Biomedical Engineering Reference
In-Depth Information
9.3.1 Coupling 3D and 2D models
Suppose we have a flow model with symmetry of revolution set up in
Ω
1
,thatisa
2D model, and a full flow model in
Γ
a
.Inthis
case, at any time
t
, the velocity fields are described by functions of the form
Ω
2
, with a circular coupling interface
u
1
→
u
1
(
r
,
z
)
∈Q
u
2
→
u
2
(
x
)
∈Q
,
(9.42)
1
2
where
r
and
z
are the radial and axial coordinates in
Ω
1
(denoted now by
Σ
), re-
spectively, and
x
is the three-dimensional position vector in
Ω
2
(see Fig. 9.2). Note
that the above form of
u
1
is equivalent to writing
u
1
(
r
,
z
)=
u
1
,
r
(
r
,
z
)
e
r
+
u
1
,
z
(
r
,
z
)
e
z
,
e
r
e
z
being the unit vectors in the cylindrical system of coordinates. In this case the
spaces
,
Q
i
and
T
Γ
a
(
Q
i
)
,
i
=
1
,
2 are (the dual spaces follow directly)
H
1
/
2
H
r
(
Σ
)
Q
1
=
T
Γ
a
(
Q
1
)=
(
∂Σ
a
)
,
(9.43)
r
H
1
H
1
/
2
Q
2
=
(
Ω
2
)
T
Γ
a
(
Q
2
)=
(
Γ
a
)
.
(9.44)
Here,
∂Σ
a
and
Γ
a
correspond to the same coupling interface viewed from the do-
Ω
2
(see again Fig. 9.2). The space
H
r
(
Σ
)
mains
is the
r
-weighted Sobolev
space defined as the set of measurable functions
v
with norm (see [2])
Σ
and
1
∑
=
0
k
=
0
∂
2
H
r
(
Σ
)
=
k
r
∂
−
k
2
L
r
(
Σ
)
2
L
r
(
Σ
)
=
v
2
v
v
and
v
(
r
,
z
)
r
d
r
d
z
.
(9.45)
z
Σ
We will express the vector
x
in terms of cylindrical coordinates over
Γ
a
,thatis
x
|
Γ
a
→
(
r
,
φ
,
z
|
Γ
a
)
. In this situation, the choice
γ
=
1 produces the following coupling equa-
tions
2π
1
2
(
,
z
|
∂Σ
a
)=
(
,
φ
,
z
|
Γ
a
)
∈
∂Σ
,
u
1
r
u
2
r
d
φ
for
r
(9.46a)
a
π
0
−
p
1
(
r
,
z
|
∂Σ
a
)
n
+
2
με
(
u
1
(
r
,
z
|
∂Σ
a
))
n
=
r
,
z
με
(
−
p
2
(
r
,
φ
,
z
|
Γ
a
)
n
+
2
u
2
(
r
,
φ
,
z
|
Γ
a
))
n
for
(
r
,
φ
)
∈
Γ
a
,
(9.46b)
s
is the symmetric part of the gradient of
u
2
,and
ε
r
,
z
(
where
ε
(
u
2
)=(
∇
u
2
)
u
1
)
is the
(
,
)
corresponding symmetric part of the gradient of
u
1
in cylindrical coordinates
r
z
.
Eq. (9.46a) condenses the dependence upon
and establishes the continuity in a
mean sense from the point of view of the angular coordinate, whereas Eq. (9.46b)
extends the value of the dual quantity from the 2D model, defined
φ
∀
r
, to all val-
r
∗
,wehave
ues of the pair
(
r
,
z
)
in the 3D model. That is, for a fixed value
r
=
r
∗
,
φ
,
r
∗
,
φ
,
−
. In other words, Eq. (9.46a)
stands for Eq. (9.30), that is the continuity of the primal variable in the sense of
H
1
/
2
p
2
(
z
|
Γ
a
)
n
+
2
με
(
u
2
(
z
|
Γ
a
))
n
constant
∀
φ
, whereas Eq. (9.46b) expresses the continuity of the dual variable like in
(9.32), in the sense of
(
∂Σ
a
)
r
T
Γ
a
(
Q
2
)
=
H
−
1
/
2
(
Γ
a
)
.
Analogously, for
γ
=
0 we obtain the following coupling equations
u
1
(
r
,
z
|
∂Σ
a
)=
u
2
(
r
,
φ
,
z
|
Γ
a
)
for
(
r
,
φ
)
∈
Γ
a
,
(9.47a)
