ECH 141
Problem Set #5
Due Wednesday 2/21/18
1.
The generalized Newtonian fluid (GNF) model provides a simple way to describe non-Newtonian
fluids in steady unidirectional flow, and the power-law model is the simplest example of a GNF.
Here
you will derive a generalized form of the Hagen-Poiseuille equation for fully developed flow of a
power-law fluid through a long, cylindrical tube.
As the flow is steady and fully developed, you may
assume that
v
z
=
v
z
r
( )
only, v
r
=
v
θ
=
0
and consequently that
η
=
m
dv
z
dr
n
−
1
and
τ
rz
=
m
dv
z
dr
n
−
1
dv
z
dr
a)
If the dynamic pressure at z=0 is P
0
and at z=L is P
L
, derive an expression for the velocity profile
v
Z
(r) of the fluid in the tube.
b)
Use your result from part (a) to derive an expression for the flow rate Q.
Show that your result
simplifies to the Newtonian, Hagen-Poiseuille equation in the limit n=1 and m=
μ
.
R

2.
A 13.5% by weight solution of polyisoprene in isopentane has the following power-law parameters
at 323 K: