Chiral swimmer with a regular arbitrary active patch

We investigate the low Reynolds number hydrodynamics of a spherical swimmer with a predominantly hydrophobic surface, except for a hydrophilic active patch. This active patch covers a portion of the surface and exhibits chiral activity that varies as a function of $\theta$ and $\phi$. Our study considers two types of active patches: (i) a symmetric active patch (independent of $\phi$) and (ii) an arbitrary active patch (depends on both $\theta$ and $\phi$). The swimming velocity, rotation rate, and flow field of the swimmer are calculated analytically. The objective of this work is to find the optimal configurations for both patch models to maximize the swimmer's velocity and efficiency. Interestingly, the maximum velocity can be controlled by adjusting the hydrophobicity, patch configuration, and strength of the surface activity. We find that for the symmetric patch model, the swimmer's velocity is $U_{SP} = 1.414 U_s$, where $U_s$ is the velocity of a swimmer whose surface is fully covered with chiral activity as a reference. For the arbitrary patch model, the velocity is $U_{AP} = 1.45 U_s$, which is higher than that of the symmetric patch model. Our results indicate that swimmers with low hydrophobicity exhibit efficient swimming characteristics. Additionally, due to the incomplete coverage of the active patch, the Stokeslet and Rotlet terms appear in the flow field generated by the swimmer, which is a deviation compared to the case of a swimmer whose surface is fully covered with chiral activity. This study provides insights useful for designing synthetic active particles, which can be applied, for example, in targeted drug delivery, chemotaxis, and phototaxis.