PHYSICAL REVIEW E 89, 052145 (2014) Colored-noise Fokker-Planck equation for the shear-induced self-diffusion process of non-Brownian particles Laura J. Lukassen * and Martin Oberlack † Graduate School of Excellence Computational Engineering, TU Darmstadt and Department of Mechanical Engineering, Chair of Fluid Dynamics, TU Darmstadt, Germany (Received 21 November 2013; published 29 May 2014) In the literature, it is pointed out that non-Brownian particles tend to show shear-induced diffusive behavior due to hydrodynamic interactions. Several authors indicate a long correlation time of the particle velocities in comparison to Brownian particle velocities modeled by a white noise. This work deals with the derivation of a Fokker-Planck equation both in position and velocity space which describes the process of shear-induced self-diffusion, whereas, so far, this problem has been described by Fokker-Planck equations restricted to position space. The long velocity correlation times actually would necessitate large time-step sizes in the mathematical description of the problem in order to capture the diffusive regime. In fact, time steps of specific lengths pose problems to the derivation of the corresponding Fokker-Planck equation because the whole particle configuration changes during long time-step sizes. On the other hand, small time-step sizes, i.e., in the range of the velocity correlation time, violate the Markov property of the position variable. In this work we regard the problem of shear-induced self-diffusion with respect to the Markov property and reformulate the problem with respect to small time-step sizes. In this derivation, we regard the nondimensionalized Langevin equation and develop a new compact form which allows us to analyze the Langevin equation for all time scales of interest for both Brownian and non-Brownian particles starting from a single equation. This shows that the Fokker-Planck equation in position space should be extended to a colored-noise Fokker-Planck equation in both position and colored-noise velocity space, which we will derive. DOI: 10.1103/PhysRevE.89.052145 PACS number(s): 05.40.−a, 05.10.Gg, 47.55.Kf , 47.27.W− I. INTRODUCTION Non-Brownian particles in shear flow have become a considerably discussed topic throughout the literature. As the name implies, non-Brownian particles do not perform the well-known Brownian motion, which is an infinitely short correlated motion due to the molecule pushes of the surrounding solvent. Still, also non-Brownian particles are found to perform diffusive motion. This type of diffusion has been found to occur for particles with negligible inertia in Stokes flow in the absence of any kind of Brownian or turbulent diffusion [1]. Indeed, the so-called shear-induced self-diffusion results from hydrodynamic interactions between particles in the flow. This remarkable phenomenon has been subject to various theoretical, experimental, and numerical works, which are summarized below. The starting point of theoretical considerations is the two-particle interaction of purely hydrodynamically interacting particles with negligible inertia in Stokes flow as described by Batchelor and Green [2], [3] and in a review by Morris [4]. Though a viscous fluid is considered, mathematically this implies fore-aft symmetry of particle pair trajectories and reversibility [2–5]. From a theoretical point of view, this fore-aft symmetry and reversibility is refracted, the reasons being manifold, cf. Refs. [4,6–8], including surface roughness [9,10], weak Brownian motion and interparticle forces [5], and inertia [11]. The underlying process of shear-induced self-diffusion was evaluated experimentally, see, e.g., Refs. [1,12,13]. Numerical considerations can be found in, e.g., Refs. [14–17] using the * lukassen@gsc.tu-darmstadt.de † oberlack@fdy.tu-darmstadt.de Stokesian dynamics method [18] and in, e.g., Ref. [7] using so-called accelerated Stokesian dynamics [19]. Further, Pine et al. [20] compare experimental data with numerical results obtained by the Stokesian dynamics method. Even though Drazer et al. [15] use a repulsive force in their simulation, they argue that the diffusive behavior should also follow in the case of purely hydrodynamically interacting particles. In the present paper, which focuses on theoretical derivations, we also regard the case of purely hydrodynamically interacting particles and follow the assumptions in Refs. [7,8] that enough many-particle interactions suffice to create diffusive behavior. The common models to describe particle flows either refer to an equation of motion (Langevin equation or Langevin-like equation) or the corresponding Fokker-Planck equation. One essential part of the present work is an asymptotics of the equation of motion in Sec. II, which includes the analysis for Brownian and non-Brownian particles for all apparent time scales, here the inertial relaxation time scale and the time scale of configurational changes, in a single starting equation. The asymptotics to be discussed here can be placed in a context of other asymptotics, e.g., Refs. [11,18,21–23], whereby we derive a new compact form which includes all other cases as special cases. The corresponding Fokker-Planck equation depends on the time scale of interest and differs from the Brownian to the non-Brownian case. The basic version of the Fokker-Planck equation derived for Brownian particles provides an equation in position-velocity space (also indicated as phase space distribution), cf. Refs. [21,24,25]. Under the assumption that the velocity relaxes to equilibrium on a smaller time scale than the position, this form can be reduced to position space (Smoluchowski equation) [24,25]. Further, the reduction of the 1539-3755/2014/89(5)/052145(15) 052145-1 ©2014 American Physical Society