**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

**9.4.2.1 Reynolds Number**

Consider an object of characteristic dimension L moving at
velocity v through a fluid of density r and viscosity
h. The object's movement is resisted by two forces
-- inertia and viscous drag. The inertial force on the object is of order F_{inertial}
~ r v^{2} L^{2}; the viscous drag
force is of order F_{viscous} ~ h v L. Thus
a slow human underwater swimmer with L ~ 1 meter, v ~ 0.1 m/sec, r
~ 1000 kg/m^{3}, and h ~ 10^{-3}
kg/m-sec must apply F_{inertial} ~ 10 N plus a minor additional F_{viscous}
~ 10^{-4} N of motive force in order to keep moving forward. Clearly,
the human swimmer lives in a world of predominantly inertial forces.

A bacterial swimmer faces entirely different challenges.^{389,1386,1387}
A bacterium of size L ~ 1 micron and velocity v ~ 10 micron/sec must apply F_{inertial}
~ 10^{-4} fN (femtonewtons; 1 fN = 10^{-15} N) but also a much
larger F_{viscous} ~ 10 fN of motive force in order to keep moving forward.
The ratio of the two forces is still 10^{5}:1, but the roles have reversed.
The bacterium (or any micron-scale medical nanorobot) lives in a world dominated
by viscosity, where, as an example, the phenomenon of "coasting" essentially
ceases to exist. For instance, if motive power to a swimming nanorobot with
radius R_{nano} = 1 micron and velocity v_{nano} = 1 cm/sec
is suddenly stopped, then the nanorobot will "coast" to a halt in a time t_{coast}
= r R_{nano}^{2} / 15 h
= 0.1 microsec and in a distance x_{coast} ~ v_{nano} t_{coast}
= 1 nm.^{1395} If the nanorobot
is rotating at a frequency n_{nano} = 100
Hz when its rotational power source is suddenly turned off, n_{nano}
decays exponentially to zero in a time t_{coast} ~ 0.1 microsec and
stops after turning q_{coast} = 2 p
n_{nano} r R_{nano}^{2}
/ 15 h ~ 40 microradians.

The ratio of inertial to viscous forces is called the Reynolds
number N_{R}, or:

which is a dimensionless number. In the examples given above,
N_{R} = 10^{5} for the human swimmer and N_{R} = 10^{-5}
for the bacterium. Purcell^{389 }notes
that for a man to be swimming at the same Reynolds number as his own sperm,
he would have to be placed in a swimming pool full of molasses and then be forbidden
to move any part of his body faster than 1 cm/min, roughly the speed of the
minute-hand of a large wall clock.

The Reynolds number has already been introduced in connection
with laminar tube flow (Section 9.2.5), and elsewhere
it has been noted that N_{R} = 100-6100 in the arteries, 200-900 in
the veins, 0.0004-0.003 in the blood capillaries (Table
8.2), and ~10^{-6}1 for lymph vessels (Table
8.5). However, these figures are relevant only when considering flow phenomenon
on a scale large enough such that the cellular graininess of human blood may
be ignored. In the case of microscopic motile cells and medical nanorobots,
this assumption is not valid.

On the contrary, nanorobotic sanguinatators will find themselves
negotiating a viscous Newtonian-fluid plasmatic environment, punctuated by numerous
closely-spaced free-floating cellular obstacles. Powered sanguinatation thus
may involve traversing opportunistic clear volumes of blood plasma between red
cells, then altering course to take advantage of the next available open space
further along, following a zigzag path generally in the desired direction. While
the sizes and shapes of these clear volumes are strongly time and position-dependent,
their characteristic size is x_{clear} ~ (MCV (100% - Hct) / Hct)^{1/3}.
Taking the Mean Cell Volume MCV = 94 micron^{3} for red cells (Section
8.2.1.2), then x_{clear} ~ 5 microns in the arteries where Hct ~
46% and x_{clear} ~ 10 microns in the capillaries where Hct = Hct_{tube}
~ 10% (Section 9.4.1.6). If paths taken through clear
volumes at a mean velocity v_{nano} average an angle q_{path}
relative to the desired direction of travel, then the net forward velocity in
the desired direction is v_{net} = v_{nano} cos(q_{path}).
Taking v_{nano} ~ 1 cm/sec giving N_{R} = 10^{-2}, then
v_{net} ~ 0.5 cm/sec through heavy traffic in the vessel lumen assuming
q_{path} ~ 60°. Brownian displacements of
RBCs are negligible by comparison -- v_{brownian} ~ DX
/ t ~ 0.1 micron/sec for DX
~ 1 micron (Eqn. 3.1) -- while shear velocities
in small arteries are typically ~1 mm/sec (Section 9.4.2.2).
For powered trajectories exclusively confined to the cell-free "plasmatic" zone
near the blood vessel wall (the vascular "express lane" for the fastest-moving
nanorobot traffic; Section 9.4.2.6), x_{clear}
is largely independent of Hct, q_{path} ~
0°, and v_{net} ~ v_{nano}.

Last updated on 21 February 2003