 Right-angle two-rod resistance matrix

For two perpendicular rods of length $2L$ joined at one endpoint, the slender-body force density gives translation resistance $CL\mathrm{diag}(3,3,4)$, rotation resistance $C{L}^3\mathrm{diag}(8/3,8/3,16/3)$, and translation–rotation block $C{L}^2\left(\begin{array}{ccc}0& 0& -2\\ 0& 0& 2\\ 2& -2& 0\end{array}\right)$. The opposite block is its transpose. Here $G$ is the torque about the joint and $F$ the force on the fluid. Integrating along each rod proves the entries and makes the symmetry and positivity of a rigid-body resistance matrix explicit.