Rheology and microstructure of magmatic emulsions: theory and experiments

Journal of Volcanology and Geothermal Research, 49 (1992) 157-174

157

Elsevier Science Publishers B.V., Amsterdam

Rheology and microstructure of magmatic emulsions: theory and experiments Daniel J. Stein and Frank J. Spera Department of Geological Sciences and Institute for Crustal Studies, University of California, Santa Barbara, CA 93106, USA (Received November 26, 1990; revised and accepted March 14, 1991 )

ABSTRACT Stein, D.J. and Spera, F.J., 1992. Rheology and microstrncture of magmatic emulsions: theory and experiments. J. Volcanol. Geotherm. Res., 49:157-174. The rheological properties of viscous emulsions composed of melt plus vapor bubbles constitute a critical but largely uninvestigated aspect of magmatic transport phenomena. In this study, the rheological behavior of dilute emulsions of GeO2 containing from 0.8 to 5.5 vol.% air bubbles has been measured experimentally between 1100 and 1175°C at 100 kPa in a rotating rod rheometer at shear rates between 0.05 and 7 s- I. At constant bubble volume fraction, when the range of shear rates examined is greater than a factor of twenty, the rheological behavior of the emulsions can be modeled by a power-law constitutive relation. The power-law emulsions are pseudoplastic (shear-thinning), having a flow index of 0.870.93. A tentative correlation between relative viscosity and bubble volume fraction is given as r/r= 1 + 13.1 ~, with r/r= t/c/ r/m (r/e is the emulsion viscosity at constant shear stress and r/m is the viscosity of the pure Newtonian melt phase.) The strong variation of relative viscosity with volume fraction of bubbles is placed in the context of current theoretical and experimental understanding of the effects of shear on viscous emulsions, and is attributed to the deformation and eventual disruption of bubbles by shear forces. Bubble deformation is promoted by shear and opposed by surface tension. Two dimensionless parameters governing bubble deformation are the capillary number Ca- )~Irnrb/trand viscosity ratio ;t---r/v/t/m determined from melt viscosity ~/m,vapor viscosity ~/v,bubble radius rb, shear rate 7, and vapor-melt interracial tension a. The capillary number is a measure of the relative importance of shear and interfacial stresses. Low-;t bubbles may attain very elongate stable shapes, and high shear rates are required before fragmentation occurs at a critical capillary number Cacnl.f. The number of daughter bubbles formed during disruption is known to depend on Ca/Camt.f and to rise steeply as this ratio increases from 1 to 20. Bubbles are deformed into prolate ellipsoids with deformation parameter D = ( 1 - b ) / ( l + b) where I and b represent the long and short axes of the ellipsoidal bubble; for small non-dimensional shear rate (Ca <--0.4), D = Ca. The bubbles undergo transient oscillation with respect to both D and long-axis orientation relative to the shear direction. This may lead to variability of torque during a single experiment even when shear rate is held constant. Bubble fragmentation has consequences for observed bubble size distributions in post-experimental counts as well as in nature. However, bubble fragmentation by the fracture mechanism is unlikely (or at least, not dominant) in most natural magmatic flows. Instead a sub-critical instability known as tip-streaming can occur at a much lower capillary n u m b e r , Cacrit,ts = 0.56. This mechanism produces much smaller daughter bubbles than that of fracture, but is much more relevant to magmatic flows which are characterized by capillary number between 0 and 100. The deformation of bubbles produces viscoelastic behavior in viscous emulsions. Normal stress differences amounting to several per cent of the total shear stress can be produced at shear rates of less than 10 s-~. In rotating rod rheometry, this leads to rod-climbing behavior (Weissenberg effect) which permits the measurement of the normal stress differences by climbing rod viscometry. A preliminary assessment of the first normal stress difference (defined N~ = roe- r=) is made in one of our experiments, and is estimated to be about 2% of the total shear stress. Inferences drawn regarding the viscosity and discharge of lava flows may be misleading if allowance is not made for the effects of vapor bubbles on magma rheology.

Introduction

The rheological properties of silicate magma 0377-0273/92/$05.00

play a central role in the transport of material, momentum and heat within the Earth and other terrestrial planets. Important magma

© 1992 Elsevier Science Publishers B.V. All rights reserved.

158

transport phenomena include, for example, the extraction of melt from regions undergoing partial fusion, the mobilization and ascent of mushy diapirs, the flow of magma through cracks and fissures, convection within crustal magma bodies and magma oceans, the settling and transport of xenoliths and crystals, the growth of lava domes, the dynamics of endogenous lava flows, and the high-speed eruption of compressible magmatic mixtures comprised of solids, melt and vapor. Many measurements of viscosity have been made both on simple binary and ternary component systems and on melts of natural composition as a function of temperature, pressure and oxidation state (e.g., see Bottinga and Weill, 1972; Shaw, 1972; Kushiro, 1976, 1978; Urbain et al., 1982; Scarfe et al., 1987; Dingwell and Virgo, 1987, 1988, and references therein). Although a reasonably good understanding now exists for the dependence of melt viscosity on these intensive variables, it is important to emphasize that natural magmas are multiphase mixtures consisting of crystals and vapor bubbles of various shapes and sizes dispersed in a continuous melt phase. Although there is experimental evidence that high-silica melts behave as shearthinning fluids in the temperature and shear rate ranges of 900-1300°C and 10-2-10 s -1 respectively (Spera et al., 1982, 1988), the deviation from Newtonian behavior for mafic liquids appears small. In contrast to singlephase melts which show only modest deviations from Newtonian behavior, magmatic suspensions exhibit a wide variety of highly non-Newtonian effects including yield phenomena, pseudoplasticity, rheological dilatancy, thixotropy and viscoelasticity (Shaw, 1969). Although there has been some systematic experimental study of crystal plus melt mixtures (Shaw et al., 1968; Shaw, 1969; McBirney and Murase, 1984; Ryerson et al., 1988) and of melt plus vapor mixtures (Spera et al., 1988), the experiments are far from definitive. The development of a constitutive relation for multiphase, multicomponent magma

D.J. STEIN AND F.J. SPERA

that incorporates yield phenomena, normal stress effects and accounts for variations in crystallinity, bubble content, melt composition, temperature and pressure remains an important goal for igneous petrologists and geochemical fluid dynamicists. The present pilot study was undertaken in order to understand better the rheological properties of a dilute mixture of melt plus vapor bubbles. Such mixtures are referred to as emulsions. Low-temperature polymeric emulsions are known to exhibit non-Newtonian characteristics such as shear-dependent viscosity, viscoelasticity and normal stress differences even when the volume fraction (¢) of the dispersed (vapor) phase is within the dilute range (0<0.03). Important parameters governing the behavior of dilute emulsions include the viscosity ratio of the dispersed (vapor) and continuous (melt) phases defined ~ . ~ T / v / g / m , the size distribution of bubbles, the melt-vapor surface tension (tr), and the viscosity of the melt phase (Y]m). When considering vapor bubbles in viscous melts of natural composition, the limit 2--,0 is most relevant; 2 > 1 may be important when considering liquid immiscibility in silicate, silicate/sulfide, or silicate/carbonatitic melts. The experiments reported on here are relevant to low-2 mixtures (2 << 1 ), the primary regime of interest to volcanologists and igneous petrologists. The melt phase studied in these experiments is molten germania (GeO2) which is quite viscous (r/m~ 105 Pa s) at the temperature and pressure conditions of the experiments. The chemical structure and mechanism of viscous flow of glassy or molten germania are considered to be similar to those of silicates (with which we make analogy in these experiments ), and its viscosity is well-characterized from below the GeO2 liquidus (supercooled liquid) to temperatures in excess of 1500°C. Air is used as the vapor phase (r/v~ 10 -5 Pa s) because of its chemical inertness and low solubility in molten germania. There is a lack of experimental information

RHEOLOGY AND MICROSTRUCTURE OF MAGMATIC EMULSIONS: THEORY AND EXPERIMENTS

on the rheometric properties of magmatic emulsions; this is surprising in view of their ubiquitous occurrence in nature. Unlike crystals suspended in a melt, vapor bubbles are highly deformable entities. Dilute emulsions of deformable bubbles differ from Newtonian melts in that they manifest nonzero normal stress differences that may be an appreciable fraction of the shear stress. Consequently, the flow field can strongly deviate from that of a Newtonian fluid. Inferences regarding the viscosity of magmas drawn from morphological observations of lava flows may be misleading or incorrect if based on the assumption of Newtonian behavior. In order to understand the dependence of the shear viscosity of an emulsion on the volume fraction of bubbles, it is imperative to evaluate the dynamics of bubble deformation and mechanisms of bubble break-up as a function of shear rate and viscosity ratio 2. There are a number of shear and capillary instabilities that operate during the flow of bubble-bearing natural magmas and it is our goal to study these instabilities by means of concentric cylinder rheometry under laboratory-controlled conditions. In this study, a preliminary experimental assessment is made of the effect of bubble content and deformation on the shear viscosity of magma and of the magnitude of the normal stresses. The remainder of this paper is organized as follows. In the next section, a summary of previous experimental and theoretical work on the rheometry and microstructure of immiscible non-silicate emulsions is presented. This is followed by description of our experiments and presentation of the laboratory results. In the final section, the experimental results are discussed in light of previous experimental work and a discussion of the significance of our rheometric studies to magma transport is given. It will be apparent that theoretically and experimentally, inquiry has barely begun regarding the non-Newtonian properties of silicate emulsions; the practical implications for pe-

159

trological studies and especially magma dynamics remain largely uncharted.

Theoretical background The experimental quantities of interest in concentric cylinder rheometry of an emulsion include the shear stress (zro) and the primary (N~), and secondary (N2) normal stress differences. These are defined: Tr0 = r/e(~,T,~) ~r0 N , - Z o o - Vrr = ~, (~',T,(~ )72rO

( 1)

iV: =Zrr-Zzz = ~v:(#,T,O)#2o where r/e is the emulsion viscosity that depends, at fixed pressure, on the shear rate (~), temperature (T), and bubble volume fraction (0) and ~v1 and ~v2 are the primary and secondary normal stress coefficients. At low rates of shear, one expects ~v and ~v: to be simple constants under isothermal and isobaric conditions. Recall that for a Newtonian fluid N~ = N 2 = 0. In the cylindrical Couette flow of the experiments, 0 is the direction of fluid flow, r is the direction of velocity variation (radial distance from the axis of rotation ) and z is the vertical coordinate. The important questions addressed in the experiments include: ( 1 ) the dependence of the shear viscosity of the emulsion on shear rate and emulsion microstructure; (2) the conditions needed for bubble break-up during viscous flow; and (3) the magnitude of normal stress differences as a function of the volume fraction of vapor in a dilute two-phase emulsion. Each of these problems is addressed below in order to provide a framework for interpretation of the rheometric experiments and to assess their significance for natural flows. Bubble deformation a n d emulsion microstructure

When freely suspended in a viscous melt undergoing simple shear, a single, low-viscos-

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D.J. STEIN AND F.J. SPERA

ity bubble undergoes deformation into a nonspherical shape. In fact, when the shear rate is large enough, the deformed bubble breaks up by one of two mechanisms: bubble fracture or bubble tip streaming (see Fig. 1 ). Because the Stokes ascent rate of a mm-sized bubble is small relative to a typical magma ascent rate, gravitational forces are generally of second order importance; this is certainly the case for the experiments undertaken in this study. Furthermore, in all petrologically significant cases, the bubble Reynolds number, Re=p~r2/rlm, defined by melt viscosity ~]m and density p, bubble radius rb, and shear rate ~ is generally small compared to unity; consequently, inertial effects can be safely ignored. There are two remaining important dimensionless parameters governing bubble deformation and the rheol(a)

0

ogy of dilute emulsions. These include the viscosity ratio: 2 = r/v/r/m

with r/v defined as the bubble viscosity and the capillary number:

Ca=Trlmrb/tr

(bl

0

0

0

0

q:::b

E=l/rb O

(4)

where l and b represent the semi-major and semi-minor axes (Fig. 2 ) of the deformed prolate spheroid bubble of volume (4/3)nlb 2. Note that D is identically zero for a spherical bubble and asymptotically approaches unity as the bubble becomes infinitely slender. For extremely elongate bubbles, such as those commonly found in viscous lava flows, it is useful to introduce an additional bubble deformation parameter. Bentley and Leal (1986) suggest the elongation ratio:

0

O

(3)

in which a is the melt-vapor interfacial tension. In melt-vapor magmatic systems, 2 is in the range 10 - 5 < 2 < 10- ~o with smaller values applicable to silicic melts and larger ones applicable to mafic ones. The capillary number, Ca, represents the ratio of viscous forces which act to deform a bubble to the interfacial forces which tend to maintain bubble sphericity. In magmatic systems, Ca ranges from 0 to about 102. In the limit 2-+0, the relevant limit for vapor-melt emulsions, experimental studies indicate that bubbles become highly elongate before fragmentation. In order to describe bubble deformation it is convenient to introduce quantitative measures. Taylor (1934) introduced the deformation parameter:

O=(l-b)/(l+b)

0

(2)

(5)

O 0

0

0

Fig. 1. (a). Bubble fragmentation due to fracture induced by shear stresses large enough to overcome cohesive capillary forces. (b). Bubble fragmentation due to the subcritical instability of tip streaming: this phenomenon is intrinsically an unsteady process.

where rb is the radius of an undeformed spherical bubble of volume ( 4 / 3 ) n r 3 . Since bubbles are deformed at constant volume, (lb2=r 3), Eqn. (5) may be written:

E--- (l/b) 2/3

(6)

RHEOLOGY AND MICROSTRUCTURE OF MAGMAT1C EMULSIONS: THEORY AND EXPERIMENTS

Fig. 2. A spherical bubble placed in a Couette flow field will distort so that the bubble long axis (l) lies at some angle a to the direction of flow. At low capillary number, a ~ 45 °. As the shear rate increases, the long axis of the bubble rotates into the direction of shear. The minor diameter of the deformed ellipsoidal bubble is b.

Similarly, the Taylor deformation number D may be written in terms of the elongation number: D=

(E 3/2- 1

)/(E3/2+ 1 )

(7)

As a bubble becomes infinitely slender note that E ~ oo; E is a more convenient measure of bubble slenderness in the large deformation limit applicable to magmatic bubbles.

Weak flows The early work of Taylor (1932, 1934) and more recent studies by Cox (1969), Torza et al. (1972) and Barth6s-Biesel and Acrivos ( 1973 ) focus experimentally and theoretically on bubble deformation for flows with Ca << 1. These are called weak flows because bubble deformation remains small due to the dominating influence of interfacial tension relative to viscous forces. Experiments (Taylor, 1934; Rumscheidt and Mason, 1961; Torza et al., 1972; Grace, 1982) and theoretical studies (Hinch and Acrivos, 1980) relevant for 2< 1 and arbitrary Ca indicate a complex behavior for the deformation, orientation and stability ofnearly inviscid bubbles in simple shear flows. In particular, there is a transient approach to an equilibrium bubble shape and orientation

161

at low Ca which, in fact, is unstable with respect to sudden changes in Ca (Hinch and Acrivos, 1980; Rallison, 1984). Starting at low shear rate (low Ca), a bubble assumes an elliptical shape with long axis oriented at 45 ° to the direction of shear (Fig. 2). As Ca increases, the equilibrium shape becomes more elongate (i.e., E increases ) and the long axis of the bubble rotates into the plane of shear. Detailed studies show that the approach to a constant bubble shape and long axis orientation is a damped oscillation. The decay time of the oscillation is roughly ~ - I C a 3/4. For one of our typical experiments, the decay time is of order 10 s. We believe this intrinsic unsteadiness contributes to experimental scatter in measured torque at a fixed shear rate, temperature and ¢ in the experiments (see Figs. 6 and 7 ). For Ca <-0.4, the studies of Taylor (1934) and Grace (1982) give a simple relationship between the bubble deformation and the capillary number:

D=Ca

(8)

That is, the equilibrium deformation is independent of r/v and depends only on Ca provided 2 < 1. For r/m= 105 Pa s, rb= 102/tm and a = 0 . 3 N m - l , the maximum shear rate for which (8) is valid is 10 -2 s -I a value at the low end of our experimental range but within the range relevant for natural flows (e.g., see table 1 in Spera et al., 1988).

Strong flows For higher rates of shear the situation becomes more complex with attendant hysteresis effects and subcritical instabilities. As Ca increases beyond the weak flow limit of 0.4, bubbles become progressively deformed from slightly elliptical prolate spheroids into long, thin, thread-like shapes with pointed tips. In the high-Ca regime, the relationship between bubble elongation and shear rate is (Hinch and Acrivos, 1980):

162

D.J. STEIN A N D F.J. SPERA

E = 3.45Ca 1/2

(9)

which may be recast in terms of the Taylor deformation parameter:

D=(6.4Ca3/a-1)/(6.4Ca3/4+I)

(10)

using Eqns. ( 5 ), ( 7 ) and (9). For example, in a flow with Ca= 20, a typical value from one of our experiments, I/b = 210. Clearly, bubbles are highly elongate at C a = 2 0 with bubble lengths being over 200 times greater than bubble widths. Eventually, a critical Ca is reached and bubble disruption by a fragmentation (bursting) mechanism occurs (Fig, la). There are both experimental (Grace, 1982) and theoretical results (Hinch and Acrivos, 1980) which give the relationship between the critical capillary number at which bubble fracture occurs (Cac_ tit,f) and the viscosity ratio 2 (Fig. 3 ). Grace ( 1982 ) found experimentally that: ( 11 )

Cacrit,f = 0.172-0.55

whereas Hinch and Acrivos (1980) analytically determined: Cacrit,f = 0.0542 - 2/3

( 12 )

as the relevant criterion for bubble fragmentation by fracture. This is reasonably adequate 103102O

101 100 I

10-5

I

t

10-3

10-1

X Fig. 3. Relationship between critical capillary number at bubble fracture (Cacrit,f) and the viscosity ratio (2). Solid line represents theoretical relationship of Hinch and Acrivos (1980). Dashed line represents experimental results of Grace ( ! 982). Lower dashed fine represents the regime of low-Ca fragmentation by the mechanism of tipstreaming, Cacnt,f= 0.56. Both dashed curves represent best fits of experimental data.

agreement given the experimental difficulties and the approximations implicit in the theoretical analysis. Typically, a parent bubble is broken into 10 to 100 daughter bubbles upon fragmentation. If N is the number of daughter bubbles, then the approximate ratio of parent to daughter bubble radii is N 1/3. Once fracture has occurred and rb has been reduced. Ca decreases due to the reduction of rb to some value of Ca < Cacdt,f. Bubble fracture is obviously a very efficient means for increasing the bubble number density of an emulsion. However, Eqns. ( 11 ) and ( 12 ) show that, in the limit of small ,~, Cacrit,f is very large. For example at 2= 10-10, Eqn. (11) gives Cac,t,f= 54,000, a value that greatly exceeds the Ca number range both in our experiments and in virtually all natural flows. Recall that the typical range for Ca in magmatic flows is between 1 and 102. An additional complexity is that the rate of change of Ca with time can strikingly influence the dynamics of bubble break-up. Experiments indicate that, starting from an equilibrium bubble shape at some Ca, if the shear rate is slowly increased, the bubble will attain a new equilibrium shape corresponding to the new Ca value. However, a sudden jump to this same higher value of Ca from the same initial value causes the bubble to fragment into many small droplets (Rallison, 1984). The break-up mode in general depends upon the history of the deformation and on the ratio Ca/Cacrit,f. When Ca/Cacrit,f,~ 1 fragmentation occurs by fracture with the generation of 10 to 102 daughter bubbles. In contrast, with Ca/Cacr~t,f around 20, the number of daughter bubbles can be as high as 104 and tip streaming as well as fracture are important (Grace, 1982 ).

Tip streaming An additional mode of bubble fragmentation is often found experimentally. This mechanism operates at subcritical values of Ca for break-up by fracture (i.e., at Ca< Cacdt,f) and

16 3

R H E O L O G Y A N D M I C R O S T R U C T U R E OF M A G M A T I C EMULSIONS: T H E O R Y A N D EXPERIMENTS

is called tip streaming. Physically, the process corresponds to the shedding of small droplets from the highly pointed tips of the parent bubble (see Fig. l b). At fixed shear rate and temperature, bubbles continue to tip stream until their consistent shrinkage reduces Ca below Cacrit,t s. Daughter bubbles produced by tip streaming are very much smaller than their parent bubbles. Grace ( 1982 ) reports the critical Ca value for tip streaming from his experiments as: Cacrit,t s =

0.56

20-

(a) = 0.015 mean rb, 80p. n = 46

i

10

0

0.5

( 13 )

which, unlike the critical capillary number for bubble fracture, is independent of 2. Unfortunately, tip streaming is not well understood theoretically. Presumably, interfacial capillary stresses in the vicinity of highly pointed bubble tips are sufficient to initiate ejection of tiny daughter droplets. This interfacial instability has been studied numerically and analytically by Buckmaster ( 1972, 1973 ) and Khakhar and Ottino (1987). Because of rotor acceleration (transient) effects and because Ca values greater than Cacrit,t s a r e commonly attained in our experiments, we believe that bubble frequency-size distributions as depicted in Figure 4 were influenced, at least in part, by the tipstreaming instability. Whether the subcritical tip streaming instability occurs seems to depend on the history of the flow, especially the time rate of change of the shear rate. The main features of bubble deformation and break-up inferred from experimental and laboratory studies reported in the literature may be summarized as follows: (1) Bubble deformation is promoted by shear and opposed by the forces of surface tension. The two dimensionless parameters that govern bubble deformation and fragmentation are the capillary number, Ca~rrlmrb/tr and viscosity ratio 2 = r/v/r/re. (2) Low-2 bubbles can attain highly elongate stable shapes and require large shear rates for break-up by fracture. The number of daughter droplets formed by fracture depends

1.0

1.5

Bubble radius, mm

30

¢ = 0.055 mean rb, 601.L n = 64

10

0 0.5

1.0

1.5

Bubble radius, mm

10-

(c) (b = 0.054

mean rb, 440 ,It n = 55

g, ===

.

.

.

.

.

.

i,il,ll,,i,l,,11,l,,.... 0.5

1.0

1.5

Bubble radius, mm

Fig. 4. (a). Histogramof bubble sizedistributionfor sample recoveredfrom JA-1, JA-2, JA-3. See Table 1Afor experimental conditions. (b). Histogramof bubble size distribution for sample recovered from MA-1. See Table 1B for experimental conditions. (c). Histogramof bubble size distribution for sample recovered from JN-I. See Table IB for experimental conditions.

164

strongly on Ca/Cacrit,f with daughter/parent bubble number ratios from a few to 10 4 a s Ca/ Cacrit.f increases from 1 to 20. (3) For weak flows ( C a < 0 . 4 ) , D=Ca; the long to short axis length ratio for bubbles (l/ b) at Ca=0.4 is 2.33. Bubbles undergo transient damped oscillatory behavior in deformation and long-axis orientation with respect to the direction of shear. The time scale of this transient damped response is ~-1Ca3/4 For a typical experiment with ~= 10 - 1 s- ~, t/m=105 Pa s, a = 0 . 3 N m -~ and r b = l mm, t is of order 102 s. This explains experimental variability of torque as a function of time in constant shear rate rheometry of magmatic emulsions. For a 1-mm bubble in a basaltic magma/~m = 10 Pa s, ~= 1 s -1, a=0.25 N m -~, this time scale is of order 10 -2 s; a 1-mm bubble in a rhyolitic magma r/m=105Pas, ~= 10-4 s-1 a=0.35 N m-~ has a time scale of order 1 0 3 S. (4) For strong flows, very low-2 bubbles can become highly elongate before fragmentation. The elongation ratio E=--l/rb= 3.45Ca 1/2 in this regime and is independent of 2. Bubble fracture occurs at a critical Ca given by Cac_ tit.r= 0.172 .0.55 which for magmatic emulsions is very large and in the range 103-105. Capillary numbers of this magnitude are not common in magmatic flows; bubble fragmentation by this mechanism is unlikely. (5) A sub-critical instability known as tip streaming can occur at a capillary number equal to Cacrit,ts=0.56. The tip streaming instability is subcritical with respect to the instability that produces break-up by fracture (i.e., Cacrit,t s < Cacrit,f ) . Shedding produces daughter droplets very small in comparison to the parent bubbles. Tip shedding rather than bubble fracture is evidently the most geologically-relevant instability promoting the reduction of bubble sizes in magmatic emulsions. High rates of change of ~ appear to induce bubble fragmentation by tip streaming. Quantifying the history of deformation is important in order to predict the relative importance of tip stream-

D.J. STEIN AND F.J. SPERA

ing and bubble fracture as fragmentation mechanisms. (6) It may prove possible to use relationships between vesicle or bubble shape and the capillary number [Eqns. (3), (4), (6), (8) and (9), for example] to estimate the shear stress, rate of shear, and velocity of magma flowing in a dike, sill or lava flow based on the shape of a typical bubble.

Relative viscosity: effect of bubbles on magma viscosity Taylor ( 1932 ) extended Einstein's theory for a dilute suspension of rigid spheres (2--,~) to the case of an emulsion of inviscid bubbles (2--, 0 ). Taylor found the constitutive relation between shear stress and shear rate for Couette cylindrical flow:

ZrO=l']e~rO~-?]m( 1 + q~)~rO

(14)

where/~m and ¢ represent the Newtonian viscosity of the melt phase and volume fraction of bubbles, respectively. In terms of the relative viscosity, defined as ?~r~/~e//~m where/~e is the shear viscosity of the emulsion, Eqn. ( 14 ) implies: ~r = 1 -1-0

(15)

That is, the viscosity of a dilute emulsion is slightly higher than the viscosity of the melt alone, due to the presence of bubbles. The Taylor theory assumes a dilute emulsion of bubbles homogeneously distributed within the mixture and that bubble shapes deviate little from sphericity. Equation ( 15 ) is valid in the low-Ca limit for a dilute emulsion (i.e. ~ << 1, Ca << 1 ). At moderate bubble fractions, bubble interactions lead to nonlinear relations between/~r and 0. A wealth of these parameterizations exist, most being empirical although some have a partial theoretical basis. Some of these models are described and summarized by Pal and Rhodes (1989). A few of the more general expressions include those suggested by Mooney ( 1951 ):

RHEOLOGY AND MICROSTRUCTUREOF MAGMATICEMULSIONS:THEORY AND EXPERIMENTS

In rb = 2 . 5 0 / ( 1 - 1.50)

(16)

o

Yro

Eilers ( 1943 ): In r/r = 2 ln[ 1 + 1 . 2 5 0 / ( 1 - 1.30) ]

TrO

7--~/'/e (0,~)) = r/m//r ( 0 , ~ ) ----m~rn0- l

(17)

Roscoe ( 1952): (18)

=

(~m (T) ) - l m (O,T)~no(o'T) - - I (21)

and most recently by Pal and Rhodes ( 1989 ): In ~r =2.5 In[1 +aO/( 1 --a0) ]

(19)

In Eqn. ( 19 ) the coefficient a is determined by fitting laboratory data. There are virtually no experimental data for silicate systems in the literature to test the validity of these relations. Furthermore, the experimental data used to obtain expressions ( 16 )- ( 19 ) (and others could be cited) are generally obtained over a large and variable range of shear rate. Relatively little is known about the shear rate dependence of rb for variable 0 values. At very low 0 (in the dilute range), the dependence of Zr0is linear with ~'rO.At higher emulsion bubble fractions, this is not the case. Information from two-phase low-temperature polymer systems indicates that the apparent emulsion viscosity /~e,app~---~'rO/~r0 becomes dependent upon both shear rate and bubble volume fraction, as 0 increases. Thus there are two sources of complexity in these systems: increasing importance of bubble-bubble interaction (e.g., bubble collisions) as 0 increases, and the dependence of r/e on shear rate due to bubble deformation. Very careful experiments are needed to separate and quantify these distinct physical effects in silicate emulsions. It is obvious that both factors are important in magmatic flows. In the regime for which the emulsion viscosity depends upon shear rate, a simple two-parameter expression may be anticipated by analogy with pseudoplastic single-phase nonNewtonian melts. The power-law expression Zro=m~ together with a generalized form of Eqn. (14) and the definition of the relative viscosity gives:

(20)

The apparent relative viscosity (r/r) may therefore be written, at any temperature under isobaric conditions, as: r/r (0,~, T)

In ~r = - - 2 . 5 In( 1 - 1.350)

165

where both m and n may depend on 0 and T. In the experiments described below, values of qr (0,5') are calculated. It is important to emphasize that experiments need span shear rates by at least a factor of 20 or so in order to fix accurately the power law parameters in Eqns. (20) and (21).

Normal stresses Schowalter et al. (1968) and Frankel and Acrivos ( 1970 ) have developed a constitutive relation for a dilute emulsion in time-dependent shearing flow. A particular result of special interest is the stress tensor in Couette cylindrical flow. The expression for the shear stress is identical to that obtained by Taylor (1932) given in Eqn. (14). However, by relaxing Taylor's assumption of bubble sphericity, they showed that the resistance to bubble deformation by interfacial tension provides an elastic-like component to the response of the emulsion. This is manifested by the appearance of non-zero normal stress differences defined by Eqn. (1). In the small deformation limit (e.g., D << l, Ca < 0.4 ), for a dilute emulsion (0<< 1) Schowalter et al. (1968) found for the first and second normal stress differences: N~ ArE=

32 t/2mr b ~ 2 0 5 a 20 r12 rb~)20 7 tr

(22a) (22b)

where rb is the radius of the undeformed bubble and a is the interfacial tension (see also

166

D.J. STEIN AND F,J. SPERA

Schowalter, 1978). In a Newtonian bubble-free melt both N~ and N2 are identically zero. Evidently, this is not the case in a dilute emulsion. Although (22) are valid only for a dilute emulsion in the small deformation limit, they demonstrate a physical basis for the rod-climbing behavior (Weissenberg effect) observed by Spera et al. (1988) and also in the experiments described below. Note that within the range of validity of Eqn. ( 15 ), Eqn. (22) may be written:

N1 =Zoo-Zr~=(32/5)Cazro(b/(l+O)

(23a)

flow and greatly influence the kinematics of the flow (Rosenblat, 1983). As an example, consider a channelized lava flow. For the typical parameters ?]m= 300 Pa s, ~ = 10 - ~ s- ~, rb=3 mm, a = 0 . 2 N m -t and 0 = 0 . 1 5 , the ratio N~/rre=(32/5)Ca~/(l+(~) is equal to 0.40. Clearly, normal stress differences can be an appreciable fraction of the shear stress. Lava flow dynamics will deviate from the situation of flow of a Newtonian fluid in such instances. Rosenblat ( 1983 ) describes free surface shapes for fluids with non-zero normal stress differences.

and

Experiments

N2 = Trr--~'zz = ( -20/7)Cazroq)/( 1+0) (23b) Note that Eqns. (23) may also be written in terms of the bubble deformation parameter D= (l-b)/(l+b). Obviously, in the low-Ca limit ( C a < l ) , normal stress differences increase as the shear stress and bubble fraction increase and as bubbles become more elongate. Finally, it is instructive to compute the ratio of the normal stress difference to the shear stress in simple shear flow. This is an important quantity because the deviation of a flow field from that for a Newtonian fluid will depend on N~/z~o and N2/zrO. For example, consider flow of Newtonian lava in a deep channel of finite width. The flow field is quite simple: lava moves down the slope, the only component of motion being the longitudinal one. The magnitude of the longitudinal velocity depends on vertical distance from the channel bottom. However, a non-Newtonian fluid with non-zero normal stress coefficients given by Eqns. (1) and (23) as: ~u, = ( 32/5

)qzmrb~O- '

(24a)

and ~2 = - (20/7)q2mrbq~O'- ~

(24b)

will exhibit a secondary helical flow in the cross-sectional plane of the flow. This secondary flow will deform the free-surface of the lava

Methodology Electronic grade GeO2 was melted in platin u m crucibles at 1600°C for 1 hour. The resuiting glass was pulverized using a tungsten carbide "shatterbox". From the powdered glass, five samples of GeO2 melt containing dilute concentrations of air bubbles were prepared at 1150°C in the crucibles eventually used in the viscometer. The bubbles formed passively as the air spaces between the glass grains coalesced into bubbles during melting. Bubble contents during the experiments were estimated from room-temperature immersion densitometry (in ethanol) following the experiments. Densities were obtained in ethanol because of the finite solubility of GeO2 glass in water. Room-temperature densities were corrected for thermal expansion of the vapor by estimating the temperature at which the quenched melt could no longer relax around the shrinking volume of vapor. This temperature was assumed to be about 800°C; consequently, measured densities constitute minim u m values. Several of the specimens were examined microscopically to make simple assessments of the bubble size distribution. Figure 4 shows the bubble frequency-size distributions for one of the low-volume-fraction samples (0 ~ 1% ) and two of the high-volume-

RHEOLOGYAND MICROSTRUCTUREOF MAGMATICEMULSIONS:THEORYAND EXPERIMENTS 6.00

by the high-alumina fire brick crucibles was detected. In order to compute relative viscosities of the emulsions, the viscosity of pure liquid GeO2 is needed. Numerous investigators have studied the viscometric properties of molten germania (Kurkjian and Douglas, 1960; Riebling, 1963; Briickner, 1964; Fontana and Plummer, 1966). Molten germania behaves as a Newtonian fluid with an Arrhenian temperature-dependence of the viscosity in the range of temperatures and shear rates of the experiments (Fig. 5). The data of Fontana and Plummet appear to be the most comprehensive for molten GeO2 and consequently their results have been used for calculation of r/m(T) and subsequently r/r (---r/e/ r/m). In a typical experiment, the emulsion was heated to the desired temperature and the rod was immersed by allowing it to sink into the emulsion under its own weight. Torque measurements were commenced within 30 minutes of immersion, starting with the lowest rotation speed at which a measurable torque could be obtained, progressing to higher speeds thereafter. Torque measurements were averaged during a five-minute period of continuous shearing at a sampling rate of I per second, after which rotation was stopped abruptly. Approximately 15 minutes followed in which no shearing occurred, then the next rotation was begun, accelerating the rod quickly (i.e., in less

-

5.00 0 Q.

4.00 O ¢.) t/3 >

LT 3.00

0

o 2.00

Fontana and Plummer (1966) 1.00

......... 4.00

,

5.00

.........

,

.........

6.00

i ......... ?.00

,

8.00

.........

167

,

9.00

IO000/T (K) Fig. 5. Viscosity versus temperature relation for molten GeO2 from Fontana and Plummet, 1966. The best fit of the data gives the relation: log r/=13638 ( _ + l l 7 ) / T - 5.340 ( + 0 . 0 7 9 ) .

fraction samples (0 ~ 6% ). Mean bubble radii and total volume fraction vapor for the sample are noted on the histograms. Rheometric measurements were performed in the rotating rod rheometer described by Spera et al. (1988). The temperatures of the experiments were between 1100 and 1175°C at shear rates ranging from 0.06 to 7.5 s- ~, although the experiments on the emulsions containing the higher-volume fraction of vapor were mostly restricted to shear rates from approximately 1 to 2 s-~. Sample volumes were about 15 cm 3. No contamination of the GeO2

TABLE IA Rheometric data for experiments over large enough range in shear rate to determine power-law parameters. Presented from left to right are run temperature (T), post-experiment density, bubble volume fraction (¢), power-law parameters m and n (and their variances, a), as well as range of shear rate (j,, computed from Eqn. ( 10 ) of Spera et al., 1988 ), and measured range of shear stress (r) for the experiments Sample

T (°C)

Density (kg/m 3)

¢

rn (Pa s")

am (Pa s")

n

a.

Range of ~ (s - l )

Range of r (Pa)

JA-I JA-2 JA-3 AP-I AP-2 AP-4

1100 1125 1150 1100 1125 1175

3621 3621 3621 3637 3637 3637

0.015 0.015 0.015 0.008 0.008 0.008

64800 34000 27300 44800 31000 13000

1800 950 950 1600 1500 800

0.93 0.87 0.87 0.90 0.92 0.90

0.03 0.03 0.05 0.04 0.05 0.08

0.06-7.41 0.06-6.96 0.24-7.11 0.23-7.04 0.21-7.11 0.23-7.09

4100-416000 3700-242000 12500-173000 13400-305000 11000-219000 7800- 97000

168

D.J. STEIN A N D F.J. 6.00

6.00

(a)

:z:

(b)

~5.50

~5.50

g

~V~ 5 " 0 0

/z

/

z J:

~-- 4 . 5 0 03

,.<, "103

JA-1

4.00

t "/~

~

5.00

~-03

4.50

~1~4.00

I =I00" C 0.015

(3 3.50

"1"

(3 3.50

q

q

3.00

5.00

2.50

.... -2.00

, ....

-1.50

F .... , .... -1.00 -0.50

, ....

0.00

, .... 0.50 t

2¢5

,

0

1.00

. . . .

-2.00

6.00

,

. . . .

-1.50

LOG ANGULAR FREQUENCY ( s - ) 6.00

(c)

. . . .

,

. . . .

0.00

,

. . . .

0.50 t

,

1.00

(d)

03 5.00

~ 4.50 1150" C = 0.015

(D 3.50

S

/

OZ

~'~4 . 0 0 "r 03

~"

L~4.00 "1" (/3

AP-I

II000 C = 0.008

(3 3.50

q

,3.00

5.~

-.,,,b .... : .... i .... -2.00 -1.50 -t.00 -0.50

i .... 0.00

LOG ANGULAR FREQUENCY 6.00

J .... 0.,5O 1

-2.00

, .... -1.50

, .... -1.00

, .... -0.50

, ....

0.00

, ....

0.50

,

1.00

LOG ANGULAR FREQUENCY (s -I)

(s-)

6.00

(e)

~..~5.50

=o

~

03 5 . 0 0

03 5.00

z"

4.50

~- 4.so tD

/

~ 4.00

AP-2

.~, 4.00

1125"

C

~, = 0.008

3.50

0

q

1175" C = 0.008

3.50

5.00

3.00

2.50

....

2.50 1.00

,._,5.50

q

]

-0.50

~=

f

~--- 4 . 5 0 03

~

. . . .

,..,5.50

03 5 . 0 0

03

]

-1.00

LOG ANGULAR FREQUENCY ( s - )

~5.50

2.50

SPERA

....

-2.00

, ....

-1.50

, ....

-1.00

, ....

-0.50

, ....

0.00

,

.... 1,00,

0.501

LOG ANGULAR FREQUENCY ( s - )

2.50

.... -2.00

i .... i .... = .... -1.50 -1.00 -0.50

i .... 0.00

i .... 0.501

i 1.00

LOG ANGULAR FREQUENCY ( s - )

Fig. 6.a-f. Plots of experimental data for power law fluids. Errorbars represent + 1a.

than 5 s) to the next desired rotation speed. Following the measurement at the highest rotation speed for the experiment, the sequence was repeated in reverse, going from high speed to low. Major hysteresis effects were not noted. Each starting emulsion was unique in terms of bubble volume fraction and in the size and

spatial distributions of bubbles. Ca, although never exceeding Cacrit,f , w a s always larger than 1, sometimes significantly so. Bubble elongation ratios as large as 200 occurred in the experiments. Bubble fragmentation due to rod acceleration, as well as tip streaming almost certainly occurred (i.e., Ca> 0.56 generally),

RHEOLOGYAND MICROSTRUCTUREOF MAGMATICEMULSIONS:THEORYAND EXPERIMENTS

169

TABLE IB Rheometric data for experiments in which power-law parameters were not determined. Same variables as Table 1A, except that apparent (Newtonian) viscosity (r/) and its variance are given instead of m and n Sample

T (°C)

Density (kg/m 3)

~

q (Pas)

a, (Pas)

Range ofp (s - l )

Range o f z (Pa)

MA-1 JN-1 JL-l

1125 1125 1125

3526 3534 3530

0.055 0.054 0.053

41000 44800 47300

2000 4600 3100

0.74-7.11 1.13-2.39 1.15-2.48

36000-296000 36000-103000 35000- 94000

probably resulting in changes in bubble size distribution during the experiment. In concentric cylinder viscometry, the measured experimental quantities are angular velocity (~, in radians s -l ) of the inner rotating cylinder or rod and the torque (M) generated by viscous coupling between the stationary outer cylinder (the cup) and the rod. Torque is converted to shear stress along the surface of the rod by:

"c=M/2rtR2 h

(25)

where Rr is the radius of the rod, h is the effective depth of immersion in the fluid, and M represents the experimentally determined torque. For data over a sufficiently large range in shear rate, the method described in Spera et al. ( 1988 ) was used to obtain the constitutive relation. This method is general in that no particular constitutive relation is assumed at the outset. In this procedure, Q-z pairs are fit to an expression of the form: l o g ~ = a 0 + a l logz+aE(lOgz)2+...

(26)

by weighted least-squares polynomial regression, weighting the data according to experimental error in shear stress. In performing the regressions, errors in ~ were considered negligible since these are more than an order of magnitude smaller than errors in torque (see Spera et al., 1988 for discussion). Using the Fstatistic, it was found unnecessary to use the second order or higher terms, implying a power law constitutive relation at fixed temperature and bubble volume fraction. The power-law parameters rn and n of Eqn. (20) are calcu-

lated in terms of the regression coefficients ao and a~ of Eqn. (26) according to:

n= 1/a~

(27a)

and m = [ 10-~°( 1 - c a ' ) / Z a l ]l/a,

(27b)

where C is the square of the ratio of the radii of cup and rod, C=-Rr/Rc. _ 2 2 Relation (27b) corrects a typographical error in Eqn. (12b) of Spera et al. (1988). Table 1A presents data for the power-law parameters m and n for those experiments with sufficient range of shear rate to permit their determination. The power law exponent for these dilute emulsions ranges from 0.87 to 0.93, indicating slight pseudoplastic ("shear-thinning") rheological behavior. This is consistent with prior results (Spera et al., 1988). Onesigma errors on the coefficient rn are between 3 and 6%; uncertainties for the coefficient n fall approximately in the same range averaging around 5%. Figures 6a-f present the experimental data indicative of the power-law rheological model. One-sigma error bars on each data point are shown, as well as the best-fit line from which the power law parameters were determined. Numerical experiments on synthetic powerlaw data with added Gaussian noise revealed that the procedure utilized by Spera et al. (1988) is incapable of recovering the powerlaw exponent (slope of the regression) to sufficient accuracy when the shear rate spans less than about a factor of 20, given typical variances of experimental torque values (Borgia

170

D.J.

AND

F.J.

SPERA

(a)

• ooooo

M

~ooooo u~

STEIN

l

,~ Comparing Eqn. (25) with (28) it is evident that apparent viscosity may be obtained from:

200000

/ I /

z o') 100000

MA-

/

o.oo

0.50 1.00 ANGULAR

I

1125" C ~ = 0.055

/

2.00 2.50 p
3.50

!.50

(b)

,ooooo

"~300000

~O 200000

JN-

I

It25" ¢ •"r. tn 100000

o

~ =

....

0.00

4oo000

0.054

~ ' ~

i ....

i ....

i ....

~ ....

, ....

J ....

0.50 1.00 1.50 2.00 2.50 ,3.00 ANGULAR FREQUENCY ( s - ' )

i

3.50

(c)

~'~----~300000

t,o P-~ 200000 tn

JL- I Y 125" C ~ = 0.053

~: 1ooooo

o .... i .... i .... , .... , .... , .... J .... i 0.00 0.50 1.00 1.50 2.00 2.50 ~.00 5.50 ANGULAR FREQUENCY ( s - )

Fig. 7.a-c. Plotsof experimentaldata for Newtonianfluids. Error bars represent _+1tr. and Spera, 1990). Consequently, when the range of shear rates over which experiments were performed was below this limit, an alternate procedure was employed in which only apparent viscosities were calculated. The apparent emulsion viscosity as defined in Eqn. (20) is simply the ratio Zr0/~rO and is determined from:

?/e,app= (r/~¢~) ( l - C )

(29)

In Table 1B apparent emulsion viscosities of three experiments each with 0 ~ 6% at 1125 ° C are tabulated. Although the span of experimental shear rates for each sample were not sufficient to enable determination of power-law parameters, sample MA-1, which was run at a higher mean shear rate than JN- 1 and JL- 1, has a smaller apparent emulsion viscosity (e.g., 41000 _+2000 Pa s versus 46050 + 3850 Pa s). This is consistent with pseudoplastic behavior as found for the more dilute emulsions of Table 1A. The raw experimental data are graphically portrayed in Figure 7a-c. In Table 2 the range and mean values of capillary number are tabulated for each of the samples. In computing capillary number for the experiments, a melt-vapor surface tension of 0.25 N m - l (Kingery, 1959) was used. The sources of variation in Ca are both the average bubble radius and the shear rate. Capillary numbers vary over two orders of magnitude for the ensemble and range from 0.7 (sample JA1 ) to 111 (AP-I). In the experiments for which bubble size statistics were not available, the average bubble radius was assumed to be 0.1 m m in calculating Ca. All values exceed the small deformation criterion of Taylor (Ca <_0.4). Most, but not all, exceed the critical capillary number for bubble fragmentation by tip-streaming. Because of the relatively large values of Ca, one may expect significant bubble deformation during Couette cylindrical flow. Values for the relative viscosity for each experiment are also tabulated in Table 2. Because qr depends on ZrOfor a power-law fluid, the convention of determining qr at a shear stress value of TrO= 105 Pa, a value close to the mean value for all experiments, was adopted. Although one-sigma errors are fairly large

17 l

RHEOLOGY AND MICROSTRUCTURE OF MAGMATIC EMULSIONS: THEORY AND EXPERIMENTS

TABLE 2 Relative viscosity (r/,) and its variance, range of capillary number and mean capillary number for each experiment, and mean bubble radius determined from post-experimental counts where available. Relative viscosity was determined at a constant shear stress of 10s Pa for the power-law emulsions Sample

T

~h*

a,

(°C) JA-I JA-2 JA-3 AP-I AP-2 AP-4 MA-I JN-1 JL-1

1100 1125 1150 1100 1125 1175 1125 1125 1125

1.60 1.11 1.28 1.04 1.07 0.86 1.57 1.71 1.81

0.09 0.06 0.10 0.05 0.07 0.11 0.10 0.19 0.14

Ca

Ca

(range)

(mean)

Mean r b (mm)

0.7- 93 0.5- 58 1.3- 40 3.5-111 2.2- 74 0.9- 28 4.6- 44 52 -109 12 - 26

18.5 17.5 15.7 40.0 30.3 14.5 13.2 77.4 18.9

0.08 0.08 0.08 0.06 0.44 -

*For power-law emulsions, relative viscosity computed at z---10s Pa. 2.00

~.8o

........ TAYLOR(EON 15)

T/ . 2 ( /l

>~-

04>(,O ~1"60 L) .1

/

-j1.20

MOONEY (EQN 16) _ _ _ EILERS (EQN 17) _ _ ROSCOE (EQN 18)

/

TTHISSTUDY

/ ~

.j

/

rl

/~

/ ~

1.00

0.80

0.00

0.02

0.04

0.06

0.08

VOLUME FRACTION

i

.........

0.10

1

0.12

(¢)

Fig. 8. Plot of relative viscosity-bubble volume fraction. Lower group of curves are plots of relationships developed by Mooney (1951), Eilers (1943), and Roscoe (1952) (see text). The upper curve is a best-fit of experimental data, r/T= 1+ 13.1 ¢. Error bars represent +_I a at each volume fraction. ( a b o u t 10%), i n c r e a s i n g 0 clearly correlates with higher relative viscosity. F u r t h e r m o r e , the d a t a d e v i a t e c o n s i d e r a b l y f r o m the " l o w - d e formation" prediction of Taylor (1934) r/r= 1 + ¢ (see Fig. 8). T h i s is n o t s u r p r i s i n g since e x p e r i m e n t a l capillary n u m b e r s greatly e x c e e d the C a < 0.4 limit for w h i c h Eqn. ( 15 ) is valid. F i g u r e 9 is a p h o t o g r a p h o f a s e c t i o n t h r o u g h the e x p e r i m e n t a l a s s e m b l y following an e x p e r i m e n t in w h i c h a v a p o r - m e l t e m u l sion was s h e a r e d briefly at a b o u t 2 s - I . N o t e

Fig. 9. Photograph of a section through the experimental assembly following an experiment in which a vapor-melt emulsion was sheared briefly at about 2 s- t. Note the range of bubble deformation and the dependence of the deformation on bubble radius. Melt viscosity (~/m) was about l0 s Pa s at 1100°C. Bubble radii from 5 to 0.05 ram, average Ca estimated to be 500. the range o f b u b b l e d e f o r m a t i o n a n d the dep e n d e n c e o f the d e f o r m a t i o n o n b u b b l e radius. Discussion T h e results o f o u r r h e o m e t r i c e x p e r i m e n t s o n dilute viscous g e r m a n i u m dioxide e m u l s i o n s at high t e m p e r a t u r e s i n d i c a t e t h a t the e m u l s i o n s m a y be d e s c r i b e d b y a p o w e r - l a w c o n s t i t u t i v e relation w h e n o b s e r v a t i o n s c o v e r a sufficient

172

range of shear rates to resolve non-linear flow. Table 2 presents data on power-law parameters, viscosities and relative viscosities for the experiments. The power-law index n lies between 0.87 and 0.93, and is consistent with prior results from our laboratory (Spera et al., 1988). Relative viscosities range from about 1.1 at the lowest bubble volume fraction to 1.7 at the highest volume fraction examined. These relative viscosities are significantly higher than those computed from expressions applicable to a wide range of concentrations including the dilute range, given by Eqns, ( 16 )- ( 19 ), which depend explicitly only on the volume fraction of the dispersed phase (0). Figure 8 is a plot of relative viscosity vs. volume fraction of dispersed phase for emulsions in the dilute range, and shows the results of our experiments in relation to the expressions given in Eqns. ( 16)( 18 ) and also for the Taylor results, r/r = 1 + 0. For the power law emulsions, relative viscosity has been computed at constant shear stress ( 105 Pa). Variation of relative viscosity at each volume fraction is shown by I a error bars. The data are tentatively correlated by the expression rlr= 1.0+13.1 0. Although the precise value of dqr/dO is not known, it appears much larger than unity, the value appropriate for small D and 0. This is perhaps the most significant result from the experiments. The much higher relative viscosities [logarithmic relative viscosities are 2-3 times higher than what is predicted by Eqns. ( 16 )- (19) ] are likely attributable to bubble deformation at high capillary number. The range and average values of Ca presented in Table 2 are necessarily based only upon the observed (final) size distributions of bubbles. The theory developed by Schowalter et al. (1968) and by Frankel and Acrivos ( 1970 ) is valid only in the limit of small bubble or drop deformation, and predicts the existence of non-zero normal stress differences. Study of suspensions of deformable droplets with elastic interfaces (Barth6sBiesel and Chhim, 1981 ) yields an explicit dependence of relative viscosity on both volume

D.J. STEIN AND F.J. SPERA

fraction and capillary number. To first order, this expression is of the form ~r 1 + ao + bO ( Ca ): where a and b are constants. From Eqn. (21 ) and the definition of Ca, one might a priori expect to find a dependence of relative viscosity on Ca of the form: =

rlr=rlm(T)-"m(O,T)(Cacr/rb) "-t

(30)

giving a Ca"- ~dependence. With n around 0.9, this would be Ca 1/1°, a rather weak dependence of r/r on shear rate. Note that shear-rate dependences of the viscosities for MA-1, JN-1 and JL-1 have not been determined. Thus, since values of 0 and Ca for each shear rate could not be determined, no attempt has been made to include Ca independently in an expression for relative viscosity. However, it is obvious that for any emulsion having an explicitly shear-rate dependent (or, implicitly, Ca-dependent) viscosity, the relative viscosity must also bear that dependence. In addition, in one experiment (MA-1), an estimate of the magnitude of the first normal stress difference was made. The measured bubble volume fraction was 5.5 vol.% and by examining the rotor after the completion of the experiment, it was determined that the emulsion had climbed 2 cm up the rotor. Spera et al. ( 1988 ) outline a procedure for estimating the first normal stress difference from measurements of rod climb based on the methods developed by Joseph and co-workers (Joseph et al., 1973, 1984; Joseph and Fosdick, 1973 ). By combining this climb height with the maxi m u m rotation rate of the rotor during the experiment, we determined the first normal stress difference to be approximately NI ~, 5500 Pa, At the temperature of the experiment (1125°C) the shear stress at the m a x i m u m shear rate ( 7.1 s- t ) was about 2.9 X 105 Pa, so the normal stress differences are 2% of the shear stress. In contrast, predictions from the dilute emulsion, small-deformation theory of Schowalter et al. (1968) are for much higher normal stress differences (of the order of 10 MPa), indicating that experimental bubble

RHEOLOGY AND MICROSTRUCTURE OF MAGMATIC EMULSIONS: THEORY AND EXPERIMENTS

deformation has far exceeded the value appropriate to the small deformation theory for which Eqns. (22) hold. Many of the observations of this study are relevant to understanding magmatic flows, especially those occurring at or near the surface, where vesiculation is invariably expected and usually found. Typical capillary numbers in nature range from 0. l to 10, or even higher. Bubble deformation is significant above Ca = 0.4, fragmentation begins at Ca = 0.56 for tip-streaming, and many of the features of bubble distribution and deformation observed in nature may aid in quantifying such phenomena as variations in shear rate across a lava flow or shallow intrusion. A significant outcome of this study is the finding that the presence of deforming bubbles has significant effects on the relative viscosity of magmatic emulsions. The plot of relative viscosity for the emulsions studied in these experiments (Fig. 8) shows a much more rapid increase with volume fraction of bubbles than that predicted by Eqns. ( 15 )- ( 18 ). In fact, the relative viscosity for low-volume-fraction suspensions of solid particles, r/r= l + 2.5 ~ falls in the trend of Eqns. ( 16 )- ( 18 ), and is significantly lower than that for the high-Ca emulsion at least up to l0 vol.%. This has significance for magmatic flows which have not cooled very far below the liquidus, in that the processes of vesiculation can have a much greater effect on the viscosity of the flow than those of crystallization. Capillary numbers for a typical basaltic lava flow fall in the range 0.1l0 [see discussion following Eqn. (24) ] and bubble deformation in such flows can be significant. Finally, the presence of deforming bubbles leads to viscoelastic phenomena embodied by normal stress differences. While the normal stress differences estimated for the highest shear rates in our experiments are not large (about 2% of the shear stress ), they are significant, and simple calculations show that they could be as great as 25-40% of the shear stress

173

under conditions entirely relevant to natural magmatic flows. Normal stress differences will produce a flow velocity field significantly different from that occurring for example, in Newtonian flow in an open channel. Inferences about lava flow rheology made from observations of morphology will be made more robust by taking these effects into account. More experiments are needed before application to volcanological systems can be made.

Acknowledgements The authors wish to express appreciation for the helpful comments of two anonymous reviewers. This research was supported by grants from the National Aeronautics and Space Administration (NASA-NAG- 1452), the National Science Foundation (NSF-EAR-8816103 ) and the United States Department of Energy (DOE-FG03-89ER 14050).

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