Biomedical Engineering Reference
In-Depth Information
Figure 2.5
Two-dimensionalconservationofmass.
This equation may be written as
ρ
ρ
é
ù
∂
ρ
∂
ρ
∂
∂ ∂
u
v
+
u
+
v
+
+
=
0
(2.4)
ê
ú
∂
t
∂
x
∂
y
∂ ∂
x
y
ë
û
and, under a vector form
D
ρ
+ Ñ
�
.
V
=
0
(2.5)
Dt
where the operator
D
/
Dt
is
D
∂
∂
∂
∂
=
+
u
+
v
+
w
(2.6)
Dt
∂
t
∂
x
∂
y
∂
z
in a three-dimensional Cartesian coordinate system. Liquids may generally be con-
sidered as incompressible and the mass conservation equation is then reduced to
�
Ñ
.
V
=
0
(2.7)
In Cartesian coordinates we have
∂
u
∂
v
∂
w
+
+
=
0
∂
x
∂
y
∂
z
and in cylindrical coordinates, the axisymmetric form of (2.7) is
∂
v
v
∂
v
r
r
z
+
+
=
0
∂
r
r
∂
z
The second equation is the momentum conservation equation (or the Navier-
Stokes equation). The change of momentum in a fluid element is equal to the bal-
ance between inlet momentum, outlet momentum, and exerted forces [2]. Figure
