Inertial focusing of small particles in wavy channels: Asymptotic analysis at weak particle inertia

The motion of tiny non-Brownian inertial particles in a two-dimensional channel flow with periodic corrugations is investigated analytically, to determine the trapping rate as well as the exact position of the attractor, and understand the conditions under which particle trapping and long-term suspension occur. This phenomenon has been observed numerically in previous works and happens under the combined effects of confinement and inertia. Starting from the particle motion equations, a Poincaré map is constructed analytically in the limit of weak inertia and weak channel corrugations. It enables to derive the equation of the attractor, if any, and the corresponding trapping rate. The attractor is close to a streamline, the so-called “attracting streamline”, and is shown to persist in the presence of transverse gravity, provided the channel Froude number is large enough. Particles which are trapped by this streamline can therefore travel over long distances, avoiding deposition. Numerical simulations confirm the theoretical results at small particle response times τ and reveal some non-linear effects at larger τ : the asymptotic attractor becomes unstable at some critical value and splits into multiple branches each with its own basin of attraction.