round its boundary (Fig. 37), then the circulation round each element will be equal, and that along all the lines common to two adjacent elements is equal and opposite, and therefore of zero value, so that circulation along the boundary alone remains. It is proved then that:—
Fig. 37.
The sum of the circulations round the boundaries of the individual elements is equal to the circulation round the boundary of the region; that is to say, the rotation of the fluid within the region is measured by the circulation round its boundary. It is evident that this result is not confined to uniform rotation. Let us suppose that the fluid contain rotation unevenly distributed amongst its parts, so that it may be in part irrotational, and in parts the sense of rotation may be opposite to that in other parts, but so that the velocity (u v) is, throughout the region, a continuous function of x y; then if we suppose it be divided as before into a number of small elements so that each element shall be indefinitely small, then the rotation within each element is uniform, and by the preceding argument is measured by the circulation round its boundary; but since u v is a continuous function of x y, the flow along the boundary of each element is in the limit equal and opposite to that of the element adjacent to it, and the two cancel out, leaving only the circulation round the boundary. Hence for any region the sum of the rotation integrated over the surface is equal to the sum of the circulation integrated along its boundary.
§ 67. Boundary Circulation Positive and Negative.—Referring again to Fig. 36, let us suppose a boundary surface to exist at e e e dividing the region into two parts, and let e e e coincide with one of the lines of flow so that it will not interfere with the motion of the fluid; the boundary e e will thus be circular, and concentric to the boundary a a.
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