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For a power-law fluid:

(3)

where Vz is the velocity in the axial direction at

radius r.

Combining equations (2) and (3) and the separation

of variables leads to:

(4)

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Integration with respect to r gives the expression for

the velocity distribution:
(5)

At r =R: Vz =0

(6)

The velocity distribution in the pipe

(7)

The velocity profile may be expressed in terms of the

average velocity, V, which is given by:
(8)

where
h Q iis theh volumetric
l i flow
fl rate off the
h liquid.
li id
On substitution for Vz from equation (7), and
integration yields,

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Therefore:

(9)

Now the velocity distribution in the pipe may be

rewritten in terms of the average velocity as:

(10)

The profile is flatter for a shear-thinning fluid (n <1)

and sharper for a shear-thickening (n>1) fluid.

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The velocity is seen to be a maximum when r = 0, i.e.

at the pipe axis. Thus:

(11)

The value of the centre-line

centre line velocity drops from 2.33V
2 33V
to 1.18V as the value of the power-law index n
decreases from 2 to 0.1.
This ratio is 2 for a Newtonian fluid.

Rewriting eq. (9) in terms of the volumetric flow rate

and the pressure gradient:

(12)

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For a given power-law fluid and fixed pipe radius,

ΔpQn , i.e. for a shear-thinning fluid (n<1), the
pressure gradient is less sensitive than for a Newtonian
fluid
u d to changes
c a ges in flow
ow rate.
ate.

The flow rate, on the other hand, shows a rather

stronger dependence on the radius of pipe. For
instance, for n=1, QR4 whereas for n= 0.5, QR5.

It is useful to rewrite equation (12) in dimensionless

form by introducing a friction factor defined as:
 w 
f  2 
 1/ 2V 
 p   D 
where w    
 L  4 
and defining a suitable Reynolds number which will
yield the same relationship as that for Newtonian
fluids, that is
16
f
Re

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The wall shear stress in terms of the friction factor

1
 w  V 2 f (13)
2

The wall shear stress in terms of the pressure gradient

p 4fV 2
 (14) Eq. 9
L D

 n   D 2fV   D 
2
V    (15)
 3n  1   4m D   2 

Which can be rearranged to give:

16
f (16)
Re PL

where the new Reynolds number RePL is defined by:

V 2 n D n
Re  n
(17)
 3n  1 
8n 1 m  
 4n 

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Example 3.1
A polymer solution (density =1075 kg/m3) is being
pumped at a rate of 2500 kg/h through a 25 mm inside
diameter pipe. The flow is known to be laminar and the
power-law
power law constants for the solution are mm=33 Pa sn and
n=0.5.
Estimate the pressure drop over a 10 m length of
straight pipe and the centre-line velocity for these
conditions.
How does the value of pressure drop change if a pipe
of 37 mm diameter is used?

SOLUTION
Volumetric flow rate

Pipe radius

Substituting in eq. (12) and solving for pressure drop

gives:

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SOLUTION
Average velocity in pipe

The centre-line velocity is obtained by using eq. (11)

For the same value of the flow rate, eq. (12) suggests
that

SOLUTION
Hence, the pressure drop for the new pipe diameter
(37mm) can be estimated:

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Bingham plastic and

yield-pseudoplastic
fluids
A fluid with a yield stress will flow only if the applied
stress (proportional to pressure gradient) exceeds the
yield stress.

There will be a solid plug-like core flowing in the

middle of the pipe where |τrz| is less than the yield
stress, as shown in Figure below:

Schematic velocity distribution for laminar flow of a Bingham plastic fluid in a pipe

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The radius of the plug, Rp, will depend upon the

1. magnitude of the yield stress
2. the wall shear stress (i.e., the value of Δp/L)
From equation (2)

0B R p
 (18)
w R

where τw is the shear stress at the wall of the pipe.

In the annular area Rp< r < R, the velocity will

gradually decrease from the constant plug velocity to
zero at the pipe wall.
The expression for this velocity distribution will now
be derived.
For the region Rp< r < R, the value of shear stress will
be greater than the yield stress of the fluid.
The Bingham fluid model for pipe flow is given by:

 dV 
rz  0B   B   z  (19)
 dr 

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 dV 
rz  0B   B   z 
 dr 
 p  r
Eq. (3): rz   
 L 2

1  p  r B
dVz       0  dr
Integration with respect to r
B  L  2 

1  p  r 2 B
Vz      r0   C (20)
B  L  4 

1  p  r 2 B
Vz      r 0 C
B  L  4 

At r  R : Vz  0

1  p  R 2 B
C    R  0  (21)
B  L  4 

 p  R  r 2  0B 
2
r
Vz    1  2   R 1   (22)
 L  4 B  R   B  R 

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C learly, equation (22) is applicable only when

|τrz|>|τ0B| and r  Rp.
The corresponding velocity, Vzp , in the plug region
(0r  Rp) is obtained by substituting r = Rp in eq. (22)
to give:
2
 p  R  R p 
2
Vzp    1  2  (0  r  R p ) (23)
 L  4 B  R 

The volumetric flow rate:

(24)

Substituting from equations (22) and (23):

( )
(25)

(26)

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0B R p
  Eq. 26
w R

Further
(27)
simplification

(28)

It is useful to rewrite equation (28) in a dimensionless

form by introducing the following definitions:
VD
Bingham Reynolds number Re B  (29)
B

w D  p L 
Friction factor f  (30)
1 2  V 2 2V 2

D 2 0B
H d
Hedstrom number
b H 
He (31)
 2B

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Q
V
R 2
Equation 28
0B 0B
 
 w 1 2  f V 2

(32)

Di idi bboth
Dividing th sides
id by V2 andd slight
b ρV li ht rearrangementt

(33)

(34)

(35)

This can now be rearranged to obtain the final form as:

(36)

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Friction factor as a function of the Reynolds number and Hedström

numbers, as predicted by equation (36)

Many authors prefer to use another dimensionless

parameter, namely, Bingham number, Bi, which is
defined as:
0B D
Bi  (37)
BV
It is easy to show that:
He  ReB Bi
Equation (36)

(38)

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Effect of Bingham number on the value of (f·ReB) as

predicted by equation (38)

The Herschel-Bulkley fluid model

(39)

(40)

(41)
0H
where 
w

22