Active particle motion in Poiseuille flow through rectangular channels

We investigate the dynamics of a point-like active particle suspended in fluid flow through a straight channel. For this particle-fluid system, we derive a constant of motion for a general unidirectional fluid flow, and apply it to an approximation of Poiseuille flow through channels with rectangular cross-sections. We obtain a $4$D nonlinear conservative dynamical system with one constant of motion and a dimensionless parameter describing the ratio of maximum flow speed to intrinsic active particle speed. Applied to square channels, we observe a diverse set of active particle trajectories with variations in system parameters and initial conditions which we classify into different types of swinging, trapping, tumbling and wandering motion. Regular (periodic/quasiperiodic) motion as well as chaotic active particle motion are observed for these trajectories and quantified using largest Lyapunov exponents. We explore the transition to chaotic motion using Poincar\'e maps and show ``sticky" chaotic tumbling trajectories that have long transients near a periodic state. We briefly illustrate how these results extend to rectangular cross-sections with width/height ratio larger than one. Outcomes of this work may have implications for dynamics of natural and artificial microswimmers in experimental microfluidic channels that typically have rectangular cross-sections.