Page 259 - Bird R.B. Transport phenomena

§8.3 Non-Newtonian Viscosity and the Generalized Newtonian Models 243

Combining Eq. 8.3-6 and 2.3-13 then gives the following differential equation for the velocity:

( 8 3 7 )
-
After taking the nth root the equation may be integrated, and when the no-slip boundary con-
dition at r = R is used, we get

Vt ( }
~\ 2mL ) (l/я) +1 L W J
for the velocity distribution (see Eq. 8.1-1). When this is integrated over the cross section of
the circular tube we get

T T R 3 P ( ( O L ) Y
--
{ ) (8 3 9)
which simplifies to the Hagen-Poiseuille law for Newtonian fluids (Eq. 2.3-21) when n = 1
and m = fji. Equation 8.3-9 can be used along with data on pressure drop versus flow rate to
determine the power law parameters m and n.

EXAMPLE 8.3-2 The flow of a Newtonian fluid in a narrow slit is solved in Problem 2B.3. Find the velocity dis-
tribution and the mass flow rate for a power law fluid flowing in the slit.
Flow of a Power Law
Fluid in a Narrow Slit 4 SOLUTION
The expression for the shear stress T as a function of position x in Eq. 2B.3-1 can be taken over
XZ
here, since it does not depend on the type of fluid. The power law formula for T from Eq. 8.3-3 is
XZ
r xz = ml - - p ) for 0 < x < В (8.3-10)

T 2 = = m
* ~ \ 7 / for-B<x<0 (8.3-11)
To get the velocity distribution for 0 < x < B, we substitute T from Eq. 8.3-10 into Eq. 2B.3-1
XZ
to get:
/ dv \ n (0>o-^ )x
m \lx) z I L 0<*<B (8.3-12)

Integrating and using the no-slip boundary condition at x = В gives

Since we expect the velocity profile to be symmetric about the midplane x = 0, we can get the
mass rate of flow as follows:
rw гв rw гв
\ I pv dx dy = 2 \ I pv dx dy
w =
Jo J-в z Jo Jo z

2WB p ((&o - &j)B\v n
2
= (8 3 14)
п7п^-2[-^Г-) - "
When n = 1 and m = /л, the Newtonian result in Problem 2B.3 is recovered. Experimental
data on pressure drop and mass flow rate through a narrow slit can be used with Eq. 8.3-14 to
determine the power law parameters.