**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

**9.4.1.5 Bloodstream
Velocity Profiles**

Consider a Newtonian fluid of viscosity h
flowing through a cylindrical tube of length l_{tube} and radius r_{tube}
with pressure differential Dp between the ends. The
average fluid velocity (v_{flow}) for laminar or Poiseuille flow* is
given by Eqn. 9.26, but imposing a no-slip condition
at the vessel wall produces a radius-dependent parabolic velocity profile:^{362}

where r is radial distance from the tube axis and k_{p}
= Dp / 4 h l_{tube}
= 2 v_{flow} / r_{tube}^{2}. Maximum flow velocity v_{max}
= 2 v_{flow}, and occurs at the centerline (r = 0).

* For well-developed turbulent flows, the
velocity profile may be expressed as v_{turb} = v_{max} (1 -
r/r_{tube})^{(1/m)} away from the laminar sublayer near the
wall, where m ~ 7 for a wide range of Reynolds numbers.^{1390}

Of course, as fluid enters a tube from a large reservoir,
there is an entrance region called the inlet length (l_{inlet}) which
lies between the entrance and a point downstream, where the parabolic profile
is in the process of being established asymptotically (Fig.
9.14). For N_{R} <~ 1, l_{inlet} ~ 1.3 r_{tube},
while for N_{R} >~** **30, l_{inlet} ~ 0.16 N_{R}
r_{tube}, producing, post-inlet, a <1% deviation from an ideal Poiseuille
(parabolic) profile.^{361,1331}
Inlet conditions prevail throughout the entire length of the aorta and most
of the major arteries, but entrance effects are minimal in the smaller vessels.

Now suppose that red blood cells are added to the plasma.
Due to the inward migration of red cells and other factors, the velocity profile
of whole blood is affected by flow rate and hematocrit, particularly in vessels
<500 microns in diameter, becoming blunted (Fig.
9.15). The degree of blunting decreases with increasing flow rate and increases
with rising hematocrit. But the ability of RBCs to deform under the influences
of cell crowding and fluid stresses in shear flow allows whole blood to continue
to flow up to Hct ~ 98%.^{1319}
It is theoretically possible that metamorphic nanorobots could approach this
level of performance, though this issue has not yet been studied extensively.

What if rigid spherical nanorobots are added to the plasma
instead of red cells? At small R_{nano} and low Nct, the flow profile
remains parabolic. Onset of velocity profile blunting generally occurs either
when Nct >~ 20% or when R_{nano} >~ 0.05 r_{tube}.^{1322}
Such blunted flow is sometimes called partial plug flow. Partial plug flow was
not observed experimentally when (Nct R_{nano}/r_{tube}) <~
0.6%. Thus, non-plug, purely Poiseuille flow can probably be maintained for
2-micron diameter bloodborne nanorobots even while passing through the smallest
d_{tube} = 4 micron human capillary by holding Nct <~ 1.2%.

Figures 9.16A,
9.16B,
9.16C,
and 9.16D
show the experimentally-determined effects of Nct, R_{nano}/r_{tube},
and fluid flow rate on the velocity profile of suspensions of rigid spheres
and disks.^{1317,1322}
As Nct and R_{nano}/r_{tube} continue to rise, complete plug
flow eventually ensues (Fig.
9.16B), wherein the entire mass of fluid moves stiffly at constant velocity,
like toothpaste squeezed from a tube. Plug flow requires much higher pumping
power (and pumping pressure) than laminar flow. Complete plug flow was observed
at Nct = 38% and R_{nano} = 0.112 r_{tube}, that is, when (Nct
R_{nano} / r_{tube}) >~ 3.8%. If complete plug flow can be
avoided by holding R_{nano}/r_{tube} < 0.38 at our assumed
maximum Nct ~ 10% (Section 9.4.2.6), then for the
smallest d_{tube} = 4 micron human capillary, R_{nano} <
0.76 microns, allowing up to D_{nano} = 1.5 micron diameter for bloodborne
nanodevices at the maximum nanocrit.

Further analysis that might consider the effects (on bloodborne nanorobot velocity profiles) of vessel geometry (including bifurcations, nozzling, and curved paths), vessel wall elasticity (including collapsible tubes), pulsatile flow, and adding nanorobots of various mixtures of sizes and shapes to whole blood rather than plasma, would be useful but is beyond the scope of this book.

Last updated on 21 February 2003