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.

23

3

44.244.2sec60

min1

25.0min

1500sec0012.04

mNPacm

cmPaw

Multiplying w by the surface area of a cell gives the force acting on that cell as a result of the fluid flow. Therefore:

pNN

pN

m

N

micron

mmicronsFcell 1220

1044.2

10500

12

22

262

The shear stress - shear rate relationship for blood may be described by the following

empirical equation known as the Casson equation.

1 2 1 2 1 2/ / y s (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.

w

0

rz2

rzrz3w

3d)(

2

1

D

4Q U (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):

( )

srz

rz y

2

21 2 1 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.

U

1

2s2

w

y w

y

w

y

4

4

7

1

84 3

4

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 P

LQ

4

8

(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.

CT

CT

HR

H

LRHLRH2

22

1

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

L

czcP P r

L

dv

dr

0

2

BC rdv

drzc

1 0 0: , (4.28)

BC r R rz c rz p

2: ,

For the plasma layer the following equations apply.

rz

L

pzpP P r

L

dv

dr

0

2

BC r R v vzc

zp3: , (4.29)

BC r R v zp4 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 r

P P R

L

r

Rfor R r Rz

p L

p

( )

0

2 2

41

(4.30)

v rP P R

L

R

R

r

R

R

Rfor r Rz

c L

c

p

c

p

c

( )( )

02 2 2 2

41 0

(4.31)

The core and plasma volumetric flowrates are given by the following equations.

Q v r r drp zp

R

R

2

( )

Q v r r drc z

cR

20

( ) (4.32)

Integration of the above equations with the values of vzp and vz

c from Equations 4.27 and

4.28 provides the following result for the volumetric flowrates of the plasma layer and the core:

QP P

LR Rp

L

p

( )0 2 2 2

8 (4.33)

QP P R

LR

R

R

R

RcL

p

p

c

p

c

( )

( )( ) ( )0

22

4

2

4

241

2 (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:

QR P P

L RL

p

p

c

40

4

81 1 1

( )

(4.35)

Notice how this compares with the Hagan- Poiseuille equation given by 4.11:

QR P P

LL

40

8

( )

apparent

R P

LQ

4

8

Solve for apparent viscosity

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

p

cR

1 1 1

4

(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