to interfere either to start or to stop the fluid in cyclic motion, we must introduce some imaginary barrier in its path. In the special case in the figure this evidently requires to extend from the central core outward to infinity in order to intercept the whole of the flux. We are at liberty to select what position we like, circumferentially, for this barrier, and in choosing such position we fix the datum for the value of If then we suppose a suitable impulse to be applied the of the impulse pressure system will be a single-valued function throughout the field, and defines for the subsequent motion. The barrier, however, cannot be maintained under the conditions of steady motion, and it is the withdrawal of the barrier that renders indeterminate. It is the complementary fact that the barrier temporarily renders the region simply connected, and on its withdrawal the cyclic conditions supervene.
The particular case of cyclic motion taken as an illustration is one of the most elementary simplicity. The degree of complexity of any cyclic system of flow depends primarily upon the boundary conditions. We shall have occasion to refer later to cyclic systems of greater complexity, but at present the complete solution of the equations of motion is only known in some few cases where the boundary conditions are simple.
Although in any case of cyclic flow, such as in the example given, the fluid is in circulation around a central island, and so as a whole possesses angular momentum and rotary motion in the ordinary acceptation of the words, such a form of flow (i.e., one that can be generated by an impulse and possesses a velocity potential) is in reality irrotational. The theory of rotation in fluids is of considerable importance, in view of the fact that it can be proved that if the motion of an inviscid fluid is irrotational at any instant of time, it will remain irrotational for all time; that is to say, it is impossible to produce or destroy rotation in an ideal fluid.
§ 65. Fluid Rotation. Conservation of Rotation.—Let us suppose a hollow circular cylindrical vessel filled with fluid to be set in
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