Biomedical Engineering Reference
In-Depth Information
is less than 0.001 in capillaries. The Womersley criterion can also be applied in
the respiratory system.
Another dimensionless number used in characterizing the oscillating and pul-
satile flow conditions is called the Strouhal number (
N
St
), named after the Czech
physicist V. Strouhal. The
N
St
represents a measure of the ratio of inertial forces
due to the unsteadiness of the flow or local acceleration to the inertial forces due to
changes in velocity from one point to another in the flow field.
N
St
is defined as the
N
α
Nfl
=
/
V
(4.16)
St
c
Flow with a very small
N
St
suggests that the flow is quasi-steady. When the
fluid encounters a solid obstacle in the flow path, the fluid flow near that obstacle
moves slower than the flow farther away. This forms what is known as the bound-
ary layer. When an adverse pressure gradient is created from the flow at the bound-
ary, the flow separates the boundary. This causes the inception of a vortex, a swirl-
ing motion in the fluid. At higher
N
Re
, the vortex on one side of the obstacle grows
larger relative to other side and continues to build until it cannot stay attached to
the boundary. The vortex separates, which is called vortex shedding. The presence
of curvatures and branching in the blood vessels can also favor the formation of
vortices. These vortices appear as regular periodic phenomena and do not exhibit
a chaotic flow pattern associated with turbulent flow conditions.
N
St
is useful in
determining the vortex shedding, which typically occur when
N
St
is approximately
0.2.
N
St
is related to
N
Re
and
N
by
α
N
2
N
=
N
α
St
2
π
Re
Using a similar derivation for the pulsatile flow, Womersley's modified formula
for shear stress at the wall is
N
μ
Q
R
τ
=
α
(4.17)
w
3
π
2
Only if the
N
value is smaller than 1, the pulsatile flow follows the Poiseuille
α
flow.
EXAMPLE 4.5
A Palmaz-Schatz stent was introduced to a patient into a 5 mm in diameter artery. The ve-
locity of blood could be estimated to be 0.1 m/s and the frequency of pulsation to be 0.02
Hz. Calculate the Womersley number and Strouhal number for the scenario assuming the
viscosity of blood to be 3 cP and density to be 1.056 g/cm
3
. What is the stress experienced
by the stent? Compare that to the steady-state level wall of shear stress.
Solution:
2
2
3
Q
* 0.25 [cm] * 10[cm/s]
1.963[cm /s]
=
π
=
