 Straight-rod resistance in a linear flow

For a straight rod $\mathbf{X}=s\mathbf{p}$, $-L<s<L$, leading slender-body theory uses $\mathbf{f}=C(I-\mathbf{pp}/2)[\Delta \mathbf{U}+s(\Delta \mathbf{\Omega }\times \mathbf{p}-E\mathbf{p})]$. Integration gives $\mathbf{F}=2CL(I-\mathbf{pp}/2)\Delta \mathbf{U}$ and $\mathbf{G}=(2C{L}^3/3)[(I-\mathbf{pp})\Delta \mathbf{\Omega }-\mathbf{p}\times E\mathbf{p}]$. Axial spin is outside this centreline approximation.