Hindered sedimentation, diffusion, and dispersion coefficients for brownian spheres in circular cylindrical pores

Abstract

Generalized Taylor dispersion theory is used to calculate the mean sedimentation velocity U ∗ and Taylor-Aris axial dispersivity D ∗ of a nonneutrally buoyant spherical Brownian particles (radius = a) settling and diffusing in a vertical, quiescent fluid-filled circular capillary tube (radius = r 0) for circumtances in which wall effects retarding the sphere motion are sensible. Owing to the dependence of the quasistatic Bromian sphere settling, velocity U[ r] upon the instantaneous distance r(t) of its center from the tube axis at time t, a Taylor dipsersion phenomenon ensues, summarized by the formula D ∗ = D ∗ + D C , where D C = Cr 0 2( U ∗) 2 D M , in which the nondimensional coefficient C depends upon the dimensionless Langevin-Peclet number parameter Pe = r 0 F/ kT and wall parameter λ = a/ r 0 ( F = net gravity forces, kT = Boltzmann factor). The dispersion contribution D C , arising indirectly from wall effects, is in addition to the direct hindred-diffusion effect of the tube walls upon the mean axial molecular diffusivity D M of the Brownian sphere. Calculations of D C (and U∗ ) are presented for all values of 0 < Pe < ∞ for both the small (λ ⪡ 1) and closely fitting (1 − λ ⪡ 1) sphere cases, as well as for cases where the sedimenting force F, rather than being constant in time, is time-periodic with zero mean. The latter results are used to calculate the enhanced, frequency-dependent, axial “diffusion” coefficient of a charged Brownian spehre in an AC field, corresponding to the circumstance that its time-average velocity through the tube is identically zero.