Magnetocaloric Fluids - Magnetocaloric Energy Conversion: From Theory to Applications - page 172

the applied eld, the magnetic moments of the particles will align together with the

eld direction. If the eld direction and the vorticity of the fl ow are collinear, the

in fl uence on the motion of the particle and therefore on the fl ow of the fl uid does

not appear [ 18 ].
However, if the vorticity and eld direction are perpendicular, there will be a

magnetic torque that will act in opposition to the torque produced by the viscous
forces. The free rotation of the particle in the fl ow will be damped and this will

increase the ow resistance and will result in increased viscosity. This effect is

associated with the so-called rotational viscosity, which is related to highly diluted
suspensions with very small particle interaction.
According to Shliomis [ 19 ], this can be de

fl uid fl

ned as follows:

3
2 / VFF g L

a tanh a
a þ tanh a

2 b

g rot ¼ sin ð 5 : 23 Þ

where ʱ is the ratio between the magnetic and thermal energy of the particles

l 0 mH
kT

a ¼ ð 5 : 24 Þ

where ʼ 0 is the vacuum permeability, m is the magnetic moment of a particle,

k represents Boltzmann ' s constant and T is the temperature. The coef cient ʲ

represents the angle between the vorticity and the eld direction, and its term in

Eq. ( 5.23 ) is shown as the spatial average [ 7 ].
More information can be obtained in Rosensweig [ 7 ], Shliomis [ 19 ], McTague
[ 20 ], Rosensweig et al. [ 21 ], Ambacher et al. [ 22 ], and Patel et al. [ 23 ].
Note again that rotational viscosity is a property of highly diluted suspensions.
There the particle

particle interactions are not so intensive (changes of viscosity
induced by the magnetic

-

eld are in the order of a few percent as a maximum [ 20 ]).
However, in concentrated ferro uids with higher volume fractions of solids (e.g. a

volume fraction of 10 %), the interactions between the particles have an important
role and cause the so-called magneto-viscous effect.
Note that the model of Shliomis [ 19 ] does not take into account the interaction
between the particles; therefore, it can be used only for calculation of the rotational
viscosity with the absence of the magneto-viscous effect.
An example of models that take into account the magneto-viscous effects of
magnetic

fl

uids can be found in Zubarev et al. [ 24 , 25 ], who assumed that the
formation of the chains of magnetic particles dominate in the magneto-viscous
effects in magnetic

fl

uids. The results of their study also show good agreement with
the experimental data from Odenbach [ 18 ]. Zubarev and Iskakova [ 26 ] also
developed a theoretical model for prediction of the magneto-viscous effect for the
drop-like aggregates in ferro

fl

fl uids.