Magnetocaloric Fluids - Magnetocaloric Energy Conversion: From Theory to Applications - page 166

(where the carrier uid,

so the effective viscosity in such cases may be used as an approximation to describe
the relation between the shear stress and the shear rate.
In Table 5.1 (which comprises Eqs. 5.4

fl uid is a Newtonian fl uid) is similar to that of a Newtonian fl

5.13 ) we show some of the expressions
that can be applied for the Newtonian suspension

-

' s viscosity as a function of the

solid ϕ V .

For higher solids concentrations, with the effect of the shape and particle size
accounted for, the suspensions usually show non-Newtonian behaviour. Figure 5.2
shows an example of the shear stress depending on the velocity gradient for dif-
ferent types of non-Newtonian

' s volume fraction

uids.
Different rheological models for laminar

fl

ow of homogeneous non-Newtonian
suspensions are shown in Table 5.2 (which comprises Eqs. 5.14

fl

5.18 ).
Since higher concentrations of a solid phase in a non-Newtonian suspension, as
well as the

-

uence the relation between the shear stress and the
shear rate, the relation in Eq. ( 5.1 ) is not linear any more. When the rheological
behaviour (or model) of a particular suspension is not known, it

fl ow rate, strongly in fl

is useful to

determine the apparent viscosity as

dv
dy ¼ g app c

s ¼ g app ð 5 : 19 Þ

where ʷ app is a function of the shear rate c . The relation in Eq. ( 5.19 ) can be simply

generalized for homogeneous non-Newtonian fluids. Figure 5.3 shows the apparent
viscosity of non-Newtonian suspensions depending on the shear rate. It is clear that
the apparent viscosity is approaching the viscosity of a carrier

uid at high shear
rates and low concentrations of solid particles. When the shear rate is lowered,
apparent viscosity rapidly increases. In order to determine the apparent viscosity,
experiments are required.
A method used to determine the shear rate from experimental data was given by
Mooney [ 3 ] and Rabinowitch [ 4 ]. Note that in this case the shear rate is de

fl

ned at
the wall of a pipe, since the experimental data gives values of the wall shear stress.
The apparent viscosity is, therefore, determined from data that show the depen-
dence of the wall shear stress on the wall shear rate. The so-called Mooney-
Rabinowitch equation can be derived from the following expression, where the plot
of the wall shear stress ˄ w versus the shear rate at the wall may help in the selection

of an appropriate rheological model:

2 3

w ¼

8v
d

dv
dy

8 v

d

3
4 þ

1
4

d ln

4 5

ð 5 : 20 Þ

ln D p d
4 L

d

If the suspension is assumed to be homogeneous, the rheological parameters are
considered constant all over the pipe cross-section. For the non-homogeneous
suspension, one has to consider the fl ow patterns [ 5 ]. However, also for such a fl ow

one may de ne the effective and apparent viscosity, as demonstrated by Kitanovski

and Poredo š [ 6 ].