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Full text of "Scientific Papers - Vi"

1914]                 ON THE STABILITY  OF  VISCOUS  FLUID  MOTION                     273
It may be added that Oseen is disposed to refer the instability observed in practice not merely to the square of the disturbance neglected in (6), but also to the inevitable unevenness of the walls.
We may perhaps convince ourselves that the infinitesimal disturbances of (6), with U = @y, tend to die out by an argument on the following lines, in which it may suffice to consider the operation of a single wall. The argument could, I think, be extended to both walls, but the statement is more complicated. When there is but one wall, we may as well fix ideas by supposing that the wall is at rest (at y = 0).
The difficulty of the problem arises largely from the circumstance that the operation of the wall cannot be imitated by the introduction of imaginary vorticities on the further side, allowing the fluid to be treated as uninterrupted. We may indeed in this way satisfy one of the necessary conditions. Thus if corresponding to every real vorticity at a point on the positive side we introduce the opposite vorticity at the image of the point in the plane y  0, we secure the annulment in an unlimited fluid of the velocity-component v parallel to y, but the component u, parallel to the flow, remains finite. In order further to annul u, it is in general necessary to introduce new vorticity at y = 0. The vorticities on the positive side are not wholly arbitrary.
Let us suppose that initially the only (additional) vorticity in the interior of the fluid is at A, and that this vorticity is clockwise, or positive, like that of 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
:?/=0    B
Fig. 2.                                             Fig. 3.
to satisfy the condition u = 0, there must be instantaneously introduced at the wall a negative vorticity of an amount sufficient to give compensation. To this end the local intensity must be inversely as the distance from A and as the sine of the angle between this distance and the wall (Helmholtz). 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 vorticity must be developed at the wall, whose diffusion over short distances rapidly annuls the negative so far
K. vi.                                                                                             18 The suggestion came from me, but the notice was (I think) drawn up by Stokes.