Environmental Engineering Reference
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¼ represents the rate of strain tensor (rate of deformation tensor). Therefore,
in the generalized Newtonian model one can simply replace the constant viscosity
by the non-Newtonian viscosity as the function of shear rate, which can be written
as the function of the magnitude of the rate of strain tensor [ 91 ]:
term
r
1
2
¼
¼
c ¼
:
ð 5 : 38 Þ
uids, both the viscosity and the potential yield stress may
depend not only on the conventional viscous forces, but also on the magnetic
Note that in magnetic
fl
eld
intensity. The magnetic component of the stress can be expressed according to
Rosensweig [ 7 ]as
¼ magnetic ¼ l 0 H 2
¼
þl 0 H H
2
ð 5 : 39 Þ
where ¼ represents the unit tensor and H represents the vector of the magnetic
eld.
Based on Eq. ( 5.39 ), Rosensweig [ 7 ] expressed the magnetic force density on the
body as
r ¼ magnetic ¼ l 0 M
r H
ð 5 : 40 Þ
For which in the case that the magnetization M is collinear with H, the force
density is de
ned by magnitudes of M and H as [ 7 , 92 ]
r ¼ magnetic ¼ l 0 M
r
H
ð 5 : 41 Þ
Based on the equations above, we now write the equation of motion for an
incompressible magnetic
fl
uid in the following form:
h
i
o
o
qðÞ ¼ r q vv r p þr ¼ viscous
þ l 0 M r H þ q g
ð 5 : 42 Þ
t
The energy equation in the form of the equation of change of internal energy is
expressed according to Bird et al. [ 91 ] for an incompressible magnetic
fl
uid as
h
i
Du
Dt
total rate of increase of
internal energy per unit
volume
rate of magnetic work
þ ¼ viscous : r v
¼ r q
rate of internal
energy addition by
heat conduction per
unit volume
þ
irreversible rate
of internal energy
increase per unit
of volume by viscous
dissipation
ð 5 : 43 Þ
According to the work of Neuringer and Rosensweig [ 92 ], the rate of magnetic
work done on the system can be expressed by the substantial derivative as
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