Stability of Viscous Fluid Motion. 617 



clockwise, or positive, like that o£ the undisturbed motion 

 (fig. 2). If! this existed alone, there would be of necessity a 

 finite velocity u along the wall in its neighbourhood. In order 

 to satisfy the condition u = 0, there must be instantaneously 



Fig. 2. 







Fig. 3. 







^zzy^o 



B 



<1 





introduced at the wall a negative vorticity of an amount 

 sufficient to give compensation. To this end the local in- 

 tensity must be inversely as the distance from A and as the 

 sine of the angle between this distance and the wall (Helm- 

 holtz). As we have seen these vorticities tend to diffuse 

 and in addition to move with the velocity of the fluid, those 

 near the wall slowly and those arising from A more quickly. 

 As A is carried on, new negative vorticities are developed 

 at those parts of the wall which are being approached. At 

 the other end the vorticities near the wall become excessive 

 and must be compensated. To effect this, new positive vor- 

 ticity must be developed at the wall, whose diffusion over 

 short distances rapidly annuls the negative so far as may be 

 required. After a time dependent upon its distance, the 

 vorticity arising from A loses its integrity by coming into 

 contact with the negative diffusing from the wall and thus 

 suffers diminution. It seems evident that the end can only 

 be the annulment of all the additional vorticity and restora- 

 tion of the undisturbed condition. So long as we adhere to 

 the suppositions of equation (6), the argument applies equally 

 well to an original negative vorticity at A, and indeed to 

 any combination of positive and negative vorticities, however 

 distributed. 



It is interesting to inquire how this argument would be 

 affected by the retention in (5j of the additional velocities 

 ?/, r, which are omitted in (6), though a definite conclusion 

 is hardly to be expected. In fig. 2 the negative vorticity 

 which diffuses inwards is subject to a backward motion due 

 to the vorticity at A in opposition to the slow forward motion 

 previously spoken of. And as A passes on, this negative 

 vorticity in addition to the diffusion is also convected in- 

 wards in virtue of the component velocity v due to A. The 



