Biomedical Engineering Reference
In-Depth Information
where
U
is the average fluid velocity,
D
is a characteristic dimension of the chan-
µ
nel (or the obstacle), and
is the kinematic viscosity (expressed in m
2
/s). The
ν
=
ρ
Reynolds number naturally appears by performing a dimensional analysis of the
NS equations. For simplicity we consider the two-dimensional Cartesian NS equa-
tion with no external forces
é
2
2
ù
∂
u
∂
u
∂
u
1
∂
p
µ
∂
u
∂
u
+
u
+
v
= -
+
+
ê
ú
2
2
∂ ∂
t
x
∂
y
ρ
∂
x
ρ
∂
x
∂
y
ê
ú
ë
û
(2.36)
2
2
é
ù
∂
v
∂
v
∂
v
1
∂
p
µ
∂
v
∂
v
+
u
+
v
= -
+
+
ê
ú
2
2
∂ ∂
t
x
∂
y
ρ
∂
y
ρ
∂
x
∂
y
ê
ú
ë
û
If
U
is a velocity reference (the average velocity), and
D
is a length reference
(a characteristic dimension of the flow, for example, the diameter of the tube for a
microflow in a capillary), we use the following scaling
u
v
x
y
t
p
*
*
*
*
*
*
u
=
,
v
=
,
x
=
,
y
=
,
t
=
,
p
=
2
U
U
D
D
D U
/
U
ρ
Note that
ρU
2
has the same unit as the pressure. Then, the system (2.36) becomes
é
ù
*
*
*
*
2 *
2 *
¶
u
¶
u
¶
u
¶
p
ν
¶
u
¶
u
*
+
u
+
v
= -
+
+
*
ê
ú
*
*
*
*
*2
*2
UD
¶
t
¶
x
¶
y
¶
x
¶
x
¶
y
ë
û
(2.37)
é
ù
*
*
*
*
2 *
2 *
¶
v
¶
v
¶
v
¶
p
ν
¶
v
¶
v
*
*
+
u
+
v
= -
+
+
ê
ú
*
*
*
*
*2
*2
UD
¶
t
¶
x
¶
y
¶
y
¶
x
¶
y
ë
û
The system (2.37) is nondimensional with only one nondimensional parameter,
the Reynolds number. Note that this conclusion agrees with Buckingham's theorem
[21], which states that if a problem depends on
N
dimensional parameters contain-
ing
M
different units, the nondimensional form depends on
N
-
M
nondimensional
numbers. In the present case, there are
N
= 4 dimensional parameters
ρ
,
η
,
D
, and
U
. The
M
units contained in these four parameters are kilos, meters, and seconds.
From Buckingham's theorem, it results that there is
N
-
M
= 1 nondimensional
number for the dimensionless system.
The characteristic scales
D
and
U
depend on the geometry of the problem. For a
flow inside a tube,
U
is the average axial velocity and
D
is the tube diameter. For an
obstacle in a fluid flow,
U
is the velocity far from the obstacle and
D
is the obstacle
characteristic dimension (usually its hydraulic diameter).
The criterion for laminar flow has the form
UD
ν
(2.38)
Re
=
<
Re
trans
Re
trans
is the transition threshold between laminar and turbulent flow. For flow
in tubes and pipes, Re
trans
is of the order of 1,000-2,000, and for a flow past an
obstacle, Re
trans
is of the order of 64-100 [22].
