The rheology of a bubbly liquid

Transcription

1 /rspa The rheology of a bubbly liquid By E. W. Llewellin 1, H. M. Mader 1 and S. D. R. Wilson 2 1 Department of Earth Sciences, University of Bristol, Wills Memorial Building, Queens Road, Bristol BS8 1RJ, UK 2 Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK Received 26 April 2001; revised 8 August 2001; accepted 4 October 2001; published online 6 March 2002 A semiempirical constitutive model for the visco-elastic rheology of bubble suspensions with gas volume fractions φ<0.5 and small deformations (Ca 1) is developed. The model has its theoretical foundation in a physical analysis of dilute emulsions. The constitutive equation takes the form of a linear Jeffreys model involving observable material parameters: the viscosity of the continuous phase, gas volume fraction, the relaxation time, bubble size distribution and an empirically determined dimensionless constant. The model is validated against observations of the deformation of suspensions of nitrogen bubbles in a Newtonian liquid (golden syrup) subjected to forced oscillations. The effect of φ and frequency of oscillation f on the elastic and viscous components of the deformation are investigated. At low f, increasing φ leads to an increase in viscosity, whereas, at high f, viscosity decreases as φ increases. This behaviour can be understood in terms of bubble deformation rates and we propose a dimensionless quantity, the dynamic capillary number Cd, as the parameter which controls the behaviour of the system. Previously published constitutive equations and observations of the rheology of bubble suspensions are reviewed. Hitherto apparently contradictory findings can be explained as a result of Cd regime. A method for dealing with polydisperse bubble size distributions is also presented. Keywords: rheology; bubble suspension; time-dependent flow; capillary number; relaxation time; visco-elasticity 1. Introduction The rheology and flow behaviour of suspensions of particles in liquids is of interest because of the enormous range and complexity of the phenomena exhibited and also because of the practical significance of this class of materials. The dispersed phase can be solid particles, immiscible liquid droplets (an emulsion), or gaseous bubbles. Commercially important suspensions include all manner of foodstuffs and cosmetics and industrial suspensions such as paint, ink and ceramics. Naturally occurring suspensions range from biological materials such as milk and blood to geophysical flows of mud and magma. Adding particles to a liquid does not just change the magnitude of the viscosity, it can introduce all known non-newtonian phenomena, including shear and time dependence of the viscosity, and elastic effects such as stress relaxation and normal stress differences. These phenomena arise because of a range of particle liquid 458, c 2002 The Royal Society

2 988 E. W. Llewellin, H. M. Mader and S. D. R. Wilson and particle particle interactions, which are reviewed by Tadros (1985) and Macosko (1994). Previous research on suspensions has focused primarily on the dilute (volume fractions of the dispersed phase φ<0.05) and highly concentrated (φ >0.75) endmembers and on suspensions of solid particles and emulsions (for reviews see Schowalter 1978; Pal & Rhodes 1989; Pinkerton & Stevenson 1992; Stein & Spera 1992; Macosko 1994). Most of the reported research on the rheology of bubble suspensions is prompted by a desire to understand the dynamics of magmatic flows, although the results are clearly more widely applicable. Magmas are Newtonian liquids over a wide range of stresses and strain rates (Webb & Dingwell 1990). The viscosity of pure magmas can be calculated as a function of temperature and composition and ranges over 15 orders of magnitude (from about 10 to Pa s (Murase & McBirney 1973)). Bubbles occur in magmas due to exsolution of dissolved volatiles. The bubbles are a crucial component of magmatic flows because it is the growth and expansion of the dispersed gaseous phase that drives many volcanic eruptions, especially the most violent. Gas volume fractions can range from zero to φ>0.9 and frequently lie in the intermediate range, in common with most suspensions of practical significance. Our aim is to produce a general model for the visco-elastic rheology of bubble suspensions that will allow the shear viscosity of the suspension to be calculated, given a knowledge of other material parameters, bubble content and the deformation rate. We seek a model that is applicable to magmatic systems and so it must cover intermediate gas volume fractions and bubble suspensions in Newtonian liquids. A determination of visco-elastic effects necessarily involves a consideration of timedependent flows. Rheological studies typically adopt one of several distinct approaches. A purely empirical approach involves formulating a constitutive law for a material on the basis of experimental data alone. Such laws can be very useful as they are couched in terms of observable parameters of real systems, nevertheless, the form of the equation generally has no physical basis. An alternative approach is to produce a physical theory in which explicit account is taken of the shape and mechanical properties of each phase to determine flow properties of the bulk. However, problems frequently arise in such theories because the simplifications necessary to render the mathematics tractable often result in a highly abstracted system that bears little resemblance to any real material. We therefore favour a semiempirical approach, which has a physical basis but involves experimentally determined constants. In the following section, we review the various constitutive equations that have been proposed to describe the rheology of bubble suspensions. The constitutive equation for a dilute emulsion of Frankel & Acrivos (1970) is recast in terms of observable parameters in the form of a linear Jeffreys model. This model is compared with observations of the visco-elastic deformation of bubble suspensions in a Newtonian liquid with 0 φ 0.5 subjected to forced oscillations. The experimental data provide an empirical determination of just one of the parameters in the model. Using this parameter, the model describes our data closely. The visco-elastic behaviour of the bubble suspensions depends strongly on frequency of oscillation; viscosity can increase or decrease with increasing φ depending on the conditions of strain. The apparent contradictions in previously reported research are revisited and discussed in the light of these findings.

3 The rheology of a bubbly liquid Constitutive equations for bubbly suspensions The general differential equation for linear visco-elasticity relates stress τ and strain γ as follows, ( 1+α 1 t +α 2 2 t 2 + +α n ) ( n t n τ = β 0 +β 1 t +β 2 2 t 2 + +β m ) m t m γ, (2.1) where n = m or n = m 1 (Barnes et al. 1989). The time derivatives are ordinary partial derivatives restricting the applicability of the equation to cases where there are only small changes in the variables (as in the present study). A constitutive equation relates the constants α 1, α 2, β 0, β 1, etc., to the physical properties of the two-phase material. For constant stress and steady shear (steady-state flow conditions), only terms up to n =0,m = 1 are necessary. For a Newtonian fluid, β 0 = 0, which leads to the familiar result τ = η γ, where η β 1. Many different constitutive equations exist which relate β 1 to the physical properties of suspensions experiencing steady-state flow and these are reviewed below. It is conventional to normalize β 1 to the fully dense viscosity η 0 and present the relationship in terms of the ratio β 1 /η 0 = η r, the relative viscosity. Steady-state conditions necessarily imply that the bubble deformation is constant; i.e. the bubbles are no longer actively changing shape and have reached what will be called their equilibrium deformation in this paper. The equilibrium deformation of bubbles subjected to steady shear is a function of the capillary number Ca, which is the ratio of viscous (deforming) to surface tension (restoring) forces, Ca = η 0r γ Γ, (2.2) where r is the radius of the undeformed bubble, η 0 is the viscosity of the fully dense liquid and Γ is the liquid bubble interfacial tension. If Ca is large, the restoring force is small compared with the deforming force and the bubble is readily deformed under shear. If Ca is small, surface tension forces dominate and the bubble attains its equilibrium deformation rapidly after changes in strain rate. In general, the Taylor deformation D = kca n, where D is a dimensionless measure of the elongation of the bubble and k and n are dimensionless parameters (Taylor 1934; Loewenberg & Hinch 1996). The deformation can also be written as a dimensionless strain rate, with the strain rate normalized by a deformation time-scale λ called the relaxation time. For small Ca, n = 1 (Taylor 1934; Loewenberg & Hinch 1996) and so D = kca = λ γ, (2.3) in which case λ = k η 0r Γ. (2.4) The parameter k increases with the volume fraction of the disperse phase (Oldroyd 1953; Oosterbroek & Mellema 1981; Loewenberg & Hinch 1996). In the limit of a solitary bubble in an infinite, pure liquid, k 1 (Taylor 1934). In steady flow, constant shear conditions have existed for some time t λ. Theory and observations that are directly applicable to bubbly suspensions are scarce. The flow of emulsions has been more thoroughly investigated. As a result,

4 990 E. W. Llewellin, H. M. Mader and S. D. R. Wilson models of bubbly flows frequently use constitutive laws derived for emulsions on the grounds that bubbles and liquid droplets are both deformable inclusions. It is instructive at this point to review the various models that have been proposed and to consider their validity for bubble suspensions. (a) Constitutive equations for steady-state conditions Einstein (1906, 1911) shows that, for a dilute suspension of solid particles, η r =1+ 5 2φ. (2.5) Taylor (1932) extends this theory to cover dilute emulsions. For the case of an inviscid dispersed phase: η r =1+φ. (2.6) The increase in viscosity with increasing φ predicted arises because flow lines are distorted around the bubbles. The relationship assumes bubbles are approximately spherical (i.e. Ca 1). As Ca increases, bubbles become increasingly elongate and the relationship is no longer valid. The majority of workers find that equation (2.6) describes their experimental data poorly. Various alternative formulations are proposed. Stein & Spera (1992) propose a modified form of Taylor s equation, η r =1+aφ, (2.7) where the value of a must be experimentally determined. They found a =13.1 based on rotating rod viscometry of suspensions of air bubbles with φ in molten glass with η Pa s and 0.5 <Ca<100. They suggest that the large difference between the value of a for their experiments and the value predicted by Taylor s theory for emulsions (unity) is due to the violation of the deformation criterion (Ca 1) in their experiments. Barthes-Biesel & Chhim (1981) attempt to take account of higher values of Ca with their equation, η r =1+(2.5 a(ca) 2 )φ, (2.8) where a 70. This equation is based on a mathematical treatment of the behaviour of liquid droplets with an elastic bounding membrane suspended in a viscous liquid (i.e. it is probably most applicable to particles such as blood cells). The constant a is a calculable parameter of the membrane material. Several authors propose nonlinear relationships between η r and φ. Roscoe (1952) presents a theoretical treatment of suspensions of solid spheres which has been applied to micro-emulsions, η r =(1 aφ) 2.5, (2.9) where a = 1 for a polydisperse suspension and a =1.35 for a monodisperse suspension. Eilers (1941, 1943) proposes a relationship based on experiments on bitumen emulsions, ( η r = φ ) 2, (2.10) 1 aφ where 1.28 <a<1.30 and is empirically determined.

5 The rheology of a bubbly liquid 991 Mooney (1951) proposes an exponential relationship for monodisperse suspensions of solid spheres, ln η r = 2.5φ 1 aφ, (2.11) where geometrical arguments give 1.35 <a<1.91. Sibree (1934) presents high Ca measurements of the viscosity of froths of a fungicide (0.52 <φ<0.73), which is stabilized by an organic colloid. He compares his results with the theoretical model for foam (φ >0.74) rheology of Hatschek (1911), 1 η r =, (2.12) 1 (aφ) 1/3 and obtains a good fit for a =1.2. The liquid phase in his experiments is strongly non-newtonian (shear-thinning) due to the presence of the organic colloid. (b) Investigations of porous glasses and field observations of magmas All of the models presented so far predict an increase in viscosity with increasing φ and the experimental data (most of which derives from observations of emulsions) support this. By contrast, experiments relating to the behaviour of porous glasses (defined by Bagdassarov & Dingwell (1992) as having η 0 > 10 9 Pa s) indicate a decrease in viscosity with increasing φ (Rahaman et al. 1987; Ducamp & Raj 1989; Sura & Panda 1990; Bagdassarov & Dingwell 1992, 1993; Lejeune et al. 1999). Experimental and theoretical studies predicting a decrease in viscosity with increasing φ are presented below. Mackenzie (1950) describes a theoretical treatment of the elastic constants of a solid containing a small volume fraction of identical spherical holes. He appeals to an analogy between viscous and elastic moduli to derive the following viscosity/gas volume fraction relationship for a dilute, monodisperse bubble suspension: η r =1 5 3φ. (2.13) It is assumed that the viscous deformation of the bubbles is unopposed by surface tension forces, i.e. λ. Mackenzie claims validity for slow shear for as long as the bubbles remain approximately spherical. Scherer (1979) uses an analogy between thermal and viscous stresses to derive a rheological model for highly viscous porous glasses. He assumes both phases are continuous and that the glass is arranged as a cubic framework of cylinders. He finds 3(1 φ) η r =. (2.14) 2(1+2φ)+(1+φ 2φ 2 ) 1/2 Ducamp & Raj (1989) carried out dilatometric measurements on glasses (η Pa s) with gas volume fractions in the range 0 φ They propose a relationship of the form, ( ) φ ln η r = a. (2.15) 1 φ This equation has no theoretical basis but is constructed to satisfy three criteria: (i) the relationship must be nonlinear, (ii) it must reduce to the form of Mackenzie s

6 992 E. W. Llewellin, H. M. Mader and S. D. R. Wilson equation at low φ, (iii) η r = 1 when φ = 0. Ducamp & Raj obtain a good fit to their data when 2.5 a 4. Rahaman et al. (1987) performed similar experiments to Ducamp & Raj on glasses bearing highly non-spherical pores and with 0.14 φ They present a purely empirical relationship based on their data, ln η r = aφ, (2.16) where they obtain a =11.2. Bagdassarov & Dingwell (1992) also carried out dilatometric measurements on porous glasses (η Pa s) over the range 0 φ 0.7. They propose an empirical relationship that satisfies two criteria: (i) the relationship reduces to the form of Mackenzie s equation at low φ, (ii) it reduces to η r = 1/φ at high φ (based on experimental results of Sura & Panda 1990), 1 η r = 1+aφ, (2.17) where they obtain a =22.4. Bagdassarov & Dingwell (1993) present, as far as we know, the only published study of the time-dependent rheology of a bubbly material, performing oscillatory rheometry on highly viscous magma (10 9 η Pa s) with gas volume fraction 0 <φ<0.3. They state that the viscosity of the material decreases with increasing gas volume fraction but do not present a constitutive rheological equation. Comparison of theory and behaviour of bubbly liquids and porous glasses is complicated for several reasons. The very high viscosity of glasses leads to very long relaxation times and so steady-state conditions are unlikely to be attained. Moreover, the high viscosity of the glasses prevents shear viscosity from being measured directly by, for example, rotating rod viscometry. Instead, samples are generally subjected to uniaxial compression. Inferring shear viscosity from compressional measurements requires a knowledge of the bulk viscosity of the material, which has not been measured. The usual approach adopted is to appeal to an analogy with elastic stress analysis (Mackenzie 1950; Scherer 1979; Sura & Panda 1990; Bagdassarov & Dingwell 1992); however, it has been shown by Wilson (1997) that this analogy is invalid. This is because in a compressible fluid the thermodynamic pressure enters the equations in an essential way via an equation of state. In the usual theory of elastic solids, on the other hand, all the states considered are thermodynamic equilibrium states. Reported observations of magmatic flows in the field are reviewed by Manga et al. (1998) and are also contradictory, with some researchers inferring that vesicular magmas have apparently higher viscosities and some lower viscosities than equivalent gas-free flows. Evidently, there are contradictions present in the literature and confusion as to even the direction of the effect of adding bubbles to a liquid. These issues may be partly explained by recent numerical work by Manga & Loewenberg (2001) and experimental work on corn syrup bubble suspensions by Rust and Manga (M. Manga 2001, personal communication), which suggests that bubble suspensions are shear thinning and that viscosity can increase with gas volume fraction when Ca<O(1) and decrease with gas volume fraction at high when Ca>O(1). We revisit this issue later in the light of our data and show that the different results can be explained for

7 The rheology of a bubbly liquid 993 constant, low Ca if the unsteady nature of the measurements on ultra-high viscosity suspensions (i.e. the porous glasses) is taken into account. (c) Time-dependent flow In unsteady flows visco-elastic effects become important and more terms from equation (2.1) are required to describe the behaviour. Constitutive equations that include the visco-elasticity of a suspension allow systems in which shear stress and strain rate change with time to be modelled. One such equation, commonly used to describe the visco-elastic behaviour of polymers, is the Jeffreys model, a specific case of equation (2.1), where n =1,m = 2 and β 0 = 0, giving τ + α 1 τ = β 1 γ + β 2 γ. (2.18) Frankel & Acrivos (1970) obtain a constitutive equation for a dilute, monodisperse emulsion in which the droplets remain approximately spherical. Their model provides one of the most comprehensive and rigorous physical analyses presented to date for such a flow. For the case of inviscid inclusions, such as bubbles, and for small amplitude oscillatory shear such that Ca 1 as in the present experimental programme, their model can be recast in terms of observable parameters in the form of a Jeffreys model (see Appendix A) τ ij λ τ 6 ij = η 0 (1 + φ) γ ij + η 0 5 λ(1 5 3 φ) γ ij. (2.19) This model forms the basis for the analysis of our measurements of visco-elasticity in bubble suspensions. (d) Emulsions versus bubble suspensions It is reasonable to expect that models derived for emulsions will be relevant to bubble suspensions to some extent because both liquid droplets and bubbles are deformable. However, bubble suspensions differ from emulsions in a number of important respects. (i) Liquid droplets are viscous whereas bubbles are inviscid and so internal flow and viscous dissipation within bubbles will be negligible. (ii) Bubble suspensions are compressible whereas emulsions are incompressible. (iii) Bubbles have much higher internal energies than liquid droplets. Surface tension is important to aggregative and breaking stability in bubble suspensions and emulsions and has been discussed by Babak (1994) and Princen & Kiss (1986, 1989) with reference to highly concentrated emulsions. Surface tension enters nonlinearly into the equation for shear viscosity presented by Princen & Kiss (1989). Moreover, the lower surface tensions of liquid droplets mean that they will deform more easily than bubbles and so bubbly liquids might be expected to behave more like suspensions of solid particles than emulsions for a given particle size. However, it is suggested that bubbles have a stronger influence on rheological properties than similar concentrations of crystals (Stein & Spera 1992; Pinkerton & Norton 1995).

8 994 E. W. Llewellin, H. M. Mader and S. D. R. Wilson ROTOR COOLING JACKET GAS INLET SYRUP INLET STATOR OUTLET Figure 1. The aerator. Cutaway of the mixing head of the Mondomix aerator showing the interlocking teeth of the stator and the rotor. Nitrogen is injected at about 4 bar into the mixing head. The rotor turns at about 1000 RPM. The cooling jacket is maintained at 10 C. The diameter of the mixing head is 15 cm. (iv) The density contrast between the dispersed phase and the liquid phase is much higher in bubble suspensions than in emulsions. Indeed, whereas it is possible to generate neutrally buoyant emulsions, bubbles are always inherently buoyant with respect to the liquid phase. This also affects the aggregative properties. For these reasons, it is unlikely that any model of emulsions can be applied without amendment to bubble suspensions. We therefore use experimental observations of the viscoelastic rheology of bubble suspensions to investigate how well the model proposed in equation (2.19) describes the data and what adjustments are necessary. 3. The bubble suspensions Suspensions of nitrogen bubbles in golden syrup are created using a commercial aerator (figure 1). Golden syrup was chosen because it has a Newtonian rheology, is water soluble and non-hazardous. Physical properties of golden syrup are given in figure 2. The suspension produced by the aerator typically has a gas volume fraction

9 The rheology of a bubbly liquid (a) viscosity, η (Pa s) (b) 1440 density (kg m 3 ) temperature (ºC) Figure 2. Properties of golden syrup. (a) Viscosity of golden syrup as a function of temperature. The viscosity measurements were made using a Haake RV20 controlled-rate, rotational viscometer with a concentric-cylinder sensor. (b) Density of golden syrup as a function of temperature. The density was determined by weighing a known volume of syrup at a range of temperatures. in the range 0.3 <φ<0.4. This is poured into a tall box (10 cm 10 cm 100 cm) and left standing for about one week at room temperature until bubble rise produces a stratified column with φ increasing from φ = 0 at the base to φ>0.6 at the top. The gas volume fraction of samples taken from different heights in the box is measured to an accuracy of 2% by weighing a known volume. Samples are then transferred to the sensor system of the rheometer, described in 4 b. Some alteration of the gas volume fraction of the samples may occur during transfer due to trapped or lost air. The total uncertainty on φ in the test material is estimated to be less than 5%. To measure the bubble size distribution, samples are pressed between two glass plates 0.5 mm apart to separate the bubbles. A digital image is taken through a microscope at 4 magnification. A typical image contains around 1000 bubbles with radii in the range mm. Figure 3 shows that the distributions of bubble number, radius, surface area and volume are distinctly different.

10 996 E. W. Llewellin, H. M. Mader and S. D. R. Wilson (a) bubble number fraction bubble surface area fraction (b) (d) (e) 0.8 bubble size, Φ bubble radius fraction bubble volume fraction (c) bubble size, Φ Figure 3. Bubble size distributions. (a) A typical image of a sample with φ =0.140 pressed between plates 0.5 mm apart. The radii of the bubbles are measured and sorted into bins according to the standard Φ scale, where Φ = log 2 (diameter in mm). Image is 3 mm across. Four different bubble size distributions are calculated: (b) number, (c) radius, (d) surface area, (e) volume. In each case, the distributions are given in terms of the fraction of the total which is present in each Φ class.

11 The rheology of a bubbly liquid 997 viscous component elastic component total stress strain γ or γ stress τ. δ ω t ω t strain rate strain Figure 4. Idealized waveforms for forced oscillations of a viscoelastic medium. The imposed stress is given by τ = τ 0 cos(ωt + δ) =τ 0 cos δ cos ωt τ 0 sin δ sin ωt, which can be viewed as a superposition of two components: the cos ωt term is in-phase with the strain, γ = γ 0 cos ωt, and so describes the elastic component; the sin ωt term is in-phase with the strain rate, γ = γ 0 sin ωt, and so gives the viscous component. Viscoelastic materials therefore have phase-shifts in the range 0 <δ<π/2. 4. Experimental method (a) The method of forced oscillations A standard technique used to investigate the visco-elastic properties of liquids is the method of forced oscillations in which a sinusoidally varying stress τ is imposed on the sample and the resultant deformation is observed. Within the linear viscoelastic region the structure of the material is stable, the observed deformation is similarly sinusoidal and measurements are repeatable. The total stress consists of two components: an elastic component in-phase with the strain γ and a viscous component in-phase with the strain rate γ (and so π/2 out-of-phase with the strain and the elastic deformation). The superposition of these components generates a sinusoidal signal that is phase-shifted with respect to the applied stress by some amount δ. Idealized wave forms are shown in figure 4 and can be described by the following system of equations, τ = τ 0 cos(ωt + δ) = Re(τ 0 e iδ e iωt ), (4.1) γ = γ 0 cos(ωt) = Re(γ 0 e iωt ), (4.2) γ = γ 0 sin(ωt) = Re(i γ 0 e iωt ), (4.3)

12 998 E. W. Llewellin, H. M. Mader and S. D. R. Wilson where ω is the angular frequency of oscillation and γ 0 = ωγ 0. It is convenient to define a complex viscosity η in terms of the moduli of amplitudes of the oscillations such that η = τ 0e iδ = η iη, (4.4) i γ 0 hence η = τ 0 sin δ (4.5) γ 0 and η = τ 0 cos δ. (4.6) γ 0 η is called the dynamic viscosity and describes the viscous deformation which is inphase with the strain rate. Conversely, η is called the loss viscosity and describes the elastic deformation which is in-phase with the strain. The phase shift δ is given by tan δ = η. (4.7) η It is immediately apparent that the phase shift is a measure of the relative contributions of elasticity and viscosity to the deformation. A purely viscous material will have no loss modulus and δ = π/2, whereas a purely elastic material will have no dynamic modulus and δ = 0. In general, for a visco-elastic material 0 <δ<π/2. The linearized Frankel & Acrivos (1970) model, as expressed in equation (2.19), has real and imaginary parts of the complex viscosity of the sample, η and η, respectively, given by (see Appendix A) η = β 1 + α 1 β 2 ω 2 1+α1 2, (4.8) ω2 η = (β 1α 1 β 2 )ω 1+α1 2, (4.9) ω2 where ω is the angular frequency of oscillation and the coefficients are related to measurable physical properties of the suspension as follows: α 1 = 6 5λ, (4.10) β 1 = η 0 (1 + φ), (4.11) β 2 = η 0 α 1 (1 5 3φ). (4.12) Note that α 1 and β 2 only enter into unsteady flow problems (see equation (2.18)) and that (4.11) is the Taylor equation for steady flow conditions (see equation (2.6)). (b) Experimental constraints and rheological tests The response of the bubble suspensions to forced oscillations is observed using a Haake RS100 rheometer fitted with a specially designed wide-gap sensor system (figure 5). All rheological tests are performed at 25.0 ± 0.1 C. Viscous heating is

13 The rheology of a bubbly liquid mm Figure 5. The wide-gap, parallel-plate sensor system. The sensor system has a circular cross-section. The lower plate is provided by a shallow dish which contains the sample. The gap is 9 mm, which is more than 30 the maximum bubble radius. The sensor is enclosed in a water jacket that holds the whole system at 25.0 ± 0.1 C (not shown). The parallel plate geometry was chosen as it is non-intrusive and so its use does not modify the bubble suspension in any way. negligible at the strain rates attained during the experiments. The RS100 applies a torque M to the top plate of the sensor and measures the resultant angular displacement Ω. For all of our experiments M = N m and Ω varies in the range Ω 0.2 rad depending on frequency of oscillation and viscosity of the sample. Torque and angular displacement are converted to stress and strain using two calibration factors, A and D, such that τ = AM and γ = DΩ. A and D are determined for the wide-gap sensor system against calibration oils with 5 <η<100 Pa s to an accuracy of 3.1%. The spatial variation of stress and strain rate within the sensor system cannot be calculated without a detailed a priori knowledge of the rheology of the fluid being tested. The calibration procedure means that the ratio of stress to strain rate gives the apparent viscosity of the sample; i.e. a Newtonian fluid of this viscosity would strain at the same rate as the sample for the imposed stress. The maximum angular displacement of the sensor which we allow is π/10 rad, ensuring that all deformations are small compared with the sensor gap. We estimate the maximum strain rate attained to be of order 0.01 s 1 for all experiments at all frequencies. The calculated relaxation times λ (equation (2.4) with k = 1) are of order 0.1 s for the largest bubbles giving a maximum bubble capillary number Ca of order This means that the bubbles remain essentially spherical, in keeping with the theoretical assumptions made earlier ( 2 c and 4 a). However, Ca can only strictly be calculated for steady flow conditions and so has little physical meaning for experiments where oscillations are occurring around a point of zero strain, except where the period of oscillation is much longer than the relaxation time. We will return to this point later ( 6 b). It is essential for the structure of the sample to remain as constant as possible for the duration of a test. The most significant problem in this regard is that of bubble 8 cm

14 1000 E. W. Llewellin, H. M. Mader and S. D. R. Wilson rise. An estimate for the rate at which bubbles rise u can be calculated from the Hadamard Rybczynski equation (Hadamard 1911; Rybczynski 1911) u = r 2 ρg/3η 0, where r is the bubble radius, ρ is the density of the suspending medium and g is the acceleration due to gravity. This equation is a generalization of Stokes s equation to arbitrary viscosity ratios. It applies for bubble rise in clear liquids. Bubble rise is likely to be impeded by the presence of other bubbles in a suspension and so the equation tends to overestimate the rise velocity. To minimize bubble rise we reject samples containing bubbles larger than Φ = 1(r = 0.25 mm). This means that even the largest bubbles take more than 20 min to rise through the sample. Since the maximum test duration is about 30 min, only the largest bubbles rise significantly during an experimental run. Samples are first subjected to a stress sweep in which the stress amplitude is varied at constant frequency. The range of stresses over which the complex viscosity η is constant defines the region of linear visco-elastic response. This range is typically 0.1 <τ 0 < 200 Pa for our samples. The visco-elasticity of the suspensions is then determined by conducting a frequency sweep. In this test, the stress amplitude τ 0 is fixed well within the linear visco-elastic region (as determined from the stress sweep) and the strain response is observed as a function of the frequency of oscillation f. The range of frequencies investigated is limited at the low end to f>0.01 Hz by the need to keep the total time taken for the experiment to under 30 min and at the high end to f<10 Hz by the sensitivity of the instrument to the very small strains produced at high frequencies. Bubble rise is minimized by restricting the bubble size and the duration of the sweeps but cannot be completely eliminated. The effect is also progressive during the course of a frequency sweep. To even out the residual effect of bubble rise, each frequency sweep is performed from low f to high f and then back down to low f. The data presented here are the averaged data from the up and down sweeps. The complex viscosity η of pure syrup (φ = 0) is expected to be independent of f. Thus the scatter in the frequency sweep data for pure syrup gives a measure of the random error on η. The 2σ error is 2%. 5. The rheological data (a) Monodisperse model The linearized Frankel & Acrivos model (equation (2.19)) describes the behaviour of a bubbly liquid in terms of three material properties: the gas volume fraction φ, the viscosity of the fully dense liquid η 0 and the relaxation time λ. φ is measured directly for each sample and is known to within 5%. η 0 for golden syrup varies over a small range from batch to batch. It is also likely to change slightly during the aeration process. It is not, therefore, possible to know η 0 for each sample to a high degree of accuracy and so we assume the true value lies somewhere in the range 40 η 0 70 Pa s for all samples. The relaxation time λ involves η 0, the liquid bubble interfacial tension Γ, the bubble radius r and the parameter k (see equation (2.4)). Strictly, the analysis of Frankel & Acrivos assumes k = 1 (case for dilute limit), however, since we are investigating non-dilute suspensions, we allow k to vary. For the surface tension of golden syrup we use Γ =0.08Nm 1. This is the value measured by P. Heath (Carl Stuart Ltd 2001, personal communication)

15 The rheology of a bubbly liquid η (Pa s) frequency (Hz) 1 10 Figure 6. Monodisperse model curve fits. Best fits to data for φ =0.140 using a monodisperse model as described in 5 a. Solid line assumes β 1 = η 0(1 + φ), dashed line assumes β 1 = bη 0, where b is a free parameter that can take any positive value. For the dashed line, η 0 =33.8 Pas and b =2.71. for golden syrup at 21 C. The value is in broad agreement with the relationship given in Sinat-Radchenko (1982) and with values measured for the surface tension of other concentrated sugar solutions, such as corn syrup (0.07 N m 1 (Manga & Stone 1995); 0.08 N m 1 (Borhan & Pallinti 1999)). There is no evidence to suggest that the surface tension of such solutions is a strong function of water content or temperature and we expect the true value to be within 10% of Γ =0.08Nm 1. Defining a single bubble radius is problematic as all of the samples are significantly polydisperse, with bubble radii varying over four orders of magnitude. We therefore allow the bubble radius to take any value within the range of radii present in the sample. The statistical analysis and model fitting package, Simfit (see Bardsley (1993) for a description), is used to determine the values of η 0, r and k that provide the best fit of the model (equation (2.19)) to each sample of known φ. η 0 and r are constrained within the ranges specified above and k can take any value. An example of a curve fit produced using this method is shown in figure 6. Although the shape of the model curve reflects the trend in the data, the fit is extremely poor. It was pointed out earlier that β 1 as given in equation (4.11) is the same as Taylor s expression, equation (2.6). Both Taylor s expression and the Frankel & Acrivos theory were derived for emulsions and so we might reasonably expect that this expression may need adjustment if it is to be applied to bubble suspensions. To this end, we start by generalizing the functional form of equation (4.11) to β 1 = bη 0, (5.1) where b is now a free parameter. We will use our data to determine the functional form that describes how b depends on gas volume fraction φ. Figure 6 also shows a curve fit using the same procedure described above, but assuming the functional form given in equation (5.1). The fit is much improved but still shows marked deviation from the data. This is primarily a consequence of the fact that our highly polydisperse

16 1002 E. W. Llewellin, H. M. Mader and S. D. R. Wilson (a) (b) η (Pa s) (c) (d ) η (Pa s) frequency (Hz) frequency (Hz) Figure 7. Polydisperse model curve fits. Best fits to data for φ =0.140 using a polydisperse model (as described in 5 b) based on the four bubble size distributions shown in figure 3. For (a) bubble number distribution fit η 0 =48.9 Pasandb =2.63, for (b) bubble radius distribution fit η 0 =44.8 Pasandb =2.91, for (c) bubble area distribution fit η 0 =51.2 Pasandb =2.45, for (d) bubble volume distribution fit η 0 =54.2 Pasandb =2.28. samples cannot be accurately modelled by a single bubble size and a single relaxation time. (b) Polydisperse model The monodisperse model described in the previous section can be accurately applied to each Φ size class (where Φ = log 2 (diameter in mm) as described in the caption to figure 3) to give a viscosity for that fraction of the sample. The gas volume fraction for each Φ class is taken to be the same as that of the bulk (which effectively means dividing up the liquid among the Φ classes according to the bubble volume distribution). The viscosity of the bulk is then taken to be the weighted sum of the individual monodisperse viscosities for the different Φ classes, where the weightings are provided by the fractional distribution of the Φ classes. This process leads to four potential polydisperse models, depending on whether the bubble number, radius, area or volume distribution is used to provide the weightings. The curve-fitting process involves finding the optimal values of b, η 0 and k that are the same for all Φ classes within a given distribution and lie within the ranges specified in 5 a.

17 The rheology of a bubbly liquid 1003 Table 1. Values of WSSQ for fits of the monodisperse model and the polydisperse model based on the four bubble size distributions (number, radius, area and volume) for all data (Low WSSQ indicates a better fit.) WSSQ { }} { φ number radius area volume monodisperse Figure 7 shows the best-fit curves produced by Simfit using the polydisperse model for each of the four distributions (based on bubble number, radius, area and volume) for a specific sample. The fits are all significantly better than that obtained by applying the monodisperse model as in figure 6. Values of the weighted sum of squares (WSSQ) of the curve fits for all datasets are given in table 1. A lower WSSQ value indicates a better fit. For all but two of the samples tested, the best fit was obtained for the model based on the bubble radius distribution, suggesting that this is the property that controls the response of a bubble to strain. This is to be expected, since the relaxation time λ (equation (2.4)) contains a first-order radius term. Figure 8 shows the complete dataset with fits obtained for each sample using the polydisperse model based on the bubble radius distribution. The values of η 0, k and b found for each sample are given in table 2. Note that the general trend in k increases with increasing φ, as expected (Oldroyd 1953; Oosterbroek & Mellema 1981; Loewenberg & Hinch 1996). However, k is significantly greater than the values obtained in the numerical calculations of Loewenberg & Hinch (1996) for emulsions in which the disperse and continuous phases have the same viscosity. 6. Semiempirical model for the rheology of a bubbly liquid We can use the data given in table 2 to determine how the free parameter b of equation (5.1) depends on the gas volume fraction φ. Figure 9 shows that the relationship is of the form b =1+aφ, where 7.5 a 9.5 with a best-fit curve of a =9.0. Thus, b is a modified form of the Taylor expression. The scatter in the data is thought to be primarily a result of residual bubble rise. The parallel plate geometry is particularly sensitive to bubble rise effects because the rising bubbles accumulate beneath the upper plate on which the torque is measured. The model curve fits to the data in figure 8 are sensitive to the exact value of the parameter a. Figure 10 illustrates the effect of varying the parameter across the uncertainty band shown in figure 9, i.e. from a =7.5 toa =9.5, for a particular sample.

18 1004 E. W. Llewellin, H. M. Mader and S. D. R. Wilson (a) (b) η (Pa s) η (Pa s) η (Pa s) (c) (d) (e) ( f ) η (Pa s) η (Pa s) (g) (h) (i) ( j ) frequency (Hz) frequency (Hz) Figure 8. Complex viscosity data. Solid lines show the best fits using the polydisperse model (as described in 5 b) based on the bubble radius distribution. Note the large variation in vertical scale between datasets. Values of φ are (a) 0.000, (b) 0.036, (c) 0.057, (d) 0.063, (e) 0.085, (f) 0.103, (g) 0.140, (h) 0.235, (i) 0.302, (j)

19 The rheology of a bubbly liquid 1005 Table 2. Parameter values of best fit of polydisperse model based on bubble radius distribution φ η 0 (Pa s) k b b gas volume fraction, φ Figure 9. Values of b from polydisperse fit based on bubble radius distribution against φ. Best fit (solid line) has equation b = φ. Upper and lower dashed lines (representing limits of acceptable fit) have equations b = φ and b = φ, respectively. For a steady flow, consideration of equations (2.18), (4.11) and (5.1) and the discussion at the start of 2 allows us to conclude that b = η r. We can therefore easily compare the functional form of b as determined from our data with the forms proposed by other researchers as reviewed in 3 a, and this is done in figure 11 and the top half of table 3. Our data are best described by the relationships proposed by Sibree (1934) and Stein & Spera (1992), who are the only other researchers, to the best of our knowledge, to propose constitutive equations specifically for bubble suspensions that are underpinned by observation. However, the Sibree data consist of five measurements of bubble suspensions in a strongly shear-thinning liquid with high gas volume fractions (0.5 <φ<0.73). Sibree s data points lie outside the graph shown in figure 11, in the region where our constitutive law would diverge strongly from his. In addi-

20 1006 E. W. Llewellin, H. M. Mader and S. D. R. Wilson 200 η (Pa s) frequency (Hz) 1 10 Figure 10. Data and polydisperse model based on bubble radius distribution for φ = Solid line shows model prediction assuming β 1 = η 0(1+9.0φ). Upper and lower dashed lines show limits of uncertainty taken from figure 9 (assuming β 1 = η 0(1+9.5φ) and β 1 = η 0( φ), respectively). tion, Sibree s equation is derived from a model specifically for highly concentrated suspensions (Hatschek 1911) and so it is likely that the agreement at low φ (well outside his data range) is entirely coincidental. The other data of interest are the three values measured in the low gas volume fraction range φ by Stein & Spera (1992). These points are shown superimposed on our data in figure 11 and are well-described by the equation η r =1+aφ using our best-fit value of a =9.0. We can now propose the following constitutive rheological equation for a bubbly liquid: τ ij λ τ ij = η 0 (1+9φ) γ ij + η λ(1 5 3 φ) γ ij. (6.1) This is a semiempirical model. Its theoretical basis is the linear Jeffreys model, parametrized according to the physical analysis of Frankel & Acrivos, with the dependence of the parameter β 1 on the gas volume fraction φ determined empirically. (a) Time-dependent rheology Time-dependent effects emerge in our experimental data due to the increasingly unsteady nature of the flow as frequency of oscillation increases. These effects manifest themselves as a frequency dependence of absolute and relative magnitudes of the viscous and elastic components of deformation. The relative importance of the viscous and elastic components is encapsulated in the phase shift δ (equation (4.7)) and is dependent on both φ and f. The random error on tan δ is conservatively estimated to be 10% from the scatter in η and η. Figure 12 shows δ data for a particular sample and compares the monodisperse and polydisperse model curve fits. The polydisperse model based on the bubble radius distribution provides the best fit to the data. The pronounced minimum in δ indicates that there is a maximum in the elastic response of the suspensions at about 1 Hz. The position of this minimum is related to the bubble size, hence the broadening and

21 relative viscosity, η r relative viscosity, η r relative viscosity, η r The rheology of a bubbly liquid 1007 (a) (b) (c) (d) (e) ( f ) gas volume fraction, φ gas volume fraction, φ Figure 11. Equations for η r(φ) for steady state flow presented by previous workers plotted against our experimental data. Solid lines represent limited fits, where parameters are restricted to limits or values imposed by the originator of the model, dashed lines represent free fits where parameters are allowed to assume any positive value. Where only one line is shown, limited and free fits are identical. The data of Stein & Spera (1992) are plotted as hollow squares with error bars. (a) Barthes-Biesel & Chhim (1981), (b) Eilers (1943), (c) Mooney (1951), (d) Roscoe (1952), (e) Sibree (1934), (f) Stein & Spera (1992). shallowing of the trough when a polydisperse model is used. The elastic maximum occurs at the frequency where ω (= 2πf) is approximately equal to the inverse of the relaxation time (λ 1 ). The φ dependence of δ is straightforward. The composite plot of all δ data in figure 13 shows that as φ increases, the suspension becomes more elastic. When φ = 0 the suspension is purely viscous (δ 90 ) and when φ =0.461 its character is more elastic than viscous (δ <45 ) at its elastic maximum. Figure 14 shows the dependence of viscosity η on φ and f. Asφ increases, η becomes increasingly dependent on f: when φ =0, η is essentially constant at η = η 0 for all f, but when φ =0.461, η decreases by a factor of 17 as f increases from 0.01 to 10 Hz. The most striking feature of the data is that at low f, η increases with increasing

22 1008 E. W. Llewellin, H. M. Mader and S. D. R. Wilson Table 3. WSSQ for fits of various models to our data for η r against φ (The column headings free and limited refer to whether the parameters in the model were allowed to vary freely or within the range set by the models originators. The top half of the table refers to models where η r increases with increasing φ, and the bottom half to models where η r decreases with increasing φ.) WSSQ { }} { model free limited Barthes-Biesel & Chhim (1981) Eilers (1943) Mooney (1951) Roscoe (1952) Sibree (1934) Stein & Spera (1992) Bagdassarov & Dingwell (1992) Ducamp & Raj (1989) Mackenzie (1950) Rahaman et al. (1987) Scherer (1979) phase shift, δ (deg) frequency (Hz) 1 10 Figure 12. δ as a function of frequency of oscillation for φ = Solid line is monodisperse model fit. Dashed line is polydisperse model fit based on bubble radius distribution. φ, whereas at high f, η decreases with increasing φ. Thus total resistance to deformation can either increase or decrease with increasing φ, depending on the frequency of oscillation. This reversal in the effect of φ on viscosity immediately brings to mind the apparently contradictory results from different groups of models discussed in 2 and from different experimental studies (see, for example, Stein & Spera 1992;

23 The rheology of a bubbly liquid phase shift, δ (deg) frequency (Hz) 1 10 Figure 13. Composite plot of δ against frequency of oscillation for all gas volume fractions. The lines are not curve fits but simple interpolations between neighbouring datapoints. The random error on each datapoint is estimated to be ±10%. δ decreases steadily as φ increases η (Pa s) frequency (Hz) Figure 14. Composite plot of η against frequency of oscillation for all gas volume fractions. Polydisperse model fits to data are plotted. At low frequency, η increases steadily as φ increases, at high frequency, the opposite is true. Bagdassarov & Dingwell 1992) and deserves careful consideration. All experiments are carried out at small Ca and bubbles remain essentially spherical throughout. Therefore, the response cannot be explained on the basis of the steady-state shearthinning rheology calculated by Manga & Loewenberg (2001), which depends upon bubbles undergoing considerable elongation. In the following sections, we show that the behaviour is a viscoelastic response that is independent of Ca. It is a result of