Debris Flows - Environmental Geomechanics - page 16

that the collisional viscosity should depend on the Coulomb number Co = ρ p a 2 γ 2 /σ
to allow for this coupling in a simple way
τ = σ tan ϕ + μ (Co) γ.
Jop et al. [JOP 05] proposed a slightly different version of this model, where both
the bulk frictional and collisional contributions collapse into a single term, which is a
function of the Coulomb number (referred to as the inertial number)
τ = σ tan ϕ (Co) .
Contrasting with other propositions, Josserand et al. [JOS 04] stated that the key
variable in shear stress was the solid concentration φ rather than the Coulomb number
τ = K ( φ ) σ + μ ( φ ) γ 2 ,
where K is the friction coefficient. Every model is successful in predicting
experimental observations for some flow conditions, but to date, none have been able
to describe the frictional-collisional regime for a wide range of flow conditions and
material properties.
When the bulk is a bimodal suspension of coarse particles within a colloidal
dispersion, it still behaves like a viscoplastic material. Sengun and Probstein
[SEN 89a, SEN 89b] proposed different arguments to explain this behavior. Their
explanation consists of two approximations. First, as this is the interstitial phase,
the dispersion resulting from the mixing of fine colloidal particles and water imparts
most of its rheological properties to the entire suspension. Second, the coarse fraction
is assumed to act independently of the fine fraction and enhance bulk viscosity.
They introduced a net viscosity η nr of a bimodal slurry as the product of the fine
relative viscosity η fr and the coarse relative viscosity η cr . The fine relative viscosity
is defined as the ratio of the apparent viscosity η f of the fine-particle suspension to
the viscosity of the interstitial fluid μ : η fr = η f /μ . The coarse relative viscosity is
defined as the ratio of the apparent viscosity η c of the coarse-particle slurry to the
viscosity of the fine-particle suspension: η cr = η c /η f . The two relative viscosities
depend on the solid concentrations and a series of generalized Péclet numbers. For
the coarse-particle suspensions, all the generalized Péclet numbers are much greater
than 1. Using a dimensional analysis, Sengun and Probstein deduced that the coarse
relative viscosity cannot depend on the shear rate. In contrast, bulk behavior in
fine-particle suspensions is governed by colloidal particles and thus at least one of
the generalized Péclet numbers is of the order of 1, implying that the fine relative
viscosity is shear-dependent. Sengun and Probstein's experiments on the viscosity of
coal slurries confirmed the reliability of this concept [SEN 89a]. Plotting log η nr and
log η fr against log γ , they found that over a wide range of concentrations, the curves
were parallel and their distance was equal to log η cr (see Figure 1.10). However, for
solid concentrations in the coarse fraction exceeding 0.35, they observed a significant