Conservation of Mass for a Differential Open System (Control Volume)
[Corrected]
Consider a fluid flowing unsteadily such that the only important spatial variables are
x
and
y
.
This is a
twodimensional, unsteady flow field as shown in the figure below.
Every intensive property of the flowing fluid is a function of
only
three variables: time and two spatial
coordinates,
x
and
y
. For example in this twodimensional, unsteady flow field the following variables
can be measured:
Scalar Field
Vector Field
Pressure, Temperature, Density
P
=
P
(
x, y, t
);
T
=
T
(
x, y, t
);
ρ
=
ρ
(
x, y, t
)
x and yvelocity
V
x
=
V
x
(
x, y, t
)
and
V
y
=
V
y
(
x, y, t
)
Velocity
V
=
V
(
x, y, t
)
=
V
x
i
+
V
y
j
Linear momentum density
ρ
V
=
ρ
(
x, y, t
)
V
(
x, y, t
)
= (
ρ
V
x
)
i
+ (
ρ
V
y
)
j
In each case, the measured variables depend on
x
,
y
, and
t
and are referred to as either scalar or vector
fields. For purposes of analysis, we will also assume that the flow field has a constant and uniform depth
∆
z
in the direction of the z

axis.
Consider writing the conservation of mass for a small, finitesized open system shown by dashed lines in
the figure. We get the standard relationship between the rate of change of the mass inside the system and