Scientific Papers - Vi

388.
FURTHER REMARKS ON THE STABILITY OF VISCOUS FLUID MOTION.
[Philosophical Magazine, Vol. xxvm. pp. 609—619, 1914.]
AT an early date my attention was called to the problem of the stability of fluid motion in connexion with the acoustical phenomena of sensitive jets, which may be ignited or unignited. In the former case they are usually referred to as sensitive flames. These are naturally the more conspicuous experimentally, but the theoretical conditions are simpler when the jets are unignited, or at any rate not ignited until the question of stability has been decided.
The instability of a surface of separation in a non-viscous liquid, i.e. of a surface where the velocity is discontinuous,' had already been remarked by Helmholtz, and in 1.879 I applied a method, due to Kelvin, to investigate the character of the instability more precisely. But nothing very practical can be arrived at so long as bhe original steady motion is treated as discontinuous, for in consequence of viscosity such a discontinuity in a real fluid must instantly disappear. A nearer approach to actuality is to suppose that while the velocity in a laminated steady motion is continuous, the rotation or vorbicity changes suddenly in passing from one layer of finite thickness to another. Several problems of this sort have been treated in various papers*. The most general conclusion may be thus stated. The steady motion of a non-viscous liquid in two dimensions between fixed parallel plane walls is stable provided that the velocity C7", everywhere parallel to the walls and a function of y only, is such that dzU"jdyz is of one sign throughout, y being the coordinate measured perpendicularly to the walls. It is here assumed that the disturbance is in two dimensions and infinitesimal. It involves
* Proc. Lend. Math. Soc. Vol. x. p. 4 (1879) ; xi. p. 57 (1880); xix. p. 67 (1887) ; xxvn. p. 5 (1895); Phil. Mag. Vol. xxxiv. p. 59 (1892); xxvi. p. 1001 (1913) ; Scientific Papers, Arts. 58, 66, 144, 216, 194. [See also Art. 377.]ep the celluloid is held firmly to the paper and the tripod is moved to the point of the scale required to give the next value of the curvature. The ordinates of the curve so drawn are given in the second and fifth columns of the annexed table. It will be seen that from x — 0 to x = 2 the curve is very flat. Fig. (1).