Ideal Collision Rate of Symmetric Hexapods in a Simple Shear Flow

An ideal collision rate (ICR) is
defined as the average
rate of
contact between two particles that translate and rotate with the imposed
fluid flow in the absence of interparticle interactions. ICR in a
simple shear flow provides an estimate of the collision rate in a
flowing dilute particle suspension, and its value is known only for
traditional convex shapes such as spheres and cylinders. In this work,
we compute the ICR for a family of symmetric hexapods (shown in Figure 1) that are particles composed of three orthogonal
cylinders with coinciding centers (forming six arms) with at least
two cylinders having the same length. We employed the finite-element
method to obtain the rotational dynamics of hexapods and Monte Carlo
simulations to calculate their ICR. Our results indicate that the
ICR for hexapods is not directly proportional to the volume of the
particle, in contrast to what is observed for convex shapes such as
spheres and cylinders. For hexapods of the same size, the ICR could
be similar for particles with order-of-magnitude differences in their
volumes and it could also vary by an order of magnitude for particles
with similar volumes. Specifically for hexapods termed branched fibers,
which have one longer cylinder, the ICR changes by an order of magnitude
even when the shorter cylinders are significantly smaller than the
longer cylinder. This change is attributed to the increase in the
tumbling frequency of the particle due to hydrodynamic forces acting
on the shorter arms, in addition to the increased probability of collision
afforded by them. Using asymptotic theory for high-aspect ratio cylinders,
we showed that the tumbling period of hexapods was proportional to
an algebraic power of the ratio of its arm lengths and has a weaker
logarithm relationship with the aspect ratio of its arms. The collision
cross-section provided the relative cross-streamline particle separations
for the most likely binary collisions in the suspensions, and its
value was sensitive to the arm lengths, as well. The collision rate
of hexapods also could not be estimated within an order of magnitude
from existing geometric models, such as the ICR of a circumscribing
convex shape, ICR for one of the individual cylinders of the hexapod,
or using the particle volume times the shear rate. Our work indicates
that collision rate for non-convex particles in shear flows critically
depends on the shape of the particle due to nontrivial changes in
the particle’s orientational dynamics, and the ICR calculation
serves as a more reliable method for estimating their true collision
rates in the suspension.