Biomedical Engineering Reference
In-Depth Information
{
}
)
(
)
(
{
}
(
)
(
)
∑
F
=
2
π
r
+ Δ
r
τ
+ Δ
τ
r
+ Δ
r
Δ
z
−
2
π τ
r
Δ
z
−
2
π
r r P
Δ
+
dP
−
2
π
r r P
Δ
.
=
0
x
rz
rz
rz
, and assuming the differential elements ap-
proach zero, the above equation is written in the form
Neglecting the product of
Δ
r
·
Δτ
dP
(
rdr
+−=
)
ττ
r rdr
dz
Rearranging the terms using chain rule for
τ
and
r
results in
()
dr
τ
dP
=
r
dr
dz
This equation applies to both laminar and turbulent flow in one-dimension,
and to Newtonian and non-Newtonian fluids. For Newtonian fluids and the lami-
nar flow condition, (4.5) is substituted to obtain
1
d V
1
P
⎛
⎞
r
=
⎜
⎟
⎝
⎠
rdr
dr
μ
dz
P
2
)/
L
where
L
is the tube length. Since
dP/dz
is not a function of
r
, integrating twice with
respect to
r
, and applying the boundary condition that
dV/dr
at
r
where
V
is the velocity in the
z
-direction.
dP/dz
can be approximated by
−
(
P
1
−
=
0 (due to sym-
metry) and
V
x
=
R
(with no slip of the fluid at the wall assumption, which
is true for the majority of the laminar flow conditions), we get
0 at
r
=
)
(
)
(
PPR r
−
2
−
2
1
2
4
(4.8)
V
=
μ
L
Equation (4.8) is similar to an equation of a parabola, where
V
is dependent
on
r
2
. Hence, plotting
V
at various
r
values using (4.8) yields a parabolic velocity
profile over the tube cross-section. The maximum velocity,
V
max
, occurs at the cen-
tral axis of the tube where
r
0. Integrating (4.8) over the cross-section provides
an expression for the volumetric flow rate as
=
π
RP
4
Δ
(4.9)
Q
=
8
μ
L
Equation (4.9) is commonly known as the Poiseuille equation or Hagen-
Poiseuille equation due to the work of Hagen in 1839 and Poiseuille in 1840.
