Biomedical Engineering Reference
In-Depth Information
z
h
v
x
x
y
Figure 13.5
Flow between two stationary parallel plates.
solution for (the components of) the velocity field
v
(
x
) and the pressure field
p
(
x
)
according to:
1
2
∂
p
∂
x
z
(
h
v
x
=−
−
z
),
v
y
=
0,
v
z
=
0
(13.37)
η
∂
p
∂
x
∂
p
∂
y
=
∂
p
∂
z
=
constant,
0,
0,
(13.38)
=
0)
and the prescribed boundary conditions. It can be observed that the inertia term
(the right-hand side of the Navier-Stokes equation) for the defined problem van-
ishes. This is quite obvious as material particles of the fluid move with constant
velocity (no acceleration) in the
x
-direction.
The situation described above is called the 'plane Poiseuille flow'. The flow is
characterized by a parabolic (in the
z
-coordinate) profile for the velocity in the
x
-direction, coupled to a constant pressure gradient in the
x
-direction. Note, that
the pressure itself cannot be determined. In order to do that the pressure should
be prescribed for a certain value of
x
, in combination with the pressure gradient
in the
x
-direction or in combination with for example the total mass flux (per unit
time) in the
x
-direction.
In Fig.
13.6
the stationary flow is depicted of a fluid between two parallel flat
plates (distance
h
) in the case where the bottom plate (
z
=
satisfies exactly the continuity equation, the Navier-Stokes equation (with
q
0) is spatially fixed and
the top plate (
z
=
h
) translates in the
x
-direction with a constant velocity
V
. There
is no pressure gradient. In this example a fully developed flow is the starting point
(plates are assumed to be 'infinitely wide' and peripheral phenomena at the inflow
and outflow are left out of consideration). Also in this case for
z
h
a
perfect adhesion between the fluid and plates occurs. Again, it is simple to verify
that for the velocity components the following solution holds:
=
0 and
z
=
z
h
V
,
v
x
=
v
y
=
0,
v
z
=
0.
(13.39)
