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We study the cross-stream inertial migration of a torque-free neutrallybuoyant spheroid, of an arbitrary aspect ratio $\kappa$, in wall-bounded planePoiseuille flow for small particle Reynolds numbers\,($Re_p\ll1$) andconfinement 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 thesemi-major axis of the spheroid and $H$ denotes the separation between thechannel walls. In the Stokes limit\,($Re_p =0)$ and for $\lambda \ll 1$, aspheroid rotates along any of an infinite number of Jeffery orbitsparameterized by an orbit constant $C$, while translating with a time dependentspeed along a given ambient streamline. Weak inertial effects stabilize eitherthe spinning\,($C=0$) or the tumbling orbit\,($C=\infty$), or both, dependingon $\kappa$. The separation of the Jeffery-rotation and orbital drift timescales, from that associated with cross-stream migration, implies that thelatter occurs due to a Jeffery-averaged lift velocity. Although the magnitudeof this averaged lift velocity depends on $\kappa$ and $C$, the shape of thelift profiles are identical to those for a sphere, regardless of $Re_c$. Inparticular, the equilibrium positions for a spheroid remain identical to theclassical Segre-Silberberg ones for a sphere, starting off at a distance ofabout $0.6(H/2)$ from the channel centerline for small $Re_c$, and migratingwallward with increasing $Re_c$. For spheroids with $\kappa \sim O(1)$, theJeffery-averaged analysis is valid for $Re_p\ll1$; for extreme aspect ratiospheroids, 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.

Inertial migration of a neutrally buoyant spheroid in plane Poiseuille flow