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Full text of "Scientific Papers - Vi"

410             ON  THE  DISCHARGE  OF   GASES  UNDEE  HIGH  PRESSUEES           [408
At the surface of the jet, but not within it, the condition is imposed that the pressure must he that of the surrounding atmosphere.
The problem of a jet in which the motion is completely steady in the hydrodynarnical sense and approximately uniform was taken up by Prandtl*, both for the case of symmetry round the axis (of z) and in two dimensions. In the former, which is the more practical, the velocity component w is supposed to be nearly constant, say W, while u and v are small. We may employ the usual Eulerian equations. Of these the third, dw dtu dw dw 1 dp
-TT  +U-j- +V   -y- + W ~j-= --   -T- ,
dt        dx       dy        dz        p dz
reduces to                  W^^^t   .............................. (2)
dz        p dz
when we introduce the supposition of steady motion and neglect the terms of the second order.    In like manner the other equations become
wdu_    I dp      wdv_    I dp                         ((,,
'*     i    —            7     j        ''7    —            ;     ................... \   /
dz        p doc           dz        p dy
Further, the usual equation of continuity, viz.
d(pu) | d(pv}    d(pw) = Q .................... (4,
dx         dy         dz         ' ........................
here reduces to
F    = 0 ...................... (5)
dy     dz)         dz
If we introduce a velocity-potential <£, we have with use of (2)
-dz~ a*  dz
where a, — \/ (dp/dp), is the velocity of sound in the jet.    In the case we are now considering, where there is  symmetry round  the  axis, this becomes
_ _^_ p r = 0, .....................(7)
and a similar equation holds for w, since w = d<p/dz.
If the periodic part of w is proportional to cos fiz, we have for this part
—    I ^    /F2 _   \
<£r2    r cZr     V a2       y^    ~~   '  ....................^ '
and we may take as the solution
w— W+Hcos/3z . J0 {i\/(W'2 — a?), fir I a],  ...............(9)
since the Bessel's function of the second kind, infinite when r = 0, cannot here appear.    The condition to be satisfied at the boundary (r = R) is that * Phys. Zeitschrift, 5 Jahrgang, p. 599 (1904).-shoot the mark when he goes on to maintain that after the critical pressure-ratio is exceeded, the escaping jet moves everywhere with the same velocity, viz. the sound-velocity; and that everywhere within it the free atmospheric pressure prevails. He argues from what happens when the motion is strictly in one dimension. It is true that then a wave can be stationary in space only when the stream moves with' the velocity of sound; but here the motion is not limited to one dimension, as is shown by the swellings between the disks. Indeed the propagation of any wave at all is inconsistent with uniformity of pressure within the jet.l as of the particles of fluid with each other, is more directly shewn by an experiment on the continuance of a column of mercury, in the tube of a barometer, at a height considerably greater than that at which it usually stands, on account oi the pressure of the atmosphere. If the mercury has been well boiled in the tube, it may be made to remain in contact with the closed end, at the height of 70 inches or more " (Young's Lectures, p. 626,1807). If the mercury be wet, boiling may be dispensed with and negative pressures of two atmospheres are easily demonstrated.