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

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

9.5.3.1 Nanoflight and Reynolds Number

It has been observed that a 30-cm paper airplane will glide slowly and stably, but a 3-cm paper airplane made from thinner paper requires a much higher velocity/size ratio to remain airborne, and with a noticeable lack of stability -- sometimes the plane flies well, sometimes not. If size is further reduced into the millimeter range, the plane almost cannot fly.¹⁵⁷³

As in the case of swimming, this transition can be explained in terms of the Reynolds number N_R (Section 9.4.2.1), the ratio of inertial to viscous forces acting on a body that is passing through a fluid such as air. Generally speaking, microscopic organisms (e.g., N_R ~ 10^-5) or flying nanorobots with N_R << 1 move by utilizing viscosity, while macroscopic objects such as aircraft (e.g., N_R ~ 10⁸) with N_R >>1 use inertia to generate lift. Millimeter-size airfoils with N_R ~0.1-100, as typified by flying insects, occupy a transitional regime. Insects make use of both inertial and viscous forces, often employing unusual wing flapping patterns^{1577,1578,1585} and elastic energy storage systems^1578,1582 to remain aloft. The smallest known flying insect¹⁵⁷⁹ that can make any use of aerodynamic lift (inertial) forces is the four-winged parasitic chalcid wasp Encarsia formosa, which has a total wingspan of ~1.4 mm.¹⁵⁷⁸ Aerobotic machines with wingspans smaller than ~100 microns probably must make almost exclusive use of viscous propulsive forces.³⁶³

A nanorobot of dimension L flying at velocity v through 20°C sea-level dry air (r_air = 1.205 kg/m³, absolute viscosity h_air = 0.0183 x 10^-3 kg/m-sec)⁷⁶³ has a Reynolds number N_R = 66,000 v L (Eqn. 9.65). Viscous forces dominate when N_R < 1, or when:

{Eqn. 9.86}

where L_micron is characteristic nanorobot size expressed in microns. Thus a 1-micron nanorobot remains in the viscous regime up to ~ 15 m/sec flight velocity (34 mph), a sufficient speed for most medical applications. Note that formulas involving viscous forces may not apply to aerobotic airfoils with L <~ l_gas (Eqn. 9.23), indicating transitional or ballistic flow (Section 9.2.4).

The main implication of Eqn. 9.86 is that conventional aeronautics technologies such as rigid wings and jets are usually inappropriate for micron-size flyers. Instead, aerial nanorobots may more profitably employ natation mechanisms as described in Section 9.4.2.5, including surface deformations (e.g., flexible oars or wings, cilia, invaginating doughnuts, rotating spheroids, traveling waves), inclined planes (e.g., screw drives, corkscrews, flagella), volume displacement, and viscous anchoring. Specific aeromotive mechanisms are of great interest but will not be considered further in this Volume. In many cases, a viscous-regime nanorobot may exit the bodily fluids and enter the atmosphere,* or vice versa, using the same propulsive mechanisms, though operated at a modified speed or pitch angle.

* From Eqn. 9.14, the capillary force will restrain a 1-micron diameter nanorobot with a force of ~440 nN in pure water at the liquid-air interface due to surface tension, but this attraction may be reduced at least to ~4 nN by discharging small aliquots of the appropriate surfactants into the local aqueous environment; Section 9.2.3.

Last updated on 22 February 2003