BUBBLE DYNAMICS AND CAVITATION IN NON-NEWTONIAN ...

E. A. Brujan and P. R. Willaims, Rheology Reviews 2005, pp 147 - 172.

BUBBLE DYNAMICS AND CAVITATION IN NON-NEWTONIAN LIQUIDS Emil A. Brujan1 and P. Rhodri Williams2 1

Department of Hydraulics, University Politehnica, 060042 Bucharest, Romania 2 Centre for Complex Fluids Processing, School of Engineering, University of Wales Swansea SA2 8PP, UK

ABSTRACT Cavitation phenomena play important roles in many areas of science and engineering. The most interesting effect of the non-Newtonian properties of the liquid is the reduction of cavitation damage and noise. This article reviews experimental and theoretical efforts to understand such phenomena. The currently available information favors a description of the observed reduction of cavitation damage caused by a reduction of the pressure inside the bubble at its minimum volume, leading to a weaker shock wave emission during bubble rebound. After a brief historical review, the authors survey the major areas of research: Section 2 describes the dynamics of cavitation bubbles oscillating in a liquid of infinite extent, while Section 3 describes research on the behaviour of bubbles collapsing near rigid walls. Section IV discusses the role of the non-Newtonian properties in determining the cavitation threshold. The review also outlines some directions for future research.

KEYWORDS: Bubble dynamics, cavitation, viscoelasticity, extensional flow, liquid tensile strength

1. INTRODUCTION Cavitation phenomena play important roles in many areas of science and engineering, including acoustics, biomedicine, botany, sonochemistry and hydraulics. They occur in numerous industrial processes such as cleaning, lubrication, printing and coating. While 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. Numerous references may be found to different types of cavitation but all may be discussed in terms of the formation and subsequent activity of bubbles (or cavities, or voids) whose contents may be gaseous, © The British Society of Rheology, 2005

(http://www.bsr.org.uk)

147

E. A. Brujan and P. R. Willaims, Rheology Reviews 2005, pp 147 - 172. vaporous, or both, in varying degrees. Before discussing the influence of fluid rheology on the dynamics of such bubbles – the principal focus of this Review - it is pertinent to briefly discuss the main types of cavitation. Acoustic cavitation refers to the formation of bubbles (or their growth from preexisting nuclei) during the negative part of an acoustic cycle, when the local pressure becomes sub-ambient. Subsequently, as the acoustic pressure becomes positive, the growth of the bubble slows and it begins to collapse. The behaviour of such an acoustic bubble, which depends on several factors in addition to the acoustic pressure, has been extensively discussed [5]. The term stable cavitation refers to the conditions under which the growth rate of a bubble during the rarefaction phase of an acoustic wave is equivalent to its rate of contraction during the compression phase. It should be noted that the formation of stable bubbles implies an absence of rectified diffusion – the process by which bubbles in an oscillating pressure field grow more during expansion than they shrink during contraction, due to unequal diffusion of gases and vapor from the bulk liquid phase into the bubble. Whereas stable cavitation may persist for many acoustic cycles, transient cavitation exists for only a few cavity cycles, during which time they grow several times larger than their initial size. Cavitation over solid surfaces can involve individual travelling bubbles (travelling cavitation) or sheets (sheet cavitation). The greatest effort in cavitation research has involved the study of bubble dynamics, a preponderance of attention deriving from the association between a collapsing bubble and instances of damage to solid surfaces in its near vicinity. The term ‘cavitation damage’ is used to describe this phenomenon. The first attempt to explain it was Lord Rayleigh’s [6] seminal analysis of the behaviour of an isolated spherical void collapsing in an incompressible liquid. Rayleigh’s results are considered in detail later but an important conclusion is that as the collapse nears completion, the pressure inside the liquid becomes indefinitely large. It is this mechanism, albeit extensively modified, which has led to the association of bubble collapse with cavitation damage. On the basis of Rayleigh's 1917 paper, cavitation damage was thought to be solely attributable to the extremely large implosive pressure generated at the moment when a vacuum cavity or bubble collapses. Kornfeld and Suvorov [7] were amongst the first to suggest that cavitation bubbles deform during collapse, and that damage is also caused by the impacts of high-speed liquid jets that strike surfaces during the collapse phase. Naudé and Ellis [8] observed the formation of a liquid jet in their classic photographic study and Benjamin and Ellis [9] provided a theoretical discussion of the asymmetric collapse. The first experimental evidence for cavitation noise of bubbles near a solid wall, is given by Harrison [10] who simultaneously measured bubble dynamics and pressure pulses at a distance from the bubble. In more recent studies, Tomita and Shima [11] and Vogel and Lauterborn [12] observed a very strong dependence of the pressure pulse on the relative distance between bubble and boundary. Shutler and Messler [13] and Tomita and Shima [11] reached the conclusion that the major cause of erosion of rigid boundaries is the pressure pulses produced by the collapse of a vortex ring bubble at its second collapse. Fujikawa and Akamatsu [14] have reported experiments in which a photoelastic material was used to observe 148

© The British Society of Rheology, 2005


E. A. Brujan and P. R. Willaims, Rheology Reviews 2005, pp 147 - 172. cavitation induced stresses, while the associated acoustic pulses were also recorded. They confirmed that impulsive stresses in the material were initiated at the same moment as the acoustic pulse and concluded that the stress waves were not due to a microjet. Vogel et al [15] consider that an initially spherical bubble should hardly be capable of eroding aluminium or harder metals, either by jet impact or by collapse induced pressure wave generation. The situations considered above may be classified as steady-state collapses under temporally static pressure fields. This may be contrasted with the collapses experienced by a cavity when a transient pressure pulse, such as a shock wave, passes over it. In this case a liquid jet forms, generally travelling in a direction perpendicular to the shock front [16]. Pressure amplification occurs due to the interaction between a shock wave and a bubble [17], the interaction between individual bubbles [18] and the collapse of a bubble cluster [19]. The finding that bubble-pressure pulse interactions are likely to occur in bubble clusters in hydrodynamic cavitation [16] may explain the damage due to liquid-jet impact observed by Coleman et al [20] in experiments involving an extracorporeal shock wave lithotripter (see Kodama and Takayama [21]). Tomita and Shima [11] report that a pressure pulse with an amplitude of 50 bar hitting a gas bubble produces jets with a velocity of 200-370 m/s while Dear and Field [22] have observed jets with velocities of 400 m/s after a disk-shaped cavity was struck by a shock wave with a strength of 2.6 kbar. Even higher velocities may result from the interaction of acoustic transients with collapsing bubbles. Although the role of the jet impact in the erosion mechanism of rigid surfaces remains controversial, it was recently demonstrated that a jet plays a major role in the damage of elastic surfaces where jets with a maximum velocity as high as 1000 m/s were observed [23]. Several books are available which serve as a valuable resource for the field of bubble dynamics and cavitation [1-3], together with the extensive reviews by Plesset and Prosperetti [24] and Blake and Gibson [25]. The latter address general aspects of bubble dynamics and cavitation in Newtonian fluids. In the present Review we focus upon cavitation 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 dynamics and cavitation arises from the dramatic increase in viscosity of polymer solutions in an extensional flow [26], such as that generated about a spherical bubble during its growth or collapse phase [27, 28]. 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 [29]. 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 [30]. 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 © The British Society of Rheology, 2005


149

E. A. Brujan and P. R. Willaims, Rheology Reviews 2005, pp 147 - 172. 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 organise it into a logical structure that provides an improved mechanistic understanding of bubble dynamics and cavitation in non-Newtonian liquids. Studies prior to 1996 were thoroughly reviewed by Fruman [31]. The present review is intended to fill the gap between Fruman [31] and the present. One main message is that the introduction of ideas from theoretical studies of non-linear acoustics and modern optical techniques has led to some major revisions in our understanding of the dynamics of cavitation bubbles in non-Newtonian fluids.

2. 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.

2.1 Experimental observations A convenient method to produce a single bubble in a liquid is to focus a short pulse of laser light into the liquid. Depending on the focal spot size, the transverse mode structure of the laser, the pulse duration, and the light intensity a small or several small volumes of liquid are rapidly heated, in nanoseconds, picoseconds or femtoseconds, according to the laser and the pulse width employed.

Figure 1: Cavitation bubble dynamics in water and the corresponding pressure signal measured at a distance of 10 mm from the laser focus, for a laser pulse energy EL = 10 mJ. 150



E. A. Brujan and P. R. Willaims, Rheology Reviews 2005, pp 147 - 172. Figure 1 shows a high-speed photographic record of the dynamics of a laserinduced bubble in water and the corresponding hydrophone signals, at a distance r = 10 mm from the laser focus, for a value of the laser pulse energy EL = 10 mJ. The first frame was taken 15 µs after the moment of optical breakdown and the frame interval is 20 µs. The laser light is incident from below. The hydrophone signal is recorded simultaneously with the images documented in the photographic record. When the laser-induced stress transients possess a sufficiently short rise time, their propagation results in the formation of shock wave. The large pressure in the laser-induced vapour bubble leads to a very rapid expansion that overshoots the equilibrium state, in which the internal bubble pressure equals the hydrostatic pressure. The increasing difference between the hydrostatic pressure and the falling internal bubble pressure then decelerates the expansion and brings it to a halt. At this point, the kinetic energy of the liquid during bubble expansion has been transformed into the potential energy of the expanded bubble. The bubble energy is related to the radius of the bubble at its maximum expansion, Rmax, and the difference between the hydrostatic pressure, p∞, and the vapour pressure, pv, inside the bubble by:

EB =

4π 3 . ( p∞ − pv ) Rmax 3

..........(1)

The expanded bubble collapses again due to the static background fluid pressure. The collapse compresses the bubble content into a very small volume, thus generating a very high pressure that can exceed 1 GPa for an approximately spherical bubble collapse [12]. The rebound of the compressed bubble interior leads to the emission of a strong pressure transient into the surrounding liquid that can evolve into a shock wave. Even a third pressure transient generated during second bubble collapse can be observed in this figure. The time from optical breakdown to the first collapse is denoted by 2Tc, where Tc is the collapse time that is proportional to the maximum bubble radius Rmax. The Rayleigh collapse time is given by Tc = 0.915Rmax

ρ∞ p∞ − pv

,

..........(2)

where ρ∞ is the liquid density, and was derived by Rayleigh for the case of an empty bubble without surface tension and viscosity, collapsing under a constant pressure. It has been found that for spherical laser-produced bubbles expanding and contracting under the action of the static ambient pressure in water under normal conditions, the expansion phase and the contraction phase are to a high degree symmetrical so that the time from generation to first collapse is twice the Rayleigh collapse time [1]. The results of numerous experiments conducted to investigate the behaviour of laser-generated bubbles in shear-thinning elastic fluids (specifically, carboxymethycellulose and polyacrylamide aqueous solutions in concentration of 0.5% ) have been described by Brujan et al. [32]. It was observed that for bubbles whose maximum radius is larger than 0.5 mm the polymer additives, even in the case of polyacrylamide for which the aqueous solution display marked viscoelastic effects, did not affect the



151


Figure 2: Oscillation time of a spherical bubble situated in a 0.5% PAM solution and 0.5% CMC solution. The solid line represents twice Rayleigh's collapse time. (adapted from reference [31]).

behaviour of bubbles in any significant way, and the duration of the oscillation time (2Tc) is equal to the Rayleigh time. However, for bubbles whose maximum radius is smaller than 0.5 mm, a prolongation of the oscillation time was observed, which increases with decreasing maximum bubble radius (Figure 2). In figure 3, the maximal bubble radius is shown as a function of the laser pulse energy. No difference between the case of water and both polymer solutions is seen, indicating that the growth phase of the bubble is not affected by the polymer additive. The scaling law for the size of the bubble oscillating in both polymer solutions is the same as that in water; namely, the maximum bubble radius is proportional to the cube root of the laser pulse energy. This scaling law applies, however, only to laser pulse energies larger than 2 mJ, well above the breakdown threshold. For lower values, the energy dependence of the bubble size is stronger. It is noteworthy that in numerous previous experimental studies, no influence of the polymer additives on spherical bubbles was observed. Ellis and Ting [33] used polyethylene oxide (PEO) and Guar Gum aqueous solutions in concentration as high as 1000 ppm; Chahine and Fruman [34] 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 [35] used different polymer solutions whose viscosity was up to 10-2 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 Rmax > 1 mm. Bazilevskii et al. [36] have investigated the growth and collapse of 152

© The British Society of Rheology, 2005 (http://www.bsr.org.uk)


Figure 3: Maximal cavitation bubble radius, Rmax, as a function of the laser pulse energy, EL. The slope of the straight line gives the scaling law for the bubble radius at energy values well above the breakdown threshold. The same scaling law applies in water and polymer solutions: Rmax ∝ E1L/ 3 .

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.

2.2 Theoretical description

Efforts to understand the physics of cavitation phenomena in non-Newtonian liquids have involved a number of theoretical studies of bubble dynamics. The earliest theoretical treatment is that of Fogler and Goddard [37] who considered the collapse of a spherical bubble in a fluid model including stress accumulation with fading memory. Later, Tanasawa and Yang [38], Yang and Lawson [39], Ting [40, 41], Tsujino et al. [42], and Ichihara et al. [43] considered an Oldroyd fluid, Shima and Tsujino [44] a Carreau fluid, Shima et al. [45] a Jeffreys fluid, Ryskin [28] the ‘yo-yo’ model of polymer dynamics, and Kim [46] and Allen and Roy [47] an upper-convected Maxwell fluid. The effect of heat and mass transfer through the bubble wall was also included in the theoretical analysis by Shulman and Levitskiy [48]. Recently, Agarwal


153

E. A. Brujan and P. R. Willaims, Rheology Reviews 2005, pp 147 - 172. [49] studied the growth of bubbles in viscoelastic fluids using the pom-pom model to describe the rheological behaviour of the fluid. In the aforementioned studies, the assumption of liquid incompressibility was based on the idea that only a small fraction of the energy of the bubble motion is radiated away as sound. However, the later stages of cavitation bubble collapse proceed so quickly that the velocity of the bubble interface can be compared to the speed of sound in the liquid and thus the compressibility of the liquid can no longer be ignored. An elegant solution can be obtained for the case of a linear viscoelastic liquid for which the sum of the normal stress components is zero. 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∞ T , where c∞ 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 RT & 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. The picture becomes considerably more intricate for a non-linear viscoelastic liquid, however [50]. The analysis leads unambiguously to the following equation for the radius of a spherical bubble situated in a linear viscoelastic liquid [51-54]:

&& + RR

∞

3&2 1 & && + 2 R & 3 = H − 1  ∂τ rr + 3τ rr dr , R − R 2&&& R + 6 RRR  2 c∞ r  ρ ∞ ∫R  ∂r

(

)

..........(3)

where H is the liquid enthalpy at the bubble wall:

H=

n ( p∞ + B )  P + B   ( n − 1) ρ∞  p∞ + B 

( n −1) / n

 − 1 . 

..........(4)

In the above equations, P must be regarded as the pressure at the position occupied by the bubble centre in the absence of the bubble. The pressure pB on the liquid side of the bubble interface is related to the internal pressure p by the usual balance of normal stresses:

p = pB −

2σ − (τ rr )r = R , R

..........(5)

in which σ is the surface tension. The striking feature of equation (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∞ ) ≈ &&R (t ) − ( R / c∞ )&&& R. A similar term arises in Lorentz' theory of electrons. Lorentz was considering periodic displacements x at frequency ω and thus set &&&x ≈ −ω 2&x and identified this term with radiation damping. Later researchers, however, were deeply puzzled by this third

154


E. A. Brujan and P. R. Willaims, Rheology Reviews 2005, pp 147 - 172. derivative although there is nothing mysterious about it [54]. For c∞ → ∞ , the incompressible formulation is recovered, namely: ∞

&& + 3 R & 2 = H − 1  ∂τ rr + 3τ rr dr . RR  r  2 ρ∞ ∫R  ∂r & ″ = α R2 R & ″ + (1 − α ) R 2 R & ″ Furthermore, if one writes R 2 R incompressible formulation in the form:

(

′

( R R& ) 2

R

(

2& 1 R R − 2 R4

)

)

2

=H−

(

)

(

)

..........(6) and uses the

∞

 ∂τ rr 3τ rr  + dr , ρ∞ ∫R  ∂r r 

1

..........(7)

to evaluate the first term and (6) to express the third derivative of the radius which appears on expanding the second term, one finds:

&& 1 − α + 1 R & + 3 R & 2 1 − 3α + 1 R &  = H 1 + 1 − α R & + R H & RR       c∞ 3c∞ c∞   2     c∞ ∞

−

1  1 − α &   ∂τ rr 3τ rr R ∫ + 1+ ρ∞  c∞ r  R  ∂r

∞

1 R d  ∂τ rr 3τ rr  +  dr − ρ∞ c∞ dt ∫R  ∂r r 

  dr 

,

..........(8)

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 α = 0 , equation (8) becomes identical to the equation proposed by Keller [55], while the value α = 1 brings it into the form suggested by Herring (see [56]). It will be noted that, by dropping terms in c∞−1 , equation (8) reduces to equation (6), which is therefore seen to have an error of the order c∞−1 . The arbitrary parameter α (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 equation (8). Because of the presence of the third time derivative of the radius, the form (3) of the radial equation is hardly more attractive than (8), if for nothing else than for the && . Actually, this is a minor difficulty since, need to prescribe an initial condition for R to the same order of accuracy in the bubble wall Mach number, an initial condition for && can be obtained by substituting the given initial conditions for R and R& in the R incompressible formulation (6). However, in view of its uniqueness [52], it is proper to consider equation (3) the fundamental form of the motion equation of a spherical bubble in a compressible linear viscoelastic liquid. With reference to equation (8) it should be noted that a related equation is that due to Gilmore [52]:

& & & & && 1 − R  + 3 R & 2  1 − R  = H 1 + R  + R 1 − R  H & RR  C 2  C  C C C −

&  ∞  ∂τ rr 3τ rr 1  R + 1 +  ∫  r ρ ∞  C  R  ∂r

∞

1 R d  ∂τ rr 3τ rr  +  dr − r ρ∞ C dt ∫R  ∂r 

  dr 

,

..........(9)

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


155

E. A. Brujan and P. R. Willaims, Rheology Reviews 2005, pp 147 - 172. 1

C = c∞2 + ( n −1) H  2 ,

..........(10)

and whose derivation relies on the Kirkwood-Bethe approximation [57, 58]. 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.

2.2.1 Non-Newtonian purely viscous fluid By ignoring the probable contribution of elasticity, a Williamson fluid has been adopted in a previous study [51] as a model of carboxymethylcelullose (CMC) and hydroxyethylcelullose (HEC) aqueous polymer solutions. The apparent viscosity of the fluids is modelled by the Williamson rheological equation: η0 − η∞

η = η∞ + 1+

(

II 2 D / k

,

)

..........(11)

n

where η0 is the zero-shear viscosity, η∞ is the infinite shear-viscosity, k and n are the Williamson model parameters and II2D is the second invariant of the rate of deformation tensor. The physical properties of polymer solutions and water are listed in Table 1. Typical results of the calculations are shown in figure 4, which illustrates the maximum velocity reached by the bubble wall during first collapse (left) and maximum pressure at the bubble wall (right) as a function of the minimum bubble radius at the end of the first collapse. They demonstrate that, for values of the maximum bubble radius smaller than 10-1 mm, the shear-thinning characteristic of liquid viscosity strongly influences the behaviour of the bubble and the rheological parameter with the strongest influence is the infinite-shear viscosity η∞. In spite of the considerable differences of the apparent viscosity of the liquid, η, the behaviour of the bubble remains the same as that of an equivalent Newtonian fluid with a viscosity η∞. 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. We also note that, in the range 10-2 mm < Rmax < 1 mm, the 1/r law of pressure attenuation through the liquid is not affected by the shear-thinning characteristic of liquid viscosity. On the other hand, for values of the initial bubble radius R0 > 10-1 mm, sound emission is the main damping mechanism in spherical bubble collapse.

Table 1: Physical properties of polymer aqueous solutions and water

156



(a)

(b)

Figure 4. Maximum dimensionless velocity of the bubble wall during first bubble collapse (a) and maximum pressure at the bubble wall (b) versus minimum bubble radius at the end of first bubble collapse. The far-right points correspond to R0 = 1 mm and the far-left ones to R0 =10-2 mm. The full symbols are the results of incompressible formulation and the open ones of compressible formulation. Circles: water, triangles: 0.5% HEC solution and squares: 0.5% CMC solution. In these calculations p∞ = 101325 Pa and U = (p∞/ρ∞)1/2(reproduced from reference [51]).


157

E. A. Brujan and P. R. Willaims, Rheology Reviews 2005, pp 147 - 172. 2.2.2 Non-Newtonian viscoelastic fluid The first theoretical treatment of spherical bubble dynamics in a compressible viscoelastic fluid is due to Brujan [52]. In this study, as in the incompressible formulations of Tanasawa and Yang [38], Yang and Lawson [39], Tsujino et al. [42] and Shima et al. [59], the three-parameter, linear Oldroyd model was employed to represent the rheological behaviour of a viscoelastic liquid. The rheological equation of this model is represented as follows [60]:

τ ii + λ1

Dτ ii Deii   = −2η  eii + λ2 , Dt Dt  

..........(12)

where D/Dt is the material time derivative, λ1 is a characteristic relaxation time (for the stress), η, the viscosity coefficient, λ2, a characteristic retardation time (i.e., relaxation time for strain) and eii are the strain rate components. It should be noted that the assumption of a single relaxation time λ1 is over simplistic, even if the polymers

Figure 5. The effect of Deborah number on the dimensionless maximum velocity attained during the first collapse and dimensionless minimum radius at the end of the collapse for χ = 10-1. The filled symbols indicate the results obtained using the incompressible formulation, the open ones using the compressible formulation. Circles: Newtonian liquid, diamonds: De = 10-2, squares: De = 10-1 , triangles (∆): De = 1, triangles ( ∇ ): De = 10 and hexagons: inviscid liquid. (reproduced from reference [52]).

158


E. A. Brujan and P. R. Willaims, Rheology Reviews 2005, pp 147 - 172. are mono-dispersed. Rather, one would expect a long chain to have a distribution of time scales, corresponding to various sub-chains 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 our fundamental desideratum of simplicity. Usually, one chooses λ1 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 fluids into the bubble dynamics analysis creates two independent sets of parameters: the Reynolds number, defined as Re = Rmaxρ∞U/η where U = (p∞ /ρ∞)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 = λ1U/Rmax. The effect of Deborah number on the behaviour of a spherical bubble is illustrated in figure 5, 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, 102, 103 and λ2 / λ1 = 10-1. Figure 6 shows the influence of the ratio λ2 / λ1 on the maximum dimensionless velocity of the bubble wall and minimum radius of the bubble, for three values of the Reynolds

Figure 6. The effect of ratio χ = λ2/λ1 on the dimensionless maximum velocity attained during the first collapse and dimensionless minimum radius at the end of the collapse for De = 10. The filled symbols indicate the results obtained using the incompressible formulation, the open ones using the compressible formulation. Circles: Newtonian liquid, diamonds: χ = 10-1, squares: χ = 10-2 , triangles χ = 0, and hexagons: inviscid liquid. (reproduced from reference [33]).



159

E. A. Brujan and P. R. Willaims, Rheology Reviews 2005, pp 147 - 172. number Re = 10, 102, 103 and De = 10. It can be seen that the liquid elasticity accelerates the bubble collapse, in agreement with the predictions of Ting [40], Shima et al. [45] and Kim [46], while 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 102 and the only significant influence is that of liquid compressibility. The noticeable effect of liquid rheology was found for Re < 102. 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. The important theoretical contributions by Ting [41] and Ryskin [28] to spherical bubble dynamics must also be noted. Ting employed an Oldroyd threeconstant model with characteristic relaxation and retardation times multiplying the covariant convected time derivatives [61] of the stress and strain rate, respectively. He allowed for thermal effects due to the phase change of water being evaporated or condensed. The resulting integro-differential equation was solved numerically for the case of a 500 ppm solution of polyethylene oxide. 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. For a study of situations where diffusive effects are important, the reader is referred to the work of Shulman and Levitsky [48] and Yoo and Han [62]. On the other hand, by incorporating the polymer-induced stress calculated using a ‘yo-yo’ model, which accounts for the unravelling of the polymer molecules, Ryskin [28] computed the growth and collapse phase of a vapour bubble. He concluded that the growth of the bubble is not affected by the polymer, but that 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.

2.2.3 Comparison between experiment and theory 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 above-mentioned 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 10-1 mm and the viscosity of the polymer solutions used as testing liquids is smaller than 10-2 Pa s, so that the Reynolds number associated with the bubble motion is larger than 102. 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

160



E. A. Brujan and P. R. Willaims, Rheology Reviews 2005, pp 147 - 172. 10-2 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-characterised fluids which can be described by more sophisticated constitutive models than those that have been used previously.

3. ASPHERICAL BUBBLE DYNAMICS While 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 neighbourhood 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. However, when the bubble is formed near a material boundary, the collapse is asymmetric and associated with the formation of one or two high-speed water jets that concentrate the bubble energy at some distance from the locus of bubble generation. When the bubble collapses in the vicinity of a rigid boundary, the jet is directed toward the boundary [63]. The bubble collapse between two rigid walls is characterised by the formation of two liquid jets that are directed toward each boundary [64].

3.1. 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 normalised distance γ = s/Rmax 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 Rmax but the same γ-value exhibit similar dynamics, thus giving the chance to specify the degree of asymmetry of bubble collapse: cavitation bubbles with a small value of γ 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 γ ≈ 0 and the bubble adopts a hemispherical shape, i.e. approaches a spherical symmetry again. Figure 7 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 γ = 3.17 [32]. 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 behaviour 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 velocity of the upper bubble wall was found to be 88 m/s, a value which represents about 78% © The British Society of Rheology, 2005


161


Figure 7. Picture sequences of the behaviour of a laser-induced bubble near a rigid wall in water and polymer solutions. γ = 3.17, frame interval 4.8 µs. Top: water, Rmax = 0.63 mm, frame size 1.7 x 2 mm; Middle: 0.5% CMC solution, , Rmax = 0.47 mm, frame size 1.7 x 2 mm; Bottom: 0.5% PAM solution, , Rmax = 0.63 mm, frame size 1.7 x 2.4 mm. (reproduced from reference [32]).

of the corresponding velocity in water (113 m/s). For the CMC solution the velocity of the upper bubble wall, 102 m/s, is almost the same as that for the case of water. Similar observations have been made by Chahine and Fruman [34], who showed that the polymer additive introduces a retardation effect over the initiation of the reentering jet developed during bubble collapse.

162




Figure 8. Pressure amplitude of the acoustic transients emitted during first bubble collapse as a function of γ. The pressure values are measured at a distance of 10 mm from the ultrasound focus. (reproduced from reference [56]).

Since 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 jet velocity and pressure amplitude of the acoustic transients emitted during bubble collapse with γ [65]. Aqueous solutions of carboxymethylcellulose and polyacrylamide in a concentration of 0.5% were used as testing fluids. Steady shear measurements were made using a Contraves Low-Shear 40 rheometer equipped with a cup and bob system, and a Brookfield R/S rheometer with coaxial cylinders. The extensional properties, in the form of an apparent Trouton ratio ( Tr = ηe / η ), for both polymer solutions were measured in uniaxial extension using a Rheometric RFX opposed-jet apparatus with 1 mm diameter nozzles. The general behavior of the polyacrylamide solution is that it is extension rate thickening, which is a general characteristic for flexible polymers. The apparent Trouton ratio for the polyacrylamide solution was initially at a value of Tr ≈ 4.5 at low extension rates and then it increased to attain a maximum of Tr ≈ 70 at extension rates of &ε ≈ 4000 s-1, indicating a strong elastic component. The apparent Trouton ratio for the carboxymethylcellulose 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.



163

E. A. Brujan and P. R. Willaims, Rheology Reviews 2005, pp 147 - 172. Figure 8 shows the amplitude of the acoustic transients emitted during first bubble collapse, pmax, as a function of γ in water and both polymer solutions. The pressure values refer to a distance r of 10 mm between the ultrasound focus and the site of measurement. 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 pmax was observed. We further note that the most pronounced reduction of the shock pressure in the PAM solution was observed for γ < 0.6 and γ > 1.5. Figure 9 shows that the velocity of the liquid jet developed during the final stage of bubble collapse range from about 10 m/s up to 50 m/s. Furthermore, the jet velocity shows a similar dependence from γ like the pressure amplitude of the acoustic transients emitted during bubble collapse: There is a minimum for values γ ≈ 1 and the jet velocity decreases with increasing the extensional viscosity of the liquid. 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

Figure 9. Maximum jet velocity as a function of the stand-off parameter γ. The jet velocity is averaged over 3 µs. (reproduced from reference [56]).

164



E. A. Brujan and P. R. Willaims, Rheology Reviews 2005, pp 147 - 172. oscillating. Uniaxial extension is associated with bubble collapse, while biaxial extension is associated with the growth phase. When the bubble is formed near a rigid boundary, the collapse is associated with the formation of a high-speed liquid jet directed towards 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 γ), only developing significant non-sphericity at the end of the pulsation. The flow is thus predominantly uniaxial in extension during most part 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 due to 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 γ-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 γ < 0.6, the bubble is nearly hemispherical and the flow is directed towards 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. This finding is in good agreement with the experimental work reported by Brujan et al. [32] who found that, below a certain value of the maximum bubble radius, a spherical bubble implodes less violently in polymer solutions than in water. 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 re-entrant liquid jet developed during bubble collapse. For γ < 0.6 and γ > 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. 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.

3.2 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



165

E. A. Brujan and P. R. Willaims, Rheology Reviews 2005, pp 147 - 172. directed towards each wall [64]. Chahine and Morine [66] conducted several tests using geometry with bubbles generated in water, and 125 and 250 ppm of polyethylene oxide, 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.

3.3. Bubbles in a shear flow Virtually all of the previous observations and analyses have focussed 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, [67]). This indicates that not only the bubble size, distance to the boundary and properties of the liquid surrounding the bubble, but also the characteristics of the flow along a rigid boundary strongly influence the bubble dynamics. Some of the early observations of individual travelling cavitation bubbles, by Knapp and Hollander [68], 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 [25], using polyacrylamide (PAM) and polyethylene oxide (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 as compared to its elasticity. Ligneul [69] also performed experiments with spark-generated bubbles in the shear layer developed by a rotating cylinder. By comparing the behaviour in water and solutions of polyethylene oxide 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 have been reported by Ashrafi et al [70] 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. Few studies have examined cavitation in extremely thin films, such as disklubricant films under high shear forces, but further work may be anticipated due to the

166



E. A. Brujan and P. R. Willaims, Rheology Reviews 2005, pp 147 - 172. increasing availability of scanning probe instruments, such as the atomic force microscope. Barrow et al [71] have described studies of thin-film cavitation in such an instrument. Ouibrahim et al. [72] have studied flows of water and a non-Newtonian viscoelastic fluid (600 ppm aqueous PEO solution) between a moving and a fixed wall separated by micron size gaps. Large pressure gradients may exist in the flow direction and cavitation can occur locally if the minimum pressure falls to, or below, the vapor pressure. Ouibrahim et al. found that the initiation of cavitation was delayed in the polymer solution and speculate that the effect is due to the development of normal stresses in the elongational flow in the space between the moving and the fixed wall. Other claimed viscoelastic effects in thin film flows include a significant displacement of the point of cavitation from the centre of contact (where film thickness is a minimum) and enhanced film thicknesses [73].

4. CAVITATION THRESHOLD: THE EFFECTIVE TENSILE STRENGTH OF LIQUIDS It is important to realise that cavitation is not necessarily a consequence of the reduction of pressure to the liquid’s vapour pressure, the latter being the equilibrium pressure, at a specified temperature, of the liquid's vapour in contact with an existing free surface. Cavity formation in a homogeneous liquid requires a stress sufficiently large to rupture the liquid. This stress represents the tensile strength of the liquid at that temperature [1-3]. Estimates of the cavitation threshold or effective tensile strength, Fc , of liquids in which cavitation involves heterogeneous nucleation of cavities are important in relation to the establishment of flow boundary conditions in (fluid) mechanical engineering and process design, such as in the prediction of bearing performance. Despite numerous studies, the precise role of non-Newtonian properties in determining cavitation threshold remains unclear. Most previous work in this area has considered polymer solutions - fluids made non-Newtonian by polymeric additives [2]. Under conditions of dynamic stressing by pulses of tension there is evidence that polymer additives can lower cavitation threshold. An example has been reported by Sedgewick and Trevena [74] who studied the cavitation properties of water containing polyacrylamide additives by the bullet-piston reflection method. Williams and Williams [75] have shown that the latter method, which involves the conversion of a compressional pulse to a rarefaction at the free surface of a column of liquid, provides realistic estimates of Fc for water and other Newtonian fluids [76]. Bullet-piston work has demonstrated a reduction of liquid effective tensile strength in non-Newtonian polymer solutions, the reduction increasing with increasing polymer concentration. However, when this system was investigated using an ab initio technique, the cavitation threshold was found to be increased by the same polymer additive (Overton et al. [77], Brown and Williams [78]). When subjected to quasi-static stressing (in a modified Berthelot tube) the presence of polymer made no discernible difference to the effective tensile strength of the liquid [2]. Such apparently contradictory results have their counterparts in work involving the assessment of cavitation erosion/damage rates. Studies in this area have © The British Society of Rheology, 2005


167

E. A. Brujan and P. R. Willaims, Rheology Reviews 2005, pp 147 - 172. usually involved acoustically-generated cavitation and its effects on solid (usually metal) target specimens. In order to explain instances of reduced engine bearing wear, some ‘mitigating’ effects of viscoelasticity on cavitation in multigrade oils have been mooted [79] but it is important to emphasise that the mechanisms of any such effects remain unclear and, in some instances, the effect of viscoelasticity appears to be to promote the likelihood of cavitation damage. Ashworth and Procter [80] have reported a decrease of the incubation time for erosion and an increase of the erosion rate with a 1000 ppm concentration of the PAM agent in (stagnant) distilled water using a ultrasonic horn apparatus. For a concentration of 100 ppm, no significant change in behavior was found but their partially degraded glycerol/distilled water mixture, which had a shear viscosity comparable to that of the 1000 ppm PAM solution, showed an incubation time which was increased by a factor of three as compared to pure water. Moreover, the erosion rate was found to be slightly lower. Similar experiments by Shima et al. [81] on polyethylene oxide solutions produced markedly different results (it is worth noting that the two polymers used are effective drag reducing agents, display analogous viscoelastic effects, and have, for equal concentrations, comparable shear viscosities in solution). The principal findings of Shima et al [81] were (i) that no incubation time seemed necessary (either in water or polymer solution); (ii) a 100 ppm solution showed a behavior similar to that of water, with a slight increase of the weight loss and (iii) for 500 and 1000 ppm solutions the weight loss was found to increase for a short time after initiation and to decrease significantly thereafter. Although the results obtained by Ashworth and Procter [80] and Shima et al. [81] are markedly different, a careful reading of the reports reveals that the position of the test specimens in the ultrasonic pressure field were different in both studies, as was the ultrasonic pressure amplitude (among other factors). Numerous other studies have reported potentially important differences in bubble behaviour in this respect (see Apfel [82]) but the interpetation of results is often not straightforward due, principally, to differences in experimental arrangements. Clearly, much further work is required in this area, as in the closely related assessment of cavitation thresholds, using well-documented and wellunderstood experimental techniques.

REFERENCES 1.

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

2.

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

3.

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

4.

Shah, Y T, Pandit, A B and Moholkar, V S, ‘Cavitation Reaction Engineering’, Kluwer Academic/Plenum Publishers New York, (1999).

5.

Leighton, T G, ‘The Acoustic Bubble’, Academic Press, London, (1994).

168



E. A. Brujan and P. R. Willaims, Rheology Reviews 2005, pp 147 - 172. 6.

Rayleigh, Lord, Phil. Mag., 34 (1917) 94-98.

7.

Kornfeld, M and Suvorov, L, J. Appl. Phys., 15 (1944) 495-506.

8.

Naudé, C F and Ellis, A T, Trans. ASME D: J. Basic Engng, 83 (1961) 648656.

9.

Benjamin, T B and Ellis, A T, Phil. Trans. R. Soc. Lond. A, 260 (1966), 221240.

10.

Harrison, M, J. Acoust. Soc. Am., 24 (1952) 776-782.

11.

Tomita, Y and Shima, A, J. Fluid Mech., 169 (1986) 535-564.

12.

Vogel, A and Lauterborn, W, J. Acoust. Soc. Am., 84 (1988) 719-731.

13.

Shutler, N D and Mesler, R B, Trans. ASME D: J. Basic Engng, 87 (1965) 511517.

14.

Fujikawa, S and Akamatsu, T, J. Fluid Mech., 97 (1980) 481-512.

15.

Vogel A, Lauterborn, W and Timm, R, J. Fluid Mech. 206 (1989) 299-338.

16.

Bourne, N K and Field, J E, J. Fluid Mech., 244 (1992) 225-240.

17.

Bourne, N K and Field, J E, J. Appl. Phys., 78 (1995) 4423-4427.

18.

Testud-Giovanneshi, P, Alloncle, A P and Dufresne, D, J. Appl. Phys., 67 (1990) 3560-3564.

19.

Hansson, I and Mǿrch, K A, J. Appl. Phys., 51 (1980) 4651-4658.

20.

Coleman, A J, Suanders, J E, Crum, L A, and Dyson, M, Ultrasound Med. Biol., 13 (1987) 69-76.

21.

Kodama, T, and Takayama, K, Ultrasound in Medicine and Biology, 24 (1998) 723-738.

22.

Dear, J P and Field, J E, J. Fluid Mech., 190 (1988) 409-425.

23.

Brujan, E A, Nahen, K, Schmidt, P and Vogel, A, J. Fluid Mech., 433 (2001) 251-281.

24.

Plesset, M S and Prosperetti, A, Annual Rev. Fluid Mech., 9 (1977) 145-185.

25.

Blake, J R and Gibson, D C, Annual Rev. Fluid Mech., 19 (1987) 99-123.

26.

Jones, D M, Walters, K and Williams, P R, Rheol. Acta, 26 (1987) 20-30.

27.

Pearson, G and Middleman, S, AIChE Journal, 23 (1977) 714-721.

28.

Ryskin G, J. Fluid Mech., 218 (1990) 239-263.

29.

Spiegelberg, S H and McKinley, G H, J. Non-Newtonian Fluid Mech., 67 (1996) 49-60

30.

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



169


Fruman, D H, in ‘Advances in the flow and rheology of non-Newtonian fluids’, (Eds: D.A. Siginer, D. De Kee, R.P. Chhabra), Elsevier, (1999).

32.

Brujan, E A, Ohl, C D, Lauterborn, W and Philipp, A, Acoustica, 82 (1996) 423-430.

33.

Ellis, A T and Ting, R Y, NASA-SP 304 (1994) 403-420.

34.

Chahine, G L and Fruman, D H, Phys. Fluids, 22 (1979) 1406-1407.

35.

Kezios, P S and Schowalter, W R, Phys. Fluids, 29 (1986) 3172-3181.

36.

Bazilevskii, A V, Meyer, J D and Rozhkov, Fluid Dyn., 38 (2003) 351-362.

37.

Fogler, H S and Goddard, J D, Phys. Fluids, 13 (1970) 1135-1141.

38.

Tanasawa, I and Yang, W-J, J. Appl. Phys., 45 (1974) 754-759.

39.

Yang, W-J and Lawson, M L, J. Appl. Phys., 45 (1974) 754-758.

40.

Ting, R Y, AIChE J., 21 (1975) 810-813.

41.

Ting, R Y, Phys. Fluids, 20 (1977) 1427-1431.

42.

Tsujino, T, Shima, A and Tanaka, J, Rep. Inst. High Speed Mech. Tohoku Univ., 55 (1989), 1-17.

43.

Ichihara, M, Ohkunitani, H, Ida, Y and Kameda, M, J. Volcanol. Geotherm. Res., 129 (2004) 37-60.

44.

Shima, A and Tsujino, T, Rep. Inst. High Speed Mech. Tohoku Univ., 42 (1980) 43-58.

45.

Shima, A, Tsujino, T and Oikawa, Y, Rep. Inst. High Speed Mech. Tohoku Univ., 55 (1989) 17-34.

46.

Kim, C, J. Non-Newtonian Fluid Mech., 55 (1994) 37-58.

47.

Allen, J S and Roy, R A, J. Acoust. Soc. Am., 108 (2000) 1640-1650.

48.

Shulman, Z P, Levitsky, S P, Int. J. Heat Mass Transfer, 35 (1992) 1077-1084.

49.

Agarwal, U S, e-Polymers, 14 (2002).

50.

Khismatullin, D B and Nadim, A, Phys. Fluids, 14 (2002) 3534-3557.

51.

Brujan, E A, Fluid Dyn. Res., 23 (1998) 291-318.

52.

Brujan, E A, J. Non-Newtonian Fluid Mech., 55 (1994) 37-58.

53.

Brujan, E A, Fluid Dyn. Res., 29 (2001) 287-294.

54.

Prosperetti, A and Lezzi, A, J. Fluid Mech., 168 (1986) 457-478.

55.

Keller, J B and Kolodner, I I, J. Appl. Phys., 27 (1956) 1152-1161.

56.

Trilling, L, J. Appl. Phys. 23 (1952) 14-17.

57.

Kirkwood, J G and Bethe, H A, OSDR Report 558 (1942).

170




Knapp, R T, Daily, J W and Hammitt, F G, ‘Cavitation’, McGraw-Hill, New York (1970)

59.

Shima, A, Tsujino, T and Nanjo, H., Ultrasonics, 24 (1986) 142-147.

60.

Williams, M C and Bird, R B, J. Phys. Fluids, 2 (1962) 1126-1128.

61.

Schowalter, W R, ‘Mechanics of Non-Newtonian Fluids’, Pergamon, Oxford, (1978).

62.

Yoo, H J and Han, C D, AIChE J., 28 (1982) 1002-1009.

63.

Brujan, E A, Keen, G S, Vogel, A and Blake, J R, Phys. Fluids, 14 (2002) 8592

64.

Chahine, G L, Appl. Sci. Res., 38 (1982) 381-387.

65.

Brujan, E A, Ikeda, T and Matsumoto, Y, Phys. Fluids, 16 (2004) 2402-2410.

66.

Chahine, G L and Morine, A K, ASME Cavitation and Polyphase Flow Forum, (1980), 7-8.

67.

Blake, W K, Wolpert, M J and Geib, F E, J. Fluid Mech., 80 (1977) 617-640.

68.

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

69.

Ligneul, P, Phys. Fluids, 30 (1987) 2280-2282.

70.

Ashrafi, N, Binding, D M and Walters, K. Chem. Eng. Sci., 56 (2001) 55655574.

71.

Barrow, M S, Bowen, W R, Hilal, N, Al-Hussany, A, Williams, P R, Williams, R L and Wright, C, Proc. Roy. Soc. A: Math and Phys Sci., 459 (2003) 28852908.

72.

Ouibrahim, A., Fruman, D.H. and Gaudemer, R, Phys. of Fluids, 8 (1996) 1964-1971

73.

Narumi, T. and Hasegawa, T, Bull. JSME, 29 (1986) 3731-3739.

74.

Sedgewick, S A and Trevena, D H, J. Phys. D: Appl. Phys, 11 (1978) 25172526.

75.

Williams, P R and Williams, R L, Proc. Roy. Soc. A: Math and Phys Sci., 456 (2000) 1321−1332.

76.

Williams, P R and Williams, R L, J. Phys. D: Appl. Phys., 35 (2002) 22222230.

77.

Overton, G D N, Williams, P R and Trevena, D H, J. Phys D: Appl Phys, 17 (1984) 979−987.

78.

Brown, S W J and Williams, P R, J. Non-Newtonian Fluid Mech, 90 (2000) 1−11.

79.

Berker, A, Bouldin, M G, Kleis, S J and Van Arnsdale, W E, J. Non-Newtonian Fluid Mech. 56 (1995) 333-347. © The British Society of Rheology, 2005


171


Ashworth, V and Procter, R P M, Nature, 258 (1975) 64.

81.

Shima, A, Tsujino, T and Nanjo, H., Ultrasonics, 24 (1986) 142-147.

82.

Apfel, R E, ‘Acoustic Cavitation’, Ch. 7 in ‘Ultrasonics’ (Ed: Edmonds, P D), Academic Press, NewYork, (1981).

172

