The almost-rigid rotation of viscous fluid between concentric spheres
The almost-rigid rotation of viscous fluid between concentric spheres
复制标题
同心球之间粘性流体的几乎刚性旋转
DOI：
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发表时间：
1956
影响因子：
3.7
通讯作者：
I. Proudman
中科院分区：
文献类型：
--
作者：
I. Proudman
两个同心球体应该以几乎相同的角速度绕同一轴旋转，因此球体表面上的粘性应力会引起流动，该流动可以通过叠加在整个流体的刚体旋转上的小扰动来表示。因此，控制方程根据扰动的大小进行线性化，并且这种线性化的有效性似乎与主旋转的雷诺数无关。然后，注意力仅限于雷诺数很大的情况，该说明的主要目的是用一个特别简单的数学模型来举例说明大雷诺数下旋转系统的一些特性。发现接触内球体的圆柱面（轴线为旋转轴）是奇异面，其速度梯度非常大。在圆柱体外部的任何地方，流体都以与外球体相同的角速度作为刚体旋转。在圆柱体内部，运动中心（无粘性）核心的速度分布由球体上边界层的速度分布决定，并且获得了所有这些速度分布的显式解。还讨论了圆柱形剪切层本身的力学，尽管在这种情况下没有获得明确的解决方案。
Two concentric spheres are supposed to rotate about the same axis with almost the same angular velocity, so that the viscous stresses over the surfaces of the spheres induce a flow which may be represented by a small perturbation superimposed upon a rigid body rotation of the fluid as a whole. The governing equations are therefore linearized in the magnitude of the perturbation, and it appears that the validity of this linearization is independent of the Reynolds number of the primary rotation. Attention is then restricted to the case in which the Reynolds number is large, the principal object of the note being to exemplify some of the properties of rotating systems at large Reynolds numbers in terms of a particularly simple mathematical model. It is found that the cylindrical surface that touches the inner sphere (the axis being the axis of rotation) is a singular surface in which velocity gradients are very large. Everywhere outside this cylinder, the fluid rotates as a rigid body with the same angular velocity as the outer sphere. Inside the cylinder, the velocity distribution in the central (inviscid) core of the motion is shown to be determined by the velocity distribution in the boundary layers over the spheres, and explicit solutions are obtained for all these velocity distributions. The mechanics of the cylindrical shear layer itself is also discussed, though no explicit solution is obtained in this case.