Cavitation bubble dynamics in non-Newtonian fluids.

INTRODUCTION

Cavitation phenomena play important roles in many areas of science and engineering, including acoustics, bio-medicine, botany, sonochemistry, and hydraulics. They occur in numerous industrial processes such as cleaning, lubrication, printing, and coating. Although much of the research effort into cavitation has been stimulated by its occurrence in pumps and other fluid mechanical machinery, cavitation is also an important factor in the life of plants and animals (1-4). The most intensively studied consequence of cavitation is the occurrence of erosion or cavitation damage to solid surfaces in the near vicinity of collapsing cavities, but other widely known effects include reduced hydraulic performance and the generation of excessive vibration and noise.

Several books that serve as a valuable resource for the field of bubble dynamics and cavitation are available (1-3), together with the extensive reviews by Plesset and Prosperetti (5) and Blake and Gibson (6). The latter address general aspects of bubble dynamics and cavitation in Newtonian fluids. In this review, we focus on bubble dynamics in non-Newtonian fluids. Such fluids occur widely in process engineering, and it is essential to understand that the effects of non-Newtonian properties on bubble dynamics and cavitation are fundamentally different from those of Newtonian fluids. Arguably, the most significant non-Newtonian effect in the context of bubble dy-namics and cavitation arises from the dramatic increase in viscosity of polymer solutions in an extensional flow (7), such as that generated about a spherical bubble during its growth or collapse phase (8), (9). Specifically, polymers, which are randomly oriented coils in the absence of an imposed flow-field, are pulled apart and may increase their length by three orders of magnitude in the direction of extension (10). As a result, the solution can sustain much greater stresses, and pinching is stopped in regions where polymers are stretched. This "extensional thickening" leads to the characteristic "beds-on-a-string" profile of polymeric jets, as seen by Yarin (11).

Despite the increasing use of non-Newtonian liquids in industrial applications, a comprehensive presentation of the fundamental processes involved in bubble dynamics and cavitation in non-Newtonian liquids has not appeared in the scientific literature. This is not surprising, as the elements required for an understanding of the relevant processes are wide-ranging. Consequently, researchers who investigate cavitation phenomenon in non-Newtonian liquids originate from several disciplines. Moreover, the resulting scientific reports are often narrow in scope and scattered in journals whose foci range from the physical sciences and engineering to medical sciences. The purpose of this review is to collect the information to be gleaned from these studies and organize it into a logical structure that provides an improved mechanistic understanding of bubble dynamics and cavitation in non-Newtonian liquids. One main message is that the introduction of ideas from theoretical studies of nonlinear acoustics and modem optical techniques has led to some major revisions in our understanding of the dynamics of cavitation bubbles in non-Newtonian fluids.

SPHERICAL BUBBLE DYNAMICS

The investigation of the dynamics of spherical cavitation bubbles is of no direct interest for the explanation of cavitation erosion, because bubbles close enough to a boundary to cause damage will always collapse aspherically. Nevertheless, it provides the basis for the interpretation of data obtained for the asymmetrical collapse of bubbles in non-Newtonian fluids and is to date the only means of comparing experimental results with theory.

General Equations of Bubble Dynamics

Consider a spherical bubble of initial radius [R.sub.0] situated in a compressible viscoelastic liquid. Until the reference time, t = 0, the pressure is uniform at [p.sub.[varies]] and the liquid is at rest. At t = 0, the pressure inside the bubble is decreased instantaneously to [p.sub.0] and the bubble begins to collapse because of the pressure difference between the inside and outside of the bubble. The bubble keeps its spherical shape throughout the motion and the centre of the bubble remains fixed and is the centre of a spherically symmetric coordinate system. In principle, the quantities associated with the bubble collapse, such as velocity and pressure, can be determined from the solution of the conservation equations of continuum mechanics inside and outside of the bubble joined together by suitable boundary conditions at the bubble interface. Neglecting the effects of gravity, gas diffusion and heat conduction through the bubble wall, the governing equations may be expressed as follows:

continuity:

[[partial derivative]p/[partial derivative]t] + [[partial derivative]([rho][v.sub.r])/[partial derivative]r] + [2[rho][v.sub.r]/r] = 0, (1)

momentum:

[[partial derivative][v.sub.r]/[partial derivative]t] + [[v.sub.r][partial derivative][v.sub.r]/[partial derivative]r] = [-1/[rho]][[partial derivative]p/[partial derivative]r] [-1/[rho]][([nabla]*[tau]).sub.r], (2)

where [v.sub.r] is the radial component of the velocity field, [rho], the liquid density, p(r, t) is the pressure in the liquid, and [tau] is the stress tensor.

Equation of state for the liquid:

A widely used equation of state for liquids is the Tait form:

p + B/[p.sub.[varies]] + B = [([rho]/[[rho].sub.[varies]]).sup.n], (3)

where the subscript [varies] refers to he values at infinity, and B and n are constants having, for water, the values n = 7.15, B = 3049.13 atm.

Equation of state for the gas inside the bubble:

[p.sub.i] = [p.sub.0][([R.sub.0]/R).sup.3k], (4)

where k is the polytropic index.

Boundary conditions at the bubble wall (r = R(t)): kinematic boundary condition:

[v.sub.r](t) = dR/dt = R, (5)

dynamic boundary condition:

pB(t) = pi(t) - [[sigma]/R - [([[tau].sub.rr]).sub.r = R], (6)

where [p.sub.B] is the pressure on the liquid at the bubble wall and [sigma] is the surface tension.

Several comments relevant to bubble dynamics in non-Newtonian liquids are appropriate here. In a compressible liquid, the stress tensor consists of two parts. The first part is the shear stress tensor [[tau].sub.s] that depends on the rate-of-strain tensor. For a purely viscous liquid, this tensor has the form

[[tau].sub.S] = 2[eta][[gamma] - tr([gamma])[IOTA]/3], (7)

where [eta] is the shear viscosity of the liquid, [IOTA] is the unit tensor, and [gamma] is the shear rate. The second part is the isotropic tensor [[tau].sub.i] [f.sub.0][IOTA] with [f.sub.0] being a function of invariants of the rate-of-strain tensor, i.e., [f.sub.0] = [f.sub.0]([I.sub.1], [I.sub.2], [I.sub.3]) where, [I.sub.1] = tr([gamma]), [I.sub.2] = {[[tr([gamma])].sup.2] - tr([[gamma].sup.2])}, and [I.sub.3] = Det([gamma]). For Newtonian and linear viscoelastic liquids, [[tau].sub.i] has the form

[[tau].sub.i] = [[gamma].sub.v]tr([gamma])[IOTA] (8)

where [[lambda].sub.v] is the second coefficient of viscosity. For nonlinear viscoelastic liquids, where the shear stress tensor has a finite trace, tr([tau]) [not equal to]0, there is an additional contribution to the mean pressure [bar.p] = -tr[-p[IOTA] + [tau]] that results in its variation from the pressure p in the liquid surrounding the bubble. We further note that Eq. 3 applies only to isentropic changes, but can be applied with reasonable accuracy in general as n is independent of entropy and B and p[varies] are only slowly varying functions of entropy. Finally, Eq. 6 assumes that the gas-liquid interface is "clean," i.e., the only molecules present are those of the gas and the surrounding liquid. However, where surfactants are adsorbed onto the bubble surface, a surface stress term needs to be added to Eq. 6, which includes the effects of surface viscosity and surface tension gradients. The latter occurs when the concentration of surfactant molecules on bubble surface is not constant resulting in an additional radial force that arise from the variation in the concentration of surface active molecules. A further approximation that was introduced in Eq. 6 is the neglect of the surface viscous term which, in the case of a spherical symmetric motion, is defined as [[tau].sub.rr.s] = 4[[alpha].sub.S]R/[R.sup.2], where [[alpha].sub.s] is the surface dilatational viscosity (12). Although this procedure is justified for dilute surfactant solutions, it may be noted here that the predictions of a pure interface model are of interest in themselves in view of the frequent use of such a model in the study of bubble dynamics in non-Newtonian liquids.

Equations of Motion for the Bubble Radius

Here, we shall restrict ourselves only to the case of linear viscoelastic liquids for which the stress tensor is traceless, i.e., the sum of the normal stress components is zero. It should be emphasized here that these models are not entirely satisfactory for the description of viscoelastic flow behavior. However, studies of idealized models may provide a qualitative insight for more realistic systems and also quantitative results about their intermediate asymptotic behavior. Moreover, these models have the main advantage of being tractable and, thus, they allow us to obtain an elegant solution by reducing the problem to a nonlinear differential equation. The "near field" is a region surrounding the bubble with typical dimension R, the bubble radius, the "far field" scales with a typical length [c.sub.[varies]]T, where [c.sub.[varies]] is the speed of sound in the liquid and T a characteristic time, such as the collapse time. If one assumes that R is of the order R T, with R a typical radial velocity of the bubble wall, the ratio of length scales is just the Mach number of the bubble wall motion. Once cast in these terms, it is clear that, to lowest order, the near-field dynamics are essentially incompressible while the far field is governed by linear acoustics. However, the picture becomes considerably more intricate for a nonlinear viscoelastic liquid (13).

The analysis leads unambiguously to the following equation for the radius of a spherical bubble situated in a linear viscoelastic liquid (14-16):

R[bar.R ]+ 3/2[R.sup.2] - 1/[c.sub.[infinity]]([R.sup.2]R + 6RRR + 2[R.sup.3]) = H - 1/[[rho].sub.[infinity]][[infinity] [integral] R]([partial derivative][[tau].sub.rr]/[partial derivative]r + 3[[tau].sub.rr]/r)dr, (9)

where H is the liquid enthalpy at the bubble wall

H = n[p.sub.[infinity]] + B)/(n - 1)[[rho].sub.[infinity]][[([p.sub.B] + B/[p.sub.[infinity]] + B).sup.(n - 1)/n] - 1]. (10)

The striking feature of Eq. 3 is the appearance of the third-order derivative of the bubble radius with respect to time. This is just a consequence of using Taylor series expansions to express retarded-time quantities, e.g., R(t - R/[c.sub.[infinity]]) [approximately equal to] R(t) - (R/[c.sub.[infinity]]) R. A similar term arises in Lorentz's theory of electrons. Lorentz was considering periodic displacements x at frequency [omega] and thus set x [approximately equal to] - [[omega].sup.2]x and identified this term with radiation damping. Later researchers, however, were deeply puzzled by this third derivative although there is nothing mysterious about it (16). For [c.sub.[infinity]] [right arrow] [infinity], the incompressible formulation is recovered, namely:

RR + 3/2[R.sup.2] = H - 1/[[rho].sub.[infinity]][[infinity] [integral] R]([partial derivative][[tau].sub.rr]/[partial derivative]r + 3[[tau].sub.rr]/r)dr. (11)

which is known as the Rayleigh-Plesset formulation. Furthermore, if one writes ([R.sup.2]R)" = [alpha] ([R.sup.2]R)" + (1 - [alpha]) ([R.sup.2]R)" and uses the incompressible formulation in the form

([R.sup.2]R)'/R - 1/2[([R.sup.2]R).sup.2]/[R.sup.4] = H - 1/[[rho].sub.[infinity]][[infinity] [integral] R]([partial derivative][r.sub.rr]/[partial derivative]r + 3[[tau].sub.rr]/r)dr. (12)

to evaluate the first term and Eq. 11 to express the third derivative of the radius, which appears on expanding the second term, one finds

RR(1 - [alpha] + 1/[c.sub.[infinity]]R) + 3/2[R.sup.2](1 - 3[alpha] + 1/3[c.sub.[infinity]]R) = H(1 + 1 - [alpha]/[c.sub.[infinity]]R) + R/[c.sub.[infinity]]H - 1/[[rho].sub.[infinity]](1 + 1 - [alpha]/[c.sub.[infinity]]R) x [[infinity] [integral] R]([partial derivative][[tau].sub.rr]/[partial derivative]r + 3[[tau].sub.rr]/r)dr - 1/[[rho].sub.[infinity]]R/[c.sub.[infinity]]d/dt[[infinity] [integral] R]([partial derivative][[tau].sub.rr]/[partial derivative]r + 3[[tau].sub.rr]/r)dr (13)

which represents an extension of the general Keller--Herring equation to the case of a bubble in a linear viscoelastic liquid. For a Newtonian liquid, by taking [alpha] = 0, Eq. 13 becomes identical to the equation proposed by Keller and Kolodner (17), whereas the value [alpha] = 1 brings it into the form suggested by Herring (18). It will be noted that, by dropping terms in [c.sub.[varies].sup.-1], Eq. 13 reduces to Eq. 11, which is therefore seen to have an error of the order [c.sub.[varies].sup.-1]. The arbitrary parameter [alpha] (which does not seem to have any physical meaning) must, of course, be of order 1 so as not to destroy the order of accuracy of the approximate Eq. 13.

Because of the presence of the third time derivative of the radius, the form Eq. 9 of the radial equation is hardly more attractive than Eq. 13, if for nothing else than for the need to prescribe an initial condition for R. Actually, this is a minor difficulty because, to the same order of accuracy in the bubble wall Mach number, an initial condition for R can be obtained by substituting the given initial conditions for R and R in the incompressible formulation Eq. 6. However, in view of its uniqueness [ 14], it is proper to consider Eq. 9 the fundamental form of the motion equation of a spherical bubble in a compressible linear viscoelastic liquid.

With reference to Eq. 13, it should be noted that a related equation is that due to Gilmore (see, for example, (19)):

RR(1 - R/C) + 3/2[R.sup.2](1 - R/C) = H(1 + R/C) + R/C(1 - R/C)H - 1/[[rho].sub.[infinity]](1 + R/C)[[infinity] [integral] R]([partial derivative][[tau].sub.rr]/[partial derivative]r + 3[[tau].sub.rr]/r)dr - 1/[[rho].sub.[infinity]]R/C d/dt[[infinity] [integral] R]([partial derivative][[tau].sub.rr]/[partial derivative]r + 3[[tau].sub.rr]/r)dr (14)

whereby C is the speed of sound at the bubble wall.

C = [[c.sub.[varies].sup.2] + (n - 1)H].sup.1/2], (15)

and whose derivation relies on the Kirkwood-Bethe approximation (20). In this approach, the speed of sound C is not constant, but depends on H. This allows one to model the increase of the speed of sound with increasing pressure around the bubble, which leads to significantly reduced Mach numbers at bubble collapse.

To close the mathematical formulation, an equation for the shear stress in terms of the rate-of-strain is necessary.

Non-Newtonian Purely Viscous Fluid. A large number of theoretical studies on the behavior of spherical bubbles in purely viscous liquids have been published. The power-law model was adopted by Yang and Yeh (21) and Shima and Tsujino (22) in their investigations. In addition, the Casson model (23), the Ellis model (24), Sisko model (25), the Carreau model (26), the Powell-Eyring model (27), the Shima model (28), the Williamson model (29), (30), and the Bueche model (31) have been applied.

Brujan [ 14] derived the equation of motion for a spherical bubble and the pressure equation in a compressible purely viscous liquid by using the Williamson model, which well represents the rheological properties of carboxymethylcellulose (CMC) and hydroxyethylcelullose (HEC) polymer aqueous solutions. It was demonstrated that, for values of the maximum bubble radius smaller than [10.sup.-1] mm, the shear-thinning characteristic of liquid viscosity strongly influences the behavior of the bubble and the rheological parameter with the strongest influence on the infinite-shear viscosity, [[eta].sub.[infinity]]. For larger bubbles, in spite of the considerable differences of the apparent viscosity of the liquid, [eta], the behavior of the bubble remains the same as that of an equivalent Newtonian fluid with a viscosity [[eta].sub.[infinity]]. The effect of polymer additives leads to a significant decrease of the maximum values of the bubble wall velocity and pressure at the bubble wall and to a prolongation of the first collapse time of the bubble. On the other hand, for values of the initial bubble radius [R.sub.0] > [10.sup.-1] mm, sound emission is the main damping mechanism in spherical bubble collapse.

Linear Viscoelastic Fluid. The earliest theoretical treatment of bubble collapse in incompressible linear viscoelastic liquids is that of Fogler and Goddard (32) who considered the collapse of a spherical bubble in a liquid model including stress accumulation with fading memory. Later, Tanasawa and Yang (33), Yang and Lawson (34), Ting (35), (36), McComb and Ayyash (37), Tsujino et al. (38), and Agarwal (39) used an Oldroyd model, and Shima et al. (40) a Jeffreys model. More recently, Ichihara et al. (41) and Ichihara (42) studied the bubble oscillation in the context of magma fragmentation using a linear Maxwell model.

A theoretical treatment of spherical bubble dynamics in a compressible viscoelastic fluid was formulated by Brujan [ 15]. In this study, the three-parameter, linear Oldroyd model was employed to represent the rheological behavior of a viscoelastic liquid. The rheological equation of this model is represented as follows (43):

[[tau].sub.ii] + [[lambda].sub.1]D[[tau].sub.ii]/Dt = - 2[eta]([e.sub.ii] + [[lambda].sub.2]D[e.sub.ii]/Dt) (16)

where D/Dt is the material time derivative, [[lambda].sub.1] is a characteristic relaxation time (for the stress), [eta], the viscosity coefficient. [[lamda].sub.2], a characteristic retardation time (i.e., relaxation time for strain), and [e.sub.ii] are the strain rate components. It should be noted that the assumption of a single relaxation time [[lamda].sub.1] is over simplistic, even if the polymers are monodispersed. Rather, one would expect a long chain to have a distribution of time scales, corresponding to various subchains that compose the polymer. In principle, there is no problem in incorporating such a distribution of time scales in the model, but it would violate the fundamental desideratum of simplicity. Usually, one chooses [[lamda].sub.1] to be some average of those time scales, but perhaps it is more reasonable to assume that strong flows will be dominated by the longest relaxation time scale of the system.

The introduction of viscoelastic liquids into the bubble dynamics analysis creates two independent sets of parameters: the Reynolds number, defined as Re = [R.sub.max][[rho].sub.[varies]] U/[eta] where U = [([p.sub.[varies]]/[[rho].sub.[varies]]).sup.1/2], and the Deborah number which is defined as the ratio of the characteristic time of the fluid and the characteristic time of the bubble collapse, De = [[lamda].sub.1]U/[R.sub.max]. The effect of Deborah number on the behavior of a spherical bubble is illustrated in Fig. 1, which shows the maximum dimensionless velocity of the bubble wall plotted as a function of the minimum bubble radius, for three values of the Reynolds number Re = 10, [10.sup.2], [10.sup.3] and [[lamda].sub.2]/[[lamda].sub.1] = [10.sup.-1]. Figure 2 shows the influence of the ratio [[lamda].sub.2]/[[lamda].sub.1] on the maximum dimensionless velocity of the bubble wall and minimum radius of the bubble, for three values of the Reynolds number Re = 10, [10.sup.2], [10.sup.3] and De = 10. It can be seen that the liquid elasticity accelerates the bubble collapse, in agreement with the predictions of Ting [ 35], Tsujino et al. (38), and Agarwal (39), whereas the effect of liquid viscosity and retardation time is to decelerate the bubble collapse. These results further indicate that, under conditions comparable to those existing during cavitation, the effect of liquid rheology on spherical bubble dynamics is negligible for values of the Reynolds number larger than [10.sup.2] and the only significant influence is that of liquid compressibility. The noticeable effect of liquid rheology was found only for Reynolds values smaller than [10.sup.2]. In both situations, as in the case of a shear-thinning fluid, the 1/r law of pressure attenuation through the liquid is not affected by the viscoelastic properties of the liquid.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

The results presented by Fogler and Goddard (32) show that fluid elasticity can have an important effect on bubble collapse. However, for conditions similar to cavitation one would not expect to be in a parameter range where differences from Newtonian response are appreciable. In fact, when the characteristic time for bubble collapse is in the microsecond range, as it is for cavitation, Rayleigh-Plesset inertial solution appears to be entirely satisfactory. An analysis of surface-tension driven oscillations of a bubble was performed by Inge and Bark (1982), who also restricted the rheology to linear viscoelasticity. They found that the effects of elasticity are small and comparable to viscous effects.

We close our discussion of spherical bubble dynamics in quiescent viscoelastic liquids with the important theoretical contribution of Ryskin (44). By incorporating the polymer-induced stress calculated using a "yo-yo" model which accounts for the unraveling of the polymer molecules, Ryskin computed the growth and collapse phase of a vapor bubble. He concluded that the growth of the bubble is not affected by the polymer, but the final stage of collapse is. He showed that there is a total arrest of the collapse, with the bubble wall velocity reduced to nearly zero when the bubble radius becomes about 10% of the radius at the initiation of collapse.

Nonlinear Viscoelastic Liquid. The numerical simulation of spherical bubble collapse in nonlinear viscoelastic liquid is complicated by the fact that the stress tensor has a finite trace. In contrast to the case of a linear viscoelastic liquid, the stress tensor has two components instead of one and, therefore, the problem cannot be reduced to a single differential equation.

Kim (45) solved the continuity and momentum equations in a Lagrangian frame for the study of the free oscillations of a spherical bubble in an Upper-Convective Maxwell liquid. He implemented the Galerkin-finite element method for solving these equations and compared some of his results with those obtained by Fogler and Goddard (32). The significant parameters of his study are the Reynolds and Deborah numbers. He noted that, for values of the Reynolds number smaller than 10, the fluid elasticity accelerates the collapse in the early stage of the collapse, whereas in the later stages, it retards the collapse. The differences between a viscoelastic and an ideal liquid become smaller and smaller as the Reynolds number or the Deborah number increases. Similar trends were reported by Brutyan and Krapivsky (46) who used an Oldyroyd model, and Shulman and Levitskiy (47) and Jimenez and Crespo (48) who investigated the behavior of spherical bubbles in an Oldroyd-B liquid and Upper Convected Maxwell liquid, respectively.

Heat and Mass Transfer Through the Bubble Wall

Ting (36) employed an Oldroyd three-constant model with characteristic relaxation and retardation times multiplying the covariant convected time derivatives of the stress and strain rate, respectively. He allowed for thermal effects because of the phase change of water being evaporated or condensed. The resulting integrodifferential equation was solved numerically for the case of a 500 ppm solution of polyethylene oxide (PEO). He concluded that viscoelasticity has a very limited retardation effect on bubble growth and collapse, provided the material constants are compatible with dilute polymer solutions properties. It also appears from the work of Ting that the effects of heat and mass transfer are not important under cavitation conditions. Zana and Leal (49) numerically solved the conservation equations of mass and momentum along with a gas diffusion equation for a single bubble collapse. A complicated constitutive equation incorporating several material parameters was employed and the results were compared with the corresponding Newtonian case. They found that viscoelastic effects coupled with gas diffusion had profoundly impacted only the dissolution of gas bubbles. For a study of some other situations where diffusive effects are important, the reader is referred to the work of Burman and Jameson (50), Yoo and Han (51), Shulman and Levitsky (52), and Venerus et al. (53).

Experimental Results

In all experimental studies, no significant influence of the polymer additives on spherical bubbles was observed. Ting and Ellis (54) used PEO and Guar Gum aqueous solutions in concentration as high as 1000 ppm, Chahine and Fruman (55) used distilled water and a 250 ppm solution of PEO (Polyox WSR 301) with a viscosity two times larger than that of water, and Kezios and Schowalter (56) used different polymer solutions whose viscosity was up to [10.sup.-2] Pa s. They indicated that the time and amplitude of the first and second rebounds were unaffected by the polymer additive. It should be noted here that the bubbles generated in their experiments were extremely large, with a maximum radius [R.sub.max] > 1 mm. The negligible effect of polymer additives on growth and collapse of spherical bubbles has also been noted by Hara (57). Brujan et al. (58) reported that for bubbles whose maximum radius is larger than 0.5 mm, the polymer additives, even in the case of polyacrylamide (PAM) for which the aqueous solution display marked viscoelastic effects, did not affect the behavior of bubbles in any significant way. However, for bubbles whose maximum radius is smaller than 0.5 mm, a slight prolongation of the oscillation time was observed, which increases with decreasing maximum bubble radius. More recently, Bazilevskii et al. (59) have investigated the growth and collapse of spherical bubbles with maximum radii of about 0.1 mm generated in polyacryamide aqueous solutions in concentrations of up to 0.6%. They noted that the growth phase of the bubble is not affected by the polymer additive and, at high polymer concentration, they also observed a slight increase of the collapse time of the bubble in comparison to the case of water.

It is worth noting here that a direct comparison between experiments and numerical results is difficult owing to the limitations in the constitutive equations used and/or in the rheological data presented in all of the aforementioned studies. It is clear, however, from the experimental work that even a strong shear-thinning component of fluid viscosity and a high degree of elasticity of the fluid surrounding the bubble cannot influence the collapse of spherical bubbles dramatically. The maximum radius of the bubbles generated in these experiments is larger than [10.sup.-1] mm, and the viscosity of the polymer solutions used as testing liquids is smaller than [10.sup.-2] Pa s, so that the Reynolds number associated with the bubble motion is larger than [10.sup.2]. Obviously, the collapse of such large bubbles is dominated by inertia, irrespective of any details of fluid rheology. It should be noted here that a significant reduction of the maximum bubble size can be obtained by using laser pulses of picosecond or femtosecond duration. Such a short pulse offers the possibility to produce bubbles with a maximum radius of the order of [10.sup.-2] mm. Using such small bubbles, it is possible to achieve small enough values of Reynolds number to detect the influence of liquid rheology even in the case of dilute polymer solutions. Numerical predictions in spherical bubble dynamics is possible, but there is a need for experimental results using well-characterized liquids, which can be described by more sophisticated constitutive models than those that have been used previously.

Bubbles in a Sound-Irradiated Liquid

A spherical bubble in a liquid can be viewed as an oscillator that can be set into radial oscillations by a sound field. For very small sound pressure amplitudes, these oscillations can be considered as being linear about the equilibrium radius of the bubble. The response then is that of a linear oscillator. Going up in the driving amplitude will bring out the effects of nonlinearity manifesting themselves in the occurrence of several resonances.

The behavior of a bubble in a sound field can be described by the theoretical models outlined in section. The theoretical description starts with bubble nuclei with radius [R.sub.0]. At time t = 0, the pressure inside the bubble nuclei, [p.sub.0]], is balanced by the static pressure in the surrounding liquid, [p.sub.0], and the surface tension, [sigma]:

[p.sub.0] = [p.sub.0] + 2[sigma]/R. (17)

Provided that the bubble in the viscoelastic liquid is subjected to a periodically varying pressure, the pressure [p.sub.[various]] far from the bubble can be expressed by

[p.sub.[infinity]] = [p.sub.0](1 + A sin 2[pi]ft), (18)

where A is the ratio of the pulsating pressure amplitude to the static pressure and f is the frequency of the pulsating pressure.

The behavior of a single spherical bubble situated in a sound field and in a purely viscous liquid was investigated by Shima et al. (60), Tsujino et al. (61). and Brujan (62). On the other hand, Shima et al. (63) studied the bubble oscillations using a linear viscoelastic relationship to describe the liquid rheology.

Figure 3 shows an example of frequency response curves of a bubble situated in a Williamson liquid, as predicted by the incompressible formulation of Brujan (62) at A = 0.4, for water and PEO solutions in concentration of up to 1.5%. Here, the normalized maximum radius, ([R.sub.max] = [R.sub.0])/[R.sub.0], during one period of the driving frequency after the solution has reached steady state is ploted as a function of the ratio between the frequency of the sound field f and the resonance frequency of the bubble [f.sub.0]. The maximum response occurs when f/[f.sub.0] is nearly equal to 1. Other peaks are seen at or near f/[f.sub.0] = 1/2, 1/3, 1/4. They are known as the harmonics of the resonance response. These peaks have been labeled with an expression m/n, known as the order of the resonance (62). The case m = 1, n = 2, 3,..., denotes the well-known harmonics, the resonances n = 1 and m = 2, 3,..., are called subharmonics of order 1/2, 1/3,... The resonances n = 2, 3,..., and m = 2, 3,..., are called ultraharmonics. It is clear from this figure that the resonances are strongly damped or even suppressed with increasing polymer concentration. Although the harmonic resonances of order 2/1 and 3/1, respectively, are found in water and in all polymer solutions, the subharmonics resonance of order 4/1 is not found in the 1% PEO solution, while the sub-harmonic resonance of order 1/2 is found only in water and in the 0.5% PEO solution. For f/[f.sub.0] = 0.641, the ultraharmonic resonance of order 3/2 was found only in water. The numerical calculations indicated that the rheological parameter which is influential in this respect is the infinite shear viscosity [[eta].sub.[varies]]. The larger the value of [[eta].sub.[varies]], the smaller are the values of the normalized bubble radius during one period of bubble oscillation leading finally to the observed damping of the resonances. We also note that the nonlinearity of the bubble oscillation has a softening effect. The values of f/[f.sub.0] at the point of primary resonance move to the low frequency side and this value increases with the polymer concentration. For example, the value of f/[f.sub.0] at the primary resonance is 0.8 for a 0.5% PEO solution, 0.855 for a 1% PEO solution, and 0.865 for a 1.5% PEO solution, respectively. It was also found that the increase of polymer concentration leads to a reduction of the maximum pressure inside the bubble. Similar observation has been made by Shima et al. (60) and Tsujino et al. (61) who considered the bubble oscillations in a Powell-Eyring liquid and in a Carreau-like liquid, respectively.

[FIGURE 3 OMITTED]

Shima et al. (63) obtained the frequency response curves of spherical bubbles using a three-parameter linear Oldroyd model. They found that the harmonic and subharmonic resonances are more easily generated in elastic liquids and the normalized maximum radius, ([R.sub.max] - [R.sub.0])/[R.sub.0], increases with the relaxation time of the liquid [[lambda].sub.1]. On the other hand, the increase of the retardation time [[lambda].sub.2] leads to a decrease of the normalized bubble radius and to a strong damping of the resonances. In particular, the subharmonic resonance of order 1/2 and the harmonic resonances of order 3/1 and 4/1 are the most affected ones. More generally, the authors noted that for [[lambda].sub.1]/[[lambda].sub.2] > 10 the values of ([R.sub.max] - [R.sub.0])/[[R.sub.0] are larger than the corresponding values in a Newtonian liquid, whereas for [[lambda].sub.1]/[[lambda].sub.2] < 1 ([R.sub.max] - [R.sub.0])/[R.sub.0] is smaller. Similar trends have been observed for the pressure at the bubble wall.

Recently, numerical investigations on the nonlinear bubble oscillations in viscoelastic liquids have been carried out by Allen and Roy (64), (65), using the linear Jeffreys model, as well as the Upper Convected Maxwell model, and Jimenez-Fernandez and Crespo (66) who used a differential constitutive equation with an interpolated time derivative which includes the Oldroyd-B model and the Upper Convected Maxwell model as particular cases. Their results confirm the previous trend quoted above on subharmonics enhancement in elastic liquids. It was also shown that the fluid elasticity produces a significant growth of the amplitude of bubble oscillations.

Up to this point, all quantities have been given a single value for each solution if, after reaching steady state, the solutions have the same period as the driving pressure. But this not always the case, especially at high pressure amplitude where nonlinear effects are more prominent. One of the most significant developments in bubble dynamics is the realization that the bubble response to a time-periodic pressure field can be chaotic, even when the bubble is assumed to remain spherical. An example of chaotic oscillations of a single spherical bubble situated in a viscoelastic liquid is given by Jimenez-Fernandez and Crespo (66). They concluded that liquid elasticity may enhance the chaotic oscillations of bubbles even at moderate values of the driving pressure field. No influence of liquid elasticity on the number of collapses in a fixed amount of time was observed.

Aspherical Bubble Dynamics

Although the events during bubble generation are not influenced by the viscoelastic properties of the surrounding fluid, the subsequent bubble dynamics is primarily influenced by the boundary conditions in the neighborhood of the bubble and the properties of the fluid. A spherical bubble produced in an unconfined liquid retains its spherical shape while oscillating and the bubble collapse takes place at the site of bubble formation. When the bubble oscillates under asymmetric boundary conditions, it is usually exposed to pressure gradients. This leads to a faster collapse of the bubble section exposed to a higher pressure and to the formation of a liquid jet even for an initially spherical bubble. When the bubble collapses in the vicinity of a rigid boundary, the jet is directed toward the boundary (see, for example, (67)). The pressure gradient causing the jet formation is due to the low-pressure region between bubble and rigid wall developing during bubble collapse. During the initial collapse phase, the bubble acquires the form of a prolate spheroid. This shape also contributes to the formation of the liquid jet. A bubble oscillating between two parallel flat rigid walls is subjected to two opposite pressure-gradient forces and the collapse is characterized by the formation of two liquid jets that are directed toward each wall (68).

Bubbles Near a Rigid Wall

Of utmost interest is the case of a bubble near a rigid boundary because bubbles are the source of cavitation erosion. The use of a normalized distance [gamma] = s/[R.sub.max] where s is the distance of the bubble inception from the boundary has proven advantageous to classify bubble dynamics near a plane rigid boundary. Bubbles with different [R.sub.max] but the same [gamma]-value exhibit similar dynamics, thus giving the chance to specify the degree of asymmetry of bubble collapse: cavitation bubbles with a small value of [gamma] are more influenced by the boundary, thus collapsing with a more pronounced shape variation than those with a large value for which collapse is more sphere-like. This statement, however, does not apply to bubbles too close to the boundary, where [gamma] [approximately equal to] as 0 and the bubble adopts a hemispherical shape, i.e., approaches a spherical symmetry again.

Figure 4 shows a series of high-speed photographic records of bubble motion in water, a 0.5% CMC solution with a weak elastic component, and a 0.5% PAM solution with a strong elastic component for the case where [gamma] = 3.17 (58). The liquid jet, which is developed on the upper side of the bubble leading to the protrusion of the lower bubble wall, can be seen in the case of bubbles situated in water (top sequence). A similar bubble shape is found in the CMC solution, but, in this case, the jet is not as strong as in the case of water. The most interesting behavior for a bubble situated in the vicinity of a rigid boundary was found for the case of the PAM solution. The liquid jet is not observed and a flat form of the bubble shape is the dominant aspect of bubble motion after the first collapse. In the case of the PAM solution, the maximum jet velocity was found to be 88 m/s, a value which represents about 78% of the corresponding velocity in water (113 m/s). For the CMC solution, the jet velocity, 102 m/s, is almost the same as that for the case of water. Similar observations have been made when [gamma] was reduced to 1.67 (see Fig. 5). At first sight, the addition of polymers into water has a less significant effect on the bubble collapse. The liquid jet inside the bubble was observed for water and both polymer solutions. However, a significant influence of the polymer additives was noted for the velocity of the reentrant jet. Although in the case of water the maximum jet velocity is 104 m/s, only 63 m/s was measured for the PAM solution.

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

In a previous experimental study, Chahine and Fruman (55) indicated that although bubble growth is not sensitive to addition of 250 ppm of PEO to water, the collapse sequence and the shape near a rigid boundary are appreciably affected. In particular, they also observed that the polymer additive introduces a retardation effect over the initiation of the reentering jet developed during bubble collapse.

Because it was substantiated that the viscoelastic properties of the surrounding liquid might affect the collapse of a cavitation bubble situated near a rigid boundary, further studies have investigated the dependence of the pressure amplitude of the acoustic transients emitted during bubble collapse with [gamma] (69). In this study, two polymer solutions were investigated, namely a PAM aqueous solution and a CMC aqueous solution, both in a concentration of 0.5%. The extensional properties, in the form of an apparent Trouton ratio (Tr = [[eta].sub.e]/[eta]), for both polymer solutions were measured in uniaxial extension using a Rheometrics RFX opposed-jet apparatus with 1 mm diameter nozzles. The general behavior of the PAM solution is that it is extension rate thickening, which is a general characteristic for flexible polymers. The apparent Trouton ratio for the PAM solution was initially at a value of Tr [approximately equal to] 4.5 at low extension rates and then it increased to attain a maximum of Tr [approximately equal to] 70 at extension rates of [epsilon] [approximately equal to] 4000 [s.sup.-1], indicating a strong elastic component. The apparent Trouton ratio for the CMC solution was relatively constant at a value of about 5 for all the extension rates investigated, indicating a relatively less elastic behavior of the polymer solution.

Figure 6 shows the amplitude of the acoustic transients emitted during first bubble collapse, [p.sub.max], as a function of [gamma] in water and both polymer solutions. It can be seen that the largest values of the maximum amplitude of the acoustic transients are obtained in water. For the relatively less elastic 0.5% CMC solution, the bubble dynamics do not differ substantially from that in water and the maximum amplitude of the acoustic transients emitted during bubble collapse is almost similar to that in water. For the elastic 0.5% PAM solution, however, a significant reduction of [p.sub.max] was observed. We further note that the most pronounced reduction of the shock pressure in the PAM solution was observed for [gamma] < 0.6 and [gamma] > 1.5. Figure 7 shows that the velocity of the liquid jet developed during the final stage of bubble collapse range from about 10 up to 50 m/s. Furthermore, the jet velocity shows a dependence on [gamma] similar to the pressure amplitude of the acoustic transients emitted during bubble collapse: There is a minimum for values [gamma] [approximately or equal to]1 and the jet velocity decreases with increasing the extensional viscosity of the liquid.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

The effect of the viscoelastic properties of the liquid on the sound emission during first bubble collapse can be understood in a heuristic manner. A spherical bubble generated in a liquid of infinite extent retains its spherical shape while oscillating. When the bubble is formed near a rigid boundary, the collapse is associated with the formation of a high-speed liquid jet directed toward the boundary. However, examination of the high-speed photographic sequences shows that the bubble remains near spherical for much of its collapse period (between 90 and 95% depending on [gamma]), only developing significant nonsphericity at the end of the pulsation. The flow is thus predominantly uniaxial in extension during most of the collapse and the viscosity of both polymer solutions is significantly larger than that of water. Therefore, a large part of the maximum potential energy of the bubble is dissipated during the collapse phase because of an increased resistance to extensional flow, which is conferred upon the surrounding liquid by the polymer additive. Consequently, less energy is available for bubble collapse, the bubble content becomes less compressed than in the case of water, and the pressure amplitude of the shock wave is diminished. For large [gamma]-values, the retarding effect of the rigid boundary on the fluid during collapse is small. Therefore, the bubble remains nearly spherical and the liquid jet develops only in a very late stage of the collapse. For [gamma] < 0.6, the bubble is nearly hemispherical and the flow is directed toward the bubble center for most parts of the bubble surface, as in the case of a spherical collapse. In both cases, the bubble assumes spherical symmetry for most part of the collapse, thus the fluid elements experience a strong uniaxial extensional flow, and therefore, the energy dissipation during bubble collapse is the largest. The explanation for the significant reduction of the jet velocity is similar as for the acoustic transients emitted during bubble collapse. The presence of the polymer additive confers on the solution an ability to sustain higher extensional stresses than its Newtonian counterpart. This enhanced resistance to extensional deformation reduces the intensity of the reentrant liquid jet developed during bubble collapse. For [gamma] < 0.6 and [gamma] > 1.5, where the spherical symmetry is preserved during most part of bubble collapse, the extensional flow becomes dominant and the reduction of the jet velocity is the largest.

Using a perturbation approach, Hara and Schowalter (70) investigated the effect of viscoelasticity on the dynamics of single nonspherical bubbles situated in a quiescent viscoelastic liquid. The constitutive equation they used is of the Maxwell type, similar to that used by Fogler and Goddard (32). They showed that the effects of fluid rheology on nonspherical bubble dynamics are larger than on spherical bubbles. Nevertheless, growth and collapse of bubbles in an initially unstressed liquid remain dominated by inertia. Their method is, however, limited only to small oscillations of the bubble and cannot describe the motion of the reentrant liquid jet.

It is well known that boundary integral methods are particularly well suited to this class of problems as they involve discretization of the boundaries only. However, application of this method is possible only in the creeping and potential flow limits. The restriction of the boundary integral methods to potential flow problems precludes an exact accounting of the role viscoelastic effects play in the dynamics of cavitation bubbles near boundaries. It is, however, possible to include weak viscoelastic effects in the boundary integral formulation if it is assumed that these effects are limited to a thin region near the interface so that the bulk of the fluid remains irrotational. Lundgren and Mansour (71) performed an analysis to include weak viscous effects in a boundary integral simulation for an oscillating drop, whereas Boulton-Stone (72) applied the boundary integral method to study the effect of surfactants on the behavior of bursting gas bubbles.

The development of computer codes that would permit the calculation of bubble collapse in a viscoelastic fluid and near a rigid boundary has been slow. Owing to the difficulties involved in implementing both moving boundaries and viscoelasticity, resolution has not been possible anywhere near the experimentally attainable limit, even with present-day computers. Numerical simulations could contribute to a better understanding of the dynamics by providing pressure contours and velocity vectors in the liquid surrounding the bubble, which are not easily accessible through experiments.

Bubbles Between Two Rigid Walls

When a bubble is initiated between two parallel rigid walls, an annular flow is developed during bubble collapse. For a sufficiently small distance between the walls, the annular flow leads to bubble splitting and the formation of two opposing liquid jets directed toward each wall. Chahine (68) and Chahine and Morine (73) conducted several tests using geometry with bubbles generated in water, and 125 and 250 ppm of PEO, respectively. They found that, although the growth phase of the bubble is unaffected by the polymer additive, the lengthening effect on the oscillation period of the bubble is significantly reduced in the case of polymer solutions, and the departure from sphericity of the bubbles is considerably delayed. No results were presented by these authors with respect to the influence of polymer additive on the velocity of the liquid jets formed after bubble splitting.

Bubbles in a Shear Flow 

Virtually, all of the previous observations and analyses have focused on bubble collapse in a quiescent liquid, despite the fact that a number of experimenters have commented on the deformation of cavitation bubbles by the flow (see, for example, (74)). Some of the early observations of individual traveling cavitation bubbles by Knapp and Hollander (75) make mention of the deformation of the bubbles by the flow.

A detailed investigation of the effect of a controlled shear flow on the deformation of laser-generated bubbles was conducted by Kezios and Schowalter (56) using PAM and PEO solutions in concentrations of up to 2000 ppm. The main purpose of their work was to understand the role played by a pre-existing stress field at the moment when cavitation bubbles are generated. They demonstrated that the departure from sphericity is significantly reduced in polymer solutions, in particular in the highly elastic PAM solutions. They also noted that increasing the concentration beyond a critical value reverses the results and they speculated that this can be caused by the relative increase of the solution viscosity when compared with its elasticity.

Ligneul (76) also performed experiments with spark-generated bubbles in the shear layer developed by a rotating cylinder. By comparing the behavior in water and solutions of PEO with 50 and 250 ppm concentration, he concluded that the influence of the polymer additive is to maintain sphericity during bubble collapse. The effect of viscoelasticity on cavitation characteristics in flow between eccentric cylinders in relative rotation has been reported by Ashrafi et al. (77) who found that for low speeds of rotation, the liquid's free surface departed progressively from the initial horizontal (rest) configuration. With further increases in rotational speed, a provocative fingering mechanism appeared, generating a series of cavities, the number of which increased with rotational speed and eccentricity. The elastic liquids were found to generate more cells than their Newtonian equivalents, the shape of the cavities exhibiting distinctive cusp-like extremities. In this study, fluid elasticity was found to promote cavitation.

Shock-Wave Bubble Interaction

The interaction of a shock wave with a bubble in a liquid is of special interest because of the shock-induced formation of a high-speed liquid jet. When a shock wave reaches a resting bubble, it will be almost completely reflected because of the sharp increase in acoustic impedance at the bubble wall. The resulting momentum transfer accelerates the bubble wall and starts the collapse from this side. Together with focusing effects during the collapse stage, this situation finally leads to the formation of a fast liquid jet in the direction of wave propagation.

Shima et al. (78) examined the shock-induced collapse of bubbles situated in water and PAM aqueous solutions. They used the streak technique to visualize the collapse phase of the bubble and restricted their investigations only to collapse time. The bubble radius in this experiment was varied between 0.01 and 1 mm. They observed that, for bubbles smaller than 0.05 mm, the collapse time in PAM solutions with concentration of 0.05 and 0.1% is shorter than that in water. They explained this result as a consequence of the relaxation effect of the polymer solutions.

Outlook and Challenges

In this review, we have collected and organized information on the dynamics of cavitation bubbles in non-Newtonian fluids that until now was widely scattered in the scientific literature. We developed a framework to identify relevant mechanisms governing the motion of cavitation bubbles situated in a liquid of infinite extent or near rigid boundaries. Nevertheless, much work remains to further improve the mechanistic understanding of bubble dynamics in non-Newtonian fluids.

On the experimental side, further investigations are needed to better characterize the final stage of bubble collapse. There have been several observations of the production of light from laser-induced cavitation bubbles collapsing in water (79-82). An estimate of the interior bubble temperature at the moment of light emission can be obtained by fitting the spectra to a blackbody form, and in general this yields results of about 8000 K. An extension of these studies to investigate the effect of viscoelastic properties of the liquid surrounding the bubble on the characteristics of the luminescence is highly desirable for a better understanding of the cavitation phenomenon in non-Newtonian fluids and associated damage to nearby boundaries.

On the modeling and computational side, work is necessary to integrate the present knowledge on the hydrodynamics of bubble growth and collapse to build a model of the entire process dynamics. The challenge here is to devise models that include both moving boundaries and viscoelasticity. It is likely that the construction of a model that accommodates this consideration necessitates a computational rather than an analytical approach. The validity of results obtained through such an approach depends critically on the accuracy with which the computational model represents liquid rheology. Sophisticated computational codes that have already been developed (71), (72) must be reevaluated and modified to faithfully represent the viscoelastic properties of the liquid surrounding the bubble.

REFERENCES

(1.) C.E. Brennen, Cavitation and Bubble Dynamics, Oxford University Press, Oxford (1995).

(2.) D.H. Trevena, Cavitation and Tension in Liquids, Adam Hilger, Bristol (1987).

(3.) F.R. Young, Cavitation, McGraw-Hill, New York (1989).

(4.) T.G. Leighton, The Acoustic Bubble, Academic Press, London (1994).

(5.) M.S. Plesset and A. Prosperetti, Ann. Rev. Fluid Mech., 9, 145 (1977).

(6.) J.R. Blake and D.C. Gibson, Ann. Rev. Fluid Mech., 19, 99 (1987).

(7.) C.G. Hermansky and D.V. Boger, J. Non-Newtonian Fluid Mech., 56, 1 (1995).

(8.) G. Pearson and S. Middleman, AIChE J., 23, 722 (1977).

(9.) G. Pearson and S. Middleman, AIChE J., 23, 714 (1977).

(10.) S.H. Spiegelberg and G.H. McKinley, J. Non-Newtonian Fluid Mech., 67, 49 (1996).

(11.) A.L. Yarin. Free Liquid Jets and Films: Hydrodynamics and Rheology, Wiley, New York (1993).

(12.) R. Aris, Vectors, Tensors and the Basic Equation of Fluid Mechanics. Dover, New York (1989).

(13.) D.B. Khismatulin and A. Nadim, Phys. Fluids, 14. 3534 (2002)

(14.) E.A. Brujan, Fluid Dyn. Res., 23, 291 (1998).

(15.) E.A. Brujan, J. Non-Newtonian Fluid Mech., 84. 83 (1999).

(16.) E.A. Brujan., Fluid Dyn Res., 29, 287 (2001).

(17.) J.B. Keller and I.I. Kolodner, J. Appl. Phys., 27, 1152 (1956).

(18.) L. Trilling, J. Appl. Phys., 23, 14 (1952).

(19.) A. Prosperetti and A. Lezzi, J. Fluid Mech., 168. 457 (1986).

(20.) J.G. Kirkwood and H.A. Bethe, "The Pressure Wave Produced by an Underwater Explosion," in OSRD Report No. 558 (1942).

(21.) W.J. Yang and H.C. Yeh, AIChE J., 12, 927 (1966).

(22.) A. Shima and T. Tsujino, Chem. Eng. Sci., 31, 863 (1976).

(23.) A. Shima and T. Tsujino, J. Appl. Mech., 100, 37 (1978).

(24.) A. Shima and T. Tsujino, Rep. Inst. High Speed Mech. Tohoku Univ., 42, 25 (1980).

(25.) A. Shima and T. Tsujino, Trans. ASME: J. Lub. Tech., 99. 455 (1977).

(26.) A. Shima and T. Tsujino, Rep. Inst. High Speed Mech. Tohoku Univ., 42, 43 (1980).

(27.) A. Shima and T. Tsujino, Chem. Eng. Sci., 36, 931 (1981).

(28.) A. Shima and T. Tsujino, Appl. Sci. Res., 38, 255 (1982).

(29.) E.A. Brujan, Chem. Eng. Sci., 48, 3519 (1993).

(30.) E.A. Brujan, Europhys. Lett., 50, 175 (2000).

(31.) E.A. Brujan, Polym. Eng. Sci., 34, 1550 (1994).

(32.) H.S. Fogler and J.D. Goddard, Phys. Fluids, 13, 1135 (1970).

(33.) I. Tanasawa and W.J. Yang, J. Appl. Phys., 41, 4526 (1970).

(34.) W.J. Yang and M.L. Lawson, J. Appl. Phys., 45, 754 (1974).

(35.) R.Y. Ting, AIChE J., 21, 810 (1975).

(36.) R.Y. Ting, Phys. Fluids, 20, 1427 (1977).

(37.) W. McComb and S. Ayyash, J. Phys. D: Appl. Phys., 13, 773 (1980).

(38.) T. Tsujino, A. Shima, and J. Tanaka, Rep. Inst. High Speed Mech. Tohoku Univ., 55, 1 (1988).

(39.) U.S. Agarwal, e-Polymers, 014, 2002 (http://www.e-polymers. org).

(40.) A. Shima, T. Tsujino, and Y. Oikawa, Rep. Inst. High Speed Mech. Tohoku Univ., 55, 17 (1988).

(41.) M. Ichihara, H. Ohkunitani, M. Ida, and M. Kameda, J. Volcanol. Geotherm. Res., 129, 37 (2004).

(42.) M. Ichihara, Earth Planet. Sci. Lett., 265, 18 (2008).

(43.) M.C. Williams and R.B. Bird, J. Phys. Fluids, 5, 1126 (1962).

(44.) G. Ryskin, J. Fluid Mech., 218, 239 (1990).

(45.) C. Kim, J. Non-Newtonian Fluid Mech., 55, 37 (1994).

(46.) M.A. Brutyan and P.L. Krapivsky, Q. J. Mech. Appl. Math., 44, 549 (1991).

(47.) Z.P. Shulman and S.P. Levitsky, J. Eng. Phys., 53, 893 (1987).

(48.) J. Jimenez-Fernandez and A. Crespo, Int. J. Multiphase Flow, 32, 1294 (2006).

(49.) E. Zana and L.G. Leal, Ind. Eng. Chem. Fundam., 14, 175 (1975).

(50.) J.E. Burman and G.J. Jameson, Int. J. Heat Mass Transfer, 21, 127 (1978).

(51.) H.J. Yoo and C.D. Han, AIChE J., 28, 1002 (1982).

(52.) Z.P. Shulman and S.P. Levitsky, Int. J. Heat Mass Transfer, 35, 1077 (1992).

(53.) D.C. Venerus, N. Yala, and B. Berstein, J. Non-Newtonian Fluid Mech., 75, 55 (1998).

(54.) R.Y. Ting and A.T. Ellis, Phys. Fluids, 17, 1461 (1974).

(55.) G.L. Chahine and D.H. Fruman, Phys. Fluids, 22, 1406 (1979).

(56.) P.S. Kezios and W.R. Schowalter, Phys. Fluids, 29, 3172 (1986).

(57.) S.K. Hara, "The Effects of Flow and Non-Newtonian Fluids on Non-Spherical Cavitation Bubbles," PhD Thesis, Princeton University (1983).

(58.) E.A. Brujan, C.D. Ohl, W. Lauterborn, and A. Philipp, Acustica, 82, 423 (1996).

(59.) A.V. Bazilevskii, J.D. Meyer, and A.N. Rozhkov, Fluid Dyn., 38, 351 (2003).

(60.) A. Shima, S. Rajvanshi, and T. Tsujino, J. Acoust. Soc. Am., 77, 1702 (1985).

(61.) T. Tsujino, A. Shima, and Y. Oikawa, J. Sound Vib., 123, 171 (1988).

(62.) E.A. Brujan, J. Sound Vib., 173, 329 (1994).

(63.) A. Shima, T. Tsujino, and H. Nanjo, Ultrasonics, 24, 142 (1986).

(64.) J.S. Allen and R.A. Roy, J. Acoust. Soc. Am., 107, 3167 (2000).

(65.) J.S. Allen and R.A. Roy, J. Acoust. Soc. Am., 108, 1640 (2000).

(66.) J. Jimenez-Fernandez and A. Crespo, Ultrasonics, 43, 643 (2005).

(67.) E.A. Brujan, G.S. Keen, A. Vogel, and J.R. Blake, Phys. Fluids, 14, 85 (2002).

(68.) G.L. Chahine, Appl. Sci. Res., 38, 381 (1982).

(69.) E.A. Brujan, T. Ikeda, and Y. Matsumoto, Phys. Fluids, 16, 2402 (2004).

(70.) S.K. Hara and W.R. Schowalter, J. Non-Newtonian Fluid Mech., 14, 249 (1984).

(71.) T.W. Lundgren and N.N. Mansour, J. Fluid Mech., 194, 479 (1988).

(72.) J.M. Boulton-Stone, J. Fluid Mech., 302, 231 (1995).

(73.) G.L. Chahine and A.K. Morine, "The Influence of Polymer Additives on the Collapse of a Bubble Between Two Solid Walls," in ASME Cavitation and Polyphase Forum, New Orleans, Louisiana, 7 (1980).

(74.) W.K. Blake, M.J. Wolpert, and F.E. Geib, J. Fluid Mech., 80, 617 (1977).

(75.) R.T. Knapp and A. Hollander, Trans. ASME, 70, 419 (1948).

(76.) P. Ligneul, Phys. Fluids, 30, 2280 (1987).

(77.) N. Ahrafi, D.M. Binding, and K. Walters, Chem. Eng. Sci., 56, 5565 (2001).

(78.) A. Shima, Y. Tomita, and T. Ohno, Phys. Fluids, 27, 539 (1984).

(79.) O. Baghdassarian, B. Tabbert, and G.A. Williams, Phys. Rev. Lett., 83, 2437 (1999).

(80.) O. Baghdassarian, H.-C. Chu, B. Tabbert, and G.A. Williams, Phys. Rev. Lett., 86, 4934 (2001).

(81.) E.A. Brujan, D. Hecht, F. Lee, and G.A. Williams, Phys. Rev. E, 72, 066310 (2005).

(82.) E.A. Brujan and G.A. Williams, Phys. Rev. E, 72, 016304 (2005).

Emil-Alexandru Brujan

Department of Hydraulics, University Politehnica Bucharest, 060042 Bucharest, Romania

Correspondence to: Emil-Alexandru Brujan; e-mail: [e-mail]

DOI 10.1002/pen.21292

Published online in Wiley InterScience (www.interscience.wiley.com).

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