Biomedical Engineering Reference
In-Depth Information
Figure 2.26
Sketchofachannelwithroughupperandlowerwalls.
2.2.6.6
SlipVelocity
It is usual to consider a velocity at the contact of a solid surface to be zero. In real-
ity, solid surfaces are not perfect, and there is usually a velocity slip at the wall of
the order of a few
m
/s. We consider here a nanoscopic roughness smaller than that
of the preceding section. At the macroscopic scale, or at a sufficiently large micro-
scopic scale, the slip can be ignored because the velocities are much larger than the
slip velocity. At the nanoscopic limit, the slip at the wall is not negligible in front
of the average velocity. The slip is linked to the geometry and chemical properties
of the surface. Velocity slip at the wall is usually related to nanobubbles trapped in
nanocavities along the wall. This is why hydrophobic surfaces have the largest slip
length, usually of the order of 1
m
m.
The slip at the wall is characterized by a slip length
b
(Figure 2.27).
The slip length is usually defined by
æ
ö
∂
v
v
=
b
(2.62)
ç
÷
wall
wall
∂
y
è
ø
For a Poiseuille flow, the correction of velocity is given by
v
-
v
6
b
real
no slip
no slip
-
=
(2.63)
v
h
-
where
h
is the channel length. For
h
= 10
m
m, the error is 60%. Usual microfluidics
applications have dimensions of the order of
h
= 100
m
m, and the error is of the
order of 6%, which is to compare to the added friction linked to roughness.
2.2.7 Bernoulli's Approach
Bernoulli's work in hydraulics dates back to 1738. However, Bernoulli's equation
is probably the most frequently used in engineering hydraulics today, and it has
recently found interesting applications in microfluidics [15]. This equation relates
Figure 2.27
Sliplengthatthewall.
