标题： 摇动一个装满理想液体的容器；一个可处理的情况，一个环形壳，表现出虚拟壁 显示英文标题

标题： Shaking a container full of perfect liquid; a tractable case, a torus shell, exhibits a virtual wall

摘要： 对装有不可压缩液体的刚性容器进行“摇晃”操作，从静止状态开始，通常会导致液体在其中的位移或混合。 如果液体的粘度也为零，即为“理想”或欧拉液体，凯尔文定理大大简化了流动分析。 响应是瞬时的；停止容器后，所有液体运动也会停止。 实际上，任意的操作都可以视为容器的无限小平移和旋转的交替进行。 相对于容器，在每次平移过程中，液体都是静止的。 无限小的旋转（沿旋转轴的一个无限小矢量）在容器坐标系中分解为三个正交分量。 每个分量都会产生其自身的无限小液体位移矢量场。 然而这些情况很少易于处理，它们的综合影响也并不明确。 与其考虑体积流动，不如考虑三维中的表面流动，液体在一个壳层中自由滑动，该壳层夹在两个嵌套的闭合曲面之间，且具有恒定的无限小间隙。 闭合性避免了额外的边界。 二维性允许通过容器角速度矢量确定一个标量流函数。 随意操纵容器的角速度，除了球体外，会导致液体无限丰富的面积保持重新配置。 即使对于球体，液体内任一点在壳容器框架下都可以移动到任何其他点，一般情况下也应如此。 然而，对于孔径足够小的环面（直径<0.195倍环面直径），存在一个虚拟壁，即一个假设的轴向圆柱体，与环面相交。 不管如何操纵环面，圆柱体内的液体始终在内部；外部的液体始终在外部。 显示英文摘要

摘要： Manipulation ('shaking') of a rigid container filled with incompressible liquid starting from stationary generally results in some displacement, or mixing, of the liquid within it. If the liquid also has zero viscosity, a 'perfect', or Euler liquid, Kelvin's theorems dramatically simplify the flow analysis. Response is instantaneous; stop the container and all liquid motion stops. In fact an arbitrary manipulation can be considered as alternating infinitesimal translations snd rotations of the container. Relative to the container, the liquid is stationary during every translation. Infinitesimal rotations (an infinitesimal vector along the rotation axis) resolve into three orthogonal components in the container frame. Each generates its own infinitesimal liquid displacement vector field. However these are rarely tractable, and their combined consequences are obscure. Rather than a volume flow, a surface flow in 3D is considerably easier, the liquid slipping freely in a shell, sandwiched between two nested closed surfaces with constant infinitesimal gap. The closedness avoids extra boundaries. The two dimensionality admits a scalar streamfunction determined by the container angular velocity vector. Manipulation of the container angular velocity at will, leads (except for a sphere) to an infinitely rich variety of area preserving re-configurations of the liquid. Even for a sphere, any chosen point of the liquid can be moved, in the shell container frame, to any other point, and one would expect the same in general. However, for torus with a small enough hole (diameter<0.195 torus diameter), there exists a virtual wall, a hypothetical axial cylinder intersecting the torus. No matter how the torus is manipulated, liquid inside the cylinder stays inside; outside stays outside.