5372
Ind. Eng. Chem. Res. 2004, 43, 5372-5379
Rheological Constitutive Equation for Bubbly Suspensions Rajinder Pal* Department of Chemical Engineering, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
A rheological constitutive model is proposed for concentrated suspensions of bubbles. The proposed model is a modification of the Oldroyd constitutive model for the rheology of dilute emulsions of two immiscible Newtonian fluids. The proposed model predicts relative viscosity, reduced first normal stress difference, and reduced second normal stress difference, of concentrated suspensions of bubbles as functions of capillary number and bubble volume fraction. The model is evaluated using the available experimental data on the relative viscosity of bubbly suspensions. Further experimental studies are needed to generate data on normal stress differences in bubbly suspensions and to test the model predictions. 1. Introduction During ascent of magma from a magma chamber to the Earth’s surface, nucleation and growth of gas bubbles occur by exsolution of volatiles (mainly water and carbon dioxide) that are initially dissolved in magma at high pressures. The rheological properties of such bubbly suspensions (magmas) are important in the analysis and modeling of the volcanic processes.1-10 The rheological behavior of pure melts (magmas without bubbles) is comparatively well-known; pure melts generally exhibit Newtonian behavior and the melt viscosity is known to be a function of its chemical composition, volatile and crystal contents, temperature, and pressure.3 By contrast, the effects of bubbles on the rheological properties of magmas are not well-understood. The addition of bubbles to a Newtonian liquid not only affects the magnitude of the viscosity but also leads to viscoelastic phenomena embodied by normal stress differences. Nonzero normal stresses can have a significant influence on the morphology and growth dynamics of lava flows and endogenous domes.2,8 According to Stein and Spera,2 the ratio of the normal stress difference to the shear stress is an important quantity as it governs the deviation of the flow field from Newtonian behavior. As an example, let us consider the flow of molten lava in a deep channel of finite width. If the lava behaves as a Newtonian fluid (zero normal stresses), the flow field is simple in that the only component of motion present is the longitudinal one and the longitudinal velocity varies with the vertical distance from the channel bottom. However, in the presence of normal stresses, the flow field becomes complicated. A fluid with nonzero normal stresses, that is, a non-Newtonian viscoelastic fluid, exhibits a secondary helical flow in the cross-sectional plane of the flow. Consequently, the free surface of the fluid is deformed and the kinematics of the flow is greatly affected. Two-phase bubbly suspensions are encountered in many other applications as well, such as in the processing of polymer melts, foods, and biological materials.11 Bubbly suspensions are also important in the petroleum industry. Previous studies on the rheology of bubbly suspensions10 have focused primarily on the effects of bubbles * Phone: (519) 888-4567, ext. 2985. Fax: (519) 746-4979. E-mail:
[email protected].
on the shear viscosity. Little attention has been given to other rheological properties (such as normal stress differences) of bubbly suspensions. Clearly, there is a need to develop a rheological constitutive equation for concentrated bubbly suspensions that incorporates viscoelastic phenomena such as normal stress differences. 2. Theoretical Background 2.1. Constitutive Equations for Dilute Bubbly Suspensions. Frankel and Acrivos12 derived a constitutive equation for an infinitely dilute emulsion of uniform (same size) droplets of a Newtonian fluid in another such fluid of different viscosity. For bubbly suspensions, where the ratio of dispersed-phase viscosity to continuous-phase viscosity is essentially zero, the Frankel and Acrivos equation becomes10
(
) ( )
ηc2R Dτ Dd τ + Λ ) 2ηc(1 + φ) d + Λ +φ × Dt Dt σ
{(
-32 Dd 24 2 + (d‚d) + (dT‚dT) - tr(d‚d)δ 5 Dt 35 3
)
( )[
]} (1)
where τ is the deviatoric stress tensor, d is the rate of deformation tensor, δ is the unit tensor, dT is the transpose of d, tr refers to the trace of a tensor, ηc is a continuous-phase (matrix) viscosity, R is the bubble radius, σ is the interfacial tension between liquid and gas phases, φ is the volume fraction of bubbles, Λ is equal to (6ηcR/5σ), and D/Dt is the Jaumann derivative. It can be shown easily that, for steady shearing flow, the Frankel and Acrivos equation (eq 1) gives the following expressions for viscosity and normal-stress differences:10
[ { }] [ ]
12 1 - NCa2 τ12 5 η) ) ηc 1 + φ 2 γ˘ 6 1 + NCa 5
(2)
(325)N φη γ˘ 6 1+( N ) 5
(3)
(
N1 ) τ11 - τ22 )
10.1021/ie040011r CCC: $27.50 © 2004 American Chemical Society Published on Web 07/09/2004
)
Ca
c
2
Ca
Ind. Eng. Chem. Res., Vol. 43, No. 17, 2004 5373
(165)φN η γ˘ 1 - 3 1 + 6N [ 28{ (5 ) }] 6 1+( N ) 5 (4)
N2 ) τ22 - τ33 )
Ca c
2
Ca
2
ηr ) η0r
Ca
where η is the shear viscosity of the bubbly suspension, τ12 is the shear stress, γ˘ is the shear rate, NCa is the capillary number (defined as Rηcγ˘ /σ), N1 is the first normal stress difference, N2 is the second normal stress difference, and τ11, τ22, and τ33 are normal stresses (here, the subscript 1 denotes the direction of flow, the subscript 2 denotes the direction perpendicular to the flow (the direction pointing to the velocity gradient), and the subscript 3 denotes the neutral direction). Equations 2-4 can be rewritten in terms of the dimensionless quantities ηr, N1r, and N2r, where ηr is the relative viscosity (defined as η/ηc), N1r is the reduced first normal stress difference (defined as N1/ηcγ˘ ), and N2r is the reduced second normal stress difference (defined as N2/ηcγ˘ ):
{ } [ ]
12 N 2 5 Ca ηr ) 1 + φ 2 6 1 + NCa 5 1-
(
(5)
)
32 N φ 5 Ca N1r ) 2 6 1 + NCa 5
( ) ( )
16 φN ( 3 5) 6 ) 1 - {1 + ( N ) }] [ 6 28 5 1+( N ) 5 -
N2r
(6)
Ca
2
Ca
2
(7)
Ca
Note that the first normal stress difference (N1 or N1r) is positive whereas the second normal stress difference (N2 or N2r) is negative. Also, the first normal stress difference is always greater in magnitude than the second normal stress difference. Oldroyd13,14 also developed a constitutive equation for emulsions of two immiscible Newtonian fluids. For bubbly suspensions, the Oldroyd model becomes
(1 + λ DtD )τ ) 2η (1 + λ DtD )d 1
0
2
(8)
where λ1 is the relaxation time, λ2 is the retardation time, η0 is the zero-shear viscosity, and D/Dt is the Jaumann derivative. Expressions for η0, λ1, and λ2 for infinitely dilute bubbly suspension (φ f 0) were found to be
η0 ) ηc(1 + φ) λ1 )
( )( ( )(
(9)
) )
6 ηc R 16 1+ φ 5 σ 15
(10)
6 ηcR 8 1- φ 5 σ 5
(11)
λ2 )
For steady shearing flow, the Oldroyd model gives the following expressions for relative viscosity and reduced normal stress differences:
N1r )
N2r )
[ [
[
2 6 1 + λ1rλ2r NCa 5 2 6 1 + λ1r2 NCa 5
(
(
)
)
]
) ] ) ]
(12)
2η0r(λ1r - λ2r)NCa 2 6 1 + λ1r2 NCa 5
(13)
-η0r(λ1r - λ2r)NCa 2 6 1 + λ1r2 NCa 5
(14)
(
(
where η0r, λ1r, and λ2r are relative zero-shear viscosity, reduced relaxation time, and reduced retardation time, respectively. Equations 9-11 give the following expressions for η0r, λ1r, and λ2r:
η0 )1+φ ηc
(15)
λ1r )
λ1 16 )1+ φ λc 15
(16)
λ2r )
λ2 8 )1- φ λc 5
(17)
η0r )
where λc is the relaxation time of a single bubble, defined as λc ) 6ηcR/5σ. The Oldroyd model gives predictions of ηr, N1r, and N2r very similar to those predicted by the Frankel and Acrivos model. For example, Figure 1 shows comparison between the rheological properties calculated from these models. The relative viscosities predicted from the two models almost overlap. A similar agreement is observed for N1r, the reduced first normal stress difference. The reduced second normal stress difference (N2r) predicted from the two models matches almost exactly up to a capillary number (NCa) of unity. At high capillary numbers (NCa > 1‚0), the N2r values predicted from the Frankel and Acrivos model deviate from the Oldroyd model. A sharp reduction in the magnitude of N2r occurs at high NCa according to the Frankel and Acrivos model. However, the Oldroyd model predicts a smoother reduction in the magnitude of N2r with the increase in NCa, at high values of capillary number. The Oldroyd model, eq 8, has the advantage of being a simpler model. It does not contain the nonlinear terms in d as found in the case of the Frankel and Acrivos model (eq 1). Furthermore, the three constants present in the Oldroyd model, namely, η0, λ1, and λ2, each have a definite physical meaning (η0 is the zero-shear viscosity and λ1 and λ2 are relaxation and retardation times, respectively, of the bubbly suspensions). 2.2. Constitutive Equations for Moderately Concentrated Bubbly Suspensions. The Frankel and Acrivos constitutive equation for bubbly suspensions (eq 1) is valid only for very dilute systems as the hydrodynamic interaction between the bubbles is ignored in its derivation. However, the Oldroyd model (eq 8) can be extended to concentrated bubbly suspensions by incorporating the effect of bubble-bubble hydrodynamic interactions in the parameters η0, λ1, and λ2. Oldroyd13 utilized a “self-consistent” scheme to derive the expressions for η0, λ1, and λ2 for moderately concentrated
5374 Ind. Eng. Chem. Res., Vol. 43, No. 17, 2004
Figure 1. Comparison between the rheological properties predicted from Oldroyd13,14 and Frankel and Acrivos12 models.
systems. According to the self-consistent scheme utilized by Oldroyd, the suspension is first treated as a homogeneous fluid, with the same macroscopic rheological properties as that of the suspension. Then a small portion of the homogeneous fluid is replaced by the actual components of the suspension. The properties of the homogeneous fluid are finally determined by insisting that if a small portion of the homogeneous fluid were replaced by the actual components of the suspension, no difference in rheological behavior could be detected by macroscopic observations. Oldroyd’s analysis leads to the following expressions for the three parameters η0, λ1, and λ2:
3 η0 1 + 5φ η0r ) ) 2 ηc 1- φ 5
(18)
2 1+ φ λ1 3 λ1r ) ) 2 6 ηc R 1- φ 5 5 σ
( )
λ2r )
λ2 1-φ ) η R 3 6 c 1+ φ 5 5 σ
( )
(19)
(20)
Thus, one can predict the relative viscosity and normal stress differences for moderately concentrated bubbly suspensions using eqs 12-14 in conjunction with eqs 18-20. Note that, in the limit φ f 0, eqs 18-20 reduce to eqs 15-17. Equations 18-20 are a significant improvement over the limiting (φ f 0) eqs 15-17, as far as the effect of bubble concentration (φ) is concerned. Nevertheless,
Ind. Eng. Chem. Res., Vol. 43, No. 17, 2004 5375
Figure 3. Relative viscosity (ηr) as a function of capillary number (NCa) predicted by the proposed model. The predictions are shown for various values of bubble volume fraction (φ). The value of φm in the model is taken to be 0.637, corresponding to random close packing of uniform spherical bubbles.
Figure 2. Comparisons between published experimental data on ηr and predictions of Oldroyd model.
they are valid only for moderately concentrated suspensions with φ less than about 0.20. At higher values of φ, significant deviations are observed between the predictions of these equations and the experimental data. As an example, Figure 2 compares the experimental relative viscosities with those predicted from eq 12, in conjunction with eqs 18-20. The experimental data shows good agreement with the predicted values only at low values of φ (φ < 0.20). At higher values of φ, the equations either underpredict the relative viscosity, when NCa f 0, or overpredict the relative viscosity, when NCa f ∞. The Oldroyd model for moderately concentrated suspensions (eq 8 in conjunction with eqs 18-20) has another limitation that it does not take into account the effects of the bubble size distribution on the rheology of suspensions. The rheological properties of suspensions are known to vary with the size distribution of the dispersed particles/bubbles.10 One way to account for the effect of the bubble size distribution on the rheological properties of bubbly suspensions is to include φm, the maximum packing volume fraction of undeformed bubbles, in the expressions for η0r, λ1r, and λ2r, that is,
η0r ) η0r(φ, φm)
(21)
λ1r ) λ1r(φ, φm)
(22)
λ2r ) λ2r(φ, φm)
(23)
The maximum packing volume fraction of undeformed
particles φm is known to be sensitive to the particle size distribution; φm for suspensions of uniform bubbles is expected to be significantly smaller as compared with the φm value for polydisperse suspensions. Choi and Schowalter15 also derived a constitutive equation for concentrated suspensions of two immiscible Newtonian fluids. They utilized a “cell model” approach to extend the Frankel and Acrivos12 analysis to concentrated systems. The full constitutive equation derived by Choi and Schwalter is quite complicated. The expressions for η0r, N1r, and N2r for moderately concentrated emulsions (φ < 0.2), obtained from the Choi and Schowalter constitutive equation,15 are as follows:
η0r ) 1 + φ(1 + 2.5φ)
( )( ) ( )( )
N1r )
32 NCaφ (1 + 2.5φ)2 5 1 + Z2
N2r ) -
(24) (25)
20 NCaφ (1 + 4.7φ) 7 1 + Z2
(26)
(56)N (1 + 203φ)
(27)
where
Z)
Ca
The cell model approach utilized by Choi and Schowalter has the inherent disadvantage that the results derived are strongly dependent on somewhat arbitrary choices of size, shape, and boundary conditions of the cell.16 3. Proposed Constitutive Model for Concentrated Bubbly Suspensions On the basis of the existing theoretical studies on the rheology of bubbly suspensions, it can be concluded that the form of the constitutive equation, given by eq 8, is valid for concentrated systems as an approximation. In fact, the Frankel and Acrivos constitutive equation (eq 1) and the Choi and Schowalter constitutive equation can also be easily recast into the form of eq 8 if the nonlinear terms in d (rate of deformation tensor) are neglected. Therefore, we propose the following constitu-
5376 Ind. Eng. Chem. Res., Vol. 43, No. 17, 2004
Figure 4. Reduced first normal stress difference (N1r) as a function of capillary number (NCa) predicted by the proposed model. The predictions are shown for various values of bubble volume fraction (φ). The value of φm in the model is taken to be 0.637.
Figure 6. Comparison between experimental viscosity data of Stein and Spera8 for bubbly suspensions and predictions of the proposed model. φm is taken to be 0.637. Figure 5. Reduced second normal stress difference (-N2r) as a function of capillary number (NCa) predicted by the proposed model. The predictions are shown for various values of bubble volume fraction (φ). The value of φm in the model is taken to be 0.637.
tive model for concentrated bubbly suspensions:
[1 + λ λ
c 1r(φ,
D φm) τ ) Dt
]
D 2ηcη0r(φ, φm) 1 + λcλ2r(φ, φm) d (28) Dt
[
]
For steady shearing flow, the equations for relative viscosity and reduced normal stress differences remain the same as eqs 12-14 but new improved expressions for η0r, λ1r, and λ2r are derived in the following section. 3.1. New Expressions for η0r, λ1r, and λ2r. New expressions for η0r, λ1r, and λ2r of concentrated suspensions of bubbles are derived using the differential effective medium approach (DEMA). According to this approach, a concentrated suspension is considered to be obtained from an initial continuous phase by successively adding infinitesimally small quantities of bubbles to the system until the final volume fraction of the bubbles is reached. At any arbitrary stage (i) of the process, the addition of an infinitesimal amount of
bubbles leads to the next stage (i + 1). The suspension of stage (i) is then treated as an equivalent “effective medium,” which is homogeneous with respect to the new set of bubbles added to reach stage (i +1). The solution of a dilute suspension is then applied to determine the increment changes in zero-shear viscosity, relaxation time, and retardation time, in going from stage (i) to stage (i + 1). The differential equations derived in this manner are integrated to obtain the final solutions for a concentrated suspension. Let us now consider a suspension of bubbles with volume fraction of bubbles φ. Into this suspension, an infinitesimally small amount of new bubbles is added. The increment changes in zero-shear viscosity, relaxation and retardation times, resulting from the addition of the new bubbles, can be calculated from the solution of a dilute system, that is, eqs 15-17, by treating the suspension into which new bubbles are added as an equivalent effective medium with zero-shear viscosity η0, relaxation time λ1, and retardation time λ2. Thus,
dη0 ) η0 dφ 16 (15 ) dφ -8 ) λ ( ) dφ 5
(29)
dλ1 ) λ1
(30)
dλ2
(31)
2
Ind. Eng. Chem. Res., Vol. 43, No. 17, 2004 5377
Figure 7. Comparison between experimental viscosity data of Rust and Manga9 and predictions of the proposed model. φm is taken to be 0.637.
Equations 29-31 assume that all the volume of the suspension before new bubbles are added is available as free volume to the new bubbles. In reality, the free volume available to disperse the new bubbles is significantly less, due to the volume pre-empted by the bubbles already present. This means that when new bubbles are added to the suspension, the increase in the actual volume fraction of the bubbles is larger than dφ. Following Krieger and Dougherty,17 the incremental increase in the volume fraction of the bubbles is taken to be dφ/(1 - φ/φm) where φm is the maximum packing volume fraction of undeformed bubbles; for random close packing of monosized spherical bubbles, φm is 0.637.10 Thus, we obtain
∫ηη
0
c
∫λ
λ1 c
dη0 ) η0
∫0φ
dλ1 16 ) λ1 15
dφ
( )∫
λ2 dλ2 -8 ) λc λ 5 2
∫
φ φm
1-
dφ
φ
0
( )∫
1-
dφ
φ
0
φ φm
1-
φ φm
(32)
(33)
Figure 8. Comparison between experimental viscosity data of Pal18,19 and prediction of the proposed model at low NCa (NCa f 0).
(34)
4. Predictions of Viscosity and Normal Stress Differences from the Proposed Constitutive Model
Equations 32-34 after integration give the following new expressions for η0r, λ1r, and λ2r:
η0r ) λ1r )
(
(
)
λ1 φ ) 1λc φm
λ2r )
)
η0 φ ) 1ηc φm
(
-φm
(-16/15)φm
)
λ2 φ ) 1λc φm
(8/5)φm
(35) (36) (37)
In the limit φ f 0, eqs 35-37 reduce to eqs 15-17. An interesting point to note from eqs 35-37 is that, at φ ) φm, η0r and λ1r become infinite whereas λ2r becomes zero. Suspensions with φ g φm are no longer bubbly liquids; they become foams. Foams are well-known to behave like solids below the yield strength. Solidlike materials are expected to exhibit infinite viscosity, infinite relaxation time, and zero retardation time.
Figure 3 shows the relative viscosities predicted from the proposed model (eq 28 in conjunction with eqs 3537). The relative viscosity versus capillary number plot, at any given value of bubble volume fraction φ, exhibits three distinct regions: constant ηr region at low values of NCa, decreasing ηr region at intermediate values of NCa, and finally, constant ηr region at high values of NCa. The relative viscosity is greater than unity at low NCa (NCa < 1) and is less than unity at high NCa (NCa > 1). With the increase in φ, ηr increases at low NCa and decreases at high NCa. The reduced normal stress differences (N1r and N2r) predicted from the proposed model (eq 28 in conjunction with eqs 35-37) are plotted in Figures 4 and 5. At any given φ, N1r and -N2r increase linearly with NCa initially. After reaching some maximum values at intermediate NCa, the values of N1r and -N2r start decreasing with further increase in NCa. With the increase in bubble volume fraction φ, the values of N1r and -N2r increase. The increase in N1r or -N2r with the increase in φ is much larger in the low NCa region (NCa < 1) as compared with the increase observed in the high NCa region (NCa > 1).
5378 Ind. Eng. Chem. Res., Vol. 43, No. 17, 2004
Figure 9. Comparison between the predictions of the proposed model and Choi and Schowalter model. φm is taken to be 0.637 for the proposed model.
5. Comparison of Model Predictions with Experimental Data To our knowledge, there is little or no experimental data available on the normal stress differences of bubbly suspensions. However, several experimental studies have been reported in the literature regarding the viscosity behavior of bubbly suspensions.8-10,18,19 Figure 6 compares the experimental viscosity data of Stein and Spera8 for bubbly suspensions with the predictions of the proposed model (eq 28 in conjunction with eqs 35-37) using a φm value of 0.637, corresponding to random close packing of uniform spherical bubbles. As can be seen, there is a good agreement between the model predictions and the experimental data. Figure 7 shows comparisons between the experimental ηr - NCa data of Rust and Manga9 and model predictions using a φm value of 0.637. Again, the model predictions are in good agreement with the experimental data. Figure 8 shows a comparison between the experimental data of Pal18,19 on oil-in-water emulsions and predictions of the proposed model. As the ratio of dispersedphase (oil) viscosity to continuous-phase viscosity for these emulsions was very small, they can be considered
as bubbly suspensions. Only the low shear data are considered and, therefore, the capillary number is small (NCa f 0). As can be seen, the experimental data can be described reasonably well with the proposed model using a φm value of 0.54. This φm value of 0.54 is close to the φm value of simple cubic packing of uniform spheres, that is, φm ) 0.52. Due to lack of experimental data on normal stress differences in bubbly suspensions, it is not possible to compare the predictions of the proposed model with the experimental data. However, some numerical studies on bubbly suspensions4 have indicated that the normal stress differences predicted by the Choi and Schowalter constitutive equation (eqs 25 and 26) are reasonably accurate at low volume fractions of the dispersed phase (bubbles). Therefore, we have compared the predictions of the Choi and Schowalter model (eqs 25 and 26) with those of the proposed model in Figure 9. At a moderate φ value of 0.15, the predictions of the Choi and Schowalter model and the proposed model are in good agreement. At a large φ value of 0.60, the proposed model predicts values of N1r and -N2r higher than those predicted by the Choi and Schowalter model. This is expected as the Choi and Schowalter model is valid only at low to moderate values of φ (φ < 0.20).
Ind. Eng. Chem. Res., Vol. 43, No. 17, 2004 5379
6. Concluding Remarks The Oldroyd constitutive model for the rheology of dilute emulsions of two immiscible Newtonian fluids has been extended to highly concentrated (φ e φm) suspensions of bubbles. New expressions for the model parameters, namely, zero-shear viscosity, relaxation time, and retardation time, have been derived using the differential effective medium approach. The model predicts relative viscosity and reduced first and second normal stress differences of bubbly suspensions as functions of capillary number and bubble volume fraction. The proposed model is evaluated using the existing experimental data on viscosity of bubbly suspensions. Further experimental work is needed on the rheology of bubbly suspensions to thoroughly evaluate the proposed model. In particular, experimental data on normal stress differences of bubbly suspensions is lacking in the existing literature. Acknowledgment Financial support from NSERC is appreciated. Literature Cited (1) Bagdassarov, N. S.; Dingwell, D. B. A rheological investigation of vesicular rhyolite. J. Volcanol. Geotherm. Res. 1992, 50, 307-322. (2) Stein, D. J.; Spera, F. J. Rheology and microstructure of magmatic emulsions: theory and experiments. J. Volcanol. Geotherm. Res. 1992, 49, 157-174. (3) Pinkerton, H.; Norton, G. Rheological properties of basaltic lavas at sub-liquidus temperatures: laboratory and field measurements on lavas from Mount Etna. J. Volcanol. Geotherm. Res. 1995, 68, 307-323. (4) Manga, M.; Castro, J.; Cashman, K. V.; Loewenberg, M. Rheology of bubble-bearing magmas. J. Volcanol. Geotherm. Res. 1998, 87, 15-28.
(5) Lejeune, A. M.; Bottinga, Y.; Trull, T. W.; Richet, P. Rheology of bubble-bearing magmas. Earth Planet. Sci. Lett. 1999, 166, 71-84. (6) Spera, F. J.; Stein, D. J. Comment on “Rheology of bubblebearing magmas” by Lejeune et al. Earth Planet. Sci. Lett. 2000, 175, 327-331. (7) Manga, M.; Loewenberg, M. Viscosity of magmas containing highly deformable bubbles. J. Volcanol. Geotherm. Res. 2001, 105, 19-24. (8) Stein, D. J.; Spera, F. J. Shear viscosity of rhyolite-vapor emulsions at magmatic temperatures by concentric cylinder rheometry. J. Volcanol. Geotherm. Res. 2002, 113, 243-258. (9) Rust, A. C.; Manga, M. Effects of bubble deformation on the viscosity of dilute suspensions. J. Non-Newtonian Fluid Mech. 2002, 104, 53-63. (10) Pal, R. Rheological behavior of bubble-bearing magmas. Earth Planet. Sci. Lett. 2003, 207, 165-179. (11) Rust, A. C.; Manga, M. Bubble shapes and orientations in low Re simple shear flow. J. Colloid Interface Sci. 2002, 249, 476480. (12) Frankel, N. A.; Acrivos, A. The constitutive equation for a dilute emulsion. J. Fluid Mech. 1970, 44, 65-78. (13) Oldroyd, J. G. The elastic and viscous properties of emulsions and suspensions. Proc. R. Soc. A 1953, 218, 122-132. (14) Oldroyd, J. G. Complicated rheological properties. In Rheology of disperse systems; Mill, C. C., Ed.; Pergamon Press: London, 1959; pp 1-15. (15) Choi, S. J.; Schowalter, W. R. Rheological properties of nondilute suspensions of deformable particles. Phys. Fluids 1975, 18, 420-427. (16) Jeffrey, D. J.; Acrivos, A. The rheological properties of suspensions of rigid particles. AIChE J. 1978, 22, 417-432. (17) Krieger, I. M.; Dougherty, T. J. A mechanism for nonNewtonian flow in suspensions of rigid particles. Trans. Soc. Rheol. 1959, 3, 137-152. (18) Pal, R. Rheological properties of emulsions of oil in aqueous non-Newtonian polymeric media. Chem. Eng. Commun. 1992, 111, 45-60. (19) Pal, R. Viscoelastic properties of polymer-thickened oilin-water emulsions. Chem. Eng. Sci. 1996, 51, 3299-3305.
Received for review January 12, 2004 Revised manuscript received May 24, 2004 Accepted May 25, 2004 IE040011R
