Inertial migration of a neutrally buoyant spheroid in plane Poiseuille flow

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 ( $Re_p\ll 1$ ) and confinement ratios ( $\lambda \ll 1$ ), 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 tumbling orbit ( $C=\infty$ ), or both, depending on $\kappa$ . The asymptotic separation of the Jeffery rotation and orbital drift time scales, from that associated with cross-stream migration, implies that migration 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 centreline 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 $Re_p\ll 1$ ; for extreme aspect ratio spheroids, the regime of validity becomes more restrictive being given by $Re_p \kappa /\ln \kappa \ll 1$ and $Re_p/\kappa \ll 1$ for $\kappa \rightarrow \infty$ (slender fibres) and $\kappa \rightarrow 0$ (flat disks), respectively.