Biomedical Engineering Reference
In-Depth Information
The main difference between macroscopic and microscopic flow is that macro-
scopic flows are most of the time turbulent whereas microscopic flows are laminar.
In biotechnology—or microchemistry—velocities are most of the time small and it
is very seldom that the flow is turbulent. In fact, the “laminarity” of the flow is usu-
ally high. Typical fluid velocities are of the order of 1 mm/s at the most in channels
of cross dimensions of 1 mm maximum. The kinematic viscosity of water being
ν
= 10
-6
m
2
/s, the Reynolds number is—at the most—of the order of 1. Typical
Reynolds numbers vary from 10
-4
to 1. Thus, the character of the flow is very lami-
nar, meaning that the streamlines are locally parallel (Figure 2.19) and that even
obstacles in the flow will not induce any turbulence. We will see in the next section
that for very small Reynolds numbers the Navier-Stokes equations may be simpli-
fied and are reduced to the Stokes approximation.
2.2.4 Stokes Equation
For a stationary low, at very low velocities, inertial forces become very small com-
pared to the viscous forces. The Reynolds number is smaller than 1 and the inertia
terms on the left of (2.36) may be neglected. In this regime, the Navier-Stokes equa-
tion reduces to the Stokes equation
é
2
2
2
ù
∂
p
∂
u
∂ ∂
u
u
F
x
=
ν
+
+
+
ê
ú
2
2
2
∂
x
x
y
z
ρ
∂
∂
∂
ë
û
é
2
2
2
ù
F
∂
p
∂
v
∂ ∂
v
v
y
(2.39)
=
ν
+
+
+
ê
ú
2
2
2
∂
y
ρ
∂
x
∂
y
∂
z
ë
û
2
2
2
é
ù
∂
p
∂
w
∂
w
∂
ww
F
z
=
ν
+
+
+
ê
ú
2
2
2
∂
z
ρ
∂
x
∂
y
∂
z
ë
û
In the case where the external force is just the gravity force, the simplification is
considerable because the system (2.39) is now linear
�
1
2
2
Ñ = D + Ñ
p
ν
V
z
(2.40)
2
where we have used the notation
D = Ñ
for the Laplacian operator. By taking the
rotational of (2.40), and using the following mathematical relations
curl grad P
(
)
= Ñ´Ñ =
p
0
�
�
(
)
(
)
curl A
curl A
D
=
D
we obtain
(
�
)
�
0
D Ñ´
V ω
= D =
(2.41)
where
ω
is the vorticity of the flow. Thus, in the Stokes formulation, vorticity is a
harmonic function [23] and the problem can be solved in the vorticity-streamline
formulation as soon as the values of the vorticity on the boundaries are known.
