398.
ON THE STABILITY OF THE SIMPLE SHEARING MOTION OF A VISCOUS INCOMPRESSIBLE FLUID.
[Philosophical Magazine, Vol. xxx. pp. 329 — 338, 1915.]
V
A PRECISE formulation of the problem for free infinitesimal disturbances was made by Orr (1907)*. It is supposed that £ (the vorticity) and v (the velocity perpendicular to the walls) are proportional to eint eikx, where n =p + iq. If V2^ = S, we have
ay
and
(1)
^ '
(2)
with the boundary conditions that v = Q, dv/dy = 0 at the walls where y is constant. Here v is the kinematic viscosity, and ft is proportional to the initial constant vorticity. Orr easily shows that the period-equation takes the form
- e~
e** dy = 0,
(3)
where 81} $2 are any two independent solutions of (1) and the integrations are extended over the interval between the walls. An equivalent equation was given a little later (1908) independently by Sommerfeld.
Stability requires, that, for no yalue.of k shall any of the q's determined by (3) be negative. In hia discussion Orr arrives at the conclusion that this condition is satisfied. Another of Orr's results may be mentioned. He shows that p + lefty necessarily changes sign in the interval between the walls f.
In the paper quoted reference was made also to the work of v. Mises and Hopf, and it was suggested that the problem might be simplified if it could be shown that q — vkz cannot vanish. If so, it will follow that q is always
_* Proc, Roy. Irish Acad. Vol. xxvn. •h Phil Mag. Vol. xxvni. p. 618 (1914).ther trials on another day gave confirmatory results.