Geology Reference
In-Depth Information
The law of conservation of energy equates the gain of energy within the surface
S
, given by the sum of (6.118), (6.120) and (6.121), to the energy entering the
system (6.123). Since the volume is arbitrary, integrands are equated, giving
ρ
T
∂
s
s
1
2
ρv
k
v
k
2
ρv
k
v
k
+
∂
∂
t
+
∂
∂
x
j
1
∂
t
+
v
·∇
−
p
∇·
v
v
j
·
k
∂
T
∂
x
j
∂
x
j
τ
ji
v
i
∂
∂
∂
x
j
−
ρ
f
j
v
j
−
=
+
H
.
(6.124)
Expanding derivatives, this equation can be rearranged to
ρ
T
∂
s
s
2
v
k
v
k
∂ρ
∂
x
j
ρv
j
+
ρv
k
∂v
k
1
∂
∂
t
+
v
j
∂v
k
∂
t
+
v
·∇
−
p
∇·
v
+
∂
t
+
∂
x
j
k
∂
T
∂
x
j
−
v
i
∂τ
ji
∂
x
j
−
τ
ji
∂v
i
∂
∂
x
j
−
ρ
f
j
v
j
∂
x
j
=
+
H
.
(6.125)
It can be further simplified using the mass conservation equation (6.90) and the
equation of motion (6.103) to give
ρ
T
∂
s
s
k
∂
T
∂
x
j
∂
∂
x
j
+
τ
ji
∂v
i
∂
t
+
v
·∇
=
+
H
+
p
∇·
v
∂
x
j
.
(6.126)
The left side of the equation of energy conservation (6.126) can be further trans-
formed using a
Tds
equation from thermodynamics,
α
ρ
Tds
=
c
p
dT
−
Tdp
,
(6.127)
where
c
p
is the specific heat at constant pressure and
∂ρ
∂
T
1
ρ
α
=−
(6.128)
p
is the volume coe
cient of thermal expansion. Then, we have
ρ
T
∂
s
·∇
s
=
ρ
c
p
∂
T
·∇
T
−
α
T
∂
p
·∇
p
∂
t
+
v
∂
t
+
v
∂
t
+
v
.
(6.129)
For an isotropic viscous fluid, the viscous stress resisting deformation is given
by (6.115), and we have
∂v
j
∂
x
i
+
j
j
τ
ji
∂v
i
∂v
i
∂
x
j
∂v
i
∂
x
j
−
2
3
∇·
i
j
i
i
∂
x
j
=
−
p
δ
+
η
v
δ
+
ζ
∇·
v
δ
∂v
j
∂
x
i
+
j
+
η
∂v
i
∂
x
j
∂v
i
∂
x
j
−
2
3
∇·
i
v
)
2
=−
p
∇·
v
v
δ
+
ζ (
∇·
.
(6.130)
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