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 and 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. Their combined consequences are usually 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, usually leads to an infinitely rich variety of area preserving re-configurations of the liquid. In particular, any chosen point of the liquid can be moved, in the shell container frame, to any other chosen point. However, for a torus (shell) 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. The analysis, solving for the streamfunction, is based on the relative vorticity, and the conformal mapping of a torus to a (periodic) rectangle, which lead to a fairly simple convolution integral formula for the flow.