Rheology of rigid particle suspensions

The case of a fluid coupled with rigid particles is treated in this subsection, although in practice some common fluids include deformable particles (again, blood constitutes a famous example).

It is assumed here that the shape of the particle and the fractionφ of the fluid they occupy are the only parameters to impact the viscosity. The particle volume fraction φ is computed as the sum of the volume of each particles divided by the volume of the suspension sample (thus, including the pure fluid volume as well as the volume of the particles):

where Vi denotes the volume of the particlei andV is the total volume of the suspension sample.

Maximum volume fraction

From the above definition of the particle volume fraction, it seems intuitive that a theoretical maximum volume fraction φ_m exists and depends on the shape of the particles and their arrangement. For instances, cubic particles can be stored in a very efficient way for which φ^∗_m = 1, where the star stands for a non-random packing. However, in nature as well as in experiments, one does not control the individual arrangement of particles, that are not necessarily stored following an optimal neither regular pattern.

The computation of the maximum random packing of cubes, for instance, is not trivial and its value is found to be φ_m ≈ 0.637 [54]. It is worth noting that for cuboids, the value of maximum random packing is impacted by the aspect ratio [54], whereasφ^∗_m is obviously not sensitive to this value.

Finding the maximum packing fraction of spheres is an old problem. In 1611, Kepler formulated a conjecture stating that, for an arrangement of equally sized spheres, the maximum density is reached when

φ^∗_m = π 3√

2 ≈0.74. (3.6)

Suspensions of rigid particles 45

Table 3.1.: The three regimes observed in suspensions of rigid spheres, along with the corre-sponding relative viscosityµ, depending on the volume fraction φ, the Einstein coefficientB, the maximum random packing fractionφm and the critical volume fraction φ_c. The presence of a Newtonian behaviour is indicated, though the dependence on the strain rate is not shown.

Regime name Behaviour Law type Validity

Dilute Newtonian µ= 1 +Bφ 0≤φ .0.02

Semi-dilute Newtonian µ= 1 +Bφ+Cφ² 0.02.φ.0.25 Concentrated Non-Newtonian µ= (1−φ/φm)^−Bφm 0.25.φ < φc

This conjecture was proven by exhaustion in 1998 and formally validated in 2017 [55].

Again, in the non-structurated case, the packing is less dense and the maximum random packing of spheres is found to be φm ≈0.64 [56,57], which is very close from the random packing fraction of cubes. From now on, the symbol φm will refer to the maximum close packing of equal spheres.

Finally, laboratory experiments show that a critical volume fractionφ_c exists above which continuous networks of interconnected particles (i.e. chains of particles) can form.

For uniform distribution of solid spheres, this value is between 0.5 and 0.55 [58]. Although less discussed than φ_m, this critical value constitute, more than φ_m, the actual limit of most numerical models that are not based on a granular media approach. Therefore, φc

should be the upper bound target, in terms of volume fraction, of a numerical model based on a fluid solver.

The case of spherical particles

At a given strain rate, it is arguable that a fluid containing particles will tend to resist shear constraints more than the same fluid without particles; for a fixed deformation of the fluid, one has to apply a larger torque when the fluid contains particles, because particles a priori introduce an additional source of friction (fluid-particle as well as particle-particle). The case of spherical particles is discussed here.

Three regimes are commonly distinguished in the literature [41,48,49]: the dilute regime, the semi-dilute regime and the concentrated regime, for which some characteristics are summarized in Tab. 3.1.

46 Suspensions of rigid particles

In the dilute limit, particles are so far from each other that the flow around a single particle can be viewed as resulting from the interaction between the fluid and the particle only. Hence, it is clear in this regime that the contribution of each particle to the viscosity simply sum up. It has been shown in 1906 [59] that the viscosity of the flow around a sphere occupying a fraction φ of the volume has the form

µ= 1 +Bφ, (3.7)

where B = 5/2 is the Einstein coefficient. This value, however, remains uncertain: since experimental results as well as theoretical approaches do not agree, with B spreading from 2.5 to 5 approximately [60].

Confronted to experimental data, the dilute limit remains valid only for very small volume fraction (φ.0.02). However, for φ.0.25 [47], a polynomial law still reflects the experiments:

µ= 1 +Bφ+Cφ², (3.8)

with 7.C .14, depending on the studies (see [61,62] for instance).

From φ ≈ 0.25, particle-particle interactions start to dominate the rheology. An obvious reason of the failure of the above model to predict viscosity for high volume fraction is that it yields finite value for φ→1. Rather, one would expect the viscosity of a completely jammed fluid (φ= φ_m) to be infinite. A popular model for high volume fraction is the so-called Krieger-Dougherty model [63]:

µ= 1− φ φ_m

!−Bφ_m

. (3.9)

As said above, a great number of phenomenological laws can also be found in the literature, although Eq. (3.9) will be the reference in this work, due to its popularity [41].

It is worth noting that the Maron-Pierce relationship [64], also commonly used, suggests thatBφ_m = 2, to be compared with the value of 1.6 yielded by the model chosen in this thesis (B = 2.5, φ_m = 0.64 for randomly packed monodisperse spheres).

Only a simple case has been discussed here. Indeed, the particles considered are spher-ical, monodisperse, and interact via hydrodynamic forces only, neglecting inertia. Studies including the importance of inertia [65], non-hydrodynamics interparticle forces [66] and the effect of Brownian motion [62] demonstrate that, depending on the composition

Suspensions of rigid particles 47

of the suspension, the models discussed above no longer reflect the behaviour of the considered fluid. Magmatic regimes are characterized in Sect.3.1.2, where it is shown how the assumptions made so far apply to some types of magmatic flows.

Eq. (3.9) establishes a relationship between the relative apparent viscosity and the particle volume fraction of the suspension; it does not tell anything about the evolution of this quantity as the strain rate changes.

It is widely admitted [47] that the non-Newtonian behaviour observed in highly concentrated suspensions is an effect of the rearrangements of the particles contained inside the fluid. These particles are near to jam and therefore begin to form chains [58,67], resulting in a macroscopic behaviour better described by the Herschel-Bulkley constitutive equation, discussed below. Hence, the so-called shear-thinning and shear-thickening rheologies, observed in nature, as well as the development of a yield stress (see below), can only be reproduced in first-principles simulations with a volume fraction close to critical volume fraction φ_c. This latter point is particularly challenging because of the aforesaid uncertainties remaining on the proper form that a collision model should take.

Here again, the shape of the particles involved plays a crucial role.

The Herschel-Bulkley relationship [68] reads

τ =τ₀+Kγ˙ⁿ, (3.10)

where τ0 is the yield stress,K is the consistency (note that, unless n = 1, K cannot be assimilated to a viscosity in the sense of Eq. (3.1)) and n is the flow index. The yield stress is the value of stress below which the suspension cannot flow (e.g. toothpaste will not move due to gravity on a slightly sloping surface; at a given angle, though, it will start to flow). The yield stress resulting from the numerical model presented here is not investigated in this thesis, though it constitutes an important feature of suspensions in concentrated regimes. As noted by [3], instead of applying a strain rate to the fluid by imposing velocity boundary conditions, one should rather impose a shear stress at the boundaries and observe the suspension velocity in order to discern the value of τ₀.

A fluid obeying Eq. (3.10) is called a power law fluid : whenn > 1, it is said to be shear-thickening, whereas it is shear-thinning for n <1. Ifn = 1, Eq. (3.10) is equivalent to the Bingham model [69]. Finally, the fluid is Newtonian if, in addition, τ₀ = 0. Fig.

3.1 depicts three different behaviors corresponding to three values of n.

48 Suspensions of rigid particles

Figure 3.1.: Example of power law fluids with yield stressτ0 = 0.5.