Open AccessThe steady broadside motion of an anchor ring in an infinite viscous liquid
Author(s) -
F. E. Relton
Publication year - 1931
Publication title -
proceedings of the royal society of london. series a, containing papers of a mathematical and physical character
Language(s) - English
Resource type - Journals
eISSN - 2053-9150
pISSN - 0950-1207
DOI - 10.1098/rspa.1931.0181
Subject(s) - curvilinear coordinates , mathematics , mathematical analysis , vorticity , differential equation , equations of motion , inertia , stream function , first order partial differential equation , physics , classical mechanics , geometry , mechanics , vortex
The ring is translated along its axis of revolution with constant velocity in an infinite viscous liquid. The motion of the liquid is due to the motion of the ring, each particle moving in a meridian plane to which the vector vorticity is perpendicular. The analysis is conducted in orthogonal curvilinear “ring coordinates” using vectors, and the condition of continuity leads to a stream function which is connected with the vorticity by a partial differential equation of the second order. The equation of steady motion, on ignoring the inertia terms, is a partial differential equation of the second order in which the dependent variable is the vorticity. The motion thus comes to depend on a fourth-order partial differential equation in which the dependent variable is the stream function. Two independent types of solution of this equation are obtained in trigonometrical series involving associated Legendre functions of degree half an odd integer, the solutions tending to zero at infinity. The arbitrary constants are determined from the boundary conditions of no slip at the surface of the ring. By means of the usual dyadics an expression, is obtained for the resistance to the motion. Numerical values are omitted in the absence of the necessary tables, a defect which it is hoped to remedy in the near future.