# Inertial migration of a neutrally buoyant spheroid in plane Poiseuille flow

## TL;DR

This study analyzes the inertial migration behavior of neutrally buoyant spheroids in plane Poiseuille flow, revealing how shape and inertial effects influence equilibrium positions and lift velocities at low Reynolds numbers.

## Contribution

It provides a Jeffery-averaged analysis of inertial migration for spheroids with arbitrary aspect ratios, extending classical sphere results to more complex particle shapes.

## Key findings

- Equilibrium positions are similar to classical sphere results, starting near 0.6H/2 from the centerline.
- Inertial effects stabilize specific Jeffery orbits depending on aspect ratio.
- Lift profiles are shape-independent and match those of spheres at low Re_c.

## Abstract

We study the cross-stream inertial migration of a torque-free neutrally buoyant spheroid, of an arbitrary aspect ratio $\kappa$, in wall-bounded plane Poiseuille flow for small particle Reynolds numbers\,($Re_p\ll1$) and confinement ratios\,($\lambda\ll1$), with the channel Reynolds number, $Re_c = Re_p/\lambda^2$, assumed to be arbitrary; here, $\lambda=L/H$ where $L$ is the semi-major axis of the spheroid and $H$ denotes the separation between the channel walls. In the Stokes limit\,($Re_p =0)$ and for $\lambda \ll 1$, a spheroid rotates along any of an infinite number of Jeffery orbits parameterized by an orbit constant $C$, while translating with a time dependent speed along a given ambient streamline. Weak inertial effects stabilize either the spinning\,($C=0$) or the tumbling orbit\,($C=\infty$), or both, depending on $\kappa$. The separation of the Jeffery-rotation and orbital drift time scales, from that associated with cross-stream migration, implies that the latter occurs due to a Jeffery-averaged lift velocity. Although the magnitude of this averaged lift velocity depends on $\kappa$ and $C$, the shape of the lift profiles are identical to those for a sphere, regardless of $Re_c$. In particular, the equilibrium positions for a spheroid remain identical to the classical Segre-Silberberg ones for a sphere, starting off at a distance of about $0.6(H/2)$ from the channel centerline for small $Re_c$, and migrating wallward with increasing $Re_c$. For spheroids with $\kappa \sim O(1)$, the Jeffery-averaged analysis is valid for $Re_p\ll1$; for extreme aspect ratio spheroids, the regime of validity becomes more restrictive being given by $Re_p\,\kappa/\ln \kappa \ll 1$ and $Re_p/\kappa^2 \ll 1$ for $\kappa \rightarrow \infty$\,(slender fibers) and $\kappa \rightarrow 0$\,(flat disks), respectively.