Rheology of Bubble-Bearing Magmas

Journal of Volcanology and Geothermal Research 87Ž. 1998 15±28

Rheology of bubble-bearing magmas

Michael Manga a,), Jonathan Castro a, Katharine V. Cashman a, Michael Loewenberg b a Department of Geological Sciences, UniÕersity of Oregon, Eugene, OR 97403 USA b Department of Chemical Engineering, Yale UniÕersity, New HaÕen, CT USA Received 30 March 1998; accepted 20 July 1998

Abstract

The rheology of bubble-bearing suspensions is investigated through a series of three-dimensional boundary integral calculations in which the effects of bubble deformation, volume fraction, and shear rate are considered. The behaviour of bubbles in viscous flows is characterized by the capillary number, Ca, the ratio of viscous shear stresses that promote deformation to surface tension stresses that resist bubble deformation. Estimates of Ca in natural lava flows are highly variable, reflecting variations in shear rate and melt viscosity. In the low capillary number limitŽ. e.g., in carbonatite flows bubbles remain spherical and may contribute greater shear stress to the suspension than in high capillary number flows, in which bubble deformation is significant. At higher Ca, deformed bubbles become aligned in the direction of flow, and as a result, contribute less shear stress to the suspension. Calculations indicate that the effective shear viscosity of bubbly suspensions, at least for Ca-0.5, is a weakly increasing function of volume fraction and that suspensions of bubbles are shear thinning. Field observations and qualitative arguments, however, suggest that for sufficiently large CaŽ Ca greater than about 1. the effective shear viscosity may be less than that of the suspending liquid. Bubbles reach their quasi-steady deformed shapes after strains of order one; for shorter times, the continuous deformation of the bubbles results in continual changes of rheological properties. In particular, for small strains, the effective shear viscosity of the suspension may be less than that of the liquid phase, even for small Ca. Results of this study may help explain previous experimental, theoretical, and field based observations regarding the effects of bubbles on flow rheology. q 1998 Elsevier Science B.V. All rights reserved.

Keywords: rheology; bubble-bearing magmas; capillary number

1. Introduction compositionsŽ. McBirney, 1993 . As a result, numer- ous studies have considered the effect of various Of all the physical properties of magmas that physico-chemical properties on magma rheology, in- affect their dynamic behaviour, the viscosity of the cluding the effects of composition, temperature, magma is most variable, varying by over 15 orders volatile content, crystallinity, and bubble content of magnitude within the range of natural magma Že.g., Bottinga and Weill, 1972; Shaw, 1972; Murase and McBirney, 1973; Spera et al., 1988; Bagdas- ) Corresponding author. Tel.: q1-541-346-5574; Fax: q1-541- sarov et al., 1994. . Attempts to determine the effec- 346-4692; E-mail: [email protected] tive viscosities in various geologic settings include

0377-0273r98r$19.00 q 1998 Elsevier Science B.V. All rights reserved. PII: S0377-0273Ž. 98 00091-2 16 M. Manga et al.rJournal of Volcanology and Geothermal Research 87() 1998 15±28 direct measurementsŽ e.g., Pinkerton and Norton, Pinkerton et al., 1996. . For example, Pinkerton et al. 1995.Ž , experimental studies e.g., Bagdassarov and Ž.1996 noted that `bubble-rich lavas appear more Dingwell, 1992.Ž , and theoretical approaches e.g., viscous than bubble-poor lavas' in carbonatite flows Lange, 1994. . Nearly all erupting lavas carry bubbles from Oldoinyo Lengai. Similar observations were at some point during their ascent and emplacement. made in some Hawaiian lavasŽ e.g., Hon et al., In this paper we use numerical calculations to deter- 1994. ; however, in some high-velocity basaltic erup- mine the effect of bubbles on magma rheology. tions, bubble-rich lavas appeared to be `very fluid' The present study is motivated by the wide range relative to bubble-free lavas in the same flowŽ Mac- of experimental results and theoretical models de- Donald, 1954; Shaw et al., 1968; Lipman and Banks, scribing the effects of bubbles on the effective vis- 1987. . While rhyolitic lavas have not been observed cosity of bubbly magmas. For example, the experi- to erupt, well-preserved structures in Holocene flows mental measurements of Bagdassarov and Dingwell in Oregon and California have been used to infer Ž.1992 suggest that the presence of bubbles in rhyo- rheological properties in obsidian and pumiceous lite significantly reduces their effective viscosity. lavasŽ Fink, 1980a, 1984; Castro and Cashman, Stein and SperaŽ. 1992 , however, find that bubbles 1996.Ž. . Fink 1980a studied the spacing of pumiceous increase the effective viscosity by an amount that diapirs in obsidian flows and concluded that the exceeds the increase that would occur for an equiva- viscosity ratio between pumice and obsidian varied lent volume fraction of solid particles. Theoretical from 50 to 100. While this result is consistent with and experimental studies of the effects of bubbles on experimental observations of froth viscositiesŽ e.g., the rheology of magmas and lavas are reviewed by Sibree, 1933. , folding relations at Big Glass Moun- Stein and SperaŽ. 1992 . tain as well as at many other flows suggest that Inferences drawn from direct observations and obsidian is more viscous than pumice with the same field relations of bubble-bearing lavas also provide compositionŽ. Castro and Cashman, 1996 . qualitative and quantitative indications of flow rheol- In this study we isolate the effects of bubbles on ogyŽ e.g., Fink, 1984; Lipman and Banks, 1987; flow rheology through numerical simulations of sus-

Table 1 Melt viscosity and capillary number estimates for viscous deformation of bubbles with a radius of 1 mm Lava and flow type Melt viscosityŽ. PaS Capillary number Ž. Ca Carbonatitea2 1±10 0.02±0.2 Mauna Loa basalt, near ventb2 10 0.2±1.0 Mauna Loa basalt, distalb2 10 1.0±5.0 Kilauea basaltc 234±548 0.8±1.8 Etna basaltd3 10 0.1±0.4 Etna basalte3 1.6=10 5.4 Mount Spurr, andesitef6 10 45 Mount St. Helens conduit daciteg7 10 350 h4 GeO2 10 13±78 Rhyolitei11 10 2±300 a Viscosity data from Dawson et al.Ž. 1994 ; strain rates from Nakada et al. Ž. 1995 . b Lipman and BanksŽ. 1987 . c Pinkerton et al.Ž. 1996 . d Polacci and PapaleŽ. 1997 . e Kilburn and GuestŽ. 1993 . f Strain rates were calculated from ascent data presented by Neal et al.Ž. 1995 . g Rutherford and HillŽ. 1993 , and Klug and Cashman Ž. 1996 . h Stein and SperaŽ. 1992 . i Bagdassarov and DingwellŽ. 1992 and Pinkerton and Stevenson Ž. 1992 . M. Manga et al.rJournal of Volcanology and Geothermal Research 87() 1998 15±28 17 pensions of bubbles. We begin by discussing two tend to keep bubbles spherical, is characterized by parameters that characterize the behaviour of bubbles the capillary number: in viscous flows: the capillary number and the vol- mGa ume fraction of bubbles. Because bubbles will de- Cas Ž.capillary number Ž. 1 g form in response to viscous stresses, the rheology of bubbly liquids depends on the shear rate G. The where g is the surface tension, a is the undeformed importance of shear stresses, which act to deform bubble radius and m is the suspending fluid viscos- bubbles, relative to surface tension stresses, which ity. We also might expect the rheology of bubbly

Fig. 1.Ž. a Distribution of bubble deformation D Žsee inset for definition . and Ž. b thin section image from 8 ka obsidian from Mayor Island, New Zealand. The thin was cut parallel to the direction of bubble elongation; the inset shows a thin image of the same sample perpendicular to the larger image. 18 M. Manga et al.rJournal of Volcanology and Geothermal Research 87() 1998 15±28 liquids to depend on the volume fraction of the flows, ranging from carbonatitesŽ. see Table 1 to dispersed bubbles: rhyoliteŽ. see Fig. 1 . We are able to explore a sufficiently large range of parameter space to f Ž.volume fraction . Ž. 2 demonstrate that bubble deformation can have a In Table 1 we list the range of Ca for various significant effect on the rheological properties of magma compositions and geologic settings assuming magmas. However, there are also many natural flows a bubble radius of 1 mm. For small Ca, the magni- in which Ca is larger than the range of values we tude of bubble deformation, DsŽ.Ž.LyB r LqB , can study numerically. In §4.2 we thus consider is approximately equal to Ca Ž.Taylor, 1934 ; here, L some field observations that provide qualitative re- and B are the major and minor axes of the deformed sults for the limit Ca41. bubble, respectively. For Ca41 the aspect ratio of bubbles scales as Ca3, and the bubbles develop pointed endsŽ. e.g., Stone, 1994 . Thus, given the 2. Model range of Ca in natural magmas and lavasŽ. Table 1 we can expect bubble shapes to range from nearly Consider a suspension of bubbles, with radius a, spherical Ž.Df0 in carbonatites and some basalts being sheared at a rate G. We assume that flow on Ž.e.g., fig. 2 in Cashman et al., 1994 to highly the length scale of individual bubbles occurs at low elongated in dacites and rhyolites. In Fig. 1 we Reynolds number: present measurements of D for vesicles in a peralka- s 2r line rhyolitic clast from the uppermost unit of the 8 Re rGa m<1,Ž. 3 ka obsidian flow from Mayor Island, New Zealand. where r is the fluid density, and m the is the The geology and history of this Mayor Island erup- suspending fluid viscosityŽ. Bentley and Leal, 1986 . tion are described by Stevenson et al.Ž. 1993 . D was The equations of motion governing flow in the sus- obtained from thin-section images made parallel to pending fluid are the Stokes equations: the elongation direction, and measurements are based y q 2 s P s on three-dimensional analysis of bubble shapes. Al- =p m= u 0 and = u 0.Ž. 4 though vesicle deformation can be large, Df0.7, Here, p is the pressure and u is the fluid velocity. the average deformation is modest, Df0.3. The For simplicity, we assume that buoyancy-driven mo- shape of the vesicles preserved in the Mayor Island tion and Brownian motion are negligible compared obsidian may actually be representative of bubble with flow associated with shearing. We also assume shapes in the molten rock: the pointed ends on the that the bubbles do not change volume. bubbles are preserved even though they will relax Across the surface of each bubble, there is a most rapidly owing to the high local curvature. The pressure jump D p arising from the surface tension, time scale for quenching of the sample must there- g : fore be much shorter than the time scale for relax- D sgksg= P ation of the deformed bubbles. Also, the inset of Fig. p n,5Ž. 1b shows that the cross-sections of bubbles are not where k is the local curvature of the bubble surface, circular but elliptical, with an aspect ratio consistent and n is a unit normal vector directed outward from with that expected for bubbles in a simple shear flow the bubble surface. Eq.Ž. 5 applies if surface tension Ž.e.g., Kennedy et al., 1994 . is a spatially constant interfacial property. Surface Here we present the results of numerical simula- tension may vary over of the surface of the bubble if tions of three-dimensional interacting and deforming there are temperature gradients, or if surfactants are bubbles in sheared suspensions that allow us to present and the concentration of surfactants on the calculate the rheological properties of bubbly liquids interface is spatially variable. Eq.Ž. 4 , subject to as a function of f and Ca. Due to numerical limita- boundary conditionŽ. 5 , apply at all points in the tions, we can only compute rheological properties for suspending fluid. 0-f-0.45 and 0-Ca-0.5. This range of Ca, We will consider three classes of models, as however, is large enough to cover many natural illustrated in Fig. 2, each of which allows us to focus M. Manga et al.rJournal of Volcanology and Geothermal Research 87() 1998 15±28 19

Fig. 2. Geometry of the three problems considered in this study:Ž. a single bubble, Ž. b ordered suspension, Ž. c random suspension. For Ž. b andŽ. c we use periodic boundary conditions in all directions. on different features of the problem. While the model accuracy of the Loewenberg and HinchŽ. 1996 code illustrated in Fig. 2c, consisting of a random suspen- for large bubble deformation by comparing the shapes sion of bubbles, is the most realistic and relevant of bubbles in an axisymmetric extensional flow with model, we are limited in the range of Ca and f that those calculated using the numerical procedures de- can be studied. Calculations for a single bubbleŽ Fig. scribed in Manga and StoneŽ. 1993 . These compar- 2a. allow us to consider larger Ca, and calculations isons are shown in fig. 4 of Loewenberg and Hinch for ordered suspensionsŽ. Fig. 2b allow us to study Ž.1996 and fig. 3 of Loewenberg and Hinch Ž. 1997 . higher f. The BIM is well-suited for problems involving deforming interfaces, in particular, those incorporat- 2.1. Numerical method ing interfacial properties such as surface tension. While the BIM allows us to study the motion, defor- The Stokes equationsŽŽ.. Eq. 4 can be expressed mation, and interaction of the bubbles by focusing as integral equations, and through the application of only on the surfaces of the bubbles, the method also the divergence theorem and the Lorenz reciprocal implicitly solves for the velocity and stresses theorem, can be rewritten as sets of equations involv- throughout the bulk of the liquid phaseŽŽ. see Eq. A9 ing only surface integrals over the surface of all the in Manga and Stone, 1995. . bubblesŽ. e.g., Rallison and Acrivos, 1978 . The resulting integral equations can be solved using stan- 2.2. Effect of bubbles on stresses in the suspension dard numerical techniquesŽ. e.g., Pozrikidis, 1992 . The effective stress tensor for the suspension This approach is known as the boundary integral ²:sijis given by the average of the actual stress s ij methodŽ. BIM , and has the advantage of involving over a volume V of the suspension: surface integrals and thus requires only a numerical 1 ²:s s s description of the shape of bubbles. Most previous ijH ijdV.6Ž. three-dimensional studies of deforming drops and V V bubbles in suspensions using the BIM approach have Following BatchelorŽ. 1970 and Pozrikidis Ž. 1993 , if been limited to drops with the same viscosity as the the suspending fluid is Newtonian and the flow surrounding fluidŽ e.g., Pozrikidis, 1993; Manga, occurs at low Reynolds numbers, then: sy - )q - ) 1997.Ž or single drops e.g., Rallison, 1984; Kennedy ²:sijd ijp 2m e ij et al., 1994. . Here we use the numerical approach of 1 q gk ym q Loewenberg and HinchŽ. 1996 which we found to be H 2 nxijŽ.un ijun ji dS, V S more accurate and computationally efficient than the b method of Manga and StoneŽ. 1995 . We verify the Ž.7 20 M. Manga et al.rJournal of Volcanology and Geothermal Research 87() 1998 15±28

where Sb indicates the surfaces of all the bubbles in random suspensionŽ. Fig. 2c . As the number of the suspension, eij is the rate-of-strain tensor, x is a triangles increases, the accuracy of the calculations position vector, n is again a unit normal vector increases; Loewenberg and HinchŽ. 1996, 1997 pro- directed outward from the bubbles, and²: indicates vide a detailed discussion of the numerical error a volume-averaged quantity. The velocity u, and the resulting from discretizing the surface of the bubbles. shape of the bubbles used to calculate k, are deter- In the random suspension we use 8 bubbles. Calcula- mined by solving the integral equation representation tions with 6 and 12 bubbles produce essentially of Stokes equationsŽ. see Section 2.1 . The last term identical results. on the right-hand sideŽ. RHS of Eq. Ž. 7 represents To describe the time-evolution of bubble shape the additional stresses contributed by the suspended and position, we time-integrate the velocity at inter- bubbles. facial grid pointsŽ. nodes of triangles on the surface We normalize stresses by the viscous shear stress of each bubble. For the single bubble problem, we mG, lengths by a, velocities by Ga, and time by run the calculations until a steady shape is reached. 1rG. The additional normalized stresses due to the For the ordered and random suspensions, rheological dispersed bubbles, hereafter denoted by the tensor properties vary periodically in time, as shown by

Sij, are given by: PozrikidisŽ. 1993 ; thus we run the calculations for a 3f sufficiently long time that we can time-average quan- S s 2 k nxy Ž.Ž.unqun dS,8 tities. Small temporal fluctuations of rheological ijp H Ca i j i j j i 4 Sb propertiesŽ. see Fig. 9 shown later are due to tempo- where all quantities on the RHS are now dimension- ral changes in the relative positions of the bubbles less. and their resulting interactions. Typical computation The shear viscosity of the suspension, relative to times on an UltraSparc 2 for a single simulation of that of the suspending liquid, is given by: each of the three problems shown in Fig. 2 are about 2, 5 and 10 h, respectively. effective viscosity of bubbly fluid m s1qS s . rel 12 viscosity of suspending fluid Ž.9 3. Results

Normal stress differences will also result from bub- r In Fig. 3a we show the added shear stress S12 f ble deformation. These non-dimensional quantities due to a single bubble, corresponding to f<1, as a are defined asŽ. e.g., p. 56 in Leal, 1992 : function of the capillary number. S12 is divided by N sS yS and N sS yS .10Ž. f in order to facilitate comparisons between results; 1112212233 r S12 f is proportional to the added shear stress per Eq.Ž. 7 is in fact valid only if the suspended bubble. As Ca increasesŽ corresponding to an in- bubbles are force-free, so that the last integral on the creasing shear rate. , S12 decreases, implying that RHS is independent of the choice of the origin. For fluids containing deformable bubbles are shear thin- the problem considered here, stresses associated with ning. As Ca increases, the magnitude of deformation the shear flow are assumed to be much greater than increases as shown in Fig. 2b. TaylorŽ. 1934 deter- those due to density differences. mined the deformation of a single bubble in a shear flow, with the measure of distortion defined as Ds 2.3. Solution details Ž.Ž.ŽLyB r LqB see inset of Fig. 3b for definitions of L and B. so that 0FD-1. For small values of In order to solve the integral equations, we de- Ca, DsCa Ž.Taylor, 1934 , in agreement with the scribe the shape of each bubble with a deforming numerical results for the full range 0-CaF0.5 mesh of trianglesŽ. Loewenberg and Hinch, 1996 . considered here. For Ca41, bubbles become highly We use 1620 triangles for single bubble calculations elongated with an aspect ratio LrB;Ca3 Že.g., Ž.Fig. 2a , 1280 triangles for the ordered suspension Stone, 1994. . Our numerical procedure, at least for Ž.Fig. 2b , and 500 triangles on each bubble for the three-dimensional bubbles, does not allow us to cal- M. Manga et al.rJournal of Volcanology and Geothermal Research 87() 1998 15±28 21

viscosity, mrel, of the suspension of spheres is given byŽ. e.g., Roscoe, 1952 : s y y5r2 mrel Ž.1 1.35f .11 Ž. Eq.Ž. 11 is similar to typical expressions for the

f-dependence of mrel for non-Brownian spheresŽ see Russel et al., 1989 Chap. 14. . As f increases and bubble±bubble interactions become more important, the contribution of each bubble to the shear stress decreases. By contrast, particle±particle interactions between solid spheres increase the shear stress associated with each sphere. In Fig. 5a we consider the effects of both f and r Ca on the shear stress S12 f for the case of a random suspension. The results are consistent with those shown in Fig. 3 in that suspensions of bubbles are shear thinning. Also, the shear stress contributed

r Fig. 3.Ž. a Shear stress, S12 f, and Ž. b bubble deformation D Ž.see inset as a function of Ca for a single bubble Ž. Fig. 2a . culate the high curvature associated with the pointed ends of highly deformed bubbles and thus limits our calculations to CaF0.5. r In Fig. 4 we show the added shear stress S12 f as a function of f for Cas0.3 in an ordered r suspension. For comparison we show S12 f for a suspension of rigid spheres assuming that the relative

r Fig. 5.Ž. a Shear stress, S12 f, as a function of Ca for a random suspension with eight bubbles, and f s0.1, 0.2 and 0.3. Points for f s0 are for the single bubble calculationŽ.Ž. Fig. 1a . b r Fig. 4. Shear stress, S12 f, as a function of f in an ordered Normal stress differences, Eq.Ž. 10 , as a function of Ca for a suspensionŽ. Fig. 2b for Cas0.3. For comparison, the shear random suspension with eight bubbles and f s0.3. Points for stress in a suspension of rigid spheres is shown. f s0 are for the single bubble calculation. 22 M. Manga et al.rJournal of Volcanology and Geothermal Research 87() 1998 15±28 by each bubble decreases as the volume fraction tributed by bubbles is due to the additional viscous increases, at least for Ca greater than about 0.1. dissipation occurring within the fluid that must flow Finally, in Fig. 5b we show computed normal over and around the bubbles. For spherical bubbles, r r stress differences, N12f and N f, along with the the distortion of streamlines and the associated vis- normalized analytical solutions of Schowalter et al. cous dissipation is greater than that for deformed and Ž.1968 valid for small Ca: elongated bubbles. ) 32 20 In Figs. 4 and 5 we noted that for Ca 0.1, the r s r sy N12f Ca and N f Ca.12Ž. contribution of each bubble to the total shear stress 57decreases as f increases. A physical explanation is In Eq.Ž. 12 , as before, stresses are normalized by illustrated in Fig. 6b. As f increases, bubbles are mG. The numerical results deviateŽ. slightly from the forced to slide over and around each other. Because Schowalter et al.Ž. 1968 results for Ca greater than the surface of the bubble is a free-slip boundary, that about 0.4. In a Newtonian fluid without bubbles is, the surface offers no shear resistanceŽ as opposed these normal stress differences would be zero. They to the no-slip surface on rigid particles. , deformable are non-zero here due to the deformation of the bubbles can slide over each other. As a result, the bubbles. In fact, their magnitude can become a sub- dissipation of energy in the region between the bub- stantial fraction of the shear stress which suggests bles is reduced. that caution should be used in modeling magmas containing deformable bubbles as Newtonian fluids, 4.1. EffectiÕe Õiscosity of bubbly magmas as noted previously by Spera et al.Ž. 1988 . For example, if fs0.3 and Cas0.3, the first normal In Fig. 7 we compare the relative shear viscosity, stress difference is about half the total shear stress. s q mrel1 S 12 , based on our calculations with values obtained from other models and experimental measurements. We show numerical results for Cas0.3 4. Discussion for both ordered and random suspensionsŽ filled and open symbols, respectively. . The curve labeled `Stein The numerical results in Figs. 3 and 5 are consis- and Spera' is the best-fit line for experimental mea- tent with experimental measurements of the shear - surements in GeO2 for Ca41 and f 0.055Ž Stein thinning behaviour of bubbly liquidsŽ e.g., Shaw et and Spera, 1992. . Three theoretical models are al., 1968. . A physical explanation for this behaviour shown: TaylorŽ. 1932 valid for f<1 and spherical is illustrated in Fig. 6a. The extra shear stress con- bubbles; DobranŽ. 1992 ; Jaupart and AllegreÁ Ž. 1991 . We find that bubbles, even for the modest deforma- tions that arise for Cas0.3, have a small effect on

the suspension's viscosity, with mrel increasing by only about 25% for a volume fraction of 0.45. The model used by Jaupart and AllegreÁ Ž. 1991 is consistent with the experimental measurements of SibreeŽ. 1933 for bubbly liquids. However, the `froths' studied by Sibree were stabilized by an organic colloid that produces absorption layers on the surface of the bubbles. It is possible that the reduced mobility of these layers made the bubbles behave more like rigid particles. Indeed, the measurements of Sibree, and the model of Jaupart and Fig. 6.Ž. a Illustration of bubble shapes and streamlines for small Allegre,Á are similar to typical models for suspensions Ca Ž.left and high Ca Žright . . Deformed bubbles result in less deformed streamlines, and thus less viscous dissipation.Ž. b De- of rigid spheres. formed bubbles can `slide' over each other more easily than The experimental measurements of Stein and spherical bubbles. SperaŽ. 1992 are more puzzling because at high M. Manga et al.rJournal of Volcanology and Geothermal Research 87() 1998 15±28 23

In summary, the theoretical results presented in Section 3 indicate that the relative viscosity of a suspension of bubbles is a weakly increasing function of volume fraction. In addition, suspensions of bubbles are shear thinning.

4.2. High shear rates() Ca41

The estimates of Ca for natural flowsŽ. Table 1 indicate that in some cases Ca is greater thanŽ and sometimes even much greater than. one. Unfortu- nately, the pointed ends that develop on bubbles for Ca greater than about 0.5 prevent us from calculat- ing rheological properties at these conditions. How- ever, in the limit that Ca41, we can develop some

scaling estimates for mrel. Consider a suspension of bubbles in a simple shear flow. Assuming that surface tension is negligible Ž.Ca41 and that streamlines are not distorted by the bubbles, viscous dissi- s pation occurs only in the liquid phase so that mrel 1-f. Thus, in contrast to the Ca-0.5 limit we ) considered numerically in which mrel 1, if Ca is - sufficiently large then mrel may be 1. Here we consider a set of observations of small scale folds in rhyolitic obsidian lavas that indicate Fig. 7. Relative viscosity, Eq.Ž. 9 , as a function of volume fraction that the relative shear viscosity may indeed be less f for various models. Results of this study for Cas0.3 are than one for Ca41. If the liquid viscosity is 1011 shown for orderedŽ. solid disks and random Ž open circles . suspen- Pa sŽ. Webb and Dingwell, 1990 , the surface tension sions. Other results are shown with solid curves: TaylorŽ. 1932 , gf0.2 NrmŽ. Murase and McBirney, 1973 , and a valid for spherical bubbles and low volume fractions; Dobran Ž.1992 ; Jaupart and Allegre Ž. 1991 ; experimental results of Stein typical bubble radius is 1 mm, then G would have to Á y9 y1 and SperaŽ. 1992 . be less than about 2=10 s for Ca to be less than 1. This shear rate is two orders of magnitude smaller than the lower bound of estimated shear rates associated with large scale surface folds in obsidian capillary numbers the effect of bubbles is reduced flowsŽ. Fink, 1980b . In fact, at such low strain rates, Ž.see Figs. 3a and 5a . Yet, the Stein and Spera it would take about 50 years to produce the folds we measurements suggest a viscosity increase that is consider in this sectionŽ. an unreasonably long time even greater than that for a suspension with the same suggesting that Ca is indeed 41. volume fraction of solid spheres. One possible expla- Centimeter-scale fold structures in late Holocene nation is that the highly deformed bubbles in the flowsŽ e.g., Big Glass Mountain, CA, and Big Obsid- Stein and Spera samples contributed to the shear ian Flow, OR. provide evidence of the relative vis- stress by behaving as long thin rigid particles, per- cosities of bubbly rhyolitic lavas. Fig. 8 depicts a haps due to quenching or crystallization on the sur- ptygmatically folded obsidian layer within a matrix face of the bubbles. The relative viscosity of a of coarsely vesicular pumice from Big Glass Moun- suspension of rods with fs0.05 and aspect ratio 20 tain, CA. Ptygmatic fold structures are typically in a simple shear flow isf1.4Ž Shaqfeh and produced by buckling of a highly viscous layer in a s Fredrickson, 1990. , similar to mrel 1.66 predicted less viscous matrixŽ. Ramsay and Huber, 1987 . As by the Stein and Spera best fit curve for fs0.05. the assemblage depicted in Fig. 8 was shortened, the 24 M. Manga et al.rJournal of Volcanology and Geothermal Research 87() 1998 15±28

Fig. 8.Ž. a Glassy obsidian layer, ff0, folded in coarsely vesicular pumice, ff0.5, from Big Glass Mountain, CA.Ž. b Trace of the fold structure showing thickness, T, and arc wavelength, w, of the folded layer. m12and m are the viscosities of the folded obsidian layer and the pumiceous matrix, respectively. Measurements of wavelength and thickness from this sample indicate that the relative viscosity of the s r f bubbly matrix is mrelm 2m 1 0.3.

f - obsidian layer Ž.f 0 buckled while the bubble-rich pumiceous medium, i.e., mrel 1. Indeed, all buckle Ž.ff0.5 matrix deformed around the fold. Thus, folds observed in obsidian flows are less vesicular the geometry of these structures is a consequence of than their surrounding matricesŽ Castro and Cash- the obsidian layer having a higher viscosity than the man, 1996. . Here we estimate the relative viscosity M. Manga et al.rJournal of Volcanology and Geothermal Research 87() 1998 15±28 25 of the vesicular and glassy lavas by applying buck- merely as a preliminary attempt to interpret the ling theoryŽ. Biot, 1961 to folded obsidian-pumice observed geometries of glassy folds from Big Glass assemblages. Mountain, CA. According to buckling theory, the dominant arc Measurements of the relationship between w and f wavelength w of a single layer fold is established at T for the fold in Fig. 8 suggest that mrel 0.3. Other the onset of deformation and is a function of the values of mrel for single layer glassy folds from Big layer thickness T and the relative viscosity of the Glass Mountain are listed in Table 2. We consider bubble-rich lava, mrel Ž.Biot, 1961 . Assuming the only samples in which folded layers are separated by folded layer is bubble-free, the relationship between many fold wavelengths so that no other folds influ- - mrel and the fold wavelength is expressed byŽ e.g., ence the folding. For all folds examined, 0.002 - Biot, 1961. : mrel 0.3. While we would be overly optimistic to assume that mrel obtained by the application of the Ž. T 3 Biot 1961 folding analysis is quantitatively infor- s mrel 42 .Ž. 13 mative, at least qualitatively, bubbly rhyolite appears ž/w to be much less viscous than bubble-free rhyolite. Such viscosity contrasts are most likely a conse-

The relationship between mrel and fold geometry quence of the presence of bubbles given that the described by Eq.Ž. 13 assumes that both the folded composition and crystallinity are indistinguishable material and surrounding matrix are Newtonian flu- between the obsidian and pumiceŽ Laidley and ids and that the bubbles do not change volume McKay, 1971; Fink, 1982; Grove and Donelly-Nolan, during deformation. In addition, folding was proba- 1986. . bly produced by a flow more closely resembling In contrast to the results of numerical calculations pure shear than the simple shear considered in this presented in Section 3, folding analysis of structures paper. Provided Ca is very large and strains are in obsidian flows suggest that the relative viscosity smallŽ limits that apply for the folds considered of bubbly rhyolite may be less than one. Experimen- here. , the surface tension stresses that produce the tal measurements of Bagdassarov and Dingwell ) - normal stresses in Fig. 5b will be small, and the Ž.1992 for Ca 1 also indicate that mrel 1, and Newtonian model may actually be a reasonable ap- that with increasing volume fraction the relative s q y1 proximation. Given the limitations of Newtonian viscosity decreases as mrel Ž.1 22.4f . The buckling theory, we will use the BiotŽ. 1961 model magnitude of the decrease, however, is much greater

Table 2 Measured and computed parameters on obsidian folds from Big Glass Mountain, CA a b Sample Thickness T Wavelength w Shortening fmrel Ž.cm Ž. cm Ž. % SBG-1 0.46 4.8 42 47 0.04 SBG-3 1.1 22.1cd 55 f50 0.005 SBG-55 0.78 7.6 51 67 0.05 SBG-41a 0.44 5.1 41 74 0.03 SBG-41b 0.11 2.9 47 74 0.002 SBG-69 1.3 16.3 54 N.A.e 0.02 SBG-59 0.37 4.0 37 N.A.e 0.03 a Based on an average of about 10 measurements. b Porosities were calculated from density measurements using the techniques of Houghton et al.Ž. 1988 and are determined from the average of 30 density measurements. cAverage of four wavelengths whose standard deviations is 3.1 cm. d Based on close proximity to SBG-1. e Folding analysis based on photographs only. 26 M. Manga et al.rJournal of Volcanology and Geothermal Research 87() 1998 15±28

5. Concluding remarks

The rheology of bubble-bearing magmas is a function of the volume fraction of bubbles and the amount of deformation. Results of three-dimensional boundary integral calculations indicate that the effective viscosity of bubble-bearing magmas is only a slightly increasing function of volume fraction, in contrast to the behaviour of suspensions of rigid particles. As the volume fraction increases, the dis- tance between bubbles decreases and bubbles can slip over each other. Calculations also indicate that r with increasing capillary number and deformation Fig. 9. Temporal change in the shear stress, S12 f, in a random suspensionŽ. Fig. 2c with 8 bubbles, Cas0.3 and f s0.3. Time bubbles become aligned in the flow direction and is normalized by the shear rate so that a time of 1 corresponds to a contribute less shear stress to the suspension when strain of 1. The thin curves are values for individual bubbles and compared to spherical bubbles. For strains of less the bold curve is the average value. than order one, the effective shear viscosity and other rheological properties vary with time due to changes in the shape of the bubbles. Steady rheological properties occur only once the bubbles have than our 1yf estimate made at the beginning of deformed to near-steady shapes. this section.

4.3. Transient effects Acknowledgements The rheological measurements presented in Figs. 3±5 and 7 are quasi-steady results which are ob- M.M. is supported by a NSF CAREER grant. tained after total strains of about 1±4, as shown in M.L. acknowledges support from NSF grant CTS- Fig. 9Ž time is normalized by the shear rate so that a 9624615 and NASA grant NAG 3-1935. K.V.C. dimensionless time of 1 implies a strain of 1. . As Ca acknowledges support from NSF award EAR9614753. Additional support was provided by increases, the strain required for mrel to reach a steady value increases. In Fig. 9, the bold curve is the Donors of the Petroleum Research Fund, admin- the average value of S12 and the thin curves are istered by the American Chemical Society, through a values of S12 for each of the eight bubbles. The grant to M.M. Reviews and comments by F. Spera oscillations of S12 are due to temporal variations of and an anonymous reviewer improved this the relative positions of the bubbles in the sheared manuscript. suspension. Our calculations thus indicate that strains of greater than order 1 are necessary to obtain steady References rheological measurements due to the transient evolution of bubble microstructure. For comparison, the Bagdassarov, N., Dingwell, D.B., 1992. A rheological investiga- Bagdassarov and DingwellŽ. 1992 measurements are tion of vesicular rhyolite. J. Volcanol. Geotherm. Res. 50, for strains of about 0.02; the relative shear viscosities 307±322. inferred from the folding analysis presented in this Bagdassarov, N.S., Dingwell, D.B., Webb, S.L., 1994. Viscoelas- paper are based on folded structures that represent ticity of crystal- and bubble-bearing melts. Phys. Earth. Planet. r Inter. 83, 83±99. strains of about 1 2. In both cases, it is thus likely Batchelor, G.K., 1970. The stress system in a suspension of that the inferred viscosities are in fact transient prop- force-free particles. J. Fluid Mech. 41, 545±570. erties. Bentley, B.J., Leal, L.G., 1986. An experimental investigation of M. Manga et al.rJournal of Volcanology and Geothermal Research 87() 1998 15±28 27

drop deformation and breakup in steady two-dimensional lin- Lange, R.A., 1994. The effect of H22 O, CO , and F on the density ear flows. J. Fluid Mech. 167, 241±283. and viscosity of silicate melts. In: Carroll, M.R., Holloway, Biot, M.A., 1961. Theory of folding of stratified viscoelastic J.R.Ž. Eds. , Volatiles in Magmas. Mineral. Soc. Am. Rev. in media and its implication in tectonics and orogenesis. Geol. Mineral. 30, pp. 331±369. Soc. Am. Bull. 72, 1595±1620. Leal, L.G., 1992. Laminar Flow and Convective Transport Pro- Bottinga, Y., Weill, D.F., 1972. The viscosity of magmatic silicate cesses. Butterworth-Heinemann. liquids. Am. J. Sci. 272, 438±475. Lipman, P.W., Banks, N.G., 1987. Aa flow dynamics, Mauna Loa Cashman, K.V., Mangan, M.T., Newman, S., 1994. Surface de- 1984Ž. Hawaii . U.S. Geol. Surv. Prof. Pap. 1350, 1527±1567. gassing and modifications to vesicle size distributions in active Loewenberg, M., Hinch, E.J., 1996. Numerical simulation of a basalt flows. J. Volcanol. Geotherm. Res. 61, 45±68. concentrated emulsion in shear flow. J. Fluid Mech. 321, Castro, J.M., Cashman, K.V., 1996. Rheologic behavior of obsid- 395±419. ian and coarsely vesicular pumice in Little Glass Mountain Loewenberg, M., Hinch, E.J., 1997. Collision of two deformable and Obsidian Dome California. EOS 77, 802. drops in shear flow. J. Fluid Mech. 338, 299±315. Dawson, J.B., Pinkerton, H., Pyle, D.M., Nyamweru, C., 1994. MacDonald, G.A., 1954. Activity of Hawaiian volcanoes during June 1993 eruption of Oldoinyo Lengai, Tanzania: exception- the years 1940±50. Bull. Volcanol. 16, 119±179. ally viscous and large carbonatite lava flows and evidence for Manga, M., 1997. Interactions between mantle diapirs. Geophys. coexisting silicate and carbonate magmas. Geology 22, 799± Res. Lett. 24, 1871±1874. 802. Manga, M., Stone, H.A., 1993. Buoyancy-driven interactions be- Dobran, F., 1992. Nonequilibrium flow in volcanic conduits and tween deformable drops at low Reynolds numbers. J. Fluid application to eruptions of Mt St Helens on May 18, 1980, and Mech. 256, 647±683. Vesuvius in AD 79. J. Volcanol. Geotherm. Res. 49, 285±311. Manga, M., Stone, H.A., 1995. Collective hydrodynamics of Fink, J.H., 1980a. Gravity instability in the Holocene Big and deformable drops and bubbles in dilute suspensions at low Little Glass Mountain rhyolitic obsidian flows Northern Cali- Reynolds numbers. J. Fluid Mech. 300, 231±263. fornia. Tectonophysics 66, 147±166. McBirney, A.R., 1993. Igneous Petrology. Jones and Bartlett Fink, J.H., 1980b. Surface folding and viscosity of rhyolite flows. Publishers, Boston. Geology 8, 250±254. Murase, T., McBirney, A.R., 1973. Properties of some common Fink, J.H., 1982. Surface structure of Little Glass Mountain. In igneous rocks and their melts at high temperatures. Geol. Soc. Guides to Some Volcanic Terranes in Washington, Idaho, Am. Bull. 84, 3563±3592. Oregon, and Northern California. U.S. Geol. Surv. Circ. 939, Nakada, S., Motomura, Y., Shimizu, H., 1995. Manner of ascent 171±176. an Unzen volcanoŽ. Japan . Geophys. Res. Lett. 22, 567±570. Fink, J.H., 1984. Structural geologic constraints on the rheology Neal, C.A., McGimsey, R.G., Gardner, C.A., Michelle, L.H., Nye, of rhyolitic obsidian. J. Non-Crystal. Solids 67, 135±146. C.J., 1995. Tephra-fall deposits from the 1992 eruptions of Grove, T.L., Donelly-Nolan, J.M., 1986. The evolution of young Crater Peak, Mount Spurr Volcano, Alaska: a preliminary silicic lavas at Medicine Lake Volcano, California: Implica- report on distribution, stratigraphy, and composition. U.S.G.S. tions for the origin of compositional gaps in calc-alkaline Bull. B 2139, 65±79. series lavas. Contrib. Mineral. Petrol. 92, 281±302. Pinkerton, H., Norton, G., 1995. Rheological properties of basaltic Hon, K., Kauahikaua, J., Denlinger, R., Mackay, K., 1994. Em- lavas at sub-liquidus temperatures: laboratory and field mea- placement and inflation of pahoehoe sheet flows: Observations surements on lavas from Mount Etna. J. Volcanol. Geotherm. and measurements of active lava flows on Kilauea Volcano Res. 68, 307±323. Hawaii. Geol. Soc. Am. Bull. 106, 351±370. Pinkerton, H., Stevenson, R.J., 1992. Methods of determining the Houghton, B.F., Wilson, C.J.N., Hassan, M., 1988. Density mea- rheological properties of magmas at sub-liquidus temperatures. surements for pyroclasts and pyroclastic rocks. New Zealand J. Volcanol. Geotherm. Res. 53, 47±66. Geol. Surv. Rec. 35, 73±76. Pinkerton, H., Herd, R.A., Kent, R.M., 1996, Field measurements Jaupart, C., Allegre,Á C.J., 1991. Gas content, eruption rate and of the rheological properties of basaltic lavas on Kilauea, instabilities of eruption regime in silicic volcanos. Earth Planet. Hawaii. Abstract from the Chapman Conf. on Long Lava Sci. Lett. 102, 413±429. Flows. Kennedy, M.R., Pozrikidis, C., Skalak, R., 1994. Motion and Polacci, M., Papale, P., 1997. The evolution of lava flows from deformation of liquid drops, and the rheology of dilute emul- ephemeral vents at Mount Etna: Insights from vesicle distribu- sions in simple shear flow. Comp. Fluids 23, 251±278. tion and morphological studies. J. Volcanol. Geotherm. Res. Kilburn, C.R.J., Guest, J.E., 1993. Aa lavas of Mount Etna, Sicily. 76, 1±17. In: Kilburn, C.Ž. Ed. , Active Lavas. pp. 73±106. Pozrikidis, C., 1992, Boundary Integral and Singularity Methods Klug, C., Cashman, K.V., 1996. Permeability development in for Linearized Viscous Flow. Cambridge Univ. Press. vesiculating magmasÐimplications for fragmentation. Bull. Pozrikidis, C., 1993. On the transient motion of ordered suspen- Volcanol. 58, 87±100. sions of liquid drops. J. Fluid Mech. 246, 301±320. Laidley, R.A., McKay, D.S., 1971. Geochemical examination of Rallison, J.M., 1984. The deformation of small viscous drops and obsidians from Newberry Caldera, Oregon. Contrib. Mineral. bubbles in shear flows. Annu. Rev. Fluid Mech. 16, 45±66. Petrol. 30, 336±342. Rallison, J.M., Acrivos, A., 1978. A numerical study of the 28 M. Manga et al.rJournal of Volcanology and Geothermal Research 87() 1998 15±28

deformation and burst of a drop in an extensional flow. J. Sibree, J.O., 1933. The viscosity of froth. Trans. Faraday Soc. 28, Fluid Mech. 89, 191±209. 325±331. Ramsay, J.G., Huber, M.I., 1987. The techniques of modern Spera, F.J., Borgia, A., Strimple, J., Feigenson, M.D., 1988. structural geology. Academic Press, London. Rheology of melts and magmatic suspensions: design and Roscoe, R., 1952. The viscosity of suspensions of rigid spheres. calibration of concentric cylinder viscometer with application Br. J. Appl. Phys. 3, 267±269. to rhyolitic magma. J. Geophys. Res. 93, 10273±10294. Russel, W.B., Saville, D.A., Schowalter, W.R., 1989. Colloidal Stein, D.J., Spera, F.J., 1992. Rheology and microstructure of Dispersions. Cambridge Univ. Press. magmatic emulsionsÐtheory and experiments. J. Volcanol. Rutherford, M.J., Hill, P.M., 1993. Magma ascent rates from Geotherm. Res. 49, 157±174. amphibole breakdownÐan experimental study applied to the Stevenson, R.J., Briggs, R.M., Hodder, A.P.W., 1993. Emplace- 1980±1986 Mount St-Helens eruptions. J. Geophys. Res. 98, ment history of a low-viscosity, fountain-fed pantelleritic lava 19667±19685. flow. J. Volcanol. Geotherm. Res. 57, 39±56. Schowalter, W.R., Chaffey, C.E., Brenner, H., 1968. Rheological Stone, H.A., 1994. Dynamics of drop deformation and breakup in behaviour of a dilute emulsion. J. Colloid Interface Sci. 26, viscous fluids. Annu. Rev. Fluid Mech. 26, 65±102. 152±160. Taylor, G.I., 1932. The viscosity of a fluid containing small drops Shaqfeh, E.S.G., Fredrickson, G.H., 1990. The hydrodynamic of another fluid. Proc. R. Soc. London A 138, 41±48. stress in a suspension of rods. Phys. Fluids 2, 7±24. Taylor, G.I., 1934. The formation of emulsions in definable fields Shaw, H.R., 1972. Viscosities of magmatic silicate liquids: an of flow. Proc. R. Soc. Lond. A 146, 501±523. empirical method of prediction. Am. J. Sci. 272, 870±893. Webb, S.L., Dingwell, D.B., 1990. Non-Newtonian rheology of Shaw, H.R., Wright, T.L., Peck, D.L., Okamura, R., 1968. The igneous melts at high stresses and strain rates: Experimental viscosity of basaltic magma: an analysis of field measurements results for rhyolite, andesite, basalt, and nephelinite. J. Geo- in Makaopuhi lava lake Hawaii. Am. J. Sci. 266, 225±264. phys. Res. 95, 15695±15701.