Biomedical Engineering Reference
In-Depth Information
The solution to the first of these differential equations, which is in fact an
ordinary differential equation, is given by
v
1
(
x
2
)
¼
c
1
x
2
þ
c
2
. The no-slip bound-
ary conditions are that
v
1
(
h
/2)
¼
0 and
v
1
(
h
/2)
¼
V
, thus
c
1
¼
V
/
h
and
c
2
¼
V
/2,
thus
v
1
(
x
2
)
h
/2). The solution to the second of these differential
equations is that p is a constant. The stress at the top plate is given by the
T
12
component of (5.11N),
¼
(
V
/
h
)(
x
2
þ
T
12
¼ m
@n
1
@
x
2
¼
m
V
h
:
Example 6.4.3 Plane Poiseuille Flow
Consider an incompressible viscous fluid of viscosity
m
in the domain between two
infinite flat solid plates at
x
2
¼
h
/2. Action-at-a-distance forces are not present
and the plates are not moving. The flow is steady and there is a constant pressure
drop in the
x
1
direction given by
x
1
. Determine the velocity distribution.
Solution
: As in the previous example, assume that the only nonzero velocity
component is in the
x
1
direction and that it depends only upon the
x
2
coordinate,
v
1
¼
p
/
∂
∂
v
1
(
x
2
). This velocity field automatically satisfies the incompressibility condi-
tion,
r
v
¼
0, and the reduced Navier-Stokes equation is
2
m
@
x
2
¼
@
n
1
p
x
1
:
@
@
The solution to this differential equation, which is again an ordinary differential
equation, is given by
1
2
@
p
x
1
x
2
þ
n
1
¼
c
3
x
2
þ
c
4
:
m
@
2
2
,
p
@x
1
@
1
2
m
h
The boundary conditions are that
v
1
(
h
/2)
¼
0, thus
c
3
¼
0 and
c
4
¼
and it follows that the velocity profile is parabolic in shape,
(
)
h
2
2
1
2
@
p
x
2
n
1
¼
;
m
@
x
1
which is written with a minus sign in front of the pressure gradient to emphasize
that the pressure gradient is negative in the direction of flow or, equivalently, the
pressure is dropping in the direction of flow. The volume flow rate per unit length
Q
is given by
(
)
d
x
2
¼
@
Z
h=
2
2
h
3
12
Z
h=
2
1
2
@
p
h
2
p
x
2
Q
¼
2
n
1
d
x
2
¼
m
:
m
@
x
1
@
x
1
h
=
h
=
2
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