Figure 1

Panels denoted by ${\mathit{v}}^{\mathcal{A}\left(l\sigma \right)}$ (${\mathit{f}}^{\mathcal{A}\left(l\sigma \right)}$) show the vector field plots of the slip (traction) due to an isolated $\left(l\sigma \right)$ mode of the expansion (4). Here, the irreducible tensors ${\mathit{V}}^{\lambda \left(l\sigma \right)}$ are naturally parametrized in terms of the TSH as follows: ${\mathbf{V}}^{\lambda \left(ls\right)}=V_{ls}^{0,\lambda }{\mathbf{Y}}^{\left(l\right)}\left(\mathit{e}\right),{\mathbf{V}}^{\lambda \left(la\right)}=V_{la}^{0,\lambda }{\mathbf{Y}}^{(l-1)}\left(\mathit{e}\right),$ and ${\mathbf{V}}^{\lambda \left(lt\right)}=V_{lt}^{0,\lambda }{\mathbf{Y}}^{(l-2)}\left(\mathit{e}\right),$ where the uniaxial parametrizations are defined in terms of the orientation vector $\mathit{e}$ of the active particle and $V_{l\sigma }^{0,\lambda }$ are the scalar strengths of the modes. From this parametrization, it follows that the ${\mathbf{V}}^{\lambda \left(l\sigma \right)}$ are either even (apolar) or odd (polar) under inversion symmetry $\mathit{e}\to -\mathit{e}$ with respect to the orientation of the particle. The figure shows the vector fields ${\mathit{v}}^{\mathcal{A}}$ and ${\mathit{f}}^{\mathcal{A}}$ due to the leading modes of apolar $\left(2s\right)$, polar $\left(3t\right)$, achiral $\left(3a\right)$, and chiral $\left(4a\right)$ symmetry. The fields have been plotted on the surface of the particle with orientation $\mathit{e}$ along the north pole. For clarity, we have lifted the vector field off the surface slightly, while its magnitude has been overlaid on the surface. It follows directly from the scalar friction coefficients of the generalized Stokes laws (8) that both slip and traction exhibit the same symmetry.