1/13/22, 4:34 PM
`
`Ferrofluids
`
`Ferrofluids - Scholarpedia
`
`Sebastian Andreas Altmeyer (2020), Scholarpedia, 15(11):55163.
`
`doi:10.4249/scholarpedia.55163
`
`revision #194779 [link to/cite this article]
`
`Prof. Sebastian Andreas Altmeyer, Universitat Politecnica de Catalunya, Barcelona, Spain
`
`Ferrofluids represent a special class of magnetic fluids and are manufactured fluids consisting of dispersions of
`magnetized nanoparticles in a variety of non-magnetic liquid carriers. They are stabilized against agglomeration by
`the addition of a surfactant monolayer onto the particles. In the absence of an applied magnetic field, the magnetic
`nanoparticles are randomly oriented (Figure 1), the fluid has zero net magnetization, and the presence of the nanoparticles
`provides a typically small alteration to the fluid’s properties such as viscosity and density. When a sufficiently strong magnetic
`field is applied, the ferrofluid flows toward regions of the magnetic field, properties of the fluid such as the viscosity are altered,
`and the hydrodynamics of the system can be significantly changed.
`
`Since the first successful production of stable ferrofluids in the early 1960s (Papell 1964) the field of ferrofluid research
`developed quickly in different branches:
`
`Physics: connected to the fundamental description and characterization.
`Chemistry: as basis for ferrofluid preparation.
`Engineering: to prepare and provide application.
`The aim of this review is to provide an overview of
`ferrofluids as one class of magnetic fluids discussing
`some historical background, important properties such
`as relaxation times, and their typical composition.
`Further special features such as magnetorotational
`viscosity will be explained and some of the difficulties in
`their theoretical modelling, e.g. internal magnetization,
`agglomeration, particle-particle interaction, and finite
`size effects will be described. Finally some results for
`ferrofluids studied in concrete prototypical systems are
`presented.
`
`As the name already says, ferrofluids are complex fluids,
`therefore the reader is encouraged to look up the other
`Scholarpedia articles on fluid mechanics and particularly
`those focusing on specific aspects which could not be
`covered here in detail.
`
`Contents
`
`1 HISTORY AND BACKGROUND
`1.1 Magnetic fluids
`1.2 Properties and characteristics of ferrofluids
`1.2.1 Structural composition and
`configuration
`1.2.2 Relaxation times
`1.2.3 Magnetoviscous effect -
`Magnetorotational viscosity
`1.2.4 Magnetic properties
`1.2.5 Internal magnetization
`
`Figure 1: Schematic illustrating a ferrofluid. Suspension of magnetic
`cores (large spheres, magnetically permeable particles) with diameters
`in the order of about 10 nm, surrounded by polymer shells (small
`spheres, surfactant) with a thickness of about 2 nm in a carrier fluid
`(background, carrier fluid). The cores have a permanent magnetic
`moment (green arrows) proportional to their volume. Without an
`externally applied magnetic field, these are statistically distributed.
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`1.2.6 Agglomeration, particle-particle
`interaction and viscoelastic effects
`1.2.7 Some ferrofluid data
`2 FERROHYDRODYNAMIC EQUATIONS
`2.1 Navier-Stokes equations
`2.2 Magnetization equations
`3 STUDIED SYSTEMS - MAGNETIC FLUID
`CHARACTERIZATION
`3.1 Couette flow as prototypical system to study
`ferrofluids
`3.1.1 Magnetorotational viscosity in the TCS
`3.1.2 Stabilization effect in the TCS
`3.1.3 Mode interaction and flow structure
`modification
`3.2 Rayleigh-Bénard system to study ferrofluids
`4 APPLICATIONS
`5 INTERDISCIPLINARY FEATURES
`6 References
`7 Internal references
`8 SEE ALSO
`
`HISTORY AND BACKGROUND
`
`The field of ferrofluid research is relatively young compared to the investigation that have been done in fluid dynamics in
`general. The famous book “Ferrohydrodynamics” by Rosensweig 1985 is one of the standard textbooks in this field which must
`be mentioned here. It covers various areas in this research field, synthesis and properties of magnetic fluids, foundation of
`ferrohydrodynamics theory, hydrodynamics in ferrofluids, as well as problems and applications. However, the term
`ferrohydrodynamics was established first by Neuringer and Rosensweig 1964. This includes the continuum description of the
`flow behavior of magnetic fluids in the presence of magnetic fields. Later Shliomis 1972 developed a theory including the
`experimental findings of magnetoviscous effects by Rosensweig et al. 1969 and McTague 1969. Further to mention is the book
`“Magnetic Fluids” by Blumes et al. 1997 which focuses on the rheology of ferrofluids in more detail, also including theoretical
`discussion of the magnetoviscous effect, rotational viscosity variation of shear rate, and many more. In this context also to
`mention is the earlier work by Blumes et al. 1986, which despite being mainly devoted to conducting fluids and the action of
`Lorenz forces, also elucidates the effects of heat and mass transfer in ferrofluids. The application of ferrofluids and magnetic
`fluids in general is summarized in the books by Berkovsky and Bashtovoy 1993 & 1996. They provide a wide overview of various
`possible uses of ferrofluids in different fields/areas, reaching from separation over mechanical positioning towards medical
`applications. Nowadays, ferrofluids are utilized in a wide variety of applications, ranging from their use in computer hard drives
`and as liquid seals in rotating systems to their use in laboratory experiments to study geophysical flows and the development of
`microfluidic devices.
`
`Magnetic fluids
`
`Any kind of fluid which can by externally controlled, e.g. by a magnetic field represents a challenging subject either for scientists
`interested in basic fluid mechanics/dynamics as well as application engineers. Consider basic research: the ability of
`introducing an artificial external but controllable force into the basic equations reaches out into a fascinating field of potential
`new phenomena. The fact that magnetic fields can be varied quite well and accurately, both in direction/orientation and field
`strength, makes them highly interesting for adding such external forces. Unfortunately (most normal) natural liquids do not
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`offer these features. However, artificial generated suspensions of magnetic nanoparticles in appropriate carrier liquids, i.e.
`ferrofluids, do so. Although various different effects have been discussed to date, by far the most famous field-induced property
`of magnetic fluids is the change of their viscosity (McTague 1969).
`
`In general, a magnetic field can be used effectively as a control or bifurcation parameter of the system, whose change can lead to
`characteristically distinct types of hydrodynamical behavior. In this regard, turbulence and transition to turbulence in
`Magnetohydrodynamic (MHD flows) play an important role in many astrophysical and geophysical problems, e.g. the
`generation of magnetic fields in heavenly bodies, in planets and (sometimes) in large-scale industrial facilities. For instance,
`Gellert et al. 2011 studied current-driven instabilities of helical fields.
`
`Properties and characteristics of ferrofluids
`
`Structural composition and configuration
`Most commercially available ferrofluids are suspensions containing magnetic nanoparticles with a mean diameter of about 10
`nm in a liquid carrier (Figure 1). Crucial for such suspensions is their colloidal stability. This means the avoidance of
`agglomeration due to magnetic interaction as well the sedimentation either in the gravitational field and in magnetic field
`gradients while in any given thermal motion. Typically, for most scenarios the thermal energy kT (k denoting Boltzmann’s
`constant and T the absolute temperature) of the particles with a diameter of about 10 nm fulfills the stability requirements
`although it cannot be guaranteed. A reason is the Van der Waals attraction that occurs as soon as particles come into contact
`and therefore try to agglomerate. To avoid such irreversible agglomeration of the particles, usually a surfactant layer is used to
`avoid a direct contact between particles. These surfactant layers have to match the dielectric properties of the carrier liquid.
`Most modern ferrofluids use magnetite (Fe3O4) or cobalt ferrite (CoFe2O4) as the magnetic component: the impact of the
`ferrofluid under magnetic field crucially depends on the magnetic component used. For instance cobalt based ferrofluids show
`significantly stronger effects than magnetite based ones (see also Table I). Carrier liquids have a more versatile spectrum. They
`range from simple water, different oils, over heptane, kerosene to some types of ester. The specification of the surfactant in
`general is more complex and versatile as it has to match the dielectric properties of the carrier liquid and therefore often
`remains a trade secret of the producers. Acetic acid would be an example for a surfactant to be used for magnetite in water.
`Common volume concentrations of the magnetic component are in the range of 5 %vol. - 15 %vol..
`
`Relaxation times
`A crucial parameter for the characterization of a specific ferrofluid is its corresponding relaxation time. Basically, two different
`mechanisms determine the relaxation and thus the relaxation time (Figure 2):
`
`the particle can rotate with its magnetic moment when it is aligned in the direction of the field - Brownian motion of the
`particles in the fluid, or
`the magnetic moment can align/relax itself within the particle without rotating it - Néelian relaxation process.
`
`The faster of both processes will be responsible for the actual (effective) relaxation.
`
`This results in an effective relaxation time (Fannin and Charles 1988)
`
`resulting from the Brownian relaxation time (Debye 1929)
`
`and the Néel relaxation time (Néel 1949)
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`
`=
`+
`1
`τ
`ef f
`1
`τ
`B
`1
`τ
`N
`=
`(D + 2
`τ
`B
`πη
`~
`2 Tk
`B
`s
`H
`)
`3
`=
`exp(
`).
`τ
`N
`1
`f
`0
`πK
`D
`3
`6 T
`k
`B
`

`

`Ferrofluids - Scholarpedia
`1/13/22, 4:34 PM
` stands for the viscosity of the carrier liquid, D for the particle
`Here
`diameter, and K for the anisotropy constant of the magnetic material, sH
` is a frequency of the order of 10
`is the thickness of the surfactant, and
`Hz. As mentioned before, the resulting effective relaxation time
`corresponds mainly to the faster of both relaxation processes. Typically
`small particles show Néelian behavior, while large ones show Brownian
`behavior. Thereby the position of the (rather narrow) transition area
`from Brownian to Néelian particles depends strongly on the anisotropy
` and shell thickness sH
`constant K of the used ferrofluid. Viscosity
` and only plays a
`essentially only affect the Brownian relaxation time,
`role in the range of Néelian relaxation.
`
`Figure 2: Schematics of Néelian and Brownian
`relaxation: the magnetic moment (red) of a
`magnetic particle aligns itself by rotating within the
`magnetic material (blue) (Néel) or by moving the
`whole particle including the non-magnetic shell
`(green).
`
`To give a better
`understanding of
`these relaxation
`processes, Figure 3
`illustrates a
`concrete example of the Brownian, Néelian and effective relaxation times
`depending on the particle size D for the Magnetite based ferrofluid
`APG933 at a temperature of T = 300 K. Curves are presented for two
`different values of the anisotropy constant K. On one hand side, for
`magnetite without shape anisotropy, the literature (Fannin and Charles
`1991, Berkovsky et al. 1993) mostly contains values of the anisotropy
`constant of about K = 15 kJ/m3 . On the other hand K = 50 kJ/m3 presents
`the upper limit of the values found in the literature. Further typical
` 109 Hz and sH = 2 nm are used for
`experimentally detected values of
`the Néel pre-factor and the shell thickness, respectively (Rosensweig 1985,
`
`Figure 3: Variation of Brownian, τB, Néelian, τN,
`and effective relaxation time, τeff, of APG933
`.
`depending on particle diameter
`
`Odenbach and Thurm 2002).
`
`Magnetoviscous effect - Magnetorotational viscosity
`A key parameter that may change in a ferrofluid under the influence of an applied magnetic field is the rotational viscosity
`(McTague 1969, Shliomis 1972, Rosensweig 1985, Holderied 1988, Berkovsky et al. 1992). This change results from hindrance of
`the free particle rotation when the fluid is subject to a shear flow under the influence of external magnetic fields (McTague 1969,
`Rosensweig 1985). It is known that for a ferrofluid in motion, the surrounding carrier liquid generates a viscous torque. Thus,
`with an additionally applied external magnetic field, the magnetic moments of the particles, which are statistically oriented
`(Figure 1) in the absence of any field, now align in the direction of this field. Therefore, having magnetic hard particles, a
`magnetic torque is exerted on the particles. If the vorticity of the fluid and direction of the magnetic field are not collinear, the
`magnetic torque counteracts the viscous torque, resulting in an increase in the apparent viscosity (Holderied 1988, Berkovsky et
`al. 1992).
`
`The schematics in Figure 4 illustrates the situation of a (hard) ferrofluid particle under the influence of shear flow. In general,
`particles (magnetic or not) will rotate in the flow due to the mechanical torque produced by viscous friction in the fluid. If a
`magnetic field Hext is applied to the fluid, the magnetic moment of the particles will align with the field direction. In the
`scenario that the field direction and vorticity of the flow are collinear (left in Figure 4) the magnetic alignment will only lead to
`the fact that the magnetic moment of the particles becomes collinear with the direction of vorticity. No further modification
`appears regarding the motion of the particle, and therefore the flow of the fluid as a whole remains unaffected. However, if the
`vorticity and field direction are perpendicular (right in Figure 4) the situation is different. Now the mechanical torque will force
`a misalignment of the magnetic moment of the particle and the field direction, consider the magnetic moment’s direction in the
`particle is fixed. This results in an imminent magnetic torque, trying to realign the magnetic moment and Hext caused by the
`angle between the mutual directions of the magnetic moment and the field. As this magnetic torque acts just opposite to the
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`η
`~
`f
`0
`η
`f
`0
`[D]
`=f
`0
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`1/13/22, 4:34 PM
`mechanical torque it hinders the free rotation of the
`particle in the flow, and therefore increases the flow
`resistance and thus the fluid exhibits an increased
`viscosity.
`
`Magnetic properties
`The fact that magnetic fluids present a combination of
`normal liquid behavior with super-paramagnetic
`properties represents one of the most important
`features. The magnetic particles (mean diameter
`typically of about 10 nm) itself can be seen as magnetic
`single domain particles (Kneller 1962). Thus, in an
`external magnetic field the alignment of the particles will
`be determined by a counteraction of thermal energy with
`the magnetic energy of the particle (described as a
`dipole). This qualitative behavior of the equilibrium
`magnetization M of a ferrofluid can be well described by
`the Langevin law (Langevin 1905)
`
`Ferrofluids - Scholarpedia
`
`Figure 4: Schematics illustrating the effect of rotational viscosity:
`local differences in the flow field lead to the rotation of the magnetic
`particles (blue). This rotation (black arrow) remains unaffected by an
`externally applied magnetic field, Hext, parallel to the axis of rotation
`(left: vertically dotted line). However, if the axis of rotation is
`perpendicular to the field (right: thick black dot), the magnetic
`moment (red arrow) is turned out of the direction of the field. The
`resulting magnetic torque hinders the rotation of the particle and thus
`also the flow of the carrier fluid. This effect manifests itself
`macroscopically in an increased viscosity.
`
`where Msat denotes the saturation magnetization of the fluid, m the magnetic moment of a single particle, H the applied
`magnetic field, kB Boltzmann’s constant, T the absolute temperature, and
` the vacuum permeability.
`
`Internal magnetization
`Although the Langevin equation (Langevin 1905) gives a good expression for the equilibrium magnetization, it does not account
`for any variation within the ferrofluid due to an external applied magnetic field. A frequently used assumption to simulate
`ferrofluidic flows is that the internal magnetic field within a ferrofluid is equal to the external applied field. However, this
`simplest assumption is only a leading-order approximation. In reality, the magnetic field H outside the ferrofluid results from
`super-imposition of two parts:
`the externally applied magnetic field Hext
`an interference magnetic field hout originating from the magnetized liquid itself: this field can be typically represented as a
`.
`gradient of a scalar potential
`Thus, by accounting for the ferrofluid’s magnetic susceptibility, χ, it has been shown that a uniform externally imposed
`magnetic field is modified by the presence of the ferrofluid within. For instance, for ferrofluidic flow in between rotating
`cylinders (Taylor-Couette flow, Taylor 1923) the modification to the magnetic field has an 1/r2 radial dependence and a
`magnitude which scales with the susceptibility (Altmeyer et al. 2012, Altmeyer 2018). For ferrofluids typically used in
`laboratory experiments, these modifications to the imposed magnetic field can be substantial with significant consequences on
`the structure and stability of the basic states, as well as on the bifurcating solutions.
`
`Agglomeration, particle-particle interaction and viscoelastic effects
`Although ferrofluids are manufactured in a way to avoid sticking together (Figure 1) by a surfactant, in the real world various
`external effects typically destroy the idealized scenario. This results in difficulties in both experimental observation and
`theoretical modeling of ferrofluids.
`
`When describing the hydrodynamics of ferrofluids, one possible assumption is that the particles aggregate to form clusters
`having the form of chains, and thus it hinders the free flow of the fluid and increases the viscosity (Odenbach and Müller
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`M =
`(coth(α) − 1/α), where α =
`M
`sat
`mHμ
`0
`Tk
`B
`(1)
`μ
`0
`∇ψ
`out
`

`

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`1/13/22, 4:34 PM
`2005). In this type of structure formation, it is also assumed that the interaction parameter is usually greater than unity
`(Rosensweig 1985), thus the strength of the grain-grain interaction can be measured in terms of the total momentum of a
`particle.
`Non-interacting magnetic particles with a small volume or an even point size are typically assumed in order to set up a
`mathematical model for the flow of complex magnetic fluids. Real ferrofluids, however, consist of a suspension of particles
`with a finite size in an almost ellipsoidal shape, as well as with particle-particle interactions that tend to form chains of
`various lengths. One approach to come close to this realistic situation for ferrofluids is the consideration of so-called
`elongational flow incorporated by the symmetric part of the velocity gradient field tensor, which could be scaled by a so-
` (Odenbach and Müller 2002, Altmeyer et al. 2013). Such a term also exists similar in the
`called transport coefficient
`dynamics of nematic liquid crystals as the flow alignment’s effect on the director field in an applied shear flow (Martsenyuk
`et al. 2074). Agglomeration effects have been proven to be evident and not negligible in ferrofluids (Peterson and Krueger
`1977, Storozhenkoa et al. 2016, Altmeyer 2019).
` can be considered as a material-dependent function of thermodynamic variables such as density,
`The transport coefficient
`concentration, and temperature, but independent of shear. It can be handled as a reactive transport coefficient which does not
`enter the expression for entropy production. By comparing experimental measurements for the magnetovortical resonance for a
`given ferrofluid and for flow-induced modification of the relaxation time, it can be argued that the shear flow induces fracture of
`dynamical particle chains, which leads to a reduced effective bipolar interaction between the particles. For instance,
`experimental works by Odenbach and Müller 2002 regarding the non-equilibrium magnetization of the ferrofluid in the Taylor-
`Couette system (for a simple stationary flow configuration subjected to a homogeneous transverse magnetic field) showed that
`the symmetric part of the velocity gradient (i.e., the elongational flow component) is not zero. Thus, this result indicates that
`significantly affects the magnetization vector in the ferrofluid on micro-structural properties of the ferrofluid.
`
`Finally, the elongational flow - the formation of elongated chainlike clusters, is also responsible for the appearance of
`viscoelastic effects in ferrofluids under the influence of magnetic fields, first described by Odenbach et al. 1999 for the
`investigation of the Weissenberg-Effect in ferrofluids exposed to magnetic fields.
`
`Some ferrofluid data
`The variation in rheological properties between different ferrofluids can be quite strong. Table I lists some experimental
`observed data for some ferrofluids; commercial frequently used magnetite-based ferrofluid from the APG series from FerroTec
`and one Co-based ferroluid.
`
`Experimental data of some ferrofluids from the APG series (magnetite base) von FerroTec and a Co-based
`ferrofluid (− not available).
`Ferrofluid
`APG933 APG934 APG935 APG936 APG513A Co
`0.5
`1
`1.5
`2
`0.128
`0.5
`
`dynamic viscosity
`
`[Pa s]
`
`[103
`kg/m3]
`
`1.06
`
`[10-3 m2/s] 0.47
`1.09
`
`[kA/m]
`
`18.15
`
`[%]
`[nm]
`
`[nm]
`
`4.1
`7
`
`8.7
`
`-
`
`-
`
`0.73
`
`16.4
`
`3.7
`7.1
`
`8.5
`
`density
`
`kinematic viscosity
`
`initial susceptibility
`
`saturation magnetization
`
`volume fraction
`average diameter (log-normal)
`
`average diameter (volume)
`
`standard deviation (log-normal)
`
`Brownian relaxation time
`(experimental)
`
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`
`ν=
`
`Msat
`Φ
`
`<D3>1/3
`σD
`
`τB
`
`1.47
`
`1.42
`
`1.44
`
`1.42
`
`[ms]
`
`0.599
`
`0.485
`
`0.355
`
`0.48
`
`0.78
`
`-
`
`6/16
`
`1.07
`
`1.07
`
`1.28
`
`1.4
`
`0.58
`
`17.89
`
`4.0
`6.3
`
`7.6
`
`1.87
`
`0.85
`
`19.01
`
`4.3
`7
`
`8.4
`
`0.1
`
`1.57
`
`32
`
`7.2
`-
`
`10
`
`-
`
`-
`
`-
`
`0.44
`
`6.51
`
`1.5
`11
`
`12
`
`1.27
`
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`
`λ
`2
`λ
`2
`λ
`2
`η
`~
`ρ
`/ρη
`~
`χ
`0
`D
`0
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`

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`FERROHYDRODYNAMIC EQUATIONS
`
`Ferrofluids - Scholarpedia
`
`Navier-Stokes equations
`The flow dynamics of an incompressible homogeneous mono-dispersed ferrofluid with kinematic viscosity ν and density ρ is
`governed by the incompressible Navier-Stokes equations (see also Partial differential equation), including magnetic terms, and
`the continuity equation.
`
` the velocity vector, H the magnetic field, and M the magnetization. Using characteristic time and length
`with
` can be scaled to 1 in Equation (2) representing the non-dimensional ferrohydrodynamic equation
`scales, the coefficients
`of motion.
`
`Magnetization equations
`This section is more specific and might be skipped by the general reader, who only should keep in mind that an equation
`describing the ferrofluid magnetization is required in order to numerically solve the whole system. For the more advanced
`reader who is more familiar with the topic, it will provide more details regarding the magnetization of ferrofluids presenting one
`concrete example (model by Niklas 1987) as well as variation with different assumptions and/or the use of different ferrofluids.
`
`Equation (2) is solved together with an equation that describes the magnetization of the ferrofluid. A variety of models which
`describe the magnetization dynamics in ferrofluids are discussed in the literature. Most common types are either to use the
`relaxation of the magnetization M into the equilibrium magnetization
`
`or to use the relaxation of an effective field
`
`into the magnetic field H both with one single relaxation time (Shliomis 1972, Rosensweig 1985, Berkovsky et al. 1993). The
`common form in the stationary case, is
`
` . Both coefficients
` the local vorticity and
`with
`material properties of the ferrofluid and model dependent.
`
` and
`
` are functions of H, M, and of some other
`
`A commonly used model reflecting the fact that real ferrofluids contain magnetic particles of different size considers the
`ferrofluid as a mixture of ideal mono-disperse paramagnetic fluids. In this case the resulting magnetization is given by M =∑ Mj
`where Mj denotes the magnetization of the particles with different diameter Dj. Each sub-magnetization Mj is assumed to follow
`simple Debye relaxation dynamics (Debye 1929)
`
`with
`
` being the equilibrium sub-magnetizations and τj the effective relaxation times of the different particle species.
`
`The model proposed by Niklas 1987 is based on the simple approximation to use the equilibrium magnetization of an
`unperturbed state with a homogeneously magnetized ferrofluid at rest with the mean magnetic moments oriented in the
` is the magnetic susceptibility of the ferrofluid, determined using
`, where
`direction of the magnetic field,
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`
`( + u ⋅ ∇)u − u + ∇p = (M ⋅ ∇)H +
`∇ × (M × H),
`∂
`t
`∇
`2
`α
`1
`1
`2
`α
`2
`(2)
`∇ ⋅ u = 0,
`u = (
`,
`,
`)
`u
`1
`u
`2
`u
`3
`,α
`1
`α
`2
`=
`(H )H/H
`M
`eq
`M
`eq
`=
`(H )H/H
`H
`ef f
`M
`−1
`eq
`(Ω + κM × H) × M = (M − H),
`γ
`τ
`γ
`χ
`(3)
`2Ω = ∇ × u
`κ = /(6Φ )
`μ
`0
`η
`~
`γ
`τ
`γ
`χ
`= Ω ×
`−
`(
`−
`),
`∂
`t
`M
`j
`M
`j
`1
`τ
`j
`M
`j
`M
`eq
`j
`(4)
`M
`eq
`j
`= χH
`M
`eq
`χ
`

`

`Ferrofluids - Scholarpedia
`1/13/22, 4:34 PM
`Langevin’s formula (Langevin 1905). This is an approximation of Equation (1) considering small values α for weak magnetic
`fields
`
` is the spontaneous magnetization of the magnetic material, and χ is the initial
` is the mean diameter of the particles,
`Here
` of about 10 nm and saturation magnetization of about
`susceptibility of the fluid. Typically, for particles with mean diameter
`Msat = 32 kA/m this approximation is valid up to H ≈ 15 kA/m. However, a ferrofluid’s magnetization is also influenced by the
`flow field itself. The model introduced by Niklas 1987, Niklas et al. 1989 consider the magnetic fluid to be incompressible,
`nonconducting, and to have a constant temperature and a homogeneous distribution of magnetic particles. Assuming a
`, and τ is the magnetic
` and small relaxation times
`stationary magnetization near equilibrium with small
`relaxation time. In the near-equilibrium approximation, Niklas determined the relationship between the magnetization M, the
`magnetic field H, and the velocity u [i.e. a simplification of Eqs. (3) and (4)] to be
`
`where cN is the Niklas coefficient
`
`where μ is the dynamic viscosity, and
` is the volume fraction of the magnetic material.
`As a first approach, assuming the internal magnetic field to be equal to the externally imposed magnetic field H = H ext,
`Equation (2) (together with an appropriate use of non-dimensionalization) can be simplified to
`
`In this approach, the magnetic field and all the magnetic properties of the ferrofluid influence the velocity field only via the
`magnetic field parameter
`
` on the magnetic field as well as the influence of the used
`Figure 5 illustrates the dependence of the parameter
`magnetization model. Here, the absolute value sN(H) is presented for the different models and used ferrofuids, respectively. (a)
` (ii) a polydisperse Debye-model (POLY) [Equation (4)],
`shows (i) a simple Debye-model (DEBYE);
`. Further, material parameters of the
`and a model introduced by Shliomis 1972 (S72);
`commercial ferrofluid APG933 and of a ferrofluid used in recent experiments [37] are considered. (b) illustrates the same
`variation for different ferrofluides APG933, APG935, APG936, and APG513A using either S72 (straight line) and DEBYE
`(dashed line). The variation between the different models is obvious and even becomes more striking with stronger magnetic
`field.
`
`The basic conclusion is that depending on the considered model describing the ferrofluid huge variation in the results may
`appear.
`STUDIED SYSTEMS - MAGNETIC FLUID CHARACTERIZATION
`
`www.scholarpedia.org/article/Ferrofluids
`
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`
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`
`M ≈
`=
`H = χH .
`M
`sat
`1
`3
`mH
`μ
`0
`T
`k
`B
`M
`sat
`3
`π
`μ
`0
`d
`¯
`3
`M
`0
`6 T
`k
`B
`d
`¯
`M
`0
`d
`¯
`||M −
`||
`M
`eq
`τ >> 1
`M −
`= Ω × H,
`M
`eq
`c
`N
`=
`for polydispersity ,
`c
`2
`N
`τ
`( +
`)
`1
`χ
`τ μ
`0
`H
`2
`6μΦ
`⎡
`⎣
`⎢
`⎢
`⎢
`⎢
`τ
`j
`( +
`)
`1
`χ
`j
`τ
`j
`μ
`0
`H
`2
`6μΦ
`⎤
`⎦
`⎥
`⎥
`⎥
`⎥
`(5)
`Φ
`( + u ⋅ ∇)u = (1 + ) u − ∇ −
`× ∇[(∇ × u) ⋅
`].
`∂
`t
`s
`2
`N
`∇
`2
`p
`M
`s
`N
`s
`N
`=
`H.
`s
`N
`c
`N
`2
`− −−
`√
`(6)
`(H )
`s
`N
`= χ, = 1/τ , κ = 0
`γ
`χ
`γ
`τ
`= χ, = 1/τ , κ = /(6Φ )
`γ
`χ
`γ
`τ
`μ
`0
`η
`~
`

`

`1/13/22, 4:34 PM
`This section provides
`insight of the study of
`ferrofluids in different
`systems. This cannot
`cover the large amount of
`literature available in
`this field, studying
`different systems,
`different ferrofluids,
`applications, and so on.
`The reader should get an
`impression of the way
`how ferrofluids in the
`presence of an external
`magnetic field can affect
`and modify ‘classical’
`flow dynamics.
`
`Ferrofluids - Scholarpedia
`
`Figure 5: Variation of the absolute value sN(H) of the magnetic field parameter s N(H) [Equations (5) and
`(6)] with H. (a) Different models DEBYE, POLY, and S72 with parameters of the commercial ferrofluid
`APG933 full line and of a ferrofluid used in recent experiments (Reindl et al. 2009) dashed. (b) Different
` (S72, straight line) and
`ferrofluides APG933, APG935, APG936, and APG513A for
`(DEBYE, dashed) (with relaxation time
`
`)
`
`Couette flow as prototypical system to study ferrofluids
`
`One prototypical system to investigate the influences of a
`magnetic field on magnetic fluids (e.g. ferrofluids) is the so-
`called Taylor-Couette flow, Taylor 1923 (Taylor-Couette system,
`TCS) (Figure 6) driven by the differential rotation of two
`concentric cylinders. Its simplicity and experimental
`accessibility has proven to be a good system to study various
`different aspects of fluid dynamics, bifurcation theory, and
`pattern formation. The geometrical system setup is particularly
`suited to studying rotating magnetic fluids, e.g. rotating
`ferrofluids. There are many theoretical and experimental
`analyses of the influence of magnetic fields in various
`configurations on the flow of a ferrofluid in Taylor-Couette
`system setup (Niklas 1987, Hart 2002, Altmeyer et al. 2010).
`
`Magnetorotational viscosity in the TCS
`As discussed above, magnetic fields may hinder of the free
`rotation of the particles in a shear flow and therefore change the
`viscosity of a magnetic fluid. The Taylor-Couette system can be
`used to determine the rotational viscosity in magnetic fluids.
`Among others, Odenbach and Müller 2002 studied the
`magnetorotational viscosity effect for different field
`configurations (radial, azimuthal, and axial) with the overall
`result that the rotational viscosity increases, independent the
`direction of the magnetic field and only varies in
`strength/amplitude (Odenbach and Pfister 2000). Further they
`proved that the investigation of field and shear rate dependent
`changes of viscosity in a ferrofluid provides an excellent tool to
`get insight into the microscopic reasons for the rheological
`properties of suspensions.
`
`Figure 6: Schematic of the Taylor-Couette system illustrating
`different external applied homogeneous magnetic fields;
`, axial
`, azimuthal
`, and transverse
`radial
`. Basic control parameters
`in TCS are the inner (index ) and outer (index ) Reynolds
`numbers (Taylor-Couette flow, Taylor 1923),
` and
`and
`, respectively
` are the non-dimensionalized inner and
`outer cylinder radii).
`
`www.scholarpedia.org/article/Ferrofluids
`
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`
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`TACTION EX2013 PAGE009
`
`κ = /(6Φ )
`μ
`0
`η
`~
`κ = 0
`τ = (<
`)
`τ
`B
`D
`3
`>
`1/3
`H
`r
`e
`r
`H
`z
`e
`z
`H
`θ
`e
`θ
`=
`(cos(θ) − sin(θ)
`)
`H
`x
`e
`x
`H
`x
`e
`r
`e
`θ
`i
`o
`R =
`d/ν
`e
`i
`ω
`i
`r
`i
`R =
`d/ν
`e
`o
`ω
`o
`r
`o
`= /( − )
`r
`i
`R
`i
`R
`o
`R
`i
`= /( − )
`r
`o
`R
`o
`R
`o
`R
`i
`

`

`1/13/22, 4:34 PM
`
`Ferrofluids - Scholarpedia
`
`Stabilization effect in the TCS
`Within all studies in the literature, either theoretical and experimental, a common observation is that any stationary external
`applied magnetic field, independent of its orientation (radial, axial, azimuth, transverse) stabilizes the basic state (Circular
`Couette flow) of the system (Niklas 1987, Altmeyer et al. 2010, Altmeyer 2018, Reindl and Odenbach 2011a, Reindl and
`Odenbach 2011b). Thus the bifurcation thresholds of primary bifurcating flow states become shifted to larger control
`parameters. How