1. Introduction
Suspensions of solid particles are encountered in a variety of industrial and biological systems. The knowledge of the viscosity and, more in general, of the rheological properties of these materials is fundamental for correctly designing the processing stage and predicting the hydrodynamic resistance. It is well-known that the addition of solid particles in a fluid increases the viscosity as compared with the suspending liquid [
1]. Particle shape and size, as well as the solid concentration strongly alter the suspension viscosity giving rise to non-Newtonian phenomena such as shear-thinning and normal stresses [
1,
2,
3].
For the simplest case of a dilute suspension of spherical particles in a Newtonian fluid, Einstein calculated that the suspension viscosity is related to the fluid viscosity
${\eta}_{\mathrm{s}}$ according to the formula
${\eta}_{\mathrm{s}}(1+B\varphi )$ where
$\varphi $ is the solid volume fraction and the factor
B, usually referred as ‘intrinsic viscosity’, is equal to 2.5 [
4,
5]. Einstein’s calculation was extended by Jeffery to spheroidal particles [
6], finding that the intrinsic viscosity depends on the particle orientation and aspect ratio. Specifically, the minimum and maximum average viscosities are obtained for particles aligned along the vorticity axis or tumbling on the flow-gradient plane, respectively. For randomly-oriented particles, integration of the instantaneous intrinsic viscosity for several initial orientations over the corresponding orbits leads to an average value of
B higher than the Einstein’s coefficient for both prolate and oblate spheroids [
7].
In many processes, however, the suspended particles have a complex and irregular shape, without symmetry axes or planes. This is, for instance, the case when primary spherical particles undergo an aggregation process and form clusters with fractal-like morphology [
8,
9,
10]. As for the spheroidal particle case, the prediction of the intrinsic viscosity requires the calculation of the orientational dynamics of the particles subjected to an external shear flow. This approach is carried out in the work by Harshe and Lattuada [
11] where the average rigid body resistance matrix of arbitrary shaped clusters made of uniform sized spheres is computed through the Stokesian dynamics method and Brownian dynamic simulations. The intrinsic viscosity was found to be similar to the spherical particle case for clusters with high fractal dimension, indicating no preferential orientation in the flow. Deviations from the Einstein’s coefficient was, instead, found for aggregates with low fractal dimension due to their more anisotropic shape.
All the aforementioned works deal with Newtonian suspensions. In several applications, however, the suspending fluid shows non-Newtonian properties such as shear-thinning and viscoelasticity. A typical example is in the tire industry where, during the processing stage, particles of carbon black or silica are added to a polymer melt and can agglomerate and form complex structures. It is well-known that fluid non-Newtonian properties alter the suspension rheology as compared to the Newtonian case [
12,
13]. For instance, for a dilute suspension of spherical particles in a power-law fluid, it has been shown that shear-thinning reduces the intrinsic viscosity [
14,
15]. Concerning more complex particle shapes, the rheology of a dilute suspension of spheroids in a generalized Newtonian fluid is recently investigated by numerical simulations [
16]. Different flows of a Carreau fluid around spheroidal particles are simulated and a homogenization procedure is adopted to obtain the intrinsic viscosity of the suspension as function of the applied rate of deformation, thinning exponent and particle aspect ratio. The results show that the intrinsic viscosity strongly depends on the particle aspect ratio along with the rheological parameters of the constitutive equation [
16]. Very recently, these calculations have been extended to a dilute suspension of rigid rods in a power-law fluid showing no similarity of the rheological coefficients between rods and spheroids with large aspect ratio [
17]. Similar studies for dilute suspensions of particles with irregular shape, such as fractal-like morphologies, are not available.
In this paper, we investigate the rheology of a dilute suspension of aggregates with complex shape suspended in a shear-thinning fluid by direct numerical simulations. Aggregates made of primary spherical particles are built through a fractal-like model. The fluid is modeled by the power-law constitutive equation. The dynamics of a single particle in an unbounded shear flow field is first computed. To this aim, finite element simulations are employed to calculate the angular velocity of the particle for several orientations on the unity sphere. The orientational dynamics is then reconstructed by integrating the kinematic equations for the orientation vector interpolating the angular velocity field. The first-order contribution to the viscosity is computed by means of a homogenization procedure and time-averaged over the particle orbits. The effect of particle morphology and power-law index on the ensemble-average intrinsic viscosity is investigated.
3. Results
In this section we present results by varying the fractal dimension, the flow index, and the number of primary particles. Specifically, we consider three values for each parameter, that is, ${D}_{\mathrm{f}}=[1.5,2.0,2.5]$, $n=[1.0,0.7,0.4]$, and ${N}_{\mathrm{p}}=[10,20,50]$. Notice that, since the intrinsic viscosity is normalized with respect to the aggregate volume, the number of primary particles defines the structure resolution (finer for many particles) and, for low fractal dimension, it is also connected to the aggregate aspect ratio.
Figure 2 shows the probability distribution of the intrinsic viscosity with respect to the same initial orientations considered to build the look-up table, for the two aggregates reported in
Figure 1 and three flow indexes. The dashed lines represent the medians of the distributions, which are all approximately unimodal. Comparing the results for different fractal dimensions, it can be seen that at lower
${D}_{\mathrm{f}}$ the distribution of the intrinsic viscosity is wider, as the particles are more anisotropic and their orientation becomes more relevant. In particular,
B is higher when the aggregate is aligned with the gradient direction (i.e.,
y), and lower when it is aligned with the flow or vorticity direction, similarly to prolate spheroids. On the other hand, the distribution variance reduces at higher
${D}_{\mathrm{f}}$ as the particles are more sphere-like. In agreement with the results for a sphere in power-law fluid [
25], shear-thinning determines a reduction of the intrinsic viscosity, but the shape of the distribution remains the same.
As stated above, to make the results independent of the specific random seed used to generate the aggregates, for every combination of the other parameters, we repeat the simulations with ten different seeds. The effect of the specific morphology on the intrinsic viscosity distribution is reported in
Figure 3 with a box plot showing the first, second (i.e., the median), and third quartile of every distribution. In all cases we find again the same trend seen in
Figure 2 with fractal dimension and flow index, whereas the effect of the random seed is of secondary importance.
Since the intrinsic viscosity depends on the particle orientation, which changes with time due to the imposed shear flow, it is important to reconstruct the particle dynamics. The rotational dynamics of an irregularly shaped aggregate can be better understood by looking at its principal axes of inertia. Specifically, let’s denote with
$\mathit{P}$ the principal axis corresponding to the smallest moment of inertia, which for an elongated particle (small
${D}_{\mathrm{f}}$) corresponds to its ‘natural’ orientation.
Figure 4 and
Figure 5 show the time evolution of the Cartesian components of
$\mathit{P}$ and the angular velocity around it
${\omega}_{\mathit{P}}={\omega}_{\mathrm{p}}\xb7\mathit{P}$, for the aggregate reported in
Figure 1b and two initial orientations. In both cases the orientational dynamics is rather complex and neither a steady-state nor a simple periodic regime is achieved. Two periods can be recognized, one shorter connected to the rotation around the vorticity axis
z, and one longer connected to the rotation around
$\mathit{P}$. When the particle starts with an orientation close to the
z-axis (see
Figure 4), it undertakes a kayaking-like dynamics with approximately elliptical orbits around the
z-axis, more elongated in the
x-direction. On the other hand, when the initial orientation is sufficiently far from the
z-axis (see
Figure 5), the particle tends to align with the shear plane, that is,
${P}_{z}$ progressively reduces, similar to a tumbling motion. At the same time also
${\omega}_{\mathit{P}}$ decreases, thus determining an extension of the rotation period around
$\mathit{P}$. The same qualitative behavior of the aggregate dynamics is observed for shear-thinning fluids.
Once the particle orientational dynamics is known, the time evolution of intrinsic viscosity
B and time-average intrinsic viscosity
$\overline{B}$ can be reconstructed.
Figure 6 shows the trend of
B and
$\overline{B}$ for the trajectories seen in previous figures. Specifically, panel
Figure 6a refers to
Figure 4 (where the particle undertakes kayaking) and panel
Figure 6b refers to
Figure 5 (where the particle tumbles close to the shear plane). In both cases, while
B continues to oscillate due to the aforementioned dynamics, after a certain time,
$\overline{B}$ reaches a steady state condition. Notice that such steady state values are different for the two initial orientations considered.
$\overline{B}$ is lower in
Figure 6a since the aggregate is mainly aligned with its longest axis towards the vorticity direction. Conversely,
$\overline{B}$ is higher in
Figure 6b since, during the aggregate tumbling motion,
$\mathit{P}$ periodically passes close to the gradient direction, which corresponds to the maximum intrinsic viscosity.
By repeating this procedure for many orientations, it is possible to obtain the evolution of the probability distribution of the time-average intrinsic viscosity, as reported in
Figure 7 for the usual two aggregates and
$n=1.0$. Notice that the initial distributions are the same of
Figure 2. As just seen in the previous figure, for low fractal dimension more than one steady state value is present, with the majority of initial orientations leading to dynamics like the one in
Figure 6b with
$\overline{B}\approx 11.5$. The two lower peaks visible for
${D}_{\mathrm{f}}=1.5$ at
$t=5000$ correspond to
$\mathit{P}$ approximately aligned with the positive (shown in
Figure 4) or negative (not shown) vorticity direction. These last two peaks are close but not equal since the aggregate has no symmetries. A similar behavior can be observed at high fractal dimension, but with a longer time needed to reach the steady state. The two peaks visible for
${D}_{\mathrm{f}}=2.5$ at
$t=30,000$ are still related to the kayaking and tumbling motion. However, those related to alignment of
$\mathit{P}$ to the vorticity direction or its opposite are not distinguishable because of the more isotropic particle shape.
Finally, averaging on all the initial orientations and random seeds we get the ensemble-average intrinsic viscosity defined in Equation (
22).
Figure 8 reports the dependency of
${\langle \overline{B}\rangle}_{\mathrm{m}}$ on the fractal dimension and flow index, parametric in the number of primary particles, together with standard deviation and trend line. In the range considered,
${\langle \overline{B}\rangle}_{\mathrm{m}}$ decreases with
${D}_{\mathrm{f}}$ and increases with
${N}_{\mathrm{p}}$. The decreasing trend of
${\langle \overline{B}\rangle}_{\mathrm{m}}$ with the fractal dimension, previously observed for Newtonian fluids [
11], is related to the aggregate shape and can be justified recalling that the intrinsic viscosity of a suspension of rod-like particles is higher than the one for a suspension of spheres [
7]. For the same reason, the number of primary particles forming the aggregate weakly affects the intrinsic viscosity at high fractal dimension. On the contrary,
${N}_{\mathrm{p}}$ has a strong influence on
${\langle \overline{B}\rangle}_{\mathrm{m}}$ for low values of
${D}_{\mathrm{f}}$. Indeed, for a sphere-like aggregate the number of primary particles only defines its resolution, whereas for a rod-like aggregate it is connected to the aspect ratio that, in turn, determines the viscosity of the suspension [
7]. As regarding the effect of the flow index, in agreement with the previous literature for suspensions of spherical and spheroidal particles [
14,
16,
25], shear-thinning reduces the intrinsic viscosity. Specifically, as visible on the right column of
Figure 8, in the investigated range the intrinsic viscosity linearly depends on the flow index. As expected standard deviation is lower for high fractal dimensions, since both initial orientation (see
Figure 2) and random seed (see
Figure 3) have a relatively minor effect.
The intrinsic viscosity shown in the previous figures accounts for the effective volume of the aggregate that, as discussed in
Section 2.1, is computed from the union of spheres and cylinders. An alternative approach is to normalize the intrinsic viscosity by the aggregate hydrodynamic volume that, for a set of spherical particles with radius
a, is the volume of a sphere with radius
${R}_{\mathrm{H}}\phantom{\rule{0.166667em}{0ex}}a$, where
${R}_{\mathrm{H}}$ is the hydrodynamic radius [
11]. The hydrodynamic radius is defined as the radius of a sphere that gives the same drag force acting on the aggregate in a uniform flow [
34], and, in a Newtonian fluid, it is related to the eigenvalues of the translational mobility tensor. Neglecting the contribution of the connecting cylinders, the ensemble-average intrinsic viscosity normalized with the hydrodynamic volume is then:
Figure 9a shows the trend of
${\langle \overline{B}\rangle}_{\mathrm{m},\mathrm{H}}$, in which the values of
${R}_{\mathrm{H}}$ are taken from the literature [
23,
34]. Some remarks are in order: (i) Equation (
23) and the used hydrodynamic radii assume that the aggregate is made of tangential spherical particles, (ii) the values of
${R}_{\mathrm{H}}$ are calculated for an aggregate suspended in a Newtonian fluid (notice that the calculation of the hydrodynamic radius for a power-law fluid is not straightforward as, due to the non-linearity of the constitutive equation, the mobility tensor cannot be used for its evaluation). Despite these approximations, the data scale fairly well with respect to the number of primary particles (symbols with the same color in figure). As a consequence of the previous normalization, at high fractal dimension
${\langle \overline{B}\rangle}_{\mathrm{m},\mathrm{H}}$ tends to the value for a spherical particle (i.e., 2.5 in the Newtonian case) [
11]. This motivates us to further normalize the data with respect to the intrinsic viscosity of a dilute suspension of spheres in a power-law liquid, given by
${B}_{\mathrm{sph}}=0.383+2.117n$ [
14,
25,
35]. As visible in
Figure 9b, all the data collapse on a single curve. Hence the viscosity of a dilute suspension of aggregates (with the same fractal parameters) in a power-law fluid is completely determined by the fractal dimension. Of course, the prediction of the intrinsic viscosity form the master trend in
Figure 9b requires the knowledge of the hydrodynamic radius of the aggregate population.