Edward W. Llewellin

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.

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 �� �

The thickness of the falling film of liquid around a Taylor bubble

We present the results of laboratory experiments that quantify the physical controls on the thickness of the falling film of liquid around a Taylor bubble, when liquid–gas interfacial tension can be neglected. We find that the dimensionless film thickness λ′ (the ratio of the film thickness to the pipe radius) is a function only of the dimensionless parameter , where ρ is the liquid density, g the gravitational acceleration, D the pipe diameter and μ the dynamic viscosity of the liquid. For , the dimensionless film thickness is independent of Nf with value λ′≈0.33; in the interval , λ′ decreases with increasing Nf; for film thickness is, again, independent of Nf with value λ′≈0.08. We synthesize existing models for films falling down a plane surface and around a Taylor bubble, and develop a theoretical model for film thickness that encompasses the viscous, inertial and turbulent regimes. Based on our data, we also propose a single empirical correlation for λ′(Nf), which is valid in the range 10−1<Nf<105. Finally, we consider the thickness of the falling film when interfacial tension cannot be neglected, and find that film thickness decreases as interfacial tension becomes more important.

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:

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 α–β phase transition in volcanic cristobalite

Volcanic cristobalite commonly contains structural substitutions of Al3+ and Na+ for Si4+. Quantifying the effect of these substitutions on the crystal structure may provide insight into volcanic processes and the variable toxicity of crystalline silica.

LBflow: An extensible lattice Boltzmann framework for the simulation of geophysical flows. Part I: theory and implementation☆

Abstract This article presents LBflow , a flexible, extensible implementation of the lattice Boltzmann method. The code has been developed with geophysical applications in mind, and is designed to be usable by those with no specialist computational fluid dynamics expertise. LBflow provides a ‘virtual laboratory’ which can be used, rapidly and easily, to obtain accurate flow data for the geometrically complex, three-dimensional flows that abound in geophysical systems. Parameters can be ‘steered’ by the user at runtime to allow efficient and intuitive exploration of parameter space. LBflow is written in object-oriented C++ and adopts a modular approach. Lattice Boltzmann algorithms for distinct classes of material are encoded in separate modules, which implement a standard interface, and which are linked to LBflow dynamically at runtime. This allows users with programming skill and expertise in the lattice Boltzmann method to create and share new LBflow modules, extending functionality. A companion application, LBview , provides a graphical user interface to LBflow and renders a user-configurable visualization of the output. LBflow s output can be piped directly to LBview allowing realtime visualization of steered flow. LBview also facilitates analysis of the data generated by LBflow . This article presents an overview of the theory of the lattice Boltzmann method and describes the design and operation of LBflow . The companion paper, ‘Part II’, describes the practical usage of LBflow and presents detailed validation of its accuracy for a variety of flows.

Sintering of viscous droplets under surface tension.

We conduct experiments to investigate the sintering of high-viscosity liquid droplets. Free-standing cylinders of spherical glass beads are heated above their glass transition temperature, causing them to densify under surface tension. We determine the evolving volume of the bead pack at high spatial and temporal resolution. We use these data to test a range of existing models. We extend the models to account for the time-dependent droplet viscosity that results from non-isothermal conditions, and to account for non-zero final porosity. We also present a method to account for the initial distribution of radii of the pores interstitial to the liquid spheres, which allows the models to be used with no fitting parameters. We find a good agreement between the models and the data for times less than the capillary relaxation timescale. For longer times, we find an increasing discrepancy between the data and the model as the Darcy outgassing time-scale approaches the sintering timescale. We conclude that the decreasing permeability of the sintering system inhibits late-stage densification. Finally, we determine the residual, trapped gas volume fraction at equilibrium using X-ray computed tomography and compare this with theoretical values for the critical gas volume fraction in systems of overlapping spheres.

Evidence for explosive silicic volcanism on the Moon from the extended distribution of thorium near the Compton‐Belkovich Volcanic Complex

We reconstruct the abundance of thorium near the Compton-Belkovich Volcanic Complex on the Moon, using data from the Lunar Prospector Gamma Ray Spectrometer. We enhance the resolution via a pixon image reconstruction technique, and find that the thorium is distributed over a larger (