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Full text of "Scientific Papers - Vi"

274                     ON  THE  STABILITY OF  VISCOUS  FLUID MOTION                     [388
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 restoration 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 (5) of the additional velocities u, v, 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 inwards in virtue of the component velocity v due to A. The effect is thus a continued passage inwards behind A of negative vorticity, which tends to neutralize in this region the original constant vorticity (Z\ When the additional vorticity at A is negative (fig. 3), the convection behind A acts in opposition to diffusion, and thus the positive developed near the wall remains closer to it, and is more easily absorbed as A passes on. It is true that in front of A there is a convection of positive inwards; but it would seem that this would lead to a more rapid annulment of A itself; and that upon the whole the tendency is for the effect of fig. 2 to preponderate. If this be admitted, we may perhaps see in it an explanation of the diminution of vorticity as we recede from a wall observed in certain circumstances. But we are not in a position to decide whether or not a disturbance dies down. By other reasoning (Reynolds, Orr) we know that it will do so if /9 be small enough in relation to the other elements of the problem, viz. the distance between the walls and the kinematic viscosity v.
A precise formulation of the problem for free infinitesimal disturbances was made by Orr (1907). We suppose that £ and v are proportional to eint ei?x, where n =*p + iq. If V2y = S, we have from (6) and (10)
d2v    .
and                              ap-'^--5'.................................(19>
with the boundary conditions that v - 0, dv/dy = 0 at the walls.    Orr easily .shows that the period-equation takes the form
(s.e^'dy. Is2e-*ydy-!81e-kydy.lsze>°vdy = (),   .........(20)
J                            J                                  J                               Jice was (I think) drawn up by Stokes.