motion in advance of the plane may be looked upon as irrotational.
The Helmholtz-Kirchhoff result shows that the distribution of pressure over the front of the plane is fairly uniform over the central part, falling off rapidly near the edges; this is evident from the fact that a maximum of .5 is associated with a mean = .440 in the case of the infinite lamina. In the case of a plane of compact outline it is probable that the Helmholtz hypothesis would give a considerably lower figure, about .40 or somewhat less; the maximum, however, will be the same as for the infinite lamina, so that it may be anticipated in this case the pressure will fall off more rapidly towards the periphery.
§ 139. Theoretical Considerations relating to the Shape of the Plane.—The influence of the shape of the plane is most conveniently studied in the two extreme cases to which we have already directed attention, i.e., the compact form (a square or circular disc) and the parallel strip or infinite lamina. The former is a symmetrical case of three-dimensional motion; in the latter the motion takes place in two dimensions only.
It has been sometimes suggested that since the pressure increases with the relative periphery, the pressure is greatest in the peripheral regions; we have already seen that such is not the case. The true reason is to be found in a complication of causes.
(1) The congestion of fluid that gives rise to the pressure region is less when the fluid can escape laterally in two dimensions than when its “spread” is confined to one dimension.
(2) The “spurting” of the lines of flow past the edges of the plane will be greater when the access of the fluid to the “hinter-land” is the more complete. Thus in an infinite strip of width the layers of fluid adjacent to the face of the plane are fed by a much greater stream area than in the case, say, of a circular disc of which is the diameter, and the spurting past the edge of the plane will be correspondingly the more vigorous;
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