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

Abstract

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.