Particle motion nearby rough surfaces

We study the hydrodynamic coupling between particles and solid, rough boundaries characterized by random surface textures. Using the Lorentz reciprocal theorem, we derive analytical expressions for the grand mobility tensor of a spherical particle and find that roughness-induced velocities vary nonmonotonically with the characteristic wavelength of the surface. In contrast to sedimentation near a planar wall, our theory predicts continuous particle translation transverse and perpendicular to the applied force. Most prominently, this motion manifests itself in a variance of particle displacements that grows quadratically in time along the direction of the force. This increase is rationalized by surface roughness generating particle sedimentation closer to or farther from the surface, which entails a significant variability of settling velocities.

Figure 1

(a) Side view of a sphere with radius $a$ near a vertical rough wall $S_w$, characterized by the shape function $H(x,y)$, a wavelength $\lambda$, and amplitude $\epsilon a$. In response to an external force $\mathbf{F}=F\mathbf{e}$ (parallel to $S_0$), the sphere translates and rotates with velocities $\mathbf{U}$ and $\mathrm{\Omega }$, respectively. The sphere is located at ${\mathbf{r}}_S$ and at a distance $h$ relative to $S_0$. (b) Top view of the sphere and the rough surface (grayscale indicates a height map), where $x,y$ are the coordinates spanning $S_0$. Here, $\mathbf{F}$ is oriented at an angle $\phi$ relative to the surface structure and $\mathbf{e}$, ${\mathbf{e}}_{\perp }$, and ${\mathbf{e}}_z$ denote the unit vectors along, transverse, and perpendicular to $\mathbf{F}$.

Figure 2

Variance of the translational, $〈{\left(\mathbf{U}-〈\mathbf{U}〉\right)}^2〉$, and rotational velocities, $〈{\left(\mathrm{\Omega }-〈\mathrm{\Omega }〉\right)}^2〉$, with respect to (a) the distance $h/a$ with $\lambda /a=2$ and (b), (c) the wavelength $\lambda /a$. (b) and (c) show the variances for different particle-surface distances, $h/a=0.1,0.5$. Here, $U_{\parallel }$ denotes the settling velocity of a sphere near a planar wall and we have used $N=50$ in Eq. (5). The variances are shown for the velocities along ($\parallel$), transverse ($\perp$), and perpendicular ($z$) to the force direction, $\mathbf{F}=F{\mathbf{e}}_x$.

Figure 3

(a) Variance of the displacements, $\langle {[\mathrm{\Delta }\mathbf{r}\left(t\right)-{\mathbf{U}}^{\left(0\right)}t]}^2\rangle$, for different wavelengths $\lambda /a$, $h\left(0\right)/a=0.2$, $\epsilon =0.1$, and $\mathbf{F}=F\mathbf{e}=F{(cos\pi /4,sin\pi /4,0)}^T$. Over the simulation horizon the sphere has translated $\sim 60\times a$ along the force. Results are obtained by averaging over ${10}^3$ trajectories along different surface realizations [i.e., Eq. (5) with random ${\alpha }_{nm},{\beta }_{nm}$ ($N=20$)]. Error bars remain smaller than the symbol size. Inset: Variance parallel ($\parallel$), transverse ($\perp$), and perpendicular ($z$) to the force for $\lambda /a=2$. (b), (c) Distributions of the displacements at subsequent times $t_i$ (blue, red, and black correspond to the smallest, intermediate, and largest $t_i$) for $\lambda /a=2$. (b) $\mathrm{\Delta }h$ and (c) $\mathrm{\Delta }{r}_{\perp }$ relative to $\mathrm{\Delta }{r}_{\parallel }-\langle \mathrm{\Delta }{r}_{\parallel }\rangle$ with $\langle \mathrm{\Delta }{r}_{\parallel }\rangle =U_{\parallel }{t}_i$. The ellipses enclose $95\%$ of the displacements.

Figure 4

(a) Schematic of a sphere diffusing near a rough surface. (b) Variance of the roughness-induced diffusivities, $\langle {\left({\mathbf{D}}_{ij}^{\left(1\right)}\right)}^2\rangle$, as a function of $\lambda /a$ for $h/a=0.1,0.5$ (solid/dashed lines). The diffusivity in bulk is $D_{\text{free}}=k_{B}T/\left(6\pi \mu a\right)$.