A colored-noise Fokker-Planck equation for non-Brownian particles in shear-induced diffusion

Abstract

The topic of the present work are non-Brownian particles in shear flow. As reported in literature, the occurring phenomenon in this context is shear-induced diffusion which takes place in the absence of the well-known Brownian diffusion. Diffusive processes can be described stochastically in terms of a stochastic differential equation (Langevin equation or Langevin-like equation) or a differential equation for the probability density, in second order referred to as Fokker-Planck equation. It is known that in contrast to Brownian diffusion, the shear-induced diffusion is a long-time diffusion which poses a challenge to the stochastic description of this phenomenon. The present work analyzes the problem of non-Brownian particles in shear-induced diffusion with regard to the Markov property of the treated variables. This concludes that the Fokker-Planck equation so far derived in pure position space may not be sufficient. In order to ensure the Markov process property, a Fokker-Planck equation extended to coupled position colored-noise velocity space is derived. Throughout the extension, the colored-noise velocity is modeled as an Ornstein-Uhlenbeck process. These first two steps were also treated in the author’s Master thesis (Lukassen 2012). A detailed substantiation of this approach is published in (Lukassen & Oberlack 2014b) including a new multiple time scale analysis and a Gaussian solution. The multiple time scale analysis results in the dimensionless form of the equation of motion which serves as a starting point for the derivation of the new colored-noise Fokker-Planck equation. As a next step, this coupled Fokker-Planck equation is integrated over velocity space and approximated to yield a reduced position-space Fokker-Planck equation. It is shown that such a reduction as in the present work is only possible under certain conditions concerning the correlation time. The resulting position-space equation is analyzed and compared to the traditional position-space models. The reduced form exhibits additional correction terms. In an outlook, possible extensions of the presented model are discussed with exemplary simulation results. Chapter 5 and 6 as a whole are based on the author’s publication (Lukassen & Oberlack 2014b).