Biomedical Engineering Reference
In-Depth Information
Fig. 5.1
Mass flow into and out of a 2D control volume element
(inflow = outflow). In other words,
out
˙
0
=
m
˙
−
m
(5.1)
in
Recalling that the mass flow rate is given by
ρ
U
A
, which is the product of
density, average velocity, and cross-sectional area normal to the flow, then the rate
at which mass enters the control volume is
ρuA
, where
A
=
y
m
˙
=
·
1 for a unit depth,
i.e.
z
= 1 (Fig.
5.1
). Thus,
m
in
=
˙
ρuA
x
=
·
ρu
(
y
1)
(5.2)
and the rate at which the mass leaves the surface at
x
+
x
may be expressed as
(
ρu
)
x
y
∂
(
ρu
)
∂x
m
out
=
˙
+
·
1
(5.3)
Similarly in the
y
-direction,
(
ρv
)
y
(
x
∂
(
ρv
)
∂y
m
in
=
˙
·
m
out
=
˙
+
·
ρv
(
dx
1)
1)
Substituting the mass in and mass out terms into Eq. (5.1) for a two-dimensional
control volume we get
(
ρu
)
x
(
y
·
∂
(
ρu
)
∂x
0
=
+
1)
−
ρu
(
d
·
1)
(
ρv
)
y
(
x
∂
(
ρv
)
∂y
+
+
·
1)
−
ρv
(
x
·
1)
Search WWH ::
Custom Search