Coupled Faxen relations for non-uniform slip Janus spheres

A non-uniform surface slip can cause a symmetry breaking in the geometry of an otherwise homogeneous spherical particle to give rise to an anisotropic hydrodynamic resistance to the particle. Here, we develop a more general theoretical framework capable of decoding the surface-pattern-dependent hydrodynamic features for single heterogeneous spheres having arbitrary non-uniform slip length distributions in small variations, especially for those of weakly stick-slip or slip-slip Janus spheres in either the two-faced or striped type. Utilizing the Lorentz reciprocal theorem in conjunction with surface spherical harmonic expansion, we derive a new coupled set of Faxen formulas for the hydrodynamic force and torque on a non-uniform slip sphere by expressing impacts of slip anisotropy in terms of surface dipole and quadrupole without solving detailed flow fields. Our results reveal not only how various additional forces/torques arise from surface dipole and quadrupole, but also that it is the anti-symmetric dipole responsible for distinctive force-rotation/torque-translation coupling. These features are very distinct from those of no-slip or uniform-slip particles, possibly spurring new means to characterize or sort Janus particles in microfluidic experiments. In addition, the coupled Faxen relations with surface moment contributions reported here may infer potential changes in the collective nature of hydrodynamic interactions between non-uniform slip spheres. Furthermore, the present framework can also be readily applied to heterogeneous self-propelled squirmers whose swimming velocities are sensitive to slip anisotropy.