Translational diffusion of axisymmetrical particles in shear flow

Abstract

The motion of axisymmetric particles undergoing translational and rotary Brownian motion and subjected to a simple shear flow is studied theoretically and experimentally. In general, translational diffusion is affected by rotary Brownian motion because the displacement of particles depends on their orientation. From solutions of the translational convective diffusion equation, calculations have been made of the mean-square displacement dyadic for particles in an unbounded simple shear flow. The components of this dyadic can be expressed in terms of integrals containing orientational averages. Analytical results for high rotary Péclet number are worked out. It is found that at large times the diffusion in the equatorial plane of the shear field can be described by an effective diffusion coefficient D eff. Because the effective diffusion constant depends only weakly on the orbit constant, it is to be expected that for arbitrary Péclet numbers, D eff does not deviate much from the value at high Péclet numbers. Similarly, diffusion in the direction perpendicular to the equatorial plane can be, at large times, described by an effective diffusion constant as well. Experiments performed with a newly designed traveling microtube apparatus confirm certain aspects of the theory. Measurements have been made of the mean-square displacement 〈 x 3 ∗2 〉 in the direction of flow of doublets consisting of touching equal-sized polystyrene latex spheres, subjected to Poiseuille flow, with a rotary Péclet number of order unity. It was found that at short times 〈 x 3 ∗2 〉 varied linearly with time, in agreement with a generalization of the early results of Einstein; at large times, the displacement was found to increase cubically with time and the diffusion could be described by an effective diffusion coefficient, which value was in good agreement with theory.