Closing the reduced position-space Fokker-Planck equation for shear-induced diffusion using the Physalis method

In the shear flow of non-Brownian particles, we describe the long-time diffusive processes stochastically using a Fokker-Planck equation. Previous work has indicated that a Fokker-Planck equation coupling the probability densities of position and velocity spaces may be appropriate for describing this phenomenon (Lukassen \& Oberlack, Phys.~Rev.~E 89, 2014). The stochastic description, integrated over velocity space to obtain a reduced position-space Fokker-Planck equation, contains unknown space diffusion coefficients. In this work, we use the Physalis method for simulating disperse particle flows (Sierakowski \& Prosperetti, J.~Comp.~Phys., 2016) to verify the colored-noise velocity space model (an Ornstein-Uhlenbeck process) by comparing the simulated long-time diffusion rate with the diffusion rate proposed by the theory. We then use the simulated data to calculate the unknown space diffusion coefficients that appear in the reduced position-space Fokker-Planck equation and summarize the results.