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# Hydrodynamics

## tube, fluid, gas, velocity, pressure, flowing, liquid and phenomena

HY'DRODYNAMICS (from Gk. Mop, hy di:r, water ± div•auic, dynamis, power, from 6iTaat3ai, dynasthai, to be able). Strictly speak ing, that branch of mechanics which is the ap plication of dynamics to liquids; it should there fore includa both the statics and the kinetics of liquids: that is, a study of their properties both when they are in equilibrium rind when not. In general. however, the former phenomena are treated separately under hydrostatics (q.v.), leaving for hydrodynamics simply the phenomena of kinetics. Further, although gases are easily eompressed, while liquids are not, yet, if a gas is flowing slowly and without great fluctuations, it will have properties closely resembling those of a flowing liquid. Consequently hydrody namics also includes most of the phenomena of the kinetics of gases. If a liquid is flowing regularly through a tube of an irregular cross section, or if a gas is flowing slowly and regu larly through such a tube, the mass of the fluid which passes each point in the tube in a given time must be the same: otherwise there would be a state of compression or rarefaction snme where in the tube; it follows. then, that where the tube is narrow the velocity must be large, and conversely, like a river flowing first through a lake and then through a narrow channel. If Clue velocity at any point in tube is greater than at another. it shows that there must be a force acting in the direction from the second point toward the first, so as to increase the velocity of the moving fluid: hut the force must always be produced by a fall in pressure, and so the pressure at the second point is greater than at the first. it follows, then, that in a fluid flowing in a steady state where the velocity is small the pressure is large, and conversely if friction is supposed to be excluded. This general principle is illustrated by the 'ball nozzle,' the injector of a boiler, the common atomizer, the 'ball in the fountain' experiment, and many others. If a fluid is flowing through a long pipe, water or gas in city mains, there is of course a great amount of friction between the moving fluid and the layer that sticks to the tube. Owing to this, the pressure decreases along the pipe, and the velocity of flow is decreased also.

If an opening is made in the side of a vessel containing a liquid, the latter will make a jet out into air. If the opening is a small one in a thin wall, it may be observed that the cross section of the jet a short distance from the wall is less than that of the opening itself: this place of smallest cross-section is called the 'versa contracta.' If a quantity of liquid of mass In

escapes having a velocity v, its kinetic energy is This energy is evidently due to the fact that the effect inside the vessel is just as if these at grams had been taken off from the free surface; and so, if the centre of the opening is at a depth h below the free surface, the in grams have lost an amount of potential energy mgh. Therefore or = 29h.

This value of the velocity of efflux was first deduced by Torricelli, the pupil of Galileo. This theorem may be stated in a slightly different way. The liquid is forced out owing to a differ ence of pressure on the two sides of the opening equal to pgh. (See HYDROSTATICS.) Calling this difference of pressure P, the formula for v' be comes v' = 2 p In this form it may be applied to the rate of escape of a gas from a vessel through a small opening in a thin wall. It is seen that the square of the velocity of efflux varies inversely as the density of the gas and directly as the pressure forcing the gas out, that is as the differ ence of the partial pressures of that particular gas on the two sides of the opening, regardless of what other gases are present or what their pressures are. (See EFFUSION.) If the escape of the fluid takes place through a thick wall or through a tube. the phenomena are entirely different. owing to friction. It is observed that the rate of escape is independent—within certain limits—of the material of the tube. showing that there is a layer of the fluid adhering to the inner walls of the tube, thus forming a tube of the fluid through which the flow takes place. The following formula has been found by experi ment to held approximately for capillary tubes: Pr' v = Gti/ where r is the internal radius of the tube, / is its length, and n is a constant.

If an obstacle is placed in a stream of liquid or of gas, it experiences certain forces. One of the most interesting cases is that of an oblong solid in a fluid stream; it will tend to place itself with its length at right angles to the current.

This is illustrated by the fact that when a small piece of paper falls it does not do so edge down, but face down; if a coin is dropped in water, it falls face down, 'wobbling' as it sinks. (See