Blood Flow in Arteries and Veins - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 140

FIGURE 5.6 The variation
in blood viscosity as a function
of shear rate and blood hemato-
crit (Hct). This figure illustrates
that without the cellular com-
ponent the blood viscosity re-
mains constant at approximately
1.2 cP. When cells are present
in the blood, the viscosity is no
longer constant. This figure is
a summary of data collected by
Chien et al.

10000

1000

100

Hct = 90%

10

Hct = 40%

Hct = 0%

1

0.1
0.01 0.1 1 10 100

Shear rate (1/sec)

rheology of blood that have supported this cause for blood to behave as a non-Newtonian
fluid. Importantly, many groups have tried to determine the yield stress for blood (remem-
ber that blood is a Bingham Plastic fluid), but have seen that it is difficult to quantify
blood motion as the shear rate approaches zero. However, using a Casson model for blood
flow combined with measured values for shear stress at given shear rates, the yield stress
for blood can be extrapolated. The Casson model for blood states that

p

ηγ p 5

p

τ y

ð 5 : 1 Þ

1

as a relationship between shear stress and shear rate, where τ y is a constant yield stress

(which varies based on hematocrit) and is an experimentally fit constant, which approxi-

mates the fluid's viscosity. Under most physiological conditions, the yield stress of blood can
be approximated as 0.05 dyne/cm 2 . Therefore, for a shear stress less than 0.05 dyne/cm 2 ,
blood will not flow or will only flow as a rigid body.
We have already learned that under normal laminar, steady, fully developed flow con-
ditions in a cylindrical tube, the shear stress at the centerline is zero. Therefore, blood at
the centerline would need to move as a rigid body. Extending this thought further, the
shear stress experienced by blood will surpass 0.05 dyne/cm 2 at a finite distance from the
vessel centerline. For modeling purposes, we state that at a distance of r y , the shear stress
will exceed

η

τ y . Therefore, blood that is beyond this location (closer to the vessel walls),
will flow as “normal” viscous fluid, leading to a pseudo-parabolic velocity profile that has
a blunted centerline velocity, and fluid that is closer to the centerline will only flow as a
rigid body ( Figure 5.7 ). This is caused by the principle that the blood viscosity will be zero
(or blood can be modeled as an invisicid fluid) when the yield stress within the flow field
has not been surpassed.
Mathematically, this phenomenon can be represented using the Casson model. We
already know that

r
2

dp
dx

τ 52