Imágenes de páginas
PDF
EPUB

measured by the momentum which is transferred in unit time across unit area, we must conclude that the pressure exerted by the gas in the direction of its motion is greater than p by pa2 in consequence of the flow, and so rises to

p + pa2 = p(22 + a2.

This increment of the pressure in the direction of the flow makes itself perceptible as stress when a surface is put in the way of the flowing stream of gas.

An equally great stress results between the surface and the gas when the gas is at rest and the surface is moved with velocity a against the gas in the direction of its normal. The force which then results and tends to stop the motion is felt as resistance, and the resistance of a gas is therefore also determined by the formula

pFa2,

which expresses the law that the resistance increases proportionally to the square of the velocity.

That this law holds not only for the resistance in a liquid, but also for the motion of a body in air, has already been proved by Newton and Hawksbee' by means of experiments on falling bodies; it has lately been found also for other gases by Cailletet and Colardeau 2 by means of observations on the gases in flow. A remarkable confirmation of the formula deduced for gaseous resistance arises from an observation made by Hirn,3 from which too Hirn himself thought he must conclude that the kinetic theory of gases is wrong. He found in fact that the resistance does not alter with the temperature if the density is kept unchanged. With this fact the theoretical formula is in perfect agreement, as it does not contain the molecular velocity G, but only that of the flow a.

The range of applicability of Newton's formula is however dependent on definite limits for the value of the

1 Newton, Principia, bk. ii. prop. 40; Hawksbee, Physico-mechan. Experiments, London 1709; Musschenbroek, Tentamina Exper. in Acad. del Cimento, Lugd. 1731, pt. ii. p. 118.

2 Comptes rendus, cxvii. 1893, p. 145.
3 Mém. de l'Acad. de Belgique, xiii. 1882.

velocity a. If this value is too great, the resulting heat (see § 19) cannot be disregarded; if it is too small, the viscosity of the air (see Chapter VII.) cannot be left out of account.

In another relation, too, the formula for the resistance

pFa2

does not exactly correspond to the results of experiment. Hutton' and Borda' long ago found that the resistance is not exactly proportional to the extent of the surface F of the moved body, and that it depends also on the curvature of the surface. For plane discs which move at right angles to their plane, Schellbach,3 G. Hagen, and Hirn 5 have shown that the factor F would be more exactly replaced by expressions of the form

AF or AF(1 + Bq),

where A, B are constants, n an exponent greater than 1, and g the circumference of the disc. The cause of this deviation is easy to indicate. Part of the air which is pushed in front of the disc turns off sideways, and the resistance is thereby diminished; the theoretical expression has therefore to be multiplied by a proper fraction A. This fraction, however, depends also on the size of the disc, since the air cannot slide aside so easily in front of a large surface as in front of a smaller; hence the value of the fraction increases with the area For with the circumference q of the disc.

35. Reaction. Cross-pressure

In a flowing gas there is also, in the direction opposite. that of the flow, a change of pressure due to the flow, which is also an increase and not a decrease. To prove this statement I might merely rely on the mechanical principle that action and reaction, and consequently pressure and counter-pressure, must be always equal. Yet I prefer

1 Trans. Roy. Soc. Edin. ii. 1790.

3 Pogg. Ann. cxliii. 1871, p. 1.

Mém. de l'Acad. de Belgique, xiii. 1882.

2 Mém. Paris 1763, 1767.

Ibid. clii. 1872, p. 95.

to repeat a proof which has been given by Clausius,' by which also the rise of pressure already described in the direction of the flow is better explained.

For this we start, as in the consideration (§ 12) of the state of equilibrium, from Joule's assumption,2 which even in this case is admissible, that the pressure caused by the motion of the gaseous molecules so operates as if a third part of the molecules move to and fro along the normal to a stressed surface, while the other two-thirds move parallel to this surface. Of the first third one-half will at every moment have a molecular velocity G in the same direction as the velocity of flow a, while the other half has a molecular velocity in the opposite direction. Therefore, of the N molecules contained in unit volume, N move with a resultant velocity G+ a in the direction of the flow, and simultaneously the same number N move in the opposite direction with the resultant velocity Ga.

The difference Ga we may take to be positive, since the mean molecular velocity G is very great, while the observed speeds of flow are for the most part considerably less. The greatest velocity which the wind attains that, for instance, of the most fearful storm-may be taken at about only one-tenth of that with which the molecules move about. But if, indeed, it should happen that a were greater than G, the argument would not be invalidated.

If, now, one-sixth of all the molecules move with the velocity Ga in the line of flow, the number which pass through a surface F in unit time in this direction is

¿NF(G + a),

and they carry with them momentum equal to

¿NmF(G + a)2.

In the backward direction there pass

NF(Ga)

1 Bulletin de l'Acad. de Belgique [3] xi. 1886, p. 180; Mech. Wärmetheorie, 2. Aufl. iii. p. 248.

2 The calculation is carried out independently of this assumption, and purely on the basis of Maxwell's law, in §§ 7* and 43* of the Mathematical Appendices.

molecules through the surface F in unit time, and these carry back the oppositely directed momentum

[blocks in formation]

The two halves into which the gaseous mass is separated by the surface F, therefore, both gain and lose momentum, and the question is, What variation in the law of distribution results? If we call that side the right towards which the flow is directed, we can say that the right side gains an amount of momentum directed towards the right which is equal to

NmF(G+ a)2,

while it loses momentum directed towards the left equal to NmF(Ga)2.

The right half thereby obtains an amount of right-directed momentum which exceeds the left-directed momentum, and this excess is equal to

[blocks in formation]

The excess of left-directed over right-directed momentum which arises in the left half is of equal amount; for this half gains left-directed momentum equal to

NmF(G — a)2,

and loses right-directed momentum equal to

NmF(G+ a)2,

so that the left-directed momentum in the left half will exceed the right-directed momentum in the left half by the amount

NmF{(G — a)2 + (G + a)2}.

These formulæ, however, do not account for the whole changes that occur. In the case of a flowing gas the other two-thirds of the molecules come also into account; for these, too, take part in the flow, and therefore possess the velocity a in the direction perpendicular to the surface F. In this direction, therefore, there pass

NFa

molecules from the left half to the right in the unit of time, and these carry with them the momentum

NmFa2.

The total excess, therefore, of right-directed over leftdirected momentum which is produced in unit time in the right half is given by

¿NmF{(G+ a)2 + (G − a)2} + §NmFa2

[ocr errors]
[blocks in formation]

Just as large is the excess of left-directed over rightdirected momentum which occurs in the left half in unit time; for the former increases by

NmF(G — a)2,

while the latter diminishes by

NmF(G+ a)2 + NmFa2.

Now, according to the kinetic theory of gases, the pressure is nothing else than the momentum carried across unit area in unit time; consequently the pressure is expressed by the formula

p = Nm (G2 + a2),

As

and this formula holds good equally well for the direction in which the gas flows and for the opposite direction. for the former direction, the added term

Nma2 = pa2

expresses the stress exerted by the stream, so for the opposite direction it represents the equal reaction-stress which is exerted by the flowing gas on the containing vessel.

For a direction at right angles to the stream the pressure will be expressed by the formula

p = NmG2,

which holds good for all directions in a gas at rest. An essential difference, however, consists in the magnitude G not having the same value for a flowing gas as it has in a gas at rest. This is, at least, the case when the gas is not brought into a state of motion by the application of energy

« AnteriorContinuar »