FLUID FLOW IN NANOTUBES - Foundations of Nanotechnology: Pore Size in Carbon-Based Nano-Adsorbents - page 213

Structural regime of fluid flow (Fig. 4.13) implies the existence of the continu-
ous laminar layer of liquid (the liquid layer in the nanotube) along the walls of a
pipe. In the central part of a pipe a core of the flow is observed, where the fluid
moves, keeping his former structure, i.e. as a solid (“quasi-solid” phase in the

v .

nanotube). The velocity slip is indicated in Fig. 4.13a through

FIGURE 4.13 Structure of the flow in the nanotube.

Let's find the velocity profile ( )

v r

of the
nanotube. We select a cylinder with radius r and length l in the interlayer, located
symmetrically to the center line of the pipe (see Fig. 4.13).
At the steady flow, the sum of all forces acting on all the volumes of fluid with
effective viscosity h , is zero.
The following forces are applied on the chosen cylinder: the pressure force
and viscous friction force affects the side of the cylinder with radius r , calculated
by the Newton formula.
Thus,

in a liquid interlayer

R ≤ r ≤ R

0

dv

( )

2

p − p π = −h r 2 π rl

(4.6)

0

dr

v =
, we obtained a formula to calculate the velocity of the liquid layers located at a
distance r from the axis of the tube:

v

Integrating Eq. (4.6) between r to R with the boundary conditions :

r = R

0

−
=− +
h

Rr

2 2

( )

( )

(4.7)

vr p p v

0 0

4 l