Fluidos

Ferrohydrodynamics: Retrospective and Issues
Mark I. Shliomis
Department of Mechanical Engineering, Ben-Gurion University of the Negev,
P.O.B. 653, Beer-Sheva 84105, Israel

Abstract. Two basic sets of hydrodynamic equations for magnetic colloids (so-called
ferrofluids) are reviewed. Starting from the quasistationary ferrohydrodynamics, we
then give a particular attention to an expandedmodel founded on the concept of
internal rotation. A specific relation between magnetic and rotational degrees of freedom of suspended grains provides a coupling of the fluid magnetization with the fluid
dynamics. Hence a complete set of constitutive equations consists of the equation of
ferrofluid motion, the Maxwell equations, and the magnetization equation. There are
three kinds of the latter. Twoof them were derived phenomenologically as a generalization of the Debye relaxation equation in case of spinning magnetic grains, while
one of them was derived microscopically from the Fokker-Planck equation. Testing the
magnetization equations, we compare their predictions about the dependence of the
rotational viscosity on the magnetic field and the shear rate.

1

QuasistationaryFerrofluid Dynamics

Physics and hydrodynamics of ferrofluids has begun from the basic work [1]
and the following long series of papers by Rosensweig and co-workers, included
later on in his monograph [2]. Those papers and the book laid a serious scientific
foundation for further research and gave an impetus to great variety of ferrofluid
applications in industry, technology, and medicine. A set ofequations describing ferrofluid dynamics (“ferrohydrodynamics” – the term of Rosensweig) first
proposed in [1] consists of the equation of ferrofluid motion
ρ

dv
= −∇p + η ∇2 v + M ∇H ,
dt

d
∂
=
+ (v · ∇) ,
dt
∂t

(1)

the magnetic state equation M = M (H, T ), for which it is natural to employ
[3,4] the Langevin formula
M = nmL(ξ ) ,

ξ = mH/kB T ,

L(ξ ) = coth ξ − ξ −1 ,

(2)(here m = Md V being the magnetic moment of a single subdomain magnetic
particle, V = πd3 /6 the particle volume, n their number density, and Md stands
for the domain magnetization of dispersed ferromagnetic material) and the equations
div v = 0,

rot H = 0,

div B = 0,

Stefan Odenbach (Ed.): LNP 594, pp. 85–111, 2002.
c Springer-Verlag Berlin Heidelberg 2002

(B = H + 4π M )(3)

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M.I. Shliomis

indicating that the ferrofluid is considered to be incompressible and nonconducting. The volume density of magnetic forces in (1), F = M ∇H , is calculated as
mag
divergence of the stress tensor of magnetic field in ferrofluid, Fi = ∂σik /∂xk ,
where
∂B
H
mag
· B−ρ
σik = 41 Hi Bk −
δik .
(4 a)
π
8π
∂ρ H ,T
According to the assumption (2), ferrofluidmagnetization is nonsensitive to the
velocity field and relaxes instantaneously to the equilibrium (Langevin) value.
Then in the low-field limit, when B = µH with a constant µ = 1+4πχ, equation
(4 a) takes the well-known form [5]
mag
σik =

H2
µHi Hk
−
µ−ρ
4π
8π

∂µ
∂ρ

T

δik .

(4 b)

For a non-magnetic fluid (µ = 1) this expression is reduced to the Maxwell stress
tensor ofmagnetic field
mag
σik =

1
4π (Hi Hk

− 1 H 2 δik ).
2

Equation (4 a) may be rewritten as
mag
σik =

1
4π (Hi Bk

− 1 H 2 δik ) −
2

H
· M −ρ
2

∂M
δik .
∂ρ H ,T

(4 c)

Its divergence consists of two terms,
F = (M · ∇)H − ∇

H
· M −ρ
2

∂M
∂ρ H ,T

,

(5)

where the first is called sometimes the Kelvin force, and the second represents
the magnetostrictiveforce. Being the potential force, the latter is always included into the pressure gradient in (1), i.e., it is equilibrated automatically by
the hydrostatic or hydrodynamic pressure. Further, since the equilibrium magnetization is collinear with the local magnetic field, M = (M/H )H , the Kelvin
force takes the form (M · ∇)H = (M/H )(H · ∇)H . Hence, using an identity
(H · ∇)H = 1 ∇H 2 − H ×...