Jeffery showed that ellipsoids of revolution rotate in a linear shear field with a period
where re is the ratio between the major and minor axis of an ellipsoid of revolution. This prediction has been validated by experiment [cf. Zia et al. (1967)]. To determine whether a DPD based code could recover this result, an ellipsoid of revolution was approximated by creating a template of DPD particles that fall within the boundaries of an prolate ellipsoid of revolution with re = 2.4. The simulation had Reynolds number Re<1 and the value of Pe (of order 1000) high so that inertial and diffusive effects could be minimized. It was found that the simulation obtained a period with less than 2% error when compared to that predicted by Jeffery's theory. Figure 11 shows a comparison of simulation results and the prediction of Jeffery, for the rotational orientation of the spheroid, as a function of time. Here, the rotational orientation of the ellipsoid of revolution is given by [Larson (1999); Eirich (1967)]:
The relative viscosity as a function of φ was then determined for three spheroid
systems: monosize spheres, oblate (re = 1/3.28) ellipsoids of
revolution, and prolate (re = 2.4) ellipsoids of revolution
(Fig. 12). For each shape particle, simulations were
carried out for 1, 3, 5, 10, 17, and 25 rigid bodies. In all the simulations, the volume
of the individual rigid bodies were nearly equal. As the system was sheared, Jeffery
orbits were clearly seen at the lowest volume fractions (a few spheroids). However, for
the case of oblate spheroids, between φ = 0.10 and 0.15 the Jeffery orbits became
suppressed and an apparent nematic phase [Larson (1999)] or
orientational order (Fig. 13) was observed. At these
solid fractions some prolate spheroids were still undergoing Jeffery orbits. It was not
until the higher solid fractions were reached that the the prolate spheroids became more
aligned. It is likely that the oblate spheroids ordered at the lower volume fraction
because they are somewhat flatter, with a relatively large and round cross section, than
the prolate spheroids, making it more difficult to "squeeze" out the fluid
between them as they try to undergo Jeffery orbits near each other. Accompanying the
nematic phase was an apparent reduction in the rate of increase of the relative
viscosity. Indeed, as can be seen in Fig. 12 the relative viscosity
for the oblate spheroids was lower then the relative viscosity of the spheres when φ
0.015. It should be
pointed out that in these simulations, lubrication forces were not included and, because
of the periodicity of the system, the ordering could have been enhanced. So, further
study is needed to see whether these results apply to larger systems and to fully
understand the effect of lubrications forces.
Figure 11. Comparison of DPD simulation (dashed line) to predictions from theory (solid line) for rotation of prolate spheroid under shear. Here, φe is the angle of orientation, t is the time, and T is the period of rotation [Eq.23].
Figure 12. Relative viscosity for spheroid systems. Shown are data for oblate (dashed line), spherical (filled circles), and prolate (solid line) spheroids. Note that at φ≈ 0.1 of rate of increase of relative viscosity with φ for the oblate spheroid decreases, indicative of the onset of an apparent nematic phase. A nematic phase for the prolate spheroids occurs at somewhat higher φ. Statistical uncertainties in the simulation data were approximately 10% or smaller.
Figure 13. Evidence of an apparent nematic phase for the oblate spheroid system with re = 1/3.28. The particles initial orientation was such that the axis of symmetry was in the vorticity direction (perpendicular to the page). Here, the Jeffery's orbits were suppressed.