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Understanding the orientation dynamics of anisotropic colloidal particles is important for suspension rheology and particle self-assembly. However, even for the simplest case of dilute suspensions in shear flow, the orientation dynamics of non-spherical Brownian particles are poorly understood. Here we analytically calculate the time-dependent orientation distributions for non-spherical axisymmetric particles confined to rotate in the flow–gradient plane, in the limit of small but non-zero Brownian diffusivity. For continuous shear, despite the complicated dynamics arising from the particle rotations, we find a coordinate change that maps the orientation dynamics to a diffusion equation with a remarkably simple ratio of the enhanced rotary diffusivity to the zero shear diffusion: $$D_{eff}^{r}/D_{0}^{r}=(3/8)(p-1/p)^{2}+1$$ , where $$p$$ is the particle aspect ratio. For oscillatory shear, the enhanced diffusion becomes orientation dependent and drastically alters the long-time orientation distributions. We describe a general method for solving the time-dependent oscillatory shear distributions and finding the effective diffusion constant. As an illustration, we use this method to solve for the diffusion and distributions in the case of triangle-wave oscillatory shear and find that they depend strongly on the strain amplitude and particle aspect ratio. These results provide new insight into the time-dependent rheology of suspensions of anisotropic particles. For continuous shear, we find two distinct diffusive time scales in the rheology that scale separately with aspect ratio $$p$$ , as $$1/D_{0}^{r}p^{4}$$ and as $$1/D_{0}^{r}p^{2}$$ for $$p\gg 1$$ . For oscillatory shear flows, the intrinsic viscosity oscillates with the strain amplitude. Finally, we show the relevance of our results to real suspensions in which particles can rotate freely. Collectively, the interplay between shear-induced rotations and diffusion has rich structure and strong effects: for a particle with aspect ratio 10, the oscillatory shear intrinsic viscosity varies by a factor of $${\approx}2$$ and the rotational diffusion by a factor of $${\approx}40$$ .
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This content will become publicly available on September 10, 2026
Time-averaged interactions between a pair of particles in oscillatory flow
We study the interaction between a pair of particles suspended in a uniform oscillatory flow. The time-averaged behaviour of particles under these conditions, which arises from an interplay of inertial and viscous forces, is explored through a theoretical framework relying on small oscillation amplitude. We approximate the oscillatory flow in terms of dual multipole expansions, with which we compute time-averaged interaction forces using the Lorentz reciprocal theorem. We then develop analytic approximations for the force in the limit where Stokes layers surrounding the particles do not overlap. Finally, we show how the same formalism can be generalised to the situation where the particles are free to oscillate and drift in response to the applied flow. The results are shown to be in agreement with existing numerical data for forces and particle velocities. The theory thus provides an efficient means to quantify nonlinear particle interactions in oscillatory flows.
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- Award ID(s):
- 2143943
- PAR ID:
- 10652504
- Publisher / Repository:
- Cambridge University Press
- Date Published:
- Journal Name:
- Journal of Fluid Mechanics
- Volume:
- 1018
- ISSN:
- 0022-1120
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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