When calculating the two-dimensional Stokes flow about two identical cylinders that counterrotate about fixed axes, it is impossible to obtain a solution that decays at large distances. When considering the corotation variant of that “Jeffery's paradox,” Watson [Mathematika 42, 105 (1995)] illustrated that the comparable impediment can be resolved by the introduction of an external torque on the cylinder pair. Both the corotation and counterrotation problems were recently revisited by Dormy and Moffatt [Phys. Rev. Fluids 9, 044102 (2024)] who focused upon the case where the separation $\varepsilon a$ between the cylinders is small compared with their radius $a$. In the corotation problem, they showed that the torque calculated using Watson's solution approaches a finite limit as $\varepsilon \to 0$. In the counterrotation problem, they introduced a remote boundary of radius $a{R}_0$, rendering the problem well-posed provided an external force is applied on the cylinder pair. Dormy and Moffatt employed a numerical scheme to solve the confined problem for finite values of $R_0$ and $\varepsilon$. Given the numerical results, they conjectured that the force decays inversely with $ln{R}_0$ as $R_0\to \infty$. We here address the zero-separation limit $\varepsilon =0$ from the outset, obtaining solutions for the unconfined problem via integral transforms. In the corotation problem, we obtain the torque as a closed-form quadrature, thus avoiding the need to extrapolate from a finite-separation solution. In the counterrotation problem we solve the flow field in the presence of an external force. This solution provides an asymptotic “inner” approximation for the respective confined problem in the limit $R_0\to \infty$, with the cylinder pair appearing as a point singularity in the “outer” approximation. Analyzing that problem using matched asymptotic expansions, we derive approximations for the force for both no-slip and shear-free boundaries, confirming the conjecture of Dormy and Moffatt.

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• Low Reynolds number flows