**Nanomedicine,
Volume I: Basic Capabilities**

**©
1999 Robert A. Freitas Jr. All Rights
Reserved.**

Robert A. Freitas Jr., Nanomedicine, Volume I: Basic Capabilities, Landes Bioscience, Georgetown, TX, 1999

**9.2.5 Pipe Flow**

Cellular and nuclear microinjection using fine glass pipette
tips as small as 200 nm in diameter is commonplace in the experimental biological
sciences.^{1191} In small tubes
or nanoinjectors with r_{cap} >> l (Section
9.2.4), nanoscale fluid flow behavior is approximated reasonably well by
the classical continuum equations. Continuum flow^{1390}
is governed by the famous Hagen-Poiseuille Law (or more commonly, Poiseuille's
Law), derived from the Navier-Stokes equations, which states that a pressure
difference of Dp between the ends of a tube of radius
r_{tube} and length l_{tube} will move an incompressible fluid
of absolute viscosity h in laminar flow at a volume
rate of:

The subsonic mean fluid velocity (v_{flow}), flow
power dissipation (P_{flow}), and the flow time of an aliquot of fluid
through the length of the entire tube (t_{flow}) then follow directly
from Eqn. 9.25 as:

Taking h = 0.6915 x 10^{-3}
kg/m-sec for pure water at T = 310 K, a nanotube with r_{tube} = 10
nm, l_{tube} = 1 micron, and Dp = 1 atm passes
fluid at 'V_{HP} = 0.6 micron^{3}/sec, v_{flow} = 2
mm/sec, with t_{flow} = 0.5 millisec. The incompressibility assumption
generally holds in liquids and in gas flows where Dp
<< p_{tube}, where p_{tube} is the head pressure at the
tube entrance.

The above equations describe the resistance to Poiseuille
(laminar) flow in a pipe, which is the minimum of resistance of all possible
flows in a pipe.^{361} If the flow
becomes turbulent, the resistance increases. The determinative parameter is
a dimensionless quantity called the Reynolds number,^{1187}
N_{R}, which is the ratio of the inertial pressure (~ r
v_{flow}^{2}) to the viscous pressure (~ h
v_{flow} / r_{tube}) in the flow of a fluid of density r,
or:

For the example of the 10-nm nanotube in the previous paragraph,
N_{R} ~ 10^{-5}. A large Reynolds number (Section
9.4.2.1) implies a preponderant inertial effect and the onset of turbulence;
a small Reynolds number implies a predominant shear effect (e.g., viscosity)
and the maintenance of laminar flow. Reynolds^{1187}
found that the transition from laminar to turbulent flow typically occurred
at N_{R} ~ 2000-13,000, depending upon the smoothness of the entry conditions.
The lowest value obtainable experimentally on a rough entrance appeared to be
N_{R} ~ 2000. However, when extreme care was taken to establish smooth
entry conditions (as is more likely in precisely structured nanotubes constructed
using molecular manufacturing techniques) the transition could be delayed to
Reynolds numbers as high as 40,000.

Even assuming the more conservative N_{R} ~ 2000 figure,
it is clear that subsonic flow in nanoscale pipes will almost always be laminar.
For a pipe with r_{tube} = 1 micron conveying water at 310 K (r
= 993.4 kg/m^{3}), the onset of turbulent flow (taking N_{R}
~ 2000) occurs at v_{turb} (= v_{flow} in Eqn.
9.29) > 1400 m/sec, very nearly the speed of sound in water. In gases
or liquids of lower density, or in pipes of narrower bore, or if a less conservative
transitional Reynolds number is available due to superior design, then v_{turb}
grows still larger. Thus turbulence is primarily a high-velocity, large-tube
phenomenon.

In such turbulent flow situations with N_{R} >~
1000, a well-known empirical formula for the volume flow rate is:^{363}

where the turbulence factor Z = 0.005 N_{R}^{3/4}.
Thus for N_{R} = 3000, 'V_{turb} ~ 0.5 'V_{HP}. Nevertheless,
even in turbulent conditions of the general flow, fluid motion nearest the tube
wall remains laminar in a thin layer of thickness x_{lam}, often estimated
as:

For flowing water at 310 K and r_{tube} = 10 micron,
l_{tube} = 100 micron, and Dp = 10 atm giving
v_{flow} ~ 200 m/sec and N_{R} ~ 3000, then x_{lam}
~ 0.5 microns.

Poiseuille's Law assumes a rigid pipe, an assumption that
may not hold for some nanotube designs and which certainly does not hold for
human blood vessels, especially the veins. A complete treatment of fluid flow
in elastic tubes is beyond the scope of this book. However, for steady laminar
flow in an elastic tube with wall thickness h_{wall}, radius r_{tube},
length l_{tube}, p_{0} and p_{1} the pressures at the
entry and exit ends of the tube, and E_{wall} = Young's modulus of a
Hookean (e.g., a linear force-distance relationship) wall material, then the
volume flow rate through the tube may be approximated by:^{361}

where c_{h} = E h_{wall} / r_{tube}.
Taking h = 0.6915 x 10^{-3} kg/m-sec for
water at 310 K , h_{wall} = 2 nm, r_{tube} = 10 nm, l_{tube}
= 1 micron, p_{1} = (0.5 p_{0}) = 1 atm, and E = 10^{7}
N/m^{2} (typical for human skin; Table
9.3), then 'V_{elastic} = 0.3 micron^{3}/sec. Care must
be taken in applying this formula because many biological tube materials such
as blood vessel walls are non-Hookean and exhibit a linear pressure-radius relationship
instead. For small elastic deformations, the volume flow rate of such non-Hookean
tubes may be approximated by:^{361}

where a is the compliance constant,
measured experimentally in feline pulmonary veins as 1.98-2.79 m^{2}/N
for r_{tube} = 50-100 microns down to 0.57-0.79 m^{2}/N for
r_{tube} = 400-600 microns.^{1190}
Other geometric nonuniformities such as regular and irregular tube tapers, embedded
resonance chambers, tube bifurcations or multitube confluences have important
effects on flow rate but are beyond the scope of this book. Poiseuille's Law
also does not strictly hold for fluids seeded with polymers in concentrations
as low as 10^{-4}-10^{-5} by weight, wherein drag may be reduced
by a factor of 23.^{2952,2953}

It bears repeating here that as pipes get very small, they
can clog more easily due to van der Waals adhesive forces,^{1152}
so it becomes increasingly important to engineer interior vessel surfaces for
minimum adhesivity with respect to all materials likely to be transported through
the pipes (Section 9.2.3). Bursting strength of fluid-filled
pressure vessels is addressed in Section 10.3.1.

Last updated on 20 February 2003