We study the inertial migration of finite-size neutrally buoyant spherical particles in dilute and semidilute suspensions in laminar square duct flow. We perform several direct numerical simulations using an immersed boundary method to investigate the effects of the bulk Reynolds number ${\text{Re}}_b$, particle Reynolds number ${\text{Re}}_p$, and duct to particle size ratio $h/a$ at different solid volume fractions $\varphi$, from very dilute conditions to $20\%$. We show that the bulk Reynolds number ${\text{Re}}_b$ is the key parameter in inertial migration of particles in dilute suspensions. At low solid volume fraction ($\varphi =0.4\%$), low bulk Reynolds number (${\text{Re}}_b=144$), and $h/a=9$ particles accumulate at the center of the duct walls. As ${\text{Re}}_b$ is increased, the focusing position moves progressively toward the corners of the duct. At higher volume fractions, $\varphi =5\%$, $10\%$, and $20\%$, and in wider ducts ($h/a=18$) with ${\text{Re}}_b=550$, particles are found to migrate away from the duct core toward the walls. In particular, for $\varphi =5\%$ and $10\%$, particles accumulate preferentially at the corners. At the highest volume fraction considered, $\varphi =20\%$, particles sample all the volume of the duct, with a lower concentration at the duct core. For all cases, we find that particles reside longer times at the corners than at the wall centers. In a duct with lower duct to particle size ratio $h/a=9$ (i.e., with larger particles), $\varphi =5\%$, and high bulk Reynolds number ${\text{Re}}_b=550$, we find a particle concentration pattern similar to that in the ducts with $h/a=9$ regardless of the solid volume fraction $\varphi$. Instead, for lower Bulk Reynolds number ${\text{Re}}_b=144$, $h/a=9$, and $\varphi =5\%$, a different particle distribution is observed in comparison to a dilute suspension $\varphi =0.4\%$. Hence, the volume fraction plays a key role in defining the final distribution of particles in semidilute suspensions at low bulk Reynolds number. The presence of particles induces secondary cross-stream motions in the duct cross section, for all $\varphi$. The intensity of these secondary flows depends strongly on particle rotation rate, on the maximum concentration of particles in focusing positions, and on the solid volume fraction. We find that the secondary flow intensity increases with the volume fraction up to $\varphi =5\%$. However, beyond $\varphi =5\%$ excluded-volume effects lead to a strong reduction of cross-stream velocities for ${\text{Re}}_b=550$ and $h/a=18$. Inhibiting particles from rotatingalso results in a substantial reduction of the secondary flow intensity and in variations of the exact location of the focusing positions.

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• Particle-laden flows
• Suspensions