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.