Some Studies on the Flow of a Gas in the Region of Transition Through the Velocity of Sound Page: 7 of 13
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The function AJi through the intermediary of the parameter t
depends on '; the function W through the intermediary of t depends
on a. In figure 1, the values of S are in the direction of
the positive horizontal axis and the values of 4 along the
positive vertical axis; on the negative horizontal axis are the
values of - 5 and along the negative vertical axis, -4. As the
velocities increase from 0 (M = 0) to a (M = 1), 9 decreases
from o to 0. The value of 4iT up to very small values of S
will be very near unity and only at S = 0 does 'Ir rapidly tend
to zero. Further, when & increases from zero to its limiting
value equal to 9 (h-l) the function ji will asymptotically tend
2 19 3 k+l
to infinity (for - =-k2 + . + - the ourve has a point of
When cp is eliminated in the subsonic region, the following
equation is eliminated
+ + o
ae2 2 ds
and in the supersonic region
2e 2 do a
On the basis of the behavior of the function ' in equa-
tions (1) 'i = constant may be set to very small values of S.
The equation of Laplace is then obtained, which is the first approxi-
mation of S. A. Christianovich (references 1 and 3). In one of his
later papers, Christianovich showed (reference 4) that for the super-
sonic region, that is, where equations (2) must be taken, starting
from small a and further on, the function '4 may be approximated
in the form of pieces of curves of the second order (first a para-
bola and then a hyperbola). Equations (2) then become the equation
of Darboux and may be simply integrated to the end.
The analysis is, however, most difficult of all for the narrow
region corresponding to the values of S and a near zero. This
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Kiebel, I. A. Some Studies on the Flow of a Gas in the Region of Transition Through the Velocity of Sound, report, April 1950; (https://digital.library.unt.edu/ark:/67531/metadc64581/m1/7/: accessed May 22, 2019), University of North Texas Libraries, Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.