Biomedical Engineering Reference
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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)
=
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