Biomedical Engineering Reference
In-Depth Information
for the radial direction
r
, and
ρ
∂v
∂t
+
u
∂v
v
r
∂v
∂θ
+
w
∂v
uv
r
∂r
+
∂z
+
∂θ
+
ρF
θ
+
μ
1
r
∂v
∂r
∂
2
v
∂θ
2
−
∂
2
v
∂z
2
1
r
∂p
∂
∂r
v
r
2
+
1
r
2
2
r
2
∂u
∂θ
+
=−
−
(4.2)
r
for the contour
θ
, and
ρ
∂w
u
∂w
v
r
∂w
∂θ
+
w
∂w
∂z
∂t
+
∂r
+
μ
1
r
r
∂w
∂r
∂
2
w
∂θ
2
∂
2
w
∂z
2
∂p
∂z
+
∂
∂r
1
r
2
=−
ρF
z
+
+
+
(4.3)
in the axial direction
z
. If we have the simplest form of axi-symmetrical flow in
a cylindrical pipeline, the Navier-Stokes equations simplify by
∂
2
∂θ
2
∂
∂θ
=
=
0 and
with the contour velocity
v
0; it follows that (
4.2
) can be omitted. Let us consider
no external forces
F
r
,F
z
. Since we have very low total pressure drop variations, i.e.
≈
=
0.1 kPa [
114
], we can divide by density parameter
ρ
. Next, we introduce the di-
mensionless parameter
y
d
d
dr
dr
R
dr
,
=
r/R
, with 0
≤
y
≤
1 in the relation
dy
=
dy
=
d
1
R
d
and
dy
. The simplifying assumptions are applied: (i) the radial velocity com-
ponent is small, as well as the ratio
u/R
and the term in the radial direction; (ii) the
terms
dr
=
∂
2
∂z
2
in the axial direction are negligible, leading to the following system:
1
yR
2
∂
2
u
∂y
2
−
∂u
∂t
=−
1
ρR
∂p
∂y
+
μ
ρ
∂u
∂y
+
1
R
2
u
R
2
y
2
(4.4)
1
yR
2
∂
2
w
∂y
2
∂w
∂t
=−
1
ρ
∂p
∂z
+
μ
ρ
∂w
∂y
+
1
R
2
(4.5)
u
Ry
+
1
R
∂u
∂y
+
∂w
∂z
=
0
(4.6)
Studies on the respiratory system using similar simplifying assumptions can be
found in [
41
,
114
,
120
]. Given that the pressure gradient is periodic, it fol-
lows that also that the pressure
p(y,z,t)
and the other velocity components
u(y,z,t),w(y,z,t)
are periodic, as in
p(y,z,t)
=
A
P
(y)e
jω(t
−
z/c)
A
U
(y)e
jω(t
−
z/c)
=
(4.7)
u(y, z, t)
A
W
(y)e
jω(t
−
z/c)
w(y,z,t)
=