Biomedical Engineering Reference
In-Depth Information
13.3.1
General equations for viscous flow
Consider the flow of an incompressible fluid through (or within) a spatially
fixed, time-independent volume
V
. For general viscous flow problems it can be
stated that, in an Eulerian description, the following physical fields have to be
determined:
•
x
in
V
and all
t
and
•
the stress field:
σ
(
x
,
t
)forall
x
in
V
and all
t
.
Starting from the incompressibility condition the velocity field has to satisfy the
continuity equation (mass balance) for all
the velocity field:
v
(
x
,
t
)forall
x
in
V
for all
t
:
∇
v
,
v
T
1
2
+ ∇
tr(
D
)
=
0 with
D
=
(13.25)
while next to that the velocity field and the stress field should be related for all
x
in
V
and for all times
t
according to the local constitutive equation (see Section
12.5
):
d
(
D
) .
σ
=−
p
I
+
σ
(13.26)
Also the local balance of momentum (see Section
11.5
for the Eulerian descrip-
tion) has to be satisfied; so for all
x
in
V
and for all
t
:
,
v
T
+
δ
v
δ
t
∇·
σ
+
ρ
∇
q
=
ρ
·
v
(13.27)
with the (mass) density
constant.
The equations above form a set of coupled partial differential equations. Conse-
quently, for a unique solution of the velocity field
v
(
x
,
t
) and the stress field
σ
(
x
,
t
)
boundary conditions and initial conditions are indispensable.
With respect to boundary conditions it can be stated that for each
t
at every
point on the outer surface of
V
three (scalar) relations have to be specified: either
completely formulated in stresses (dynamic boundary conditions), or completely
expressed in velocities (kinematic boundary conditions) or in a mixed format.
A detailed description of the interpretation of the initial conditions is not
considered.
ρ
13.3.2
The equations for a Newtonian fluid
For a Newtonian fluid, see Section
12.5
, the stress tensor can be written as
σ
=−
p
I
+
2
η
D
,
(13.28)
