Biomedical Engineering Reference
In-Depth Information
forces that act on particles within the fluid. A common solution for the force and the kine-
matic trajectory of a particle is
3
ρju
p
2ujC
D
4
F
D
5
ρ
p
d
p
and:
dx
i
p
dt
5u
i
p
If the shear stress is a critical parameter in the flow scenario, then the accumulation of
shear stress with time can be calculated from
X
t
max
ðε
i
2ε
i
1
1
Þ
2
t
i
Þ 1KE
i
3ρ
3 ðμ
i
1μ
3Δt
i
t5t
0
13.2
A more accurate model can incorporate the movement of the blood vessel wall and the
deformability of particles within the fluid. This type of modeling is termed fluid structure
interaction model, and in these types of models, mass, momentum, and energy can be trans-
ferred between the wall, the particles, and the fluid. The solution of these types of simula-
tions makes use of the partial differential equation:
2
2
2
2
Φ2
@
@t
2
2u
@
Φ
@x
2
1
@
Φ
@y
2
1
@
Φ
Φ
@z
2
2
r
5
0
which can be solved for any fluid parameter of interest.
13.3
To make the model the most accurate possible, the models should be dynamically similar,
which means that the length scales and the forces that act within the model are the same. A
method to ensure that a model is similar is the Buckingham Pi Theorem, which devises a
number of functions between a parameter of interest and various fluid properties. In bio-
fluid mechanics, there are a number of dimensionless parameters that can be considered
when making models. These are the Reynolds number, the Womersley number, the
Strouhal number, the cavitation number, the Prandtl number, the Weber number, and the
capillary number. These are defined, respectively, as
5
ρvL
μ
vL
ν
Re
5
1
=
2
α5L
ω
ν
fL
v
St
5
p2p
v
1
Ca
5
=
2
ρv
2
5
μC
P
k
Pr
5
ρv
2
L
σ
C5
μ
u
σ
We
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