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Analytical calculation of the Stokes drag of the spherical particle in a nematic liquid crystal

As an approach to the motion of particles in an anisotropic liquid, we analytically study the Stokes drag of spherical particles in a nematic liquid crystal. The Stokes drag of spherical particles for a general anisotropic case is derived in terms of multipoles. In the case of weak anchoring, we use the well-known distribution of the elastic director field around the spherical particle. In the case of strong anchoring, the multipole expansion may be also used by modifying the size of a particle to the size of the deformation coating. For the case of zero anchoring (uniform director field) we found that the viscosities along the director $η_{\parallel}$ and perpendicular direction $η_{\perp}$ are almost the same, which is quite reasonable because in this case the liquid behaves as isotropic. In the case of non-zero anchoring, the general ratio $η_{\parallel}/η_{\perp}$ is about 2 which is satisfied by experimental observations.