Acoustics of a Nonhomogeneous Moving Medium Page: 12 of 202
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NACA TM 1399
tensor Mik, and the flow of energy N, which, like the flow of substance,
can be written in vector form. This determination will be such that the
divergence of the flow, taken with inverse sign, is equal to the deriva-
tive with respect to the time of the density of the corresponding mag-
nitude. In this manner from equation (1.1) for the flow of substance
(equal to the flow of momentum) the following is obtained:
S-pS (1.9)
From equation (1.3), substitution of the value of Sik from equa-
tion (1.5), gives the tensor of the momentum flow
2
Mii - pv + p + - div v - 2p vii
Mik = Pvivk - 2 vik = Mki; i / k (1.10)
where, as before, i and k = 1, 2, and 3.
The terms of the form pv , Pvivk give the momentum flow due to
the transport of momentum by the motion of the fluid, and the terms
containing P, , and 7 give the flow of momentum due to the action
of the pressure forces and the viscous stresses.
Finally, from equation (1.4), substitution of Sik from equation
(1.5) yields the energy flow
N p + pE + + pvv2 + rot vx +
y . div v v (1.11)
The first term gives the energy flow due to the transport of energy
by the fluid, the second (5) gives the heat flow, and the term2 Pv
and the terms with t and 7 give the part of the energy flow due to
the work of the pressure forces and the viscous stresses.
The fundamental equations can also be written in vector from, by
substitution of the value of the tensor Tik from equations (1.2) and
(1.5) in equations (1.3) and (1.4). Equation (1.1) may, however, be as
Sdiv() 0 (1.12)
2The vector N = p 2- + v, representing the flow of energy for
an ideal incompressible liquid, is called the N. Umov vector (ref. 3).4
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Blokhintsev, D. I. Acoustics of a Nonhomogeneous Moving Medium, report, February 1956; (https://digital.library.unt.edu/ark:/67531/metadc65701/m1/12/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.