Inertial migration of a sphere in plane Couette flow

Abstract

We study the inertial migration of a torque-free neutrally buoyant sphere in wall-bounded plane Couette flow over a wide range of channel Reynolds numbers, $Re_c$, in the limit of small particle Reynolds number\,($Re_p\ll1$) and confinement ratio\,($\lambda\ll1$). Here, $Re_c = V_\text{wall}H/\nu$ where $H$ denotes the separation between the channel walls, $V_\text{wall}$ denotes the speed of the moving wall, and $\nu$ is the kinematic viscosity of the Newtonian suspending fluid; $\lambda = a/H$, $a$ being the sphere radius, with $Re_p=\lambda^2 Re_c$. The channel centerline is found to be the only (stable)\,equilibrium below a critical $Re_c\,(\approx 148)$, consistent with the predictions of earlier small-$Re_c$ analyses. A supercritical pitchfork bifurcation at the critical $Re_c$ creates a pair of stable off-center equilibria, symmetrically located with respect to the centerline, with the original centerline equilibrium simultaneously becoming unstable. The new equilibria migrate wallward with increasing $Re_c$. In contrast to the inference based on recent computations, the aforementioned bifurcation occurs for arbitrarily small $Re_p$ provided $\lambda$ is sufficiently small. An analogous bifurcation occurs in the two-dimensional scenario, that is, for a circular cylinder suspended freely in plane Couette flow, with the critical $Re_c$ being approximately $110$.