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

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

9.4.2.3 Disturbed Flows, Hydrodynamic Interactions, and Entropic Packing

Two kinds of fluid flow have already been described -- laminar or Poiseuille streamline flow, and turbulent or random flow. However, in branching or nonuniform-diameter blood vessels, nanorobots may also be required to negotiate an additional flow regime that is not observed in straight tubes of uniform diameter. This third regime, called "disturbed flow," involves secondary fluid motions in directions away from that of the primary flow, often with separation of the streamlines from the vessel walls to form a vortex or a recirculation zone between the forward flowing mainstream and the wall.¹³⁵⁸ Disturbed flow is most common in the larger blood vessels.

Studies have been conducted to observe the flow patterns in various vessels of simple and complex geometries, using flow visualization and cinemicrographic techniques.^1358-1366 In one extensive series of experiments,¹³⁵⁹ tracer polystyrene microspheres were photographed at various flow rates as they traveled through glass tube models or through chemically-transparentized natural blood vessels. The developed movie films were then projected on a drafting table and analyzed frame by frame. Figure 9.18 illustrates some results for progressively higher angle blood vessel bifurcations. The cross-streamline and vortex patterns are quasi-stable; in pulsatile flow, vortices vary periodically in size and intensity, with the axial location of the vortex center and reattachment point oscillating in phase with the upstream fluid velocity between maximum and minimum positions about a mean.¹³⁶¹ A model study using red cells¹³⁶¹ found that over time periods long in comparison with the orbital period, single cells and small aggregates <20 microns in diameter migrate outward across the closed streamlines and exit the vortex after describing a series of spiral orbits of continually increasing diameter until rejoining the mainstream. Erythrocyte rouleaux >30 microns in size remain trapped within the vortex, either assuming equilibrium orbits or staying at the center. The vortex also provides favorable conditions for the spontaneous aggregation of normal human platelets through shear-induced collisions of particles circulating in their orbits.¹³⁶² An intimate knowledge of disturbed flow patterns in all human blood vessels will be essential in any nanomedical mission design requiring sanguinatating nanorobots, but further detailed discussion is beyond the scope of this book.

A variety of early studies attempted to model the hydrodynamics of micron-scale swimmers, including descriptions of the hydrodynamics of an individual swimming cell,^{1367,1368,3583} the hydrodynamic interaction of two magnetotactic bacteria,¹³⁶⁹ and the interaction of two parallel swimming cells.¹³⁷⁰ Wall effects are well-known.^{1393,3584,3585} For instance, the velocity of a flagellar swimmer decreases as it moves closer to a solid boundary; the boundary effect is strong at low N_R, with speed reduced 5% even at 10 object radii from the wall.¹³⁹¹ Flagellates swim in curved paths near a solid boundary.³³⁸ Another effort to model the hydrodynamic interaction of pairs of flagellar-driven 1-micron diameter bacterial swimmers found that cells are attracted toward each other when they are swimming side by side and are repelled when swimming one behind the other.³³⁶ The researchers also modeled two microswimmers approaching each other from opposite directions along parallel but noncoaxial paths.³³⁶ As the two cells pass, they move closer together, causing mutual cell rotations which lead to a significant change in swimming orientation. Eventually the cells move away from each other along a straight, now-coaxial path whose deflection angle to the original path is determined by the initial off-axis separation of the two cells (Fig. 9.19) -- a motion vaguely reminiscent of the "planetary slingshot" maneuver often performed by NASA spacecraft to change direction. This unusual cell-cell hydrodynamic interaction has now been observed experimentally in E. coli traversing fluid in a glass microcapillary.¹³⁷¹ Hydrodynamic interactions among bacterial flagellar swimmers are greatest when cell separation distances are less than the overall length of the cells including the length of the flagella, or <~10 microns.³³⁶

In 1998, relatively little was known about the likely hydrodynamic behavior of nanorobots in the close company of large numbers of other nanorobots and natural blood cells. Differing sizes, shapes, and surface characteristics may lead to subtle and unexpected hydrodynamic interactions. For example,^846,3579 leukocyte axial velocity in capillaries is slightly lower than that of erythrocytes because the white cell surface undergoes a smaller deformation during flow due to the higher stiffness and larger volume of a leukocyte -- e.g., the relative velocity ratio is 0.88 in a 6.8-micron diameter capillary, is somewhat closer but still under 1.00 in larger vessels, and differs among leukocyte types.¹³⁵³ This small velocity differential causes an axial redistribution of a train of erythrocytes in the presence of a white cell. Upstream from the leukocyte, red cells bunch up, forming a region of elevated hematocrit; downstream, a plasma gap develops, increasing linearly with time and with the velocity differential between the white cell and the next red cell down the line.^1354,1356 Upon entering a postcapillary venule, the configuration breaks up as the erythrocyte immediately upstream of the leukocyte passes the white cell and pushes it laterally against the vessel wall, often causing the white cell to attach to the endothelial surface of the venule. (Without red cells, leukocytes make no regular attachment to the venous endothelium^1356,1357). The remaining red cells then close up the gap in the flow.¹³⁵⁶ Medical nanorobots may exhibit equally unusual flow behaviors.

Self-organizing spatial patterns may also arise in mixtures of nanorobot-size particles of different shapes and sizes, a process which minimizes entropy.²¹⁶⁸ Entropic forces become significant at scales of a few tens of nanometers to several microns.²¹⁶⁸ For example, micron-diameter spheres mixed with micron-long, 10-nm thick rods in water solidify into two arrangements as water is removed -- one a "cake" with layers of vertical rods alternating with a thin frosting of balls (a stacked, lamellar arrangement akin to cell plasma membranes) and the other a lattice of vertical columns of clustered spheres embedded in a horizontal sea of parallel rods (a columnar arrangement often found in glues).²¹⁶⁹ Other less-stable patterns such as ropes with lamellar order and chains of rod packets interspersed with spheres are also observed. In another series of experiments, small spheres <~100 nm in diameter were mixed with large spheres ~500 nm in diameter in a ~1000:1 ratio, and the small spheres pushed the larger ones against the hard, flat container walls. Large balls were also forced against the most curved sections of the inner walls of pear-shaped rigid vessels.²¹⁷⁰ These entropic forces are attractive at low concentrations of small spheres, but at higher concentrations the force alternates between repulsion and attraction.²¹⁷⁰

Last updated on 21 February 2003