Biomedical Engineering Reference
In-Depth Information
Fig. 5.10
Acceleration profile a(s) along the stagnation streamline
5.2.2.3
Interpreting the Diffusion Term
Diffusion is a physical process regarding flow properties that are transported by the
random motion of gas molecules. Fluid momentum diffusion is caused by the mo-
lecular friction between fluid particles that move at different velocities. Its influence
on the flow field is primarily dependent on its diffusion coefficient,
µ
which is the
blood viscosity. The physical behaviour of viscosity was discussed in Chap. 4, where
its main characteristic is to resist fluid deformation thereby inhibiting fluid flow. In
contrast the convection term promotes fluid deformation and momentum. These two
contrasting terms compete with each other to conserve momentum transfer.
In fluid dynamics, the concept of
dynamic similarity
is frequently adopted. This
involves normalizing the mathematical equations to yield the non-dimensional gov-
erning equations. The non-dimensional form of the momentum equation is achieved
by dividing all the dependent and independent flow variables by relevant constant
quantities. For lengths the variable can be divided by a characteristic length
D
(e.g.
diameter of blood vessel), all velocities by a reference velocity
u
in
(e.g. inlet veloc-
ity), and pressure by
2
in
ρ
(twice the dynamic pressure),
x
u
v
P
(5.18)
x
*
=
,
u
*
=
,
v
*
=
,
P
*
=
D
u
u
ρ
in
in
in
where the asterisks denote the non-dimensional variables. Introducing these vari-
ables into the
x
-momentum equation produces
2
2
∂
u
*
∂
u
*
∂
P
*1
∂
u
*
∂
u
*
(5.19)
u
*
+
v
*
=−
+
+
2
2
∂
x
*
∂
y
*
∂
x
*
Re
∂
x
*
∂
y
*
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