Shear-induced self-diffusivity in dilute suspensions with repulsive interactions

Abstract

In a dilute non-Brownian suspension undergoing simple shear, pairwise hydrodynamic interactions are fore-aft symmetric at zero Reynolds number and produce no net cross-streamline displacement. A weak central repulsive force between particles breaks this symmetry, deflecting trajectories and generating irreversible transverse displacements that cumulatively yield a shear-induced self-diffusivity. We derive, via matched asymptotic expansions in the limit of weak repulsion, closed-form scaling laws for the gradient and vorticity components of this diffusivity. The gradient component exhibits a logarithmic enhancement relative to the vorticity component, a structural anisotropy that persists for all monotonically decaying repulsive potentials. The specific interaction enters only through integral functionals of the force profile weighted by hydrodynamic mobility functions, establishing that the scaling is universal across physically distinct mechanisms, such as electrical double-layer repulsion, steric interactions, or any other short-range central force. We validate the asymptotic predictions against full numerical trajectory integration for the representative case of electrostatic repulsion, modelled using the Gouy-Chapman description of the electrical double layer, and find excellent agreement in the expected regime.

Figures (11)

• Figure 1: The schematic representation of binary interactions. '1' indicates the satellite sphere of radius $a_1$ and '2' indicates the reference sphere of radius $a_2$.
• Figure 2: Schematic of in-plane trajectories for (a) $\varepsilon = 0$ and (b) $\varepsilon > 0$ in the presence of a repulsive interaction. Blue curves denote open trajectories, green curves denote closed trajectories in (a) and spiral trajectories in (b), and the red curve represents the limiting trajectory (separatrix) that separates the two types. The sphere at the center represents the reference sphere, and the black circle represents the collision sphere.
• Figure 3: Schematic of in-plane open trajectories in the upper half-plane for $\varepsilon = 0$ (blue, dashed) and $\varepsilon > 0$ (blue, solid), both originating with the same $\mathcal{O}(1)$ upstream offset $\overline{x}_2^{-\infty}$. In the far-downstream, the $\varepsilon>0$ trajectory is displaced relative to the zero-$\varepsilon$ trajectory by an amount $\upDelta \overline{x}_2$.
• Figure 4: Schematic of an in-plane trajectory for finite $\varepsilon$ ($0 < \varepsilon \ll 1$), with $\mathcal{O}(\varepsilon^{1/2})$ initial offset, shown as a solid curve. The dashed trajectories 1 and 2 denote the corresponding zero-$\varepsilon$ reference trajectories that match the far-upstream and far-downstream offsets of the solid trajectory, and serve as reference paths in the perturbation analysis. For clarity, displacements are shown exaggerated.
• Figure 5: Schematic of representative off-plane relative trajectories in the first quadrant ($\overline{x}_2>0$ and $\overline{x}_3>0$). Dashed curves correspond to $\varepsilon = 0$ and solid curves to $\varepsilon > 0$. Blue curves denote open trajectories, green curves denote closed (for $\varepsilon = 0$) or spiral (for $\varepsilon > 0$) trajectories, and the red curve represents the limiting trajectory (separatrix) for the off-plane case. The magenta curve is shown for reference and represents the in-plane limiting trajectory for $\varepsilon > 0$. The parallelograms at the left and right ends indicate the upstream and downstream offsets of the trajectories, shown for visual clarity.