Environmental Engineering Reference
In-Depth Information
A literature review shows that nowadays, molecular dynamics and mechanics
of the continuum in are the main methods of research of fluid flow in nanotubes.
Although the method of molecular dynamics simulations is effective, it at the
same time requires enormous computing time especially for large systems. There-
fore, simulation of large systems is more reasonable to carry out nowadays by the
method of continuum mechanics [52, 53, 76, 77, 81-83].
In the work of Morten Bo Lindholm Mikkelsen et al. [48], the fluid flow in the
channel is considered in the framework of the continuum hypothesis. The Navier-
Stokes equation was used and the velocity profile was determined for Poiseuille
flow.
In the work of Thomas John and McGaughey Alan [73], the water flow by
means of pressure differential through the carbon nanotubes with diameters rang-
ing from 1.66 to 4.99 nm is researched using molecular dynamics simulation
study. For each nanotube the value enhancement predicted by the theory of liquid
flow in the carbon nanotubes is calculated. This formula is defined as a ratio of
the observed flow in the experiments to the theoretical values without consider-
ing slippage on the model of Hagen-Poiseuille. The calculations showed that the
enhancement decreases with increasing diameter of the nanotube.
Important conclusion of the Thomas John and McGaughey Alan [73] is that
by constructing a functional dependence of the viscosity of the water and length
of the slippage on the diameter of carbon nanotubes, the experimental results in
the context of continuum fluid mechanics can easily be described. The aforemen-
tioned is true even for carbon nanotubes with diameters of less than 1.66 nm.
The theoretical calculations use the following formula for the steady velocity
profile of the viscosity
h
of the fluid particles in the CNT under pressure gradient
z
∂ /
p
∂
:
2
R
r
2
2
L
p
vr
1
4
RR z
2
(4.1)
ç
The length of the slip, which expresses the speed heterogeneity at the boundary of
the solid wall and fluid is defined as by Joseph [33] and Sokhan [67]:
/
vr
L
(4.2)
S
dv
dr
rR
Then the volumetric flow rate, taking into account the slip
Q
S
is defined as:
4
3
R
ð
d
/2
4
d
/2
.
L
p
S
(4.3)
Q
2
ð
r v
r dr
.
S
z
8
ç
0
Equation (4.3) is a modified Hagen-Poiseuille equation, taking into account slip-
page. In the absence of slip L
s
= 0 (the Eq. (4.3)) coincides with the
Hagen-Poiseuille flow (the Eq. (3.5)) for the volumetric flow rate without slip
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