Ratcheting of Brownian swimmers in periodically corrugated channels: A reduced Fokker-Planck approach

We consider the motion of self-propelling Brownian particles in two-dimensional periodically corrugated channels. The point-size swimmers propel themselves in a direction which fluctuates by Brownian rotation; in addition, they undergo Brownian motion. The impermeability of the channel boundaries in conjunction with an asymmetry of the unit-cell geometry enables ratcheting, where a nonzero particle current is animated along the channel. This effect is studied here in the continuum limit using a diffusion-advection description of the probability density in a four-dimensional position-orientation space. Specifically, the mean particle velocity is calculated using macrotransport (generalized Taylor-dispersion) theory. This description reveals that the ratcheting mechanism is indirect: swimming gives rise to a biased spatial particle distribution which in turn results in a purely diffusive net current. For a slowly varying channel geometry, the dependence of this current upon the channel geometry and fluid–particle parameters is studied via a long-wave approximation over a reduced two-dimensional space. This allows for a straightforward seminumerical solution. In the limit where both rotational diffusion and swimming are strong, we find an asymptotic approximation to the particle current, scaling inversely with the square of the swimming Péclet number. For a given swimmer-fluid system, this limit is physically realized with increasing unit-cell size.

Figure 1

Schematic depicting the unit cell and the angles $\theta$ and $\varphi$.

Figure 2

Schematic of the sawtooth shape.

Figure 3

Variation of $A_0\left(x\right)$ for $l=0.1$ and $M=0.9$, for the indicated values of $\mathrm{Pe}$. (a) $\alpha =10$. (b) $\alpha =100$.

Figure 4

$U$ as a function of $\mathrm{Pe}$ for the indicated values of $\alpha$, for $l=0.1$ and $M=0.9$.

Figure 5

$U$ as a function of $l$ for the indicated values of $M$, with $\mathrm{Pe}=10$ and $\alpha =100$.

Figure 6

$U$ as a function of $\mathrm{Pe}$ for $M=0.9,l=0.1$, and $\alpha =0.1{\mathrm{Pe}}^2$. The solid line is the seminumerical solution. The dashed line is the large-cell approximation (34),(35).