Biomedical Engineering Reference
In-Depth Information
flow conditions, shear rates are expected to be on order of 100 s
−
1
, and for this rea-
son the Newtonian approximation is widely justified. In regions of slow, recircu-
lating flow, however, shear rates can fall below 10 s
−
1
, raising questions about the
Newtonian approximation for such complex flow conditions.
An early attempt to make sense of the effects of non-Newtonian rheology under
physiological flow conditions was that of Ballyk et al. [31]. Employing the same
CFD techniques and 2D models used in author's investigations of wall distensibil-
ity effects, we showed that the Newtonian approximation was reasonable, insofar
as non-Newtonian effects on the wall shear stress patterns were minor, and indeed
comparable to the effects of wall distensibility. Perhaps more importantly, our study
explained why earlier investigations of steady flows were likely to have overesti-
mated non-Newtonian effects: under pulsatile flow conditions, inertial (Womersley)
flow effects dominate throughout much of the velocity profile, with viscous effects
confined to the thin viscous (Stokes) layer near the wall, where shear rates tend to
be above 100 s
−
1
anyway. In fact, we argued that the importance of non-Newtonian
effects could be anticipated by consideration of the Stokes layer thickness relative
to the lumen radius (i.e., the Womersley number) rather than the lumen radius itself,
ideas that were later demonstrated for more realistic, 3D coronary artery flows by
Johnston et al. [32, 33].
Another important and related insight was provided by Gijsen et al. [34], who
demonstrated that the effects of non-Newtonian rheology could be approximated by
considering a Newtonian viscosity based not on its value at the high shear limit but
rather its value at the characteristic or effective average shear rate for that partic-
ular flow. This recognizes that, and as Fig. 1.5 shows, even around 100 s
−
1
small
differences in shear rate can produce potentially non-negligible changes in viscosity
(i.e., effective Reynolds number). In other words, care should be taken to choose an
appropriate reference against which non-Newtonian effects are compared.
This issue of how to contextualize the effects of non-Newtonian vs. Newtonian
models is at the heart of any such attempts to evaluate the impact of relaxing assump-
tions in hemodynamic modelling. For example, there have been many studies on the
effects of shear-thinning in large artery flows, yet they have often drawn opposite
conclusions despite observing similar levels of effect. One reason for this may be
that, without some objective reference as to what constitutes a significant or impor-
tant effect, authors can only draw subjective conclusions.
With this in mind, in 2007 we published a study that attempted to contextual-
ize the impact of non-Newtonian effects on carotid bifurcation hemodynamics [30].
As Fig. 1.6 summarizes, we found that use of a shear-thinning model has only a
minor effect on the computed wall shear stress patterns, especially when compared
to the effects of image-based CFD (i.e., geometric) uncertainty that we had previ-
ously demonstrated on the same models [15]. Moreover, we showed that the already
modest non-Newtonian effects could be captured by using a characteristic rather
than shear-limit viscosity, as Gijsen et al. had suggested. Finally, by varying the
Reynolds number in a way that roughly mimicked the effect of RBC concentration
(i.e., hematocrit) we demonstrated that the choice of hematocrit could have an ef-
fect comparable to shear-thinning. In other words, efforts to improve the “accuracy”
