Civil Engineering Reference
In-Depth Information
be in balance with the net mass flow rate into the volume. Total mass inside the
volume is
d
V
, and since d
V
is constant, we must have
∂
∂
(mass flow in
−
mass flow out)
d
V
=
t
and the partial derivative is used because density may vary in space as well as
time. Using the velocity components shown, the rate of change of mass in the
control volume resulting from flow in the
x
direction is
d
x
d
y
d
z
+
∂
(
u
)
m
x
˙
=
u
d
y
d
z
−
u
(8.3a)
∂
x
while the corresponding terms resulting from flow in the
y
and
z
directions are
d
y
d
x
d
z
+
∂
(
v
)
m
y
=
˙
v
d
x
d
z
−
v
(8.3b)
∂
y
d
z
d
x
d
y
+
∂
(
w
)
m
z
=
˙
w
d
x
d
y
−
w
(8.3c)
∂
z
The rate of change of mass then becomes
∂
d
x
d
y
d
z
∂
∂
(
u
)
+
∂
(
v
)
+
∂
(
w
)
d
V
=˙
m
x
+˙
m
y
+˙
m
z
=−
(8.4)
t
∂
x
∂
y
∂
z
Noting that
d
V
=
d
x
d
y
d
z
, Equation 8.4 can be written as
∂
∂
∂
u
∂
∂
v
∂
∂
w
∂
∂
u
∂
v
y
+
∂
w
(8.5)
t
+
x
+
y
+
z
+
x
+
=
0
∂
∂
∂
z
Equation 8.5 is the
continuity equation
for a general three-dimensional flow
expressed in Cartesian coordinates.
Restricting the discussion to steady flow (with respect to time) of an incom-
pressible fluid, density is independent of time and spatial coordinates so Equa-
tion 8.5 becomes
∂
u
∂
v
y
+
∂
w
∂
x
+
z
=
0
(8.6)
∂
∂
Equation 8.6 is the continuity equation for three-dimensional, incompressible,
steady flow expressed in Cartesian coordinates. As this is one of the most funda-
mental equations in fluid flow, we use it extensively in developing the finite
element approach to fluid mechanics.
8.2.1 Rotational and Irrotational Flow
Similar to rigid body dynamics, consideration must be given in fluid dynamics
as to whether the flow motion represents translation, rotation, or a combination
of the two types of motion. Generally, in fluid mechanics,
pure rotation
(i.e.,
