of the fluid the pressure difference applied between the points and requires to be the same as that between and , so that the normals and to the field of flow are equipotentials ( constant).
This demonstration may be taken as applied to every small element of the field, so that the proposition is proved.
Corollary: When a fluid has velocity potential its motion is irrotational.
§ 71. Physical Interpretation of Lagrange's Proposition.—The foregoing proposition, taken in conjunction with that relating to the conservation of rotation, constitutes a demonstration of Lagrange's theorem that “If a velocity potential exist at any one instant for any finite portion of a perfect fluid in motion under the action of forces which have a potential, then, provided the density of the fluid he either constant or a function of the pressure only, a velocity potential exists for the same portion of the fluid at all instants before or after.”
This statement, save to a mathematician, is not very clear, as it is difficult to obtain a sufficiently close conception of velocity potential to be able to attach any physical meaning to its conservation.[1] The inversion of the statement, however, obviates all difficulty; it then becomes: If the motion of any portion of a perfect fluid he irrotational at any instant of time, then, provided the density of the fluid he either constant or a function of the pressure only, the motion of the same portion of the fluid will he irrotational at all instants before and after.
§ 72. A Case of Vortex Motion.—The case of cyclic motion resulting from an interchange of the functions and in the source or sink system is one of particular interest. If (Fig. 35) we suppose the origin circumscribed by a line of flow, then we
- ↑ The velocity potential may fall to zero in a portion of the fluid in the course of its motion without that portion of the fluid losing the attribute of velocity potential in the sense of Lagrange's theorem.
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