Bifurcation phenomena on the inertial focusing of a neutrally buoyant spherical particle suspended in square duct flows

Spherical particles in dilute suspension flows through a square duct are known to focus at certain points in the downstream cross-section. When the blockage ratio of a particle diameter to the duct width equals approximately 0.1, previous experimental and numerical studies showed three types of particle equilibrium points: channel-face equilibrium positions located at the center of the duct faces, channel-corner equilibrium positions near the corners on diagonals, and intermediate equilibrium positions located at intermediate positions symmetric with respect to the diagonals. Several patterns of the particle focusing over the duct cross-section were observed according to the appearance or disappearance of these equilibrium points depending on the Reynolds number. In the present study, we numerically investigate the inertial focusing patterns of a neutrally buoyant spherical particle suspended in square duct flows. We describe the transitions between the particle focusing patterns in terms of bifurcation of the particle equilibrium positions. Saddle-node bifurcations occur at two different critical Reynolds numbers corresponding to the appearance and disappearance of the intermediate equilibrium positions, respectively. Pitchfork bifurcation occurs at another Reynolds number where stability of the channel-corner equilibrium positions changes from saddle to stable. Channel-face equilibrium positions are always stable for all the investigated Reynolds numbers at this blockage ratio.