Brownian motion, the random motion of particles suspended in a medium, was first studied mathematically by Einstein and Smoluchowski. They obtained the particle’s mean square displacement (MSD) in a quiescent medium. Compared to the classical Brownian diffusion described by the Einstein-Smoluchowski equation, where the MSD is proportional to time, the anomalous Brownian diffusion, where the MSD has a non-linear function of time, is studied based on the Langevin equations. Contrary to previous studies in this field, we investigate Brownian diffusion using stochastic calculus instead of classical calculus.

First, we investigate the stochastic dynamics of free Brownian particles in a sinusoidal wave flow. It indicates that flows in which the velocity depends only on time will not affect the particle’s diffusion. Then, the stochastic dynamics of free Brownian particles in two-dimensional laminar flows in which the velocity profiles are polynomial functions of the transverse coordinate are studied. Our new method is validated for two different flow cases without boundary effects: Couette flow and Plane Poiseuille flow. We show that for time scales much smaller than the particle relaxation time scales, the particle almost travels at its initial velocity and that the Brownian diffusion is virtually the same as in quiescent or uniform fluids. On the other hand, to leading order and for long time scales, the time dependency of the variance is . This is due to Brownian diffusion affected by the flow’s velocity gradients, where the particle may be carried by the flow to a wide range in the streamwise direction. It reveals that Brownian motion in transverse directions and the velocity profile of the flow make a significant difference to the particle’s diffusion in the streamwise direction.

Finally, we explore the response of Brownian particles to a flow propagating by a horizontally oscillating plate. We found an analytical solution for the particle dynamics. A new exact solution is provided for the cases in which Brownian motion is negligible. We show that the diffusion in the horizontal direction is greatly affected by Brownian motion in the vertical direction and the function of the flow’s velocity, which also verifies our study on Brownian diffusion of the polynomial case.