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Full text of "Scientific Papers - Vi"

272                     ON THE STABILITY OF VISCOUS FLUID MOTION                    [388
"The curves  = const, constitute a system of coaxal and similar ellipses, whose centre at * = 0 coincides with the point fc 17, and then moves with the velocity #7 parallel to the a-axis. For very small values of * the eccentricity of the ellipse is very small and the angle which the major axis makes with the #-axis is about 45. With increasing t this angle becomes smaller. At the same  time the eccentricity becomes larger. For infinitely great values of t, the angle becomes infinitely small and the eccentricity infinitely great."
When = 0 in (12), we fall back on Fourier's solution. Without loss of generality we may suppose |= 0, ?? = 0, and then (r2 = a? + if)
representing the diffusion of heat, or vorticity, in two dimensions. It may be worth while to notice the corresponding tangential velocity in the hydro-dynamical problem. If -v/r be the stream-function,
*         z
1 \
da;2     dy2     r dr \   dr y ' so that                              r -^- = - (1  e~r*IM),...........................(15)
the constant of integration being determined from the known value of dty/dr when r = oo .    When r is small (15) gives
*     .. __                                                    / J l\ \
dr     4nrvt' ..................................
becoming finite when r = 0 so soon as t is finite.
At time t the greatest value of dty/dr occurs when
r = 1-256 x 4>vt............................(17)
On the basis of his solution Oseen treats the problem of the stability of the shearing motion between two parallel planes and he arrives at the conclusion, in accordance with Kelvin, that the motion is stable for infinitesimal disturbances. For this purpose he considers "the specially unfavourable case " where the distance between the planes is infinitely great. I cannot see myself that Oseen has proved his point. It is doubtless true that a great distance between the planes is unfavourable to stability, but to arrive at a sure conclusion there must be no limitation upon the character of the infinitesimal disturbance, whereas (as it appears to me) Oseen assumes that the disturbance does not sensibly reach the walls. The simultaneous evanescence at the walls of both velocity-components of an otherwise sensible disturbance would seem to be of the essence of the question.    dy