It is intuitive that the diffusivity of an isolated particle differs from those in a monodisperse suspension, in which hydrodynamic interactions between the particles are operative. Batchelor [J. Fluid Mech. 74, 1–29 (1976) and J. Fluid Mech. 131, 155–175 (1983)] calculated how hydrodynamic interactions influenced the diffusivity of a dilute suspension of spherical particles, and Russel et al. [Colloidal Dispersions (Cambridge University Press, 1991)] and Brady [J. Fluid Mech. 272, 109–134 (1994)] treated nondilute (higher particle volume fraction) suspensions. Although most particles lack perfect sphericity, little is known about the effects of hydrodynamic interactions on the diffusivity of spheroidal particles, which are the simplest shapes that can be used to model anisotropic particles. Here, we calculate the effects of hydrodynamic interactions on the translational and rotational diffusivities of spheroidal particles of arbitrary aspect ratio in dilute monodisperse suspensions. We find that the translational and rotational diffusivities of prolate spheroids are more sensitive to eccentricity than for oblate spheroids. The origin of the hydrodynamic anisotropy is that found in the stresslet field for the induced-dipole interaction. However, in the dilute limit, the effects of anisotropy are at the level of a few percent. These effects have influence on a vast range of settings, from partially frozen colloidal suspensions to the dynamics of cytoplasm.
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Batchelor1 obtained a slightly different value of −1.83 for the O(ϕ) correction by using a mobility matrix which was exact for any arbitrary separation between two spheres, whereas we use a mobility matrix which is approximate since only two reflections are considered. However, even with the approximate mobility matrix, we obtain a good estimate for the correction at e = 0.
The net change in the reflected velocity field along a spheroidal axis is the integral over the spheroid of the gradient of the velocity field along that direction, and it is averaged over all possible configurations of the suspension with the test spheroid fixed. This underlies the strength of the induced dipole.