Transcript Thixotropy is the property of some non-Newtonian pseudoplastic fluids to show a time-dependent change in viscosity; the longer the fluid undergoes shear.

Thixotropy is the property of some non-Newtonian
pseudoplastic fluids to show a time-dependent change in
viscosity; the longer the fluid undergoes shear stress, the
lower its viscosity. A thixotropic fluid is a fluid which takes a
finite time to attain equilibrium viscosity when introduced to
a step change in shear rate. However, this is not a
universal definition; the term is sometimes applied to
pseudoplastic fluids without a viscosity/time component.
Many gels and colloids are thixotropic materials, exhibiting
a stable form at rest but becoming fluid when agitated.
Example 4.2 Nutrient media is flowing at the rate of 1.5 liters min-1 in a tube that is 5 mm in diameter. The walls of the
tube are covered with antibody-producing cells and these cells are anchored to the tube wall by their interaction with a
special coating material that was applied to the surface of the tube. If one of these cells has a surface area of 500 microns2
what is the amount of the force that each cell must resist as a result of the flow of this fluid in the tube? Assume the
viscosity of the nutrient media is 1.2 cP = 0.0012 Pa sec.
SOLUTION We can use Equation 4.14 to find the shear stress acting on the cells that cover the surface of the tube.
3
1 min
 4  0.0012 Pa sec  1500 cm
w  

 2.44 Pa  2.44 N m 2

3

60 sec

 min 0.25 cm
Multiplying w by the surface area of a cell gives the force acting on that cell as a result of the fluid flow. Therefore:
10
 500 m icrons 
6
Fcell
2

2
m
N 1012 pN

2
.
44

 1220 pN
N
m icron2
m2
The shear stress - shear rate relationship for blood may be described by the following
empirical equation known as the Casson equation.
 1/ 2   y 1 2  s  1/ 2
(4.19)
In this equation, y is the yield stress and s is a constant, both of which must be determined from
viscometer data. The yield stress represents the fact that a minimum force must be applied to
stagnant blood before it will flow. This was illustrated earlier in Figure 4.3. The yield stress for
normal blood at 37 C is about 0.04 dynes/cm2. It is important to point out however, that the effect
of the yield stress on the flow of blood is small as the following example will show.

4Q
1 w
2

U


(

)

d rz
rz
rz
3
3 
D
2 w 0
(4.21)
This equation predicts that the reduced average velocity should only be a function of the wall shear
stress and this result is generally supported by the data presented
in Figure 4.8. This result also applies whether or not the fluid is Newtonian or non-Newtonian.
The shear rate or  ( rz ) may be written as follows for the Casson equation (4.17):
 ( rz ) 

12
rz
 y
s2
12

2
(4.22)
This equation may be substitued into Equation 4.19 and the resulting equation integrated to
obtain the following fundamental equation that describes tube flow of the Casson fluid. This
equation depends on only two parameters, y and s, that may be determined from experimental
data.
This equation depends on only two parameters, y and s, that may be determined from
experimental data.
4
y 
1 w 4
1 y
U 2

  
 
3
2s  4 7 y w 84  w 3
(4.23)
Merrill et al. (1965) provides the following values of these parameters at about 20 C:
y = 0.0289 dynes/cm2 and s = 0.229 (dynes-sec/cm2)1/2. It is left as an exercise to show that the
Casson equation with these values of the parameters provides an excellent fit to the data shown in
Figure 4.8.
Tube flow of blood at high shear rates ( > 100/sec ) shows two anomolous effects that involve the
tube diameter. These are the Fahraeus effect and the Fahraeus-Lindquist effect
Since the rheology of blood can also depend on the tube diameter, the simple relationship
(Equation 4.11) developed to describe tube flow of blood at high shear rates no longer applies.
However, since for tube flow of blood we can still measure both Q and P / L, we can use these
measurements to calculate for a given tube, the apparent viscosity of blood. This apparent viscosity
can be written
in terms of the observed values of Q and  P/L by rearranging Equation 4.11 (assuming the flow
of blood is laminar).
 apparent
R 4 P

8LQ
(4.27)
To explain the Fahraeus effect, it has been shown both by in vivo and in vitro experiments that
the cells do not distribute themselves evenly across the tube cross section. Instead, the RBCs tend
to accumulate along the tube axis forming, in a statistical sense, a thin cell-free layer along the
tube wall. This is illustrated in Figure 4.11. Recall from Equation 4.13 that the fluid velocity is
maximal along the tube axis. The axial accumulation of RBCs, in combination with the higher fluid
velocity, maintains the RBC balance (HF = HD) even though the hematocrit in the tube is reduced.
The thin cell-free layer along the tube wall is called the plasma layer. The thickness of the plasma
layer depends on the tube diameter and the hematocrit and is typically on the order of several
microns.
2
2
H T  R L  H C R    L 
 
H T  1   H C
 R
2
Because of the Fahraeus effect, it is found that as the tube diameter decreases below about
500 microns, the viscosity of blood also decreases. The decreased viscosity is a direct result of the
decrease in the tube hematocrit. The reduction in the viscosity of the blood is known as the
Fahraeus-Lindquist effect.
As shown in Figure 4.11, the blood flow within a tube or vessel is divided into two regions;
a central core that contains the cells with a viscosity c, and the cell-free marginal or plasma layer
that consists only of plasma with a thickness of  and a viscosity equal to that of the plasma given
by the symbol p. In each region the flow is considered to be Newtonian and Equation 4.12
applies to each. For the core region, we may then write:
 rz 
P
0
 PL  r
2L
dv cz
  c
dr
dv cz
BC1: r  0,
0
dr
BC2: r  R  ,  rz
(4.28)
c
  rz
p
For the plasma layer the following equations apply.
 rz 
P
0
 PL  r
2L
dv pz
  p
dr
BC3: r  R  , v cz  v pz
(4.29)
BC4: r  R, v pz  0
Equations 4.25 and 4.26 may be readily integrated to give the following expressions for the
axial velocity profiles in the core and plasma regions.
v z (r) 
p
P
0
 PL  R 2   r  2 
1     for R    r  R
4 p L   R  
(4.30)
2
2
2
2 



(
P

P
)
R
R


r
R








p
p
0
L
c
     
  for 0  r  R  
1  
v z (r) 
4 c L   R  c  R  c  R  
(4.31)
The core and plasma volumetric flowrates are given by the following equations.
R
Q p  2  v pz ( r ) r dr
R 
R 
Q c  2
v
c
z
( r ) r dr
0
(4.32)
Integration of the above equations with the values of vzp and vzc from Equations 4.27 and
4.28 provides the following result for the volumetric flowrates of the plasma layer and the core:
Qp 
 ( P0  P L ) 2
2 2
R   R   

8p L
(4.33)
  p  ( R  ) 4  p ( R  ) 4 
 ( P0  PL ) R 2 
2
Qc 


( R  )  1 

2
4 p L
 c 2R 2 
 c  R

(4.34)
The total flowrate of blood within the tube would equal the sum of the flowrates in the core and
plasma regions. After adding the above two equations, we obtain the following expression for the
total flowrate:
4
 R 4 ( P0  PL )       p 
Q
1  1   1  
8p L
  R    c 
Solve for apparent viscosity
(4.35)
Notice how this compares with the Hagan- Poiseuille equation given by 4.11:
R 4 ( P0  PL )
Q
8L
 apparent
R 4 P

8LQ
Comparison of this equation with Equation 4.27 allows one to develop a relationship for the
apparent viscosity based on the marginal zone theory. We then arrive at the following expression
for the apparent viscosity in terms of  , R, c, and p.
 apparent 
p
4
   p 

1  1   1  
 R   c 
(4.36)
As /R  0, then apparent  c   which is the bulk viscosity of blood in a large tube at
high shear rates, i.e. a Newtonian fluid, as one would expect.
Example 4.4