Inertial migration of slender prolate and thin oblate spheroids in plane Poiseuille flow

A neutrally buoyant rigid spheroid suspended in a wall-bounded plane Poiseuille flow undergoes cross-stream migration, driven by inertia. We examine theoretically the migration of a spheroid of aspect ratio

in the limit of small but finite particle Reynolds number (

italic Re Subscript p Re p ${\textit{Re}}_{\!p}$

), and for small confinement ratios (

lamda λ $\lambda$

), with the channel Reynolds number,

italic Re Subscript c Baseline equals italic Re Subscript p Baseline divided by lamda squared Re c = Re p / λ 2 ${\textit{Re}}_c = {\textit{Re}}_{\!p}/\lambda ^2$

, assumed arbitrary; here,

lamda equals upper L divided by upper H λ = L / H $\lambda =L/H$

with

upper L L $L$

being the semimajor axis of the spheroid and

upper H H $H$

denoting the separation between channel walls. For small

lamda λ $\lambda$

, the asymptotic separation between the rotation-cum-orbital-drift and migration time scales implies that, to begin with, inertia rapidly drives the spheroid towards the tumbling orbit (orbit constant,

upper C equals normal infinity C = ∞ $C=\infty$

) with negligible migration; migration is subsequently driven by a time-averaged lift velocity, the average being over the orientations sampled in the inertially stabilized tumbling orbit. While spheroids with

kappa tilde upper O left parenthesis 1 right parenthesis κ ∼ O ( 1 ) $\kappa \sim O(1)$

rotate with the Jeffery angular velocity to a very good approximation, deviations from Jeffery rotation, for both large and small

kappa κ $\kappa$

, lead to a time-averaged inertial lift profile with equilibrium locations (zero crossings) that differ from the classical Segre–Silberberg predictions for a sphere. Beyond a threshold

italic Re Subscript c Re c ${\textit{Re}}_c$

, both slender spheroids and thin disks attain a steady orientation in the neighbourhood of the walls, and with increasing

italic Re Subscript c Re c ${\textit{Re}}_c$

, these rotation-arrested regions grow in extent, moving in towards the channel centreline. In contrast to spheres, but consistent with experiments, onset of rotation arrest causes the aforementioned equilibrium positions to move towards the centreline; further, for thin disks alone, the equilibrium positions themselves become rotation-arrested beyond a threshold

italic Re Subscript c Re c ${\textit{Re}}_c$

. The

kappa κ $\kappa$

-dependence of the equilibrium positions can be leveraged towards developing passive shape-sorting protocols on microfluidic platforms.