482 HYDRO MECHANICS [HYDRAULICS. were trustworthy, when p. 2 was taken to be the general external pressure but that, if ^ <" 5 , then the pressure at the contracted Pi section was independent of the external pressure and equal to O o^. Hence in such cases the constant value 5 should be substituted in the formula for the ratio of the internal and external pres It is easily deduced from Weisbach s theory that, if the pressure external to an orifice is gradually diminished, the weight of air discharged per .second increases to a maximum for a value of the ratio Pi = 527 for air = 58 for dry steam. For a further decrease of external pressure the discharge diminishes, a result no doubt improbable. The new view of "VVeisbach s formula is that from the point where the maximum is reached, or not greatly differing from it, the pressure at the contracted section ceases to diminish. Fliegner has shown (Civilingenieur, xx., 1874) that for air flow ing from well-rounded mouthpieces there is no discontinuity of the law of flow, as Napier s hypothesis implies, but the curve of flow bends so sharply that Napier s rule may be taken to be a good approximation to the true law. The limiting value of the ratio p ~ a -, for which Weisbach s formula, as originally understood, ceases Pi to apply, is for air 5767 ; and this is the number to be substituted for 2* j u t} ie formulae when Ss falls below that value. For later Pi Pi researches on the flow of air, reference may be made to Zeimer s paper (Civilingenieur, 1871), and Fliegner s papers (ibid., 1877, 1878). VII. FRICTION OF LIQUIDS. 63. When a stream of fluid flows over a solid surface, or con versely when a solid moves in still fluid, a resistance to the motion is generated, commonly termed fluid friction. It is due to the vis cosity of the fluid, but generally the laws of fluid friction are very different from those of simple viscous resistance. It would appear that at all speeds, except the slowest, rotating eddies are formed by the roughness of the solid surface, or by abrupt changes of velocity distributed throughout the fluid ; and the energy expended in pro ducing these eddying motions is gradually lost in overcoming the viscosity of the fluid in regions more or less distant from that where they are first produced. The laws of fluid friction are generally stated thus : 1. The frictional resistance is independent of the pressure between the fluid and the solid against which it flows. This may be verified by a simple direct experiment. Coulomb, for instance, oscillated a disk under water, first with atmospheric pressure acting on the water surface, afterwards with the atmospheric pressure removed. No difference in the rate of decrease of the oscillations was observed. The chief proof that the friction is independent of the pressure is that no difference of resistance has been observed in water mains and in other cases, where water flows over solid surfaces under widely different pressures. 2. The frictional resistance of large surfaces is proportional to the area of the surface. 3. At low velocities of not more than 1 inch per second for water, the frictional resistance increases directly as the relative velocity of the fluid and the surface against which it flows. At velocities of 4 foot per second and greater velocities, the frictional resistance is more nearly proportional to the square of the relative velocity. In many treatises on hydraulics it is stated that the frictional resistance is independent of the nature of the solid surface. The explanation of this was supposed to be that a film of fluid remained attached to the solid surface, the resistance being generated between this fluid layer and layers more distant from the surface. At extremely low velocities the solid surface does not seem to have much influence 6n the friction. In Coulomb s experiments a metal surface covered with tallow, and oscillated in water, had exactly the same resistance as a clean metal surface, and when sand was scattered over the tallow the resistance was only very slightly increased. The earlier calculations of the resistance of water at higher velocities in iron and wood pipes and earthen channels seemed to give a similar result. These, however, were erroneous, and it is now well understood that differences of roughness of the solid surface very greatly influence the friction, at such velocities as are common in engineering practice. Darcy s experiments, for instance, showed that in old and incrusted water mains the resistance was twice or sometimes thrice as great as in new and clean mains. 64. Ordinary Expressions for Fluid Friction at Velocities not Extremely Small Let /be the frictional resistance estimated in pounds per square foot of surface at a velocity of one foot per second ; w the area of the surface in square feet ; and v its velocity in feet per second relatively to the water in which it is immersed. Then, in accordance with the laws stated above, the total resistance, of the surface is K=>0 3 (1), where/ is a quantity approximately constant for any given surface. If (2), where is, like/, nearly constant for a given surface, and is termed the coefficient of friction. The following are average values of the coefficient of friction for water, obtained from experiments on large plane surfaces moved in an indefinitely large mass of water. Coefficient of Friction, f Frictional Resistance in It) per sq. ft. / Xe w well-painted iron plate 00489 00473 Painted and planed plank (Beaufoy) Surface of iron ships (Rankine) 00350 00362 00339 00351 Varnished surface (Froude) . 00258 00250 Fine sand surface , , 00418 00405 Coarser sand surface ,, 00503 00488 The distance through which the frictional resistance is overcome is v feet per second. The work expended in fluid friction is therefore given by the equation Work expended =/cot 3 foot-pounds per second _ (3). The coefficient of friction and the friction per square foot of surface can be indirectly obtained from observations of the discharge of pipes and canals. In obtaining them, however, some assumptions as to the motion of the water must be made, and it will be better therefore to discuss these values in connexion with the cases to which they are related. Many attempts have been made to express the coefficient of friction in a form applicable to low as well as high velocities. The older hydraulic writers considered the resistance termed fluid fric tion to be made up of two parts, a part due directly to the distor tion of the mass of water and proportional to the velocity of the water relatively to the solid surface, and another part due to kinetic energy imparted to the water striking the roughnesses of the solid surface and proportional to the square of the velocity. Hence they proposed to take in which expression the second term is of greatest importance at very low velocities, and of comparatively little importance at veloci ties over about foot per second. Values of | expressed in this and similar forms will be given in connexion with pipes and canals. All these expressions must at present be regarded as merely empirical expressions serving practical purposes. The frictional resistance will be seen to vary through wider limits than these expressions allow, and to depend on circumstances of which they do not take account. 65. Coulomb s Experiments. The first direct experiments on fluid friction were made by Coulomb, who employed a circular disk suspended by a thin brass wire and oscillated in its own plane. His experi ments were chiefly I made at very low , velocities. When the disk is rotated to any given angle, it oscillates under the action of its inertia and the torsion of the wire. The os cillations diminish gradually in conse quence of the work- done in- overcoming the friction of the disk. The diminution fur nishes a means of determining the friction. Fig. 78 shows Coulomb s apparatus. LK supports the wire and disk ; ag is the brass wire, the torsion of which causes the oscilla-
K
