Flow induced by the rotation of two circular cylinders in a viscous fluid

The low-Reynolds-number Stokes flow driven by rotation of two parallel cylinders of equal unit radius is investigated by both analytical and numerical techniques. In Part I, the case of counterrotating cylinders is considered. A numerical (finite-element) solution is obtained by enclosing the system in an outer cylinder of radius $R_0\gg 1$, on which the no-slip condition is imposed. A model problem with the same symmetries is first solved exactly, and the limit of validity of the Stokes approximation is determined; this model has some relevance for ciliary propulsion. For the two-cylinder problem, attention is focused on the small-gap situation $\varepsilon \ll 1$. An exact analytic solution is obtained in the contact limit $\varepsilon =0$, and a net force $F_c$ acting on the pair of cylinders in this contact limit is identified; this contributes to the torque that each cylinder experiences about its axis. The far-field torque doublet ("torquelet") is also identified. Part II treats the case of corotating cylinders, for which again a finite-element numerical solution is obtained for $R_0\gg 1$. The theory of Watson [Mathematika 42, 105 (1995)] is elucidated and shown to agree well with the numerical solution. In contrast to the counterrotating case, inertia effects are negligible throughout the fluid domain, however large, provided Re $\ll 1$. In the concluding section, the main results for both cases are summarized, and the situation when the fluid is unbounded ($R_0=\infty$) is discussed. If the cylinders are free to move (while rotating about their axes), in the counterrotating case they will then translate relative to the fluid at infinity with constant velocity, the drag force exactly compensating the self-induced force due to the counterrotation. In the corotating case, if the cylinders are free to move, then they will rotate as a pair relative to the fluid at infinity and the net torque on the cylinder pair is zero; the flow relative to the fluid at infinity is identified as a "radial quadrupole." If, however, the cylinder axes are held fixed, then the Stokes flow in the counterrotating case extends only for a distance $r\sim {\mathrm{Re}}^{-1}log\left[{\mathrm{Re}}^{-1}\right]$ from the cylinders, and it is argued that the cylinders then experience a (dimensionless) force ${\hat{F}}_y\sim 1/log[{\mathrm{Re}}^{-1}log\left[{\mathrm{Re}}^{-1}\right]]$; in the corotating case, the cylinder pair experiences a (dimensionless) torque $\hat{T}$, which tends to 17.2587 as $\varepsilon \downarrow 0$; this torque is associated with a vortex-type flow $\sim r^{-1}$ that is established in the far field. Situations that can be described by the condition $\varepsilon <0$ are treated for both counter- and corotating cases in the Supplemental Material.