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 ${\widehat{F}}_y\sim 1/log[{\mathrm{Re}}^{-1}log\left[{\mathrm{Re}}^{-1}\right]]$; in the corotating case, the cylinder pair experiences a (dimensionless) torque $\widehat{\mathcal{T}}$, 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.

中文翻译：

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粘性流体中两个圆柱体旋转引起的流动

通过分析和数值技术研究了由两个单位半径相等的平行圆柱体旋转驱动的低雷诺数斯托克斯流。在第一部分中，考虑反向旋转气缸的情况。通过将系统封闭在半径为 的外圆柱体中获得数值（有限元）解${右}_0\gg 1$，对其施加无滑移条件。首先精确求解具有相同对称性的模型问题，并确定斯托克斯近似的有效性极限；该模型与纤毛推进有一定的相关性。对于两缸问题，重点关注小间隙情况$\epsilon \ll 1$。在接触极限内获得精确的解析解$\epsilon =0$，和净力$F_C$作用在该对气缸上的接触极限被识别；这有助于每个气缸绕其轴线承受扭矩。远场扭矩双峰（“torquelet”）也被识别。第二部分讨论同转圆柱体的情况，再次获得了有限元数值解：${右}_0\gg 1$。 Watson 的理论 [ Mathematika 42 , 105 (1995)] 得到了阐明，并且与数值解非常吻合。与反向旋转情况相反，只要 Re 足够大，整个流体域的惯性效应可以忽略不计，无论惯性效应有多大。$\ll 1$。在结论部分，总结了两种情况的主要结果，以及流体无界时的情况（${右}_0=\mathrm{无穷大}$）进行了讨论。如果圆柱体可以自由移动（同时绕其轴旋转），则在反向旋转的情况下，它们将以恒定速度相对于无限远的流体平移，拖曳力精确地补偿由于反向旋转而产生的自感力。在同向旋转的情况下，如果气缸自由移动，那么它们将作为一对相对于流体无限远旋转，并且气缸对上的净扭矩为零；相对于无穷远处流体的流动被识别为“径向四极杆”。然而，如果圆柱轴保持固定，则反向旋转情况下的斯托克斯流仅延伸一段距离$r～{\mathrm{关于}}^{-1}日志\left[{\mathrm{关于}}^{-1}\right]$来自气缸，并且有人认为气缸会受到（无量纲）力${\widehat{F}}_y～1/日志[{\mathrm{关于}}^{-1}日志\left[{\mathrm{关于}}^{-1}\right]]$;在同向旋转的情况下，气缸对承受（无量纲）扭矩$\widehat{\mathcal{时间}}$，趋向于 17.2587$\epsilon \downarrow 0$;该扭矩与涡流型流动相关$～r^{-1}$这是在远场建立的。可以用条件描述的情况$\epsilon <0$在补充材料中对反向旋转和同向旋转情况进行了处理。