Biomedical Engineering Reference
In-Depth Information
is simply unstable flow, characterized by vortex-shedding phenomena at frequencies
substantially higher than the heart rate (e.g., [20, 45, 46, 47]).
Setting aside the question of whether these pathological flow conditions are tur-
bulent, transitional or merely unstable, an important question arises regarding the
physical scale of the flow structures. Remember that the assumption that blood can
be modelled as homogeneous fluid rests on the assumption that the length scales of
the arteries (
∼
mm) are orders of magnitude larger than the length scales of the RBC
∼
μ
(
m). Now consider that both numerical and experimental studies of turbulent
blood flow observe, or at least infer, length scales of viscous eddies on the order of
tens of microns (e.g., [41, 48]). This raises the question of whether the homogeneity
assumption - upon which, ironically, these estimates of turbulent eddy length scales
are invariably based - is valid under pathological conditions of flowing
blood
.
The assumption that blood can be treated as a homogeneous fluid in the presence
of such small-scale flow features seems to have been only recently questioned by
Antiga, who in collaboration with the author described what might occur physically
if this assumption is indeed violated [49]. Specifically, we hypothesized that turbu-
lent kinetic energy must at some point be transferred to individual RBC, which,
be-
ing closely packed
, must dissipate their energy through laminar cell-cell interactions
mediated by the plasma. An order of magnitude analysis revealed that these laminar
shear stresses could approach the level of the Reynolds stresses.
15
This is a crucial
observation, for it potentially resolves a ongoing paradox in turbulent blood flow
research, namely, that “fictional” Reynolds stresses, which are not actual stresses on
the RBC, are excellent predictors of RBC damage (hemolysis), whereas the “actual”
viscous stresses experienced by RBC in turbulent flow are found to be one to two
orders of magnitude less (e.g., [50, 51]). Such findings, however, tend to be based
on CFD models of homogeneous fluids upon which an individual RBC is superim-
posed, which presumes that the RBC are far enough apart so as to not affect the flow
field or each other. In our thought experiment, on the other hand, the close packing
of RBC is a central and important feature.
That the homogeneity assumption for blood has not been questioned before in this
way (or at least not as overtly) might be attributed to it being all too easy to forget
that RBC are so closely packed together, even at a “half-full” concentration of 45 %.
Consider Fig. 1.7, which presents a common picture of close packing, namely the
∼
90 % optimal packing of circles in two dimensions. Next to this, 50 % packing,
representative of a normal blood hematocrit, looks positively sparse, with inter-cell
spacing on the order of a cell diameter. In three dimensions, where intuition tends to
fail us, optimal packing of ellipsoids, representing RBC under high shear conditions
preceding hemolysis, is on the order of 70 %. It can then be shown that the spacing
of ellipsoidal RBC at 50 % packing implies an inter-cell spacing on the order of one
micron [49], which is broadly consistent with the simulation of 50 % deformable
droplets shown in Fig. 1.7.
15
Reynolds stresses are terms that arise from time averaging of the Navier-Stokes equation after
decomposition of the velocity into mean and fluctuating components. They are not stresses in a
physical sense, but rather embody the effects of the fluctuating velocities on the mean flow.
