= Solution
At negligible <Reynolds number>, the <Stokes flow>[Stokes equations] and their no-slip boundary conditions are linear in velocity. Scaling or superposing a prescribed rigid translation therefore scales or superposes the resulting stress and total hydrodynamic force. Hence, for a fixed rod orientation,
$$
\boxed{\mathbf F=-\mathbf R\cdot\mathbf U}.
$$
The minus sign records that viscous drag opposes motion, and $\mathbf R$ is the translational <hydrodynamic resistance matrix>.
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