Heidy M Mader

The rheology of suspensions of solid particles

We present data for the rheology of suspensions of monodisperse particles of varying aspect ratio, from oblate to prolate, and covering particle volume fractions φ from dilute to highly concentrated. Rheology is characterized by fitting the experimental data to the model of Herschel & Bulkley (Herschel & Bulkley 1926 Kolloid Z. 39, 291–300 (doi:10.1007/BF01432034)) yielding three rheometric parameters: consistency K (cognate with viscosity); flow index n (a measure of shear-thinning); yield stress τ0. The consistency K of suspensions of particles of arbitrary aspect ratio can be accurately predicted by the model of Maron & Pierce (Maron & Pierce 1956 J. Colloid Sci. 11, 80–95 (doi:10.1016/0095-8522(56)90023-X)) with the maximum packing fraction φm as the only fitted parameter. We derive empirical relationships for φm and n as a function of average particle aspect ratio rp and for τ0 as a function of φm and a fitting parameter τ*. These relationships can be used to predict the rheology of suspensions of prolate particles from measured φ and rp. By recasting our data in terms of the Einstein coefficient, we relate our rheological observations to the underlying particle motions via Jeffery’s (Jeffery 1922 Proc. R. Soc. Lond. A 102, 161–179 (doi:10.1098/rspa.1922.0078)) theory. We extend Jeffery’s work to calculate, numerically, the Einstein coefficient for a suspension of many, initially randomly oriented particles. This provides a physical, microstructural explanation of our observations, including transient oscillations seen during run start-up and changes of rheological regime as φ increases.

The rheology of a bubbly liquid

A semiempirical constitutive model for the visco-elastic rheology of bubble suspensions with gas volume fractions φ < 0.5 and small deformations (Ca ≪ 1) is developed. The model has its theoretical foundation in a physical analysis of dilute emulsions. The constitutive equation takes the form of a linear Jeffreys model involving observable material parameters: the viscosity of the continuous phase, gas volume fraction, the relaxation time, bubble size distribution and an empirically determined dimensionless constant. The model is validated against observations of the deformation of suspensions of nitrogen bubbles in a Newtonian liquid (golden syrup) subjected to forced oscillations. The effect of φ and frequency of oscillation f on the elastic and viscous components of the deformation are investigated. At low f, increasing φ leads to an increase in viscosity, whereas, at high f, viscosity decreases as φ increases. This behaviour can be understood in terms of bubble deformation rates and we propose a dimensionless quantity, the dynamic capillary number Cd, as the parameter which controls the behaviour of the system. Previously published constitutive equations and observations of the rheology of bubble suspensions are reviewed. Hitherto apparently contradictory findings can be explained as a result of Cd regime. A method for dealing with polydisperse bubble size distributions is also presented.

Subsurface ice as a microbial habitat

We determine the physicochemical habitat for microorganisms in subsurface terrestrial ice by quantitatively constraining the partitioning of bacteria and fluorescent beads (1–10 m) between the solid ice crystals and the water-filled veins and boundaries around individual ice crystals. We demonstrate experimentally that the partitioning of spherical particles within subsurface ice depends strongly on size but is largely independent of source particle concentration. Although bacteria are shown consistently to partition to the veins, larger particles, which would include eukaryotic cells, become trapped in the crystals with little potential for continued metabolism. We also calculate the expected concentrations of soluble impurities in the veins for typical bulk concentrations found in natural ice. These calculations and scanning electron microscope observations demonstrate a concentrated chemical environment (3.5 M total ions at 10 C) in the veins, where bacteria were found to reside, with a mixture of impurities that could sustain metabolism. Our calculations show that typical bacterial cells in glacial ice would fit within the narrow veins, which are a few micrometers across. These calculations are confirmed by microscopic images of spherical, 1.9-m-diameter, fluorescent beads and stained bacteria in subsurface veins. Typical bacterial concentrations in clean ice (102–103 cells/mL) would result in concentrations of 106–108 cells/mL of vein fluid, but occupy only a small fraction of the total available vein volume (0.2%). Hence, bacterial populations are not limited by vein volume, with the bulk of the vein being unoccupied and available to supply energy sources and nutrients.

The constitutive equation and flow dynamics of bubbly magmas

[1] A generalized constitutive equation for bubbly liquids is presented which successfully reproduces the expected viscosity response for both steady flows with varying capillary number Ca (a measure of the bubble deformation) and unsteady flows with varying dynamic capillary number Cd (a measure of the steadiness of the flow) previously given in separate studies. The constitutive equation is given in terms of observable material and flow parameters and is �� �

Coupling of viscous and diffusive controls on bubble growth during explosive volcanic eruptions

Abstract The coupling of viscosity and diffusivity during explosive volcanic degassing is investigated using a numerical model of bubble growth in rhyolitic melts. The model allows melt viscosity and water diffusivity to vary spatially and temporally with water content. We find that the system is highly sensitive to the distribution of volatiles around the bubble, primarily as a consequence of the great sensitivity of melt viscosity to water content at low water concentrations. The dehydrated portion of the magma near the bubble exerts the major control on the effective viscosity of the melt; however the values of effective viscosity are lower than previously reported values calculated using an approximated concentration profile. Degassing is found to be highly sensitive to the choice of the solubility law, which controls the volatile concentration near the bubble, but insensitive to the equation of state. The form of the concentration profile in the case of concentration-dependent diffusivity is such that the effective viscosity is substantially lower (by as much as an order of magnitude) than for constant diffusivities of similar magnitude. This leads to unexpectedly high bubble growth rates, particularly for cases of highly non-equilibrium degassing. This is because low values of diffusivity are more than compensated by low melt viscosity due to higher volatile concentrations. This complex interplay between viscosity and diffusivity means that models of bubble growth must take into account the concentration-dependent nature of both these parameters; approximating either of these with a constant value will lead to significant errors in estimations of bubble growth rates.

Static and flowing regions in granular collapses down channels : Insights from a sedimenting shallow water model

A two layer model for the collapse and spreading of a granular column is presented. This model builds upon that of Larrieu et al. [J. Fluid Mech. 554, 669 (2006)] where the free fall collapse of the column and subsequent flow of material onto a plane is represented by a “raining” mass source term into a thin flowing layer of constant density. These modified shallow water equations with Coulomb friction capture the free surface of the flows and key scaling laws for initial sand columns of aspect ratios up to a<10. However, unrealistically high coefficients of friction of μ=0.9 are required to reproduce run-outs observed. Key scaling laws for high aspect ratio columns are also not captured. We thus extend the model of Larrieu (2006) to include an estimation for the interface between the static and flowing regions observed within granular collapses in the laboratory by Lube et al. [Phys. Fluids 19, 043301 (2007)]. An empirical sedimentation term Ls and the instantaneous removal of a static deposit wedge, see...

Correction to “The constitutive equation and flow dynamics of bubbly magmas”

[1] In the paper ‘‘The constitutive equation and flow dynamics of bubbly magmas’’ by E. W. Llewellin, H. M. Mader, and S. D. R. Wilson (Geophys. Res. Lett., 29(24), 2170, doi:10.1029/2002GL015697, 2002) errors were introduced to Paragraph 4 and equation 16. The corrected paragraph and equation appear below. [2] Paragraph 4, final sentence should read: The free-slip surfaces are, therefore, more important, causing a reduction in the suspension viscosity as f increases. [3] Equation 16 should read:

Strain‐induced outgassing of three‐phase magmas during simple shear

A major factor determining the explosivity of silicic eruptions is the removal of volatiles from magma through permeability-controlled outgassing. We studied the microstructural development of permeability during deformation of highly viscous magma by performing simple shear experiments on bubble (0.12–0.36 volume fraction) and crystal-bearing (0–0.42 volume fraction) silicate melts. Experiments were performed under torsion, at high temperature and pressure (723–873 K and 150–200 MPa) in a Paterson deformation apparatus at bulk shear strains between 0 and 10. The experimental setup allows for gas escape if bubble connectivity is reached on the sample periphery. Three-dimensional imaging and analysis of deformed bubbles was performed using X-ray tomography. The development of localized deformation in all samples, enhanced by crystal content, leads to brittle fracture at bulk strains > 2 and sample-wide fracturing in samples deformed to strains > 5. A decrease in both bubble fraction and dissolved volatile content with increasing strain, along with strain-hardening rheological behavior, suggests significant shear-induced outgassing through the fracture networks, applicable to shallow conduit degassing in magmas containing crystal fractions of 0–0.42. This study contributes to our understanding of highly viscous magma outgassing and processes governing the effusive-explosive transition.

The rheology of three-phase suspensions at low bubble capillary number.

We develop a model for the rheology of a three-phase suspension of bubbles and particles in a Newtonian liquid undergoing steady flow. We adopt an ‘effective-medium’ approach in which the bubbly liquid is treated as a continuous medium which suspends the particles. The resulting three-phase model combines separate two-phase models for bubble suspension rheology and particle suspension rheology, which are taken from the literature. The model is validated against new experimental data for three-phase suspensions of bubbles and spherical particles, collected in the low bubble capillary number regime. Good agreement is found across the experimental range of particle volume fraction (0≤ϕp≲0.5) and bubble volume fraction (0≤ϕb≲0.3). Consistent with model predictions, experimental results demonstrate that adding bubbles to a dilute particle suspension at low capillarity increases its viscosity, while adding bubbles to a concentrated particle suspension decreases its viscosity. The model accounts for particle anisometry and is easily extended to account for variable capillarity, but has not been experimentally validated for these cases.

The role of laboratory experiments in volcanology

Experiment is central to scienti¢c methodology. Within science in general, experiments are used for four primary purposes, i.e. as a tool to explore novel phenomena and to provide systematic observations of processes, to determine the values of key parameters, to test hypotheses and theoretical models, and to validate computational models, which themselves can be considered as numerical experiments. The study of volcanic eruptions and magmatic processes is becoming increasingly systematic, quantitative and rigorous. The drive is towards providing detailed quantitative descriptions and models of the £ow phenomena in terms of fundamental physico^chemical processes. This is motivated in no small way by an increasing need for reliable and quantitative predictions of volcanic eruptions on which to base rational hazard and risk management decisions during volcanic crises. A natural consequence of this development of the scienti¢c study of volcanic £ow phenomena is that rigorous laboratory experimentation is becoming an increasingly important feature within volcanic research. The contribution of experimental research to our understanding of volcanic processes is di⁄cult to overstate. The determination of the values of key parameters has traditionally taken place in the laboratory. However, more recently experiments have become much more widely applied to the problem of understanding complex dynamical processes and their underlying mechanisms. Without models (experimental, theoretical, numerical) we are limited to qualitative interpretations of ¢eld observations and remote sensing. Field data are of course essential and provide the measure against which ultimately the applicability of observations from other sources must be assessed. However, there are several reasons why the scienti¢c study of volcanic phenomena cannot rely solely on ¢eld data. Direct observations of eruption processes in the ¢eld are limited to those parts of an eruption that are accessible. This restricts detailed observations to a limited range of above-surface processes. Moreover, logistical constraints mean that the systematic collection of data will usually only be possible for volcanic eruptions near an established observatory and for activity for which there has been some forewarning. As a result of these constraints, direct ¢eld-based observations of natural volcanic eruption processes are generally incomplete and uncertain. Another source of natural data derives from observations made on volcanic deposits. In principle, processes can be inferred from the features