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 ( $Re_p\ll 1$ ) and confinement ratio ( $\lambda \ll 1$ ). Here, $Re_c = V_{wall}H/\nu$ , 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. Also, $\lambda = a/H$ , where $a$ is the sphere radius, with $Re_p=\lambda ^2 Re_c$ . The channel centreline 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-centre equilibria, located symmetrically with respect to the centreline, with the original centreline equilibrium becoming unstable simultaneously. The new equilibria migrate wall-ward with increasing $Re_c$ . In contrast to the inference based on recent computations, the aforementioned bifurcation occurs for arbitrarily small $Re_p$ provided that $\lambda$ 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 $110$ .