Biomedical Engineering Reference
In-Depth Information
as a consequence of the term on the right-hand side of the last equation; this has
a seriously complicating effect on the solution process. Exact analytical solutions
can only be found for very simple problems.
13.3.4
Boundary conditions
In the present section the attention is focussed on the formulations of simple
boundary conditions with respect to an arbitrary point on the outer surface of
the considered (fixed) volume
V
, with local outward unit normal
n
. A number of
different possibilities will separately be reviewed.
•
Locally prescribed velocity
v
along the boundary, i.e. the component
v
·
n
in normal
direction as well as the component
n
in tangential direction. A well-known
example of this, is the set of boundary conditions for 'no slip' contact of a fluid with a
fixed wall:
v
−
(
v
·
n
)
=
0. In fact, the impermeability of the wall (there is no flux through the
outer surface, also see Fig.
7.9
) is expressed by
v
·
n
=
0, while suppressing of slip is
expressed by
v
=
0.
•
A locally prescribed stress vector
σ
·
n
along the boundary, i.e. the component
n
·
σ
·
n
in normal direction as well as the components of
v
−
(
v
·
n
)
n
n
in tangential direc-
tion. Boundary conditions of this type can be transformed into boundary conditions
expressed in
σ
·
n
−
(
n
·
σ
·
n
)
v
and
p
by means of the constitutive equations. A known example of
this is the set of boundary conditions at a free surface: the normal component of the
stress vector is related to the atmospheric pressure (equal with opposite sign) and the
tangential components are equal to zero.
•
For a frictionless flow along a fixed wall it should be required that
v
·
n
=
0 combined
with
=
0. Again the last condition can, by using the constitutive
equation, be expressed in
v
and
p
.
σ
·
n
−
(
n
·
σ
·
n
)
n
13.3.5
Elementary analytical solutions
Figure
13.5
visualizes a stationary flow of a fluid between two 'infinitely
extended' stationary parallel flat plates (mutual distance
h
). The flow in the pos-
itive
x
-direction is activated by means of an externally applied pressure gradient.
We consider that part between the plates (the specific domain with
x
- and
z
-
coordinates) where the flow is fully developed. This means that no influence is
noticeable anymore from the detailed conditions near the inflow or outflow of the
fluid domain: for all relevant values of
x
, the velocity profile is the same. For
z
0 and
z
=
h
the fluid adheres to the bottom and the top plate respectively
('no slip' boundary conditions, see previous section). It is simple to verify that the
=
