Taylor dispersion in systems of sedimenting nonspherical Brownian particles: II. Homogeneous ellipsoidal particles

Abstract

An Aris-type moment scheme is applied to calculate the Taylor-Aris dispersion tensor for the sedimentation of small homogeneous ellipsoidal (and other orthotropic) particles settling under the influence of gravity in a quiescent viscous fluid and undergoing rotational and translational Brownian motions. This generalizes to triaxial particles a prior dispersion result for centrally symmetric bodies of revolution, such as spheroids. An independent Langevin-type dispersivity calculation is shown to yield results identical to those obtained by the moment scheme. The components D ̄ |∗ and D ̄ ⊥∗ of the transversely isotropic dispersion dyadic, parallel and perpendicular, respectively, to the direction of the gravitational field, are shown to be given by the Taylor—Aris-type formulas D ∥ ∗ = D+ γ U ∗2 15 d , D ⊥ ∗ = D+ γ U ∗2 20d . Here, U ̄ ∗ is the mean settling velocity of the particle, and D ̄ = ( 1 3 )(D 1 + D 2 + D 3) and d ̄ = ( 1 3 ) × (d 1 + d 2 + d 3) are respectively the mean translational and rotational diffusivities of the Brownian particle, with (D 1,D 2,D 3) and (d 1,d 2,d 3) respectively the appropriate diffusivity components along the principal axes of the particle. The dimensionless coefficient γ, which is of order unity, is given by the formula γ= d[( D 1 D−1 ) 2d 1+( D 2 D−1 ) 2d 2+( D 3 D−1 ) 2d 3] d 1d 2+d 2d 3+d 3d 1 , This anisometric parameter vanishes identically for spherical particles and other hydrodynamically isotropic particles (e.g., cubes, tetrahedra, octahedra, etc.) whose translational and rotational hydrodynamic resistances are independent of the orientation of the particle relative to the directions of its linear and angular velocity vectors. Upon utilizing the translational and rotational Stokes-Einstein equations, explicit numerical values of D ̄ |∗ and D ̄ ⊥∗ are furnished for ellipsoids of revolution of various aspect ratios and sizes when settling in water. Physical restrictions pertaining to sedimentation-vessel apparatus size and the requirement of reasonable sedimentation times greatly restrict the range of particle sizes whose anisometric properties may be experimentally investigated by this new particle-shape characterization technique.