Fluctuation hydrodynamics of the brownian motion of a particle in a fixed dispersed layer

Abstract

The influence of the perturbation exerted by a grid of fixed spherical particles, randomly distributed in space, on the Brownian diffusion of particles suspended in the flow of a fluid which penetrates the grid is disucssed. The fixed particles affect the coefficient of diffusion that is transverse to the flow in two ways: on the one hand they reduce it in accordance with the Stokes coefficient, and on the other they increase it because of the influence of a random, velocity field which is generated by the flow past the randomly distributed particles. A convective diffusion equation is derived on the basis of the Fokker-Planck equation for a distribution function. A stochastic diffusion equation (of Langevin's type) obtained with a random velocity field is solved by the method of Green's function, whence the desired diffusion coefficient is found. The errors allowed when solving a similar problem in /1/ are indicated. The fluctuation hydrodynamics of Brownian motion in a homogeneous viscous fluid was discussed in /2/ where, in particular, an expression for the coefficient of the particle resistance was obtained in terms of the fluctuation characteristics of the fluid. Later, the influence of hydrodynamic fluctuations on the diffusion of a particle in a homogeneous fluid was examined in /3/: it was shown that the diffusion coefficient of a particle that is large with respect to intermolecular distances is determined entirely by the thermal fluctuations of the fluid velocity field. This result was also confirmed by the microscope kinetic theory of Brownian motion in /4, 5/, where an expression similar to Kubo's formula, for the coefficient of resistance of a large particle in terms of the fluctuation of the stress tensor of the fluid was obtained, and it was shown that for an incompressible fluid the well-known Stokes formula follows from this expression. In the present paper we consider the influence of fluctuation hydrodynamics on the diffusion of a suspended particle. The hydrodynamic fluctuations are generated by a fluid flow with velcoity v 0 through a rarefied system of randomly distributed fixed particles. These fluctuations are superimposed on the thermal fluctuations of the fluid velocity, and result in an increase in the diffusion coefficient of a suspended particle. The purpose of the present paper is to analyse this effect.