Simulation of the rheological properties of suspensions of oblate spheroidal particles in a Newtonian fluid

Abstract

A simulation algorithm was developed to predict the rheological properties of oblate spheroidal suspensions. The motion of each particle is described by Jeffery’s solution, which is then modified by the interactions between the particles. The interactions are considered to be short range and are described by results from lubrication theory and by approximating locally the spheroid surface by an equivalent spherical surface. The simulation is first tested on a sphere suspension, results are compared with known experimental and numerical data, and good agreement is found. Results are then presented for suspensions of oblate spheroids of two mean aspect ratios of 0.3 and 0.2. Results for the relative viscosity η _r, normal stress differences N ₁ and N ₂ are reported and compared with the few available results on oblate particle suspensions in a hydrodynamic regime. Evolution of the orientation of the particles is also observed, and a clear alignment with the flow is found to occur after a transient period. A change of sign of N ₁ from negative to positive as the particle concentration is increased is observed. This phenomenon is more significant as the particle aspect ratio increases. It is believed to arise from a change in the suspension microstructure as the particle alignment increases.

Appendix 1: Comparison between the local sphere approximation and Cox’s results

We have estimated the difference between the values of the short-range hydrodynamic interactions obtained by our approximate method with results given by Cox (1973) who derived solutions for the resistance functions between two surfaces separated by a viscous fluid at small distance. In Cox’s work, similar to our assumption, the local curvature is characterized by two principal radii of curvature on each surface which can be given by r _m and r _t, the meridional and transverse radii. The angular orientation θ (Fig. 13) between each surface principal direction about the surface normal is also specified and Cox’s results are given depending on θ. This orientation is not the relative orientation between the particles defined below but simply the relative orientation between the principal radii of each surface. Since our approximation does not account for that relative orientation, results for different relative orientation are calculated with Cox’s method for comparison.

Fig. 13

The two interacting surfaces i and j defined respectively by two radii r _r and r _t. θ is the relative orientation between the principal radii of each particle

To obtain an estimate of the error introduced by approximating the surfaces by equivalent sphere, the characteristics of the interacting surfaces were sampled during a simulation for a 20% suspension of oblate spheroids of aspect ratio a _r = 0.2. Information such as particle orientations, local surface curvatures, and interparticle distances and velocities were recorded and grouped depending on the particle relative orientations. Indeed, for similar relative orientations, pairs of particles should show similar interacting surfaces characteristics and this facilitates the estimation of the error introduced by the local-sphere approximation. The particle relative orientation is given by α = cos^− 1 (pp), where pp is the inner product of p the spheroidal particle orientation vector and eighteen groups were created covering a range of relative orientation from 0° to 90° with intervals of 5°.

As observed in the results presented in “Orientation tensor,” a suspension of oblate spheroidal particles shows a strong particle alignment when subject to a shear flow. It is possible to observe the distribution g(α) of the relative orientation α of particle pairs as shown on Fig. 14, where a large fraction of the particles shows alignment.

Fig. 14

Evolution of the distribution function for different particle relative orientation

For each particle-relative orientation group, we use the average interacting surface characteristics to compute the lubrication forces and torques given by the approximate sphere method and those given by Cox. Since Cox’s results depend on the principal radii relative orientation θ, lubrication interactions are calculated for angles θ = 0°, 45°, and 90°. The values found at θ = 45° are found to always be between those found at θ = 0° and 90° which are angles where the minimum and maximum errors given in the total error (Table 5) are found. Once the error for a particle relative orientation group is estimated, it is weighted by the corresponding value of the distribution function g(α) to account for the probability of this particle relative orientation to be found. For a representative estimation of the error on the global behavior of the suspension, the errors are also scaled by the maximum force found in the suspension. It was observed that as the particle relative orientation angle α increases, so does the error, but since the probability to find such orientation decreases as well as the lubrication forces between such surfaces (decrease of the surface radii of curvature), the relative error decreases.

Table 5 Results for the error estimations

Using the comparison method detailed above, the errors found for the normal and tangential lubrication forces and torques on the particles are estimated to be respectively 4.866%, 0.394%, and 2.098%.