Phoretic self-propulsion of Janus disks in the fast-reaction limit

Due to the net interfacial consumption of solute, the two-dimensional problem of phoretic swimming is ill posed in the standard description of diffusive transport, where the solute concentration satisfies Laplace's equation. It becomes well posed when solute advection is accounted for. We consider here the case of weak advection, where solute transport is analyzed using matched asymptotic expansions in two separate asymptotic regions, a near-field region in the vicinity of the swimmer and a far-field region where solute advection enters the dominant balance. We carry out the analysis for a standard Janus configuration, where half of the particle boundary is active and the other half is inert. Our main focus lies in the limit of fast reaction, which leads to a mixed boundary-value problem in the near field. That problem is solved using conformal mapping techniques. Our asymptotic scheme furnishes an implicit equation for the particle velocity $s$ in the direction of the active portion of its boundary, $2s\left(8ln\frac{8D}{\left|s\right|a}-\gamma \right)=b{c}_{\infty }/a$, wherein $a$ is the particle radius, $D$ the solute diffusivity, $c_{\infty }$ its far-field concentration, $b$ the diffusio-osmotic slip coefficient, and $\gamma$ the Euler-Mascheroni constant. The nonlinear dependence of $s$ upon $b{c}_{\infty }$ is a signature of the nonvanishing effect of solute advection.

Figure 1

The nonlinearly coupled boundary-value problem.

Figure 2

The particle-scale transport problem in the fast-reaction limit.

Figure 3

The conformal map $f\left(\zeta \right)$ transplants the upper unit $\zeta$ disk to the unbounded region exterior to a unit disk. The conformal map $g\left(\zeta \right)$ transplants the upper unit $\zeta$ disk to the unbounded region exterior to a unit disk with a slit.