Enhanced Taylor Dispersion of an Axisymmetric Brownian Particle with Center Offset

Taylor dispersion, the gravity-induced enhancement of translational diffusion during the steady settling of a Brownian particle, has so far been analyzed only for torque-free bodies (H. Brenner, J. Colloid Interface Sci., 71(2): 189-208, 1979). In this work, we extend the theory to a non-centrosymmetric, axisymmetric particle whose centers of mass and buoyancy are offset from its hydrodynamic center, so that the gravitational torque acts in addition to the net gravitational force. We study the Taylor dispersion of such particles as a function of non-dimensional parameter $\alpha$ representing the strength of the gravitational torque: $\alpha$ is zero when the centers of mass and buoyancy are at the hydrodynamic center and increases when they are offset from it. Analytical calculations show that for small $\alpha$, the Taylor dispersion is always amplified: the effective diffusivities created by sedimentation always increase as ${\alpha }^2$, but they start to decrease at certain values of $\alpha$ and approach to zero for large values of $\alpha$. We further analyze the transient regime of the mean-square displacement (MSD). At short times, the MSD grows quadratically with time before crossing over to the diffusive regime. The ballistic regime persists to relatively high sedimentation Péclet numbers (the ratio of the rotational relaxation time to the sedimentation time), even in the presence of a gravitational torque, indicating that such a torque does not considerably alter sustained ballistic motion.