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1914] ON THE STABILITY OF VISCOUS FLUID MOTION 267
a slipping at the walls, but this presents no inconsistency so long as the fluid is regarded as absolutely non-viscous.
The steady motions for which stability in a non-viscous fluid may be inferred include those assumed by a viscous fluid in two important cases, (i) the simple shearing motion between two planes for which d2U/df = 0, and (ii) the flow (under suitable forces) between two fixed plane walls for which d2 U/dy- is a finite constant. And the question presented itself whether the effect of viscosity upon the disturbance could be to introduce instability. An affirmative answer, though suggested by common experience and the special investigations of 0. Reynolds*, seemed difficult to reconcile with the undoubted fact that great viscosity makes for stability.
It was under these circumstances that " the Criterion of the Stability and Instability of the Motion of a Viscous Fluid," with special reference to cases (i) and (ii) above, was proposed as the subject of an Adams Prize essayf, and shortly afterwards the matter was taken up by Kelvin I in papers which form the foundation of much that has since been written upon the subject. His conclusion was that in both cases the steady motion is wholly stable for infinitesimal disturbances, whatever may be the value of the viscosity (//-); but that when the disturbances are finite, the limits of stability become narrower and narrower as /A diminishes. Two methods are employed: the •first a special method applicable only to case (i) of a simple shear, the second (ii) more general and applicable to both cases. In 1892 (I.e.) I had occasion to take exception to the proof of stability by the second method, and Orr§ has since shown that the same objection applies to the special method. Accordingly Kelvin's proof of stability cannot be considered sufficient, even in case (i). That Kelvin himself (partially) recognized this is shown by the following interesting and characteristic letter, which I venture to give in full.
July 10 (? 1895).
" On Saturday I saw a splendid illustration by Arnulf Mallock of our ideas regarding instability of water between .two parallel planes, one kept moving and the other fixed. (Fig. 1) Coaxal cylinders, nearly enough planes for our illustration. The rotation of the outer can was kept very accurately uniform at whatever speed the governor was set for, when left to itself. At one of the speeds he shewed rne, the water came to regular regime, quite smooth. I dipped a disturbing rod an inch or two down into the water and immediately the torque increased largely. Smooth regime could only be
* Phil. Trans. 1883, Part in. p. 935.
•I- Phil. Mag. Vol. xxiv. p. 142 (1887). The suggestion came from me, but the notice was (I think) drawn up by Stokes.
J Phil. Mag. Vol. xxiv. pp. 188, 272 (1887); Collected Papers, Vol. iv. p. 321. § Orr, Proc. Roy. Irish A cad. Yol. xxvii. (1907).