We present analytical solutions for particle transport and deposition over a horizontally oscillating plate. The dynamics and deposition of particles with negligible Brownian forces are resolved, and the distribution of particle deposition and the effects of Stokes number, Froude number, and the particle's initial height on the deposition length are analyzed. The analysis reveals nonlinear oscillatory behavior of the particle deposition length in response to variations of the flow and particle properties. Then, following the method previously developed by Wang and Dagan “Brownian particle diffusion in generalized polynomial shear flows,” Phys. Rev. E 110, 024117 (2024), the dynamics and anomalous diffusion of Brownian particles are studied by solving the Langevin equation using stochastic calculus. The anomalous diffusion predicted by the analytical formulation is then validated by high-fidelity numerical simulations. We demonstrate that particle diffusion in the streamwise direction is significantly altered due to the coupling between the flow velocity gradient and Brownian motion in the transverse direction. The dynamical response of Brownian particles to both the horizontal periodic flow and the vertical body force is examined, revealing the relative significance of the coupling between Brownian motion, external forcing, and the carrier flow in realizing particle diffusion.
Wang et al. (2026) studied this question.