= Solution
A rigid body obeys the <no-slip boundary condition>. Its outer <potential flow> generally has an $O(U)$ tangential slip velocity at the surface, so the boundary layer must make an $O(U)$ velocity correction across thickness $\delta$. Its strain is therefore $O(U/\delta)$, and in two dimensions its dissipation per unit axial length is
$$
\mathcal D_{\rm layer}
\sim\mu\left(\frac U\delta\right)^2a\delta
=\mu U^2\frac a\delta,
$$
which exceeds the $O(\mu U^2)$ outer dissipation by $a/\delta\sim\operatorname{Re}^{1/2}$.
The dominant drag consequently depends on the detailed boundary-layer velocity field and possibly on <boundary-layer separation>. It cannot be obtained by integrating the outer irrotational strain alone.
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