Nanomedicine, Volume I: Basic Capabilities

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 Finertial ~ r v2 L2; the viscous drag force is of order Fviscous ~ h v L. Thus a slow human underwater swimmer with L ~ 1 meter, v ~ 0.1 m/sec, r ~ 1000 kg/m3, and h ~ 10-3 kg/m-sec must apply Finertial ~ 10 N plus a minor additional Fviscous ~ 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 Finertial ~ 10-4 fN (femtonewtons; 1 fN = 10-15 N) but also a much larger Fviscous ~ 10 fN of motive force in order to keep moving forward. The ratio of the two forces is still 105:1, 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 Rnano = 1 micron and velocity vnano = 1 cm/sec is suddenly stopped, then the nanorobot will "coast" to a halt in a time tcoast = r Rnano2 / 15 h = 0.1 microsec and in a distance xcoast ~ vnano tcoast = 1 nm.1395 If the nanorobot is rotating at a frequency nnano = 100 Hz when its rotational power source is suddenly turned off, nnano decays exponentially to zero in a time tcoast ~ 0.1 microsec and stops after turning qcoast = 2 p nnano r Rnano2 / 15 h ~ 40 microradians.

The ratio of inertial to viscous forces is called the Reynolds number NR, or:

{Eqn. 9.65}

which is a dimensionless number. In the examples given above, NR = 105 for the human swimmer and NR = 10-5 for the bacterium. Purcell389 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 NR = 100-6100 in the arteries, 200-900 in the veins, 0.0004-0.003 in the blood capillaries (Table 8.2), and ~10-6­1 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 xclear ~ (MCV (100% - Hct) / Hct)1/3. Taking the Mean Cell Volume MCV = 94 micron3 for red cells (Section 8.2.1.2), then xclear ~ 5 microns in the arteries where Hct ~ 46% and xclear ~ 10 microns in the capillaries where Hct = Hcttube ~ 10% (Section 9.4.1.6). If paths taken through clear volumes at a mean velocity vnano average an angle qpath relative to the desired direction of travel, then the net forward velocity in the desired direction is vnet = vnano cos(qpath). Taking vnano ~ 1 cm/sec giving NR = 10-2, then vnet ~ 0.5 cm/sec through heavy traffic in the vessel lumen assuming qpath ~ 60°. Brownian displacements of RBCs are negligible by comparison -- vbrownian ~ 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), xclear is largely independent of Hct, qpath ~ 0°, and vnet ~ vnano.

Last updated on 21 February 2003