Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 355 / 2 / b / Solution

Choose the handedness for which the helix tangent is $\mathbf{t}=\cos \theta {\mathbf{e}}_z+\sin \theta {\mathbf{e}}_{\varphi }$. Its local rigid velocity is $\mathbf{u}=\tilde{U}{\mathbf{e}}_z+R\tilde{\Omega }{\mathbf{e}}_{\varphi }$, so $\mathbf{u}\cdot \mathbf{t}=\tilde{U}\cos \theta +R\tilde{\Omega }\sin \theta$. Integrating the axial force and torque from the drag law gives the symmetric hydrodynamic resistance matrix in the question, withReversing helical handedness reverses the sign of $B$ but leaves $A$ and $D$ unchanged.