Yamaguchi H. Engineering Fluid Mechanics (Fluid Mechanics and Its Applications)

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496

Magnetic fluid, or alternatively called ferrofluid, is a colloidal suspension

system of nano-sized ferro- or ferri-magnetic particles stably dispersed in a

carrier (or base) liquid. Magnetic fluid is an artificial material rather than

formed naturally. It has prominent character of the fluidity and also pos-

sesses magnetic properties. Magnetic fluid can be manipulated to position

or forced to flow by means of a magnetic force.

Magnetic fluid and its concept of usage appeared in a 1965 research

product for space technology in NASA. Since the time of the first appear-

ance of magnetic fluid, much progress has been made in producing various

types of high quality magnetic fluids associated with applications in many

fields of technology, medicine and science. Many efforts have been also

made to establish thermo-mechanical equations of magnetic fluids in order

to deal with colloidal magnetic particles in a carrier liquid in an applied

magnetic field, where the local moment of momentum exchange between

particles and a fluid must be taken into account. The intrinsic angular mo-

mentum of particles under the influence of magnetic field introduces a

volumetric force that couples into the governing equations of fluid motion.

A quasi-stationary approximation is, however, possible in many practical

many processes, so that magnetic fluids can be only assumed as single-

phase, homogeneous isotropic continuous media with magnetic forces,

which is in many respects similar to the effects of the gravitational body

force.

There are many striking phenomena in the physical behavior of mag-

netic fluids that are activated by imposing magnetic fields. Those re-

sponses include the normal field instability represented by the appearance

of spikes on a fluid surface; the labyrinthine instability formed in a thin

layer; the self-levitation of an immersed magnet; magnetocaloric effects,

and so on. These phenomena are now well understood by the fluid dynam-

ics and the thermodynamics of magnetic fluids based on the continuum

mechanical approach via microscopic description. The complexity of the

system with its chemical composition requires distinct knowledge in a

physico-chemistry in order to synthesize the fluids. In addition, the utiliza-

tion of the system requires a firm knowledge of continuum mechanics and

8. Magnetic Fluid and Flow

situations, where the magnetization can be treated as an equilibrium in

497

thermodynamics with an understanding of magnetic field theory. The field

of study in magnetic fluids is recognized as “Ferrohydrodynamics,” as at-

tributed to R. Rosensweig, (1985). The overall field of study in magnetic

fluid has a highly interdisciplinary character, including physics, chemistry,

engineering, mathematics and even medicine in practical applications.

In this chapter, in view of engineering fluid mechanics, fundamental

aspects of continuum mechanics and some basic properties of magnetic

fluids are introduced. Also, some typical technological applications are de-

scribed in consideration with magnetoviscous effects.

The magnetic fluids available today are liquids containing small magnetic

particles, in a so-called suspension (a colloidal system). In manufacturing

magnetic fluids, mainly three components are required: namely such as a

base liquid (or a carrier liquid), magnetic particles of a colloidal size and

stabilizer (or repelling electric charge) to disperse the magnetic particles

from aggregation. The size of magnetic particles must be sufficiently small

enough since the stability of a magnetic fluid, such as a colloidal system, is

ensured by thermal motion of the particles, preventing agglomeration and

precipitation. The particle material must have a high level of magnetizabil-

ity. Most common magnetic fluids are usually composed of 3 to 15 nm

sized particles of solid, magnetic, single-domain particles coated with a

molecular layer of dispersant (surfactant, such as oleic acid), where the

particles are suspended in a liquid carrier. The basic mechanism of the sus-

pension is such that thermal agitation of Brownian motion keeps the parti-

cles suspended and the coatings of the particles from sticking to each other.

It should be kept in mind that the particles must not to be too small, since

at sizes less than 1 to 2 nm their magnetic properties tend to disappear.

Figure 8.1 shows a schematic diagram of magnetic fluids: (a) with a sur-

factant coating (separating of particles by the effect of steric repulsion); (b)

with ionic surface charge (separating of particles by the effect of electric

repulsion); and (c) also shows an electron micrograph of a magnetic fluid.

A typical thermophysical property of magnetic fluid, a commercially avail-

able magnetic fluid, is tabulated in Table 8.1.

In typical thermomagnetic properties of magnetic fluids, Fig. 8.2

shows: (a) equilibrium magnetization (M ) vs. magnetic field strength (H),

and (b) equilibrium saturation magnetization (M

) vs. temperature (T ). The

particles in a magnetic fluid (each with the magnetic moment m) are

analogous to the molecules of a paramagnetic gas. The magnetization law

8 Magnetic Fluid and Flow

8.1 Thermophysical Properties

498

8.1 Thermophysical properties

for the paramagnetic gas is described by the Langevin function



]

L with

the following formula

Fig. 8.1 Magnetic Fluid

Table 8.1 Magnetic fluids

(

(MK-340; Courtesy of Ferro Tech. Corp.)

Appearance Black-brown or red-brown viscous fluid

Carrier liquid Synthetic hydrocarbon

Saturation magnetization(

M )

11.0

Viscosity (at 27

)

100 smPa 

Density

0.94

mkg

Pour point

 56 C

Flash point

>200

Thermal conductivity (at 38

)

150



KmmW 

Surface tension (at 25

)

32 mN/m

Coefficient of thermal expansion

/K107.5

4

499

Fig. 8.2 Magnetization characteristic



]

LMNmM



coth

(8.1.1)

where

N is the number of particles per unit volume, TkmH

]

(Langevin argument),

k is the Boltzmann constant and m is the

8 Magnetic Fluid and Flow 500

8.1 Thermophysical properties

magnetic moment. It should be mentioned that the saturation magnetiza-

tion

M can be written in terms of the colloid density

, which

N , where

is the mass of a particle with attached carrier liquid

molecules, in such a manner as to propose

(8.1.2)

The character of this magnetization by Eq. (8.1.1) is called superparamag-

netism as displayed in Fig. 8.2(a).

In narrow ranges of magnetic fluid strength and temperature, the equa-

tion for the magnetic state of an incompressible magnetic fluid with the

equilibrium magnetization of a magnetic fluid





TH,M is approximated

with an accuracy to the first order term as follows



 

MH,M 





(8.1.3)

and

are some mean values which satisfy



T,HM=M . The quan-

tity





HM is called the differential magnetic susceptibility of a

magnetic fluid (for a ferromagnetic solid

is the magnetic sus-

ceptibility). The other quantity appears in Eq. (8.1.3) is



TMK ww ,

which is called the pyromagnetic coefficient. Using the linear relationship

in Eq. (8.1.3), the curve



1

]

of magnetization of a magnetic fluid is

well approximated as representatively displayed in Fig. 8.2(b).

The field dependence of the equilibrium magnetization is the key pa-

rameter in discussing the dynamics of the magnetic fluid, and which also

provides valuable information about the constituent elements of the fluid.

The physical prerequisite for the existence of equilibrium magnetization is

the assumption that the equilibrium is achieved by the relocation of the

orientation of elementary magnetic moments along the applied magnetic

field. There are two different relaxation processes after the applied field

has been changed.

In the first mechanism we can assume that the magnetic moment of the

particle is fixed with respect to its crystal structure and the relocation takes

place by a rotation of the whole particle. This process is characterized by a

respective time, called a Brown rotational diffusion time

, Brown

(1963), which is given where

501

(8.1.4)

is the hydrodynamic volume of the particle, including the surfactant

layer,

is the viscosity of the carrier liquid.

In the second mechanism, the magnetic moment may rotate inside the

particle relative to the crystal structure. This process, called Néel relaxa-

tion, Néel (1949), is characterized by a respective relaxation time

which is given where

(8.1.5)

is the anisotropy constant of the particles,

is the Larmor frequency of

the magnetization having the approximate value of 10

Hz and V is the

volume of the magnetic core of the particle . The Néel relaxation takes

place if the thermal energy

is high enough to overcome the energy

barrier

, which is given by the crystallographic anisotropy of the mag-

netic particle. The relaxation of magnetization in a nano-dispersed suspen-

sion will follow the process with a shorter relaxation time,

 or

!! . When



, the magnetic material is said to

have the extrinsic superparamagnetism; oppositely where

!! , it is

said to have the intrinsic superparamagnetism. In either case, the magnetic

fluid has an apparent superparamagnetism described in Eq. (8.1.1) and

shown in Fig. 8.2(a). In comparison with Eqs. (8.1.4) and (8.1.5), for big-

ger particle sizes,

becomes smaller than

and the Brownian relaxa-

tion will take place due to a rotation of the whole particle. Note that in a

real magnetic fluid the relaxation process may be very complicated due to

the particle distribution, where a part of the particles relaxes by the Néel

process, while another part of the particles relaxes by the Brownian proc-

ess, depending on the size of each constituent particle. For example, the

critical diameter

crit

, where the condition

is met, is

nm18

for

particle (with

mJ14 K and smkg10





), Odenbach

(2002). The important aspect in the equilibrium magnetization is that the

mean value of magnetization is achieved almost instantaneously within a

time much shorter than the characteristic time scale of a macroscopic proc-

ess, such as the fluid dynamical motion, where the magnetization vector

M becomes paralleled to the vector of a magnetic field of intensity H at a

given instant, i.e.

M//H .

8 Magnetic Fluid and Flow 502

8.1 Thermophysical properties

Another important property of magnetic fluids, particularly with regard

to the magnetoviscous effect, is the apparent viscosity

in the absence of

the magnetic field.

may vary from that of the carrier liquid due to the

presence of the suspended particles. In the first approach, we may be able

to use a well-known theoretical prediction by Einstein (1906) for diluted

suspensions



IKK

(8.1.6)

where

is the volume fraction of all suspended particles, including the

surfactant layer, given as



(8.1.7)

In Eq. (8.1.7),

is the volume fraction of the magnetic material, d is the

average diameter of particle and

is the thickness of the surfactant layer.

For example, for typical values of

nm10

d and nm2|s of a magnetic

fluid 10

would be calculated as 270.

. It should be aware

that Einstein’s formula, i.e. Eq. (8.1.6) is, however, no longer valid for

higher concentration suspensions, such as

10.!

and this fact must be

kept in mind in dealing with actual magnetic fluids.

A realistic correlation is proposed by Rosensweig, (1985) written

where





IIKK

(8.1.8)

b is given by a function such that



(8.1.9)

in Eq. (8.1.9) is a critical volume fraction of suspended material, which

is obtained under the assumption that the suspension’s apparent viscosity

diverges at a given value of

. For example, 740.

would give a good

estimate for

for a kerosene-based magnetic fluid, Odenbach (2002).

503

Exercise

Exercise 8.1.1 Langevin Argument

The equilibrium magnetization of magnetic fluids is described by the

Langevin formula in Eq. (8.1.1), when the saturation dipole moment

m of

the bulk of a magnetic solid is related to the saturation magnetization

through the volume fraction

of a magnetic material

dmS

(1)

Write the Langevin argument

]

in terms of the particle diameter d .

Ans.

The saturation magnetization

M of the magnetic fluid is given in

terms of the particle dipole moment m as defined in Eq. (8.1.2) by

NmM

(2)

while the Langevin argument

]

is given where

]

(3)

Using Eqs. (1) and (2), Eq. (3) becomes

(4)

Since

, where V is the volume of the magnetic core of a particle,

Eq. (4) can be reduced to the following expression:

]

(5)

Note that

is given by 6

in Eq. (5)

Exercise 8.1.2 Relaxation of Magnetization

Consider two relaxation mechanisms by which the magnetization of mag-

netic fluid suspensions relax after the applied magnetic field has been

8 Magnetic Fluid and Flow 504