[ 459 ] 



LII. On the Stability of Steady and of Periodic* Fluid Motion. 

 By Sir William Thomson f. 



1. rilHE fluid will be taken as incompressible; but the 

 J- results will generally be applicable to the motion of 

 natural liquids and of air or other gases when the velocity is 

 everywhere small in comparison with the velocity of sound in 

 the particular fluid considered. I shall first suppose the fluid 

 to be inviscid. The results obtained on this supposition will 

 help in an investigation of effects of viscosity which will follow. 

 2. I shall suppose the fluid completely enclosed in a con- 

 taining vessel, which may be either rigid, or plastic so that 

 we may at pleasure mould it to any shape, or of natural solid 

 material and therefore viscously elastic (that is to say, return- 

 ing always to the same shape and size when time is allowed, 

 but resisting all deformations with a force depending on the 

 speed of the change, superimposed upon a force of quasi- 

 perfect elasticity). The whole mass of containing-vessel and 



* By steady motion of a system (whether a set of material points, or a 

 rigid body, or a fluid mass, or a set of solids, or portions of fluid, or a 

 system composed of a set of solids or portions of fluid, or of portions of 

 solid and fluid), I mean motion which at any and every time is precisely 

 similar to what it is at one time. By periodic motion I mean motion 

 which is perfectly similar, at all instants of time differing by a certain 

 interval called the period. 



Example 1. Every possible adynamic motion of a free rigid body, 

 having two of its principal moments of inertia equal, is steady. So also 

 is that of a solid of revolution filled with irrotational inviscid incompres- 

 sible fluid. 



Example 2. The adynamic motion of a solid of revolution filled with 

 homogeneously rotating inviscid incompressible fluid is essentially periodic, 

 and is steady only in particular cases. 



Example 3. The adynamic motion of a free rigid body with three un- 

 equal principal moments of inertia is essentially periodic, and is only 

 steady in the particular case of rotation round one or other of the three 

 principal axes ; so also, and according to the same law, is the motion of a 

 rigid body having a hollow or hollows filled with irrotational inviscid 

 incompressible fluid, with the three virtual moments of inertia unequal. 



Example 4. The adynamic motion of a hollow rigid body filled with 

 rotationally moving fluid is essentially unsteady and non-periodic, except 

 in particular cases. Even in the case of an ellipsoidal hollow and homo- 

 geneous molecular rotation the motion is non-periodic. The motion, 

 whether rotational or irrotational, of fluid in an ellipsoidal hollow is fully 

 investigated in a paper under this title published in the Proceedings of 

 the Royal Society of Edinburgh for December 7, 1885. Among other 

 results it was proved that the rotation, if initially given homogeneous, 

 remains homogeneous, provided the figure of the hollow be never at any 

 time deformed from being exactly ellipsoidal. 



t Communicated by the Author, having been read before the Royal 

 Society of Edinburgh on April 18, 1887. 



