Relaxation methods applied to engineering problems XII. Fluid motions characterized by 'free' stream-lines

Abstract

Abstract This paper deals with problems which are like the percolation problems of Part VII (Shaw & Southwell 1941) in that a double condition, imposed at a boundary initially unknown, replaces the more usual single condition at a specified boundary. They relate to ‘free’ stream-lines in the hydrodynamical theory of inviscid fluids. For plane two-dimensional (steady) motions, the device of conformal transformation has led in the hands of Helmholtz, Kirchhoff and Rayleigh to a variety of solutions; but up to the present it has not taken account of gravity, and it would not seem capable of extension to motions characterized by axial symmetry. Relaxation Methods, in virtue of their tentative approach, here deal successfully with some problems hitherto unsolved.