#### A.2. General Balance
Equation

Consider the integral

F(
t
)=
∫
V(
t
)
d
3
x
f(
x,t
)
(A.13)

which gives the total amount of a quantity *F* for a fixed set of fluid particles
occupying a volume *V*(*t*).
Here, *f* is called the density
of *F*.

The convective derivative of *F* is

DF
Dt
=
D
Dt
[
∫
V(
t
)
d
3
x
f(
x,t
)
]
(A.14)

and gives the rate of change of *F* in the reference frame moving with *V*(*t*). Using the results from A.1.2, we have

DF
Dt
=
∂
∂t
[
∫
V(
0
)
d
3
z
J f[
x(
z,t
),t
]
]
[
(A.5) used ]

=
∂
∂t
[
∫
V(
0
)
d
3
z
J f(
z,t
)
]
where
f(
z,t
)=f[
x(
z,t
),t
]

=
∫
V(
0
)
d
3
z
∂
∂t
[
J f(
z,t
)
]

=
∫
V(
0
)
d
3
z
[
f(
z,t
)
∂J
∂t
+J
∂f(
z,t
)
∂t
]

=
∫
V(
0
)
d
3
z
J[
f(
z,t
)
∇
x
⋅v +
∂f(
z,t
)
∂t
]
[ (A.12) used ]

=
∫
V(
0
)
d
3
z
J[
f(
x,t
)
∇
x
⋅v +
Df(
x,t
)
Dt
]

=
∫
V(
t
)
d
3
x
[
f(
x,t
)
∇
x
⋅v +
Df(
x,t
)
Dt
]
(A.16)

Now, the change in *F* can be caused by a source with density generating rate
σ
F
,
or by a flux
J
F
so that

DF
Dt
=
∫
V(
t
)
d
3
x
σ
F
−
∮
S(
t
)
dS⋅
J
F
(A.17)

=
∫
V(
t
)
d
3
x
(
σ
F
−
∇
x
⋅
J
F
)

where *S*(*t*) is the surface bounding *V*(*t*)
and we've used the Green's theorem.

Since (A.16,17) are valid for arbitrary *V*(*t*),
we must have

f
∇
x
⋅v +
Df
Dt
=
σ
F
−
∇
x
⋅
J
F

Þ
Df
Dt
=−f
∇
x
⋅v +
σ
F
−
∇
x
⋅
J
F
(A.18)

which is known as the **balance equation**. Using

Df
Dt
=
∂f
∂t
+v⋅
∇
x
f

we have

∂f
∂t
+v⋅
∇
x
f+f
∇
x
⋅v =
σ
F
−
∇
x
⋅
J
F

or

∂f
∂t
+
∇
x
⋅(
vf
) =
σ
F
−
∇
x
⋅
J
F

∂f
∂t
+
∇
x
⋅(
J
C
+
J
F
)=
σ
F
(A.19)

∂f
∂t
+
∇
x
⋅J=
σ
F

where
J
C
=vf
is called the **convective current** of *F*
and the total current is
J=
J
C
+
J
F
. Integrating (A.19) over a volume
V
f
fixed in space, we have

∫
V
f
d
3
x
∂f
∂t
=−
∫
V
f
d
3
x
∇
x
⋅(
vf+
J
F
)+
∫
V
f
d
3
x
σ
F
(A.20)

Þ
d
dt
∫
V
f
d
3
x
f=−
∮
S
f
dS⋅
(
vf+
J
F
)+
∫
V
f
d
3
x
σ
F
(A.21)