[ 188 ] 



XXI. Stability of Fluid Motion (continued from the May and 

 June numbers). — Rectilineal Motion of Viscous Fluid between 

 two Parallel Planes*. By Sir W. Thomson, LL.D., F.R.S. 



27. O INCE the communication of the first of this series of 

 ^ articles to the Royal Society of Edinburgh in April, 

 and its publication in the Philosophical Magazine in May and 

 June, the stability or instability of the steady motion of a 

 viscous fluid has been proposed as subject for the Adams 

 Prize of the University of Cambridge for 1888 f. The pre- 

 sent communication (§§ 27-40) solves the simpler of the two 

 cases specially referred to by the Examiners in their announce- 

 ment, and prepares the way for the investigation of the less 

 simple by a preliminary laying down, in §§ 27-29, and equa- 

 tions (7) to (12) below, of the fundamental equations of 

 motion of a viscous fluid kept moving by gravity between 

 two infinite plane boundaries inclined to the horizon at any 

 angle I, and given with any motion deviating infinitely little 

 from the determinate steady motion which would be the 

 unique and essentially stable solution if the viscosity were 

 sufficiently large. It seems probable, almost certain indeed, 

 that analysis similar to that of § § 38 and 39 will demonstrate 

 that the steady motion is stable for any viscosity, however 

 small ; and that the practical unsteadiness pointed out by 

 Stokes forty-four years ago, and so admirably investigated 

 experimentally five or six years ago by Osborne Reynolds, is 

 to be explained by limits of stability becoming narrower and 

 narrower the smaller is the viscosity. 



Let OX be chosen in one of the bounding planes, parallel 

 to the direction of the rectilineal motion ; and OY perpen- 

 dicular to the two planes. Let the x-, y-, z-, component 

 velocities, and the pressure, at (x, y, z, t), be denoted by 

 U + u, v , w, and p respectively ; U denoting a function of (y, t) . 

 Then, calling the density of the fluid unity, and the viscosity 

 fj,, we have, as the equations of motion J, 



du + M + dw =0 fl> 



dx dy dz v n 



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

 Society of Edinburgh, July 18, 1887. 

 t See Phil. Mag. July 1887, p. 142. 

 X Stokes's Collected Papers, vol. i. p. 93. 



