§ 166. Resistance of a Plane moving through a Fluid, or Pressure of a Current on a Plane.—When a thin plate moves through the air, or through an indefinitely large mass of still water, in a direction normal to its surface, there is an excess of pressure on the anterior face and a diminution of pressure on the posterior face. Let v be the relative velocity of the plate and fluid, Ω the area of the plate, G the density of the fluid, h the height due to the velocity, then the total resistance is expressed by the equation
R = fGΩv2 / 2g pounds = fGΩh;
where f is a coefficient having about the value 1.3 for a plate moving in still fluid, and 1.8 for a current impinging on a fixed plane, whether the fluid is air or water. The difference in the value of the coefficient in the two cases is perhaps due to errors of experiment. There is a similar resistance to motion in the case of all bodies of “unfair“ form, that is, in which the surfaces over which the water slides are not of gradual and continuous curvature.
The stress between the fluid and plate arises chiefly in this way. The streams of fluid deviated in front of the plate, supposed for definiteness to be moving through the fluid, receive from it forward momentum. Portions of this forward moving water are thrown off laterally at the edges of the plate, and diffused through the surrounding fluid, instead of falling to their original position behind the plate. Other portions of comparatively still water are dragged into motion to fill the space left behind the plate; and there is thus a pressure less than hydrostatic pressure at the back of the plate. The whole resistance to the motion of the plate is the sum of the excess of pressure in front and deficiency of pressure behind. This resistance is independent of any friction or viscosity in the fluid, and is due simply to its inertia resisting a sudden change of direction at the edge of the plate.
Experiments made by a whirling machine, in which the plate is fixed on a long arm and moved circularly, gave the following values of the coefficient f. The method is not free from objection, as the centrifugal force causes a flow outwards across the plate.
| Approximate Area of Plate in sq. ft. | Values of f. | ||
| Borda. | Hutton. | Thibault. | |
| 0.13 | 1.39 | 1.24 | .. |
| 0.25 | 1.49 | 1.43 | 1.525 |
| 0.63 | 1.64 | .. | .. |
| 1.11 | .. | .. | 1.784 |
There is a steady increase of resistance with the size of the plate, in part or wholly due to centrifugal action.
P. L. G. Dubuat (1734-1809) made experiments on a plane 1 ft. square, moved in a straight line in water at 3 to 61⁄2 ft. per second. Calling m the coefficient of excess of pressure in front, and n the coefficient of deficiency of pressure behind, so that f = m + n, he found the following values:—
m = 1; n = 0.433; f = 1.433.
The pressures were measured by pressure columns. Experiments by A. J. Morin (1795-1880), G. Piobert (1793-1871) and I. Didion (1798-1878) on plates of 0.3 to 2.7 sq. ft. area, drawn vertically through water, gave f = 2.18; but the experiments were made in a reservoir of comparatively small depth. For similar plates moved through air they found f = 1.36, a result more in accordance with those which precede.