rotation, about its axis: then if the fluid possessed viscosity it would, in course of time, acquire sensibly the same speed of rotation as the vessel, so that the whole system would revolve en bloc. With an ideal fluid, however, the rotation of the vessel might be continued indefinitely without imparting any motion to its contents.
Fig. 36.
If we suppose substituted for the circular cylinder one of square or other irregular section, it might be imagined that rotation would be imparted to the fluid by the irregularity of the boundary surfaces; such, however, is not the case. An inviscid fluid offers no resistance to distortion, and consequently the containing vessel, however irregular its form, is unable to acquire a "purchase" on the fluid contents, and the fluid is not set in rotation. Conversely if we suppose the fluid to be in a state of rotation in a vessel or region, no matter what its form, such rotation will persist and the fluid will continue to rotate for an indefinite time.
The foregoing reasoning, although touching the essence of the matter, can hardly be regarded as rigid proof.[1]
§ 66. Boundary Circulation the Measure of Rotation.—The study of rotation may be confined to two dimensions. Let a a (Fig. 36) represent a circular cylindrical vessel of radius r within which the fluid possesses a motion of pure uniform rotation.
Now, such rotation is shared uniformly over the whole area; therefore, if we suppose the area divided into a number of equal small elements, and represent the rotation of each by a circulation
- ↑ The mathematical demonstration of this important fact will be found in "Hydrodynamics" (H. Lamb, Cambridge), or reference may be made to the original investigation (Lagrange, "Oeuvres," T. IV., p. 714).
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