Lagrangian Self-Diffusion of Brownian Particles in Periodic Flow Fields

The steady transport of Brownian particles convected by a periodic flow field is studied by following the motion of a randomly chosen tagged particle in an otherwise uniform solute concentration field. A nonlocal, Fickian constitutive relation is derived, in which the steady mass flux of Brownian particles equals a convolution integral of the concentration gradient times a (tensorial) diffusion function. In turn, the diffusion function is uniquely determined via the n-th diffusivities, which are determined analytically in terms of the nth cumulants of the probability distribution by exploiting the translational symmetry of the velocity field. The Lagrangian, long-time self-diffusion function is shown to be equal to the symmetric part of the Eulerian, gradient diffusion function. Since the latter characterizes the dissipative steady-state mass transport, while the former describes the fluctuations of the concentration field about its uniform equilibrium value, the equality between the two types of diffusivities can be seen as an aspect of the fluctuation–dissipation theorem. Finally, the present results are applied to study the transport of solute particles immersed in a fluid flowing in rectilinear pipes and through periodic fixed beds of spheres at low Péclet number. In the first case, the first six nth diffusivities are determined; in the second, the first two diffusivities are calculated, showing that the enhancement to the second diffusivity due to convection is eight times larger in the direction parallel to the fluid flow than in the transversal direction.