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\,($R{e}_p\ll 1$) and confinement ratios\,($\lambda \ll 1$), with the channel Reynolds number, $R{e}_c=R{e}_p/{\lambda }^2$, 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\,($R{e}_p=0)$ 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 $R{e}_c$. 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 $R{e}_c$, and migrating wallward with increasing $R{e}_c$. For spheroids with $\kappa \sim O(1)$, the Jeffery-averaged analysis is valid for $R{e}_p\ll 1$; for extreme aspect ratio spheroids, the regime of validity becomes more restrictive being given by $R{e}_p\kappa /\ln \kappa \ll 1$ and $R{e}_p/{\kappa }^2\ll 1$ for $\kappa \to \infty$\,(slender fibers) and $\kappa \to 0$\,(flat disks), respectively.