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entropy,
p
is the pressure and v
1/ρ is the volume per unit mass or the specific
volume. The rate of increase of the total internal energy in the volume
=
V
is then
T
Ds
1
ρ
d
p
D
Dt
Dt
−
V
ρ
V
ρ
T
∂
s
s
d
∂ρ
∂
t
+
d
p
ρ
=
∂
t
+
v
·∇
V+
v
·∇
ρ
V
V
V
ρ
T
∂
s
s
v
d
=
∂
t
+
v
·∇
−
p
∇·
V
,
(6.118)
V
on using the equation of continuity (6.92). To get the total rate of increase of
internal energy of the fluid contained in
, we must add the rate of gain of kin-
etic energy associated with the fluid motion,
S
1
2
ρv
k
v
k
d
V+
1
2
ρv
k
v
k
∂
∂
t
v
j
n
j
d
S
.
(6.119)
V
S
Using the theorem of Gauss to transform the surface integral, this becomes
∂
∂
t
1
2
ρv
k
v
k
2
ρv
k
v
k
d
∂
∂
x
j
1
+
v
j
·
V
.
(6.120)
V
S
At the same time, the fluid contained in
loses energy by doing work against the
body force and on its surroundings at the rate
∂
x
j
τ
ji
v
i
d
ρ
f
j
v
j
+
∂
−
ρ
f
j
v
j
d
V−
τ
ji
n
j
v
i
d
S=−
V
,
(6.121)
V
S
V
on transforming the surface integral by Gauss's theorem. In addition, the fluid con-
tained by the surface,
S
, gains heat energy by thermal conduction relative to the
fluid motion at the rate
k
∂
T
∂
x
j
d
k
∂
T
∂
∂
x
j
S=
V
,
∂
x
j
n
j
d
(6.122)
S
V
with
k
representing the thermal conductivity, and where once again the surface
integral has been transformed to a volume integral by Gauss's theorem. Ignoring
radiative transfer, internal heat sources add energy at the rate
H
per unit volume,
giving the total rate of energy entering the system as
∂
∂
x
j
k
∂
T
∂
x
j
H
d
+
V
.
(6.123)
V
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