Environmental Engineering Reference
In-Depth Information
v
has the core of the nanotube
Maximum speed
and is equal
0
≤
r
≤
R
0
to:
2
2
0
−
=− +
h
RR
(
)
v
pp
v
r
0
0
(4.8)
4
l
Such structure of the liquid flow through nanotubes considering the slip is
similar to a behavior of viscoplastic liquids in the tubes. For viscoplastic fluids
a characteristic feature is that they are to achieve a certain critical internal shear
stresses
τ
and behave like solids. Meanwhile, when internal stress exceeds a
critical value begin to move as normal fluid. In the work researchers the liquid
behaves in the nanotube the similar way. A critical pressure drop is also needed to
start the flow of liquid in a nanotube.
Structural regime of fluid flow requires existence of continuous laminar layer
of liquid along the walls of pipe. In the central part of the pipe is observed flow
with core radius
R
00
, where the fluid moves, keeping his former structure (i.e., as
a solid).
The velocity distribution over the pipe section with radius
R
of laminar layer
of viscoplastic fluid is expressed as follows:
= −− −
h
p
τ
(
)
( )
(
)
2
2
vr
R
r
0
R r
(4.9)
4
l
h
The speed offlow corein
is equal
0
rR
00
D
= −−−
h
p
τ
(
)
2
2
v
(
R R
)
0
RR
r
00
00
4
l
h
(4.10)
Let's calculate the flow or quantity of fluid flowing through the nanotube cross-
section
S
at a time unit. The liquid flow
dQ
for the inhomogeneous velocity field
flowing from thecylindrical layer ofthickness
dr
, which is located at a distance
r
from the tube axis is determined from the relation,
( )
( )
2
(4.11)
dQ
=
v r dS
=
v r
π
rdr
where
dS
-the area of the cross-section of cylindrical layer (between the dotted
lines in
Fig. 4.13)
.
Let's place Eq. (4.7) in Eq. (4.11), integrate over the radius of all sections from
R
0
to
R
and take into account that the fluid flow through the core flow is deter-
mined from the relationship
. Then we get the formula for the flow of
Q
=
π
R
2
0
v
ÿ
ÿ
liquid from the nanotube:
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