Magnetocaloric Fluids - Magnetocaloric Energy Conversion: From Theory to Applications - page 178

moments and strong magnetic interaction between the particles, which will tend to
create chain-like structures. In magnetorheologic uids the so-called coupling

constant represents the characteristics upon which the particles tend to form chains
or agglomerates under the applied magnetic

fl

eld. In the absence of the magnetic

eld, the coupling constant will be very small. This is especially so because the
particles have no permanent magnetic moment without an applied magnetic eld

[ 18 ]. Therefore, only the remanence will keep the magnetic moment; however, in
most applications this will be very small. As a result, in the absence of a magnetic

uid will behave as an ordinary Newtonian or non-
Newtonian, solid-liquid suspension or slurry. The latter is a strong function of the
particles

eld, an magnetorheological fl

uid. The viscosity
can be determined by viscosity of the carrier liquid and the volume fraction of the
suspended material, i.e. the effective viscosity [ 18 ]. However, the magnetic

' volume fraction as well as the properties of the carrier fl

eld will
induce an interparticle interaction and will strongly affect the rheological behaviour.

5.2.2.1 Rheologic Models Applied for Magnetorheological Fluids

The eld,

will lead to non-Newtonian behaviour. This is particularly the case when the for-
mation of particle aggregates occurs. In most cases of magnetorheological fluids, the
Bingham constitutive model was applied (see, e.g. Bica et al. [ 46 ], Carlson and Jolly
[ 14 ], Odenbach [ 18 ], Olabi and Grunwald [ 40 ], Park et al. [ 47 ], Engin et al. [ 48 ],
Jiang et al. [ 49 ], Iglesias [ 50 ], Omidbeygi and Hashemabadi [ 51 ], Serano et al. [ 52 ]).
However, in certain cases also the Herschel - Bulkey model was applied to

fl ow of a magnetorheologic fl uid, especially in the presence of a magnetic

t post-
yield shear thinning and shear thickening (see, e.g. Bica et al. [ 46 ], Burguera [ 53 ],
Wang and Gordaninejad [ 54 ], Resiga et al. [ 55 ], Yamanaka et al. [ 56 ], Mrlik et al.
[ 57 ]). Also, the Casson model was applied for magnetorheological

uids (Bica et al.
[ 46 ], Sidpara et al. [ 58 ], Gabriel and Laun [ 59 ], Kim et al. [ 60 ]). An interesting
rheological model is that given by Papanastasiou (Eq. 5.18). This model was also
applied for MR

fl

uids (see, for e.g. Farjoud et al. [ 61 ], Resiga et al. [ 62 ]). An
advantage of using such a model could be the fact that it can cover both domains;
those with a yield stress and those without (i.e. a constitutive equation for a mag-
netic-

fl

eld- and particle-size-dependent model). This also provides the possibility to

de ne the rheological transition from ferro fl uids to magnetorheologic fl uids.

1996 Ginder et al. [ 63 , 64 ] proposed a relation for the yield stress in a
magnetorheologic

In 1995

-

fl uid. For very low applied magnetic elds, where the relation-

ship between the magnetization M and the magnetic eld intensity H is considered

to be linear, the relationship between the yield stress and applied eld was given by

the following proportion:

s 0 / / V l 0 H 2 ð 5 : 26 Þ

where ϕ V is the particle volume fraction, H is the applied eld and ʼ 0 is the

permeability of free space (vacuum). For magnetic fl ux densities that are above