518 HYDROMECHANICS [HYDRAULICS. horizontal and L the vertical component of the normal pres sure. In experiments with the whirling machine it is the resistance to motion, R, which is directly measured. Let P be the pressure on a plane moved "X normally through a fluid. NN S> r Then, for the same plane flX J^(h inclined at an angle <p to its __ " " ^r direction of motion, the re sistance was found by Hutton to be A simpler and more con venient expression given by Colonel Duchemin is R=P 2 sinV 1+ sin d Consequently, the total pressure between the fluid and plane is 2 sin ft 2P 1 + sin 2 cosec <J> + sin and the lateral force is 2 sin - - : - - . 1 + 8111*0 In 1872 some experiments were made for the Aeronautical Society on the pressure of air on oblique planes. These plates, of 1 to 2 feet square, were balanced by ingenious mechanism designed by Mr Wenham and Mr Spencer Browning, in such a manner that both the pressure in the direction of the air current and the lateral force were separately measured. These planes were placed opposite a blast from a fan issuing from a wooden pipe 18 inches square. The pressure of the blast varied from & to 1 inch of water pres sure. The following are the results given in pounds per square foot of the plane, and a comparison of the experimental results with the pressures given by Duchemin s rule. These last values are obtained by taking P = 3 31, the observed pressure on a normal surface : Angle between Plane and Direction of Blast. 1 r i 20 60 90 Horizontal pressure R 0-4 0-61 2-73 3-31 Lateral pressure L 1-6 T96 T26 formal pressure VLr + lt 2 . ... 1-65 2-05 3-01 3-31 Normal pressure by Duchemin s ) rule 1-605 2-027 3-276 3-31 RESISTANCE OF SHIPS. 156. Down to a recent period the resistance of ships was supposed to be due to a difference between the pressure on the bow and stern, caused by the pushing aside of the water, precisely as in the case of the "unfair" bodies whose resistance has just been discussed. Hence the resistance was supposed to be proportional to the immersed mid ship sectional area of the vessel. It will be shown immediately, however, that in a " fair" body, completely immersed, there is no resistance of this kind, the pressure of the water closing in behind exactly balancing the excess of pressure on the bow. In such a body, therefore, the resistance is almost entirely due to the friction al drag of the water on the surface of the body, and is proportional to its skin area. In a ship, which is only partially immersed, a further resistance, which in some cases becomes very large, is due to the alteration of the surface level of the water causing a dissipation of energy in producing waves. 157. Stream Line Motion of a Fluid past a submerged Body. Consider a shipshape body, or body of fair- form, that is, bounded by surfaces of continuous curvature, moving below the surface of a fluid, and for the moment let the friction of the fluid against the surface of the body be supposed absent. In such conditions, the particles of fluid are gradually deviated sideways as the body passes, and gradually close together again behind it. They are left after the operation in their original position with momentum unchanged; there is, therefore, in this case no resistance due to the direct action of the inertia of the water. The nature of the action is more conveniently studied by supposing the body at rest and the fluid flowing past it. Let S, fig 173, repre sent the immersed body surrounded by fluid which is flowing past it. The fluid particles, arriving at A in the direction shown by the arrow, are gradually deviated as they approach S, gradually unite again after passing it ; and, if the body is of fair form, that is, if it presents no abrupt changes of section or discontinuity of curvature, the stream lines or paths of the particles will be continuous lines, which take, at a sufficient distance B stermvards of S, their original direction of motion. The fluid surrounding S may then be con ceived to be divided into an infinite number of elementary streams of continuous curvature. Suppose, for simplicity, S is a solid of revolution. Then, from the similarity of conditions in all directions, the elementary streams will be in planes drawn through the axis ofS. Each elementary stream may be conceived as a mass of fluid flow ing steadily in an infinitely thin frictionless pipe. But it has already been shown that in a tortuous pipe, the ends of which are in the same direction, there is no resultant force due to the motion of the fluid which tends to displace the pipe, either due to its curva ture or its changes of section. Consequently the whole mass of fluid exerts no resultant pressure on the body S past which it is flowing. Nor, if the fluid is at rest, will there be any resistance to the uniform motion of the body S through it. The resistance of the ship therefore cannot be due, like that of an unsh.ipsh.ape body, to the forward momentum impressed directly on the fluid. In a frictionless fluid, and for a uniformly moving and wholly immersed body of fair form, the resistance would be nil. With a fluid which is not perfectly frictionless, however, a resistance may be generated in this way. The particles of water exert a drag on the surface of the body over which they slide. They receive, either in consequence of their adhesion to the surface, or in consequence of impact on the roughnesses which project from it, a forward momentum, and the velocity at B is no longer, as in 3 frictionless fluid, the same as the velocity at A. In the case of a ship which is only in part immersed thei e is another source of resistance. Considering the elementary streams already defined as flowing along indefinitely thin frictionless pipes, it is obvious that there would be greater pressure in those parts where the cross section was large and the velocity small, and less pressure where the section was small and the velocity high. It will be seen from the diagram that the streams are large in cross section in the neighbourhood of the bow and stern, and small along the sides. There will therefore be an excess of pressure at bow and stern, and a diminution at the sides. But the free surface of the water in which the ship floats is a surface of uniform pressure. Hence the water will be forced up at the bow and stern, and sink down in the space between, the variation of the hydrostatic pressure due to depth balancing the variation of pressure in the stream lines. There are thus formed waves accompanying the ship. So far as the ship in its passage through the water has to supply the waste of energy due to the diffusion of this wave motion in the surround ing liquid, it suffers a resistance which may be termed the wave- making resistance. This resistance would arise even in a frictionless 1 fluid. It will be seen from the foregoing that the two principal causes of the resistance to the motion of a ship are the skin friction and the production of waves. The Motional resistance depends on the immersed surface of the ship, its roughness, and the velocity of the water relatively to the surface. Mr Froucle concludes that no sensible error is committed if the frictional resistance is taken to be equivalent to that of a rectangular surface of equal area and of length (in the line of motion) equal to that of the ship and mov ing at the same speed. For such a rectangular surface Mr Fronde s experiments already described furnish the means of calculating the resistance. Experiments made on H.M. ship "Greyhound" appear to show that in well-formed, clean-bottomed ships, at speeds not exceeding 8 knots per hour, the frictional resistance is from 80 to 90 per cent, of the whole resistance, and that at the greatest speeds of the quickest ships the frictional resistance is from 60 to 70 per cent, of the whole resistance. For ships with foul bottoms the frictional resistance is a still larger fraction of the whole resistance. The wave-making resistance is not yet fully understood, and involves considerations beyond the scope of the present article. For any given length of ship, with given proportions of entrance, middle body, and run, there is a limit of speed beyond which the resistance due to dissipation of energy in waves rapidly increases. Below that limit the resistance, being chiefly due to friction, in creases nearly as the square of the speed. Above that limit the resistance increases as a higher power of the speed. In the trials of the " Greyhound " the resistance varied nearly as the square of the speed up to 8 knots per hour, as the cube of the speed at 10 knots, and as the fourth power of the speed at 12 knots. 158. Ratio of the Resistance of Models and of Actual Ships. It will be understood from the foregoing explanations that the laws of
the resistance of ships are complicated and at present imperfectly
