sophisticated computational codes that have aleady been developed [71, 72] must be eevaluated and modified to faithfully epesent the viscoelastic popeties of the liquid suounding the bubble. EFEENCES 1. C.E. Bennen, Cavitation and Bubble Dynamics, Oxfod Univesity Pess, Oxfod (1995). 2. D.H. Tevena, Cavitation and Tension in Liquids, Adam Hilge, Bistol (1987). 3. F.. Young, Cavitation, McGaw-Hill, New Yok (1989). 4. T.G. Leighton, The Acoustic Bubble, Academic Pess, London (1994). 5. M.S. Plesset and A. Pospeetti, Ann. ev. Fluid Mech., 9, 145 (1977). 6. J.. Blake and D.C. Gibson, Ann. ev. Fluid Mech., 19, 99 (1987). 7. C.G. Hemansky and D.V. Boge, J. Non-Newtonian Fluid Mech., 56, 1 (1995). 8. G. Peason and S. Middleman, AIChE J., 23, 722 (1977). 9. G. Peason and S. Middleman, AIChE J., 23, 714 (1977). 10. S.H. Spiegelbeg and G.H. McKinley, J. Non-Newtonian Fluid Mech., 67, 49 (1996). 11. A.L. 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