Biomedical Engineering Reference
In-Depth Information
studied: anatomically realistic, idealized with tube side-branch, and idealized with
hole side-branch. For abbreviation these geometries will be referred to as “real,”
“idealized with branch,” and “idealized with hole,” respectively.
3
The Mathematical Models
Hemodynamics in the cardiovascular system is modeled through the time-dependent
equations for incompressible fluids, derived from the conservation of momentum
and mass. They describe a homogeneous fluid in terms of the velocity and the
pressure fields. Considering an open and bounded domain
3
, the system of
Ω
⊂
R
equations representing such fluid is given by
⎧
⎨
∂
u
1
ρ
t
+(
u
·
∇
)
u
−
div
σ
(
p
,
u
)=
f
,
∂
(1)
in
Ω
,∀
t
>
0
,
⎩
div
u
=
0
,
=
where
f
represents the body forces (that will be neglected,
f
0
, for the case study at
σ
(
,
)
hand),
depends
on the unknown fluid pressure,
p
, and velocity,
u
, and may be generally represented
as the sum of the so-called spherical,
p
I
, and deviatoric,
ρ
is the fluid constant density, and the Cauchy stress tensor
p
u
τ
(
(
))
D
u
,parts[
21
]
σ
(
p
,
u
)=
−
p
I
+
τ
(
D
(
u
))
.
(2)
In the spherical part,
p
is the Lagrange multiplier associated to the incompress-
ibility constraint
div
(
u
)
, which defines the mechanical pressure for incompressible
fluids,
p
=
p
(
x
,
t
)
,and
I
is the unitary tensor. Concerning the deviatoric tensor,
τ
,it
depends on the strain rate tensor,
D
(
u
)
, which is the symmetric part of the velocity
gradient
2
∇
T
.
1
D
(
u
)=
u
+(
∇
u
)
3.1
Newtonian Fluids
The definition of a constitutive relation for
is related to the rheological
properties of the fluid. Under the assumption of incompressible Newtonian fluids,
the Cauchy stress tensor is a linear isotropic function of the components of the
velocity gradient, and it is given by
τ
(
D
(
u
))
σ
(
u
,
p
)=
−
p
I
+
2
μ
D
(
u
)
,
(3)
τ
(
where
μ
>
0 is the fluid constant Newtonian viscosity and
u
)=
2
μ
D
(
u
)
.
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