Motion of a small flat plate in a viscous flow

Abstract

A small flat plate of arbitrary shape moving in a viscous flow is investigated analytically under the Stokes approximation. It is shown that the unit normal n(t) of the plate at position x rotates, irrespective of the plate shape, with angular velocity Ω(t) = −(n × ∇)(u·n), where u(x,t) is the velocity field at time t. This angular velocity is identical to that of material surface element at the same position and time. In contrast, the angular velocity of the spinning motion (around n(t)) of the plate depends on the shape. A circular disc spins with the same angular velocity of the local fluid. The translational velocity of the centre of gravity of the plate coincides with the local fluid velocity. This result explains the perfect reproduction by direct-numerical simulation using a flat-plate model of the bright pattern observed experimentally with mica particles for the flow in a precessing spherical cavity (Goto and Kida 2011 J. Fluid Mech.683 417).