Conditional stability of particle alignment in finite-Reynolds-number channel flow

Finite-size neutrally buoyant particles in a channel flow are known to accumulate at specific equilibrium positions or spots in the channel cross-section if the flow inertia is finite at the particle scale. Experiments in different conduit geometries have shown that while reaching equilibrium locations, particles tend also to align regularly in the streamwise direction. In this paper, the Force Coupling Method was used to numerically investigate the inertia-induced particle alignment, using square channel geometry. The method was first shown to be suitable to capture the quasi-steady lift force that leads to particle cross-streamline migration in channel flow. Then the particle alignment in the flow direction was investigated by calculating the particle relative trajectories as a function of flow inertia and of the ratio between the particle size and channel hydraulic diameter. The flow streamlines were examined around the freely rotating particles at equilibrium, revealing stable small-scale vortices between aligned particles. The streamwise inter-particle spacing between aligned particles at equilibrium was calculated and compared to available experimental data in square channel flow (Gao {\it et al.} Microfluidics and Nanofluidics {\bf 21}, 154 (2017)). The new result highlighted by our numerical simulations is that the inter-particle spacing is unconditionally stable only for a limited number of aligned particles in a single train, the threshold number being dependent on the confinement (particle-to-channel size ratio) and on the Reynolds number. For instance, when the particle Reynolds number is $\approx1$ and the particle-to-channel height size ratio is $\approx0.1$, the maximum number of stable aligned particles per train is equal to 3. This agrees with statistics realized on the experiments of (Gao {\it et al.} Microfluidics and Nanofluidics {\bf 21}, 154 (2017)).