Magnetocaloric Fluids - Magnetocaloric Energy Conversion: From Theory to Applications - page 183

5.3 Ferrohydrodynamics and Heat Transfer
in Magnetic Fluids

Most of the work that concerns the motion of a magnetic fl uid under a static,

“ ”

rotating or varying magnetic of

ferrohydrodynamics. Today, a number of studies have been performed and we
address some of them in this subsection. We start this subsection by providing the
basic governing equations for the homogeneous

eld was performed by Rosensweig [ 7 ], the father

fl ow of Newtonian or non-New-

tonian magnetic uids. These are based on the available literature (Bird et al. [ 91 ],

Neuringer and Rosensweig [ 92 ], Rosensweig [ 7 , 93

fl

95 ]). Furthermore, since we
deal in such equations with the homogeneous distribution of solid particles in the
base (nano

-

fl uids) or the carrier fl uid (particle suspensions), we also present a few

models that relate to the de nition of the effective thermal conductivity.

The equation of continuity can be written as

o q
o

t þrq~

ð v Þ ¼ 0 ð 5 : 33 Þ

where v represents the velocity of the magnetic fl uid and ˁ represents the density of

the magnetic uid, which may be calculated with help of the solids volume fraction

ϕ V or the solids mass fraction

fl

ϕ m as

1

q ¼ / V q solid þ ð 1 / V Þ q liquid ¼ ð 5 : 34 Þ

/ m ð 1 / m Þ

q solid þ

q liquid

The equation of motion for a (non)Newtonian magnetic fluid can be written
as [ 91 ]

h i

D v

Dt ¼ r

~

þr ¼ q p þ q~ g ð 5 : 35 Þ

where ¼ represents the total stress tensor and may be divided into the viscous part
and the magnetic part:

¼ ¼ ¼ viscous þ ¼ magnetic

ð 5 : 36 Þ

In order to include the non-Newtonian constitutive model, the viscous stress
tensor may be applied for an incompressible fl uid with the generalized Newtonian

model [ 91 ]:

¼ ¼ viscous ¼ g r~ v T

v þr~ ¼ g ð 5 : 37 Þ

o x i v j and

v T where r~ v is the velocity gradient tensor with components

o

r~ is the

o
o x j v i , and where the