© 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

{Eqn. 9.62}

where r is radial distance from the tube axis and k_p = Dp / 4 h l_tube = 2 v_flow / r_tube². 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