## 1. Introduction and Theoretical Background

[2] The flow of particle suspensions is central to volcanic and magmatic processes throughout the volcanic system; examples include the mingling of phenocryst-bearing magmas, the eruption of microlite-bearing magma, and the flow of crystal-bearing lava down a volcano's flank. Accurate modelling of such flows depends on detailed knowledge of the suspension's rheology. It is well-known from laboratory rheometry of magmatic suspensions that the volume fraction of suspended crystals exerts a first-order control on the effective viscosity of a magma [e.g., *Lejeune and Richet*, 1995; *Caricchi et al.*, 2007; *Ishibashi and Sato*, 2007; *Lavallee et al.*, 2007], hence on eruption dynamics [*Melnik and Sparks*, 2005]. In this article, we demonstrate that the influence of crystal shape is similarly strong, and should not be neglected. We perform rheometric experiments using pre-sheared analogue particle suspensions; whilst analogue experiments do not capture processes such as crystal breakup and pressure solution, they have the advantage over experiments with magmatic suspensions that they are better characterized and allow more thorough exploration of parameter space. Based on our data, we define a relationship between the shape of suspended particles, expressed as their mean aspect ratio, *r*_{p}, and the microstructural properties of the suspension, encapsulated by the maximum packing fraction of particles, ϕ_{m}. Since the average aspect ratio of the dominant crystal phase in a magma is often known, or can be estimated on the basis of petrological models, the relation proposed here will facilitate the use of rheological models in which ϕ_{m} is a critical parameter and will improve their accuracy by accounting for crystal shape.

[3] The rheology of a material is usually described by a constitutive equation which relates the driving stress *τ* to the resulting strain rate . For a Newtonian liquid, this equation is *τ* = *μ*, where *μ* is the Newtonian viscosity, which is independent of strain rate. Particle-bearing liquids typically develop a more complex, non-Newtonian rheology as particle concentration increases, becoming first shear-thinning at moderate volume fractions, then developing a yield stress at high volume fractions [*Mueller et al.*, 2010], though we note that yield stress is not universally observed in experiments on natural samples [e.g., *Caricchi et al.*, 2007]. This non-Newtonian behaviour is well-described by the Herschel–Bulkley model [*Herschel and Bulkley*, 1926]:

where the consistency *K* is cognate with viscosity and has the units Pa s^{n}, the flow index *n* defines the degree of non-Newtonian behaviour (*n* < 1 for shear thinning behaviour), and *τ*_{0} is the yield strength. For non-Newtonian liquids, the viscosity is commonly expressed as an ‘apparent viscosity’ *η* = *τ*/ at a specified strain rate; expressed in these terms, the Herschel–Bulkley model becomes:

For dilute and moderately-concentrated suspensions, where yield stress is negligible [*Mueller et al.*, 2010], it can be seen that *K* is equivalent to *η* evaluated at = 1 s^{−1}. For weakly shear-thinning materials, i.e., when *n* is close to 1, *K* can often be sufficiently well approximated by *η*. For suspensions, *η* is commonly non-dimensionalized by the Newtonian viscosity of the suspending liquid *μ*_{0} to give the relative viscosity *η*_{r} = *η*/*μ*_{0}; by analogy a relative consistency can be defined as *K*_{r} = *K*/*μ*_{0}.

[4] Particles affect the rheology of a suspension because additional energy is dissipated due to fluid–particle and particle–particle interactions. As particle volume fraction ϕ increases, these interactions become more common, leading to a dramatic, non-linear increase in relative viscosity until a maximum packing fraction ϕ_{m} is reached [e.g., *Einstein*, 1906, 1911; *Roscoe*, 1952; *Maron and Pierce*, 1956], at which point the suspension becomes jammed. If the particles are perfectly rigid, flow ceases when ϕ_{m} is reached; in magmatic suspensions, however, processes such as plastic deformation or creep may take place at ϕ > ϕ_{m} [e.g., *Lavallee et al.*, 2007; *Caricchi et al.*, 2007].

[5] Particle shape affects rheology by changing the nature of fluid–particle and particle–particle interactions, introducing two additional considerations: 1) anisometric particles are orientable, so their influence depends on their orientation with respect to the flow; 2) anisometry enhances particle–particle interactions because the orbit of a rotating non-spherical particle encloses a greater volume for potential interactions than that of a spherical particle. Previous theoretical and analogue experimental work on the rheology of suspensions of anisometric particles is less extensive than for spherical particles, and focusses primarily on dilute suspensions [e.g., *Jeffery*, 1922; *Brenner*, 1974; *Powell*, 1991; *Pabst et al.*, 2006]. An exception is our earlier study [*Mueller et al.*, 2010] in which we consider suspensions of anisometric particles with volume fractions spanning the range 0 ≤ ϕ ≲ ϕ_{m}; we found that the following relationship holds across that range:

and that ϕ_{m} varies with particle aspect ratio. Equation (3) is based on the widely-applied relationship of *Maron and Pierce* [1956] (with *K*_{r} substituted for *η*_{r}).

[6] For monodisperse spherical particles, the theoretical maximum packing fraction ϕ_{m} ≈ 0.74 for an ordered packing and ϕ_{m} ≈ 0.64 for a random (i.e., disordered) packing [*Bernal and Mason*, 1960]. The effect of particle shape on maximum packing fraction in the absence of hydrodynamic shearing has been investigated experimentally [e.g., *Milewski*, 1973; *Parkhouse and Kelly*, 1995; *Rahli et al.*, 1999], geometrically [*Evans and Gibson*, 1986], and numerically [e.g., *Williams and Philipse*, 2003; *Gan et al.*, 2004; *Donev et al.*, 2004]. In general, these studies show that ϕ_{m} tends to decrease with increasing particle anisometry. There have been comparatively few studies of ϕ_{m} for anisometric particles in shearing flows; we are aware of just two, *Kitano et al.* [1981] and *Pabst et al.* [2006], who both infer ϕ_{m} from rheological measurements of sheared suspensions and find that ϕ_{m} decreases for increasingly prolate particles.

[7] In the present study we define the aspect ratio *r*_{p} = *l*_{a}/*l*_{b}, where *l*_{a} is the particle's axis of rotational symmetry and *l*_{b} is its maximum diameter perpendicular to that axis. Various empirical and semi-empirical models have been proposed to describe the dependence of ϕ_{m} on *r*_{p}: *Evans and Gibson* [1986], *Rahli et al.* [1999] and *Mueller et al.* [2010] propose inverse relationships between the two parameters; *Parkhouse and Kelly* [1995] propose a logarithmic relationship; and *Kitano et al.* [1981] and *Pabst et al.* [2006] suggest a linear decrease in ϕ_{m} with *r*_{p}. In this work, we use high-resolution rheometry to investigate the relationship between *r*_{p} and ϕ_{m} for shearing flows, and compare existing models with our experimental data.