# Ferrofluids

Post-publication activity

Curator: Sebastian Andreas Altmeyer

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.
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.

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.

## 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 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.

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).

This results in an effective relaxation time (Fannin and Charles 1988) $\frac{1}{\tau_{eff}}= \frac{1}{\tau_{B}} + \frac{1}{\tau_{N}}$ resulting from the Brownian relaxation time (Debye 1929) $\tau_B = \frac{\pi\tilde{\eta}}{2 k_B T} (D + 2s_H)^3$ and the Néel relaxation time (Néel 1949) $\tau_N = \frac{1}{f_0} exp{\left(\frac{\pi K D^3}{6k_B T}\right)}.$ Here $$\tilde{\eta}$$ stands for the viscosity of the carrier liquid, D for the particle diameter, and K for the anisotropy constant of the magnetic material, sH is the thickness of the surfactant, and $$f_0$$ is a frequency of the order of 10 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 constant K of the used ferrofluid. Viscosity $$\eta$$ and shell thickness sH essentially only affect the Brownian relaxation time, $$f_0$$ and only plays a role in the range of Néelian relaxation.

Figure 3: Variation of Brownian, τB, Néelian, τN, and effective relaxation time, τeff, of APG933 depending on particle diameter $$[D]$$.

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 experimentally detected values of $$f_0=$$ 109 Hz and sH = 2 nm are used for the Néel pre-factor and the shell thickness, respectively (Rosensweig 1985, 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).

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.

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 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) $\tag{1}M=M_{sat}(\coth(\alpha)-1/\alpha), \,\,\, \text{where} \,\,\, \alpha=\frac{\mu_0mH}{k_B T}$ 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 $$\mu_0$$ 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 $$\nabla \psi_{out}$$.

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 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-called transport coefficient $$\lambda_2$$ (Odenbach and Müller 2002, Altmeyer et al. 2013). Such a term also exists similar in the 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).

The transport coefficient $$\lambda_2$$ can be considered as a material-dependent function of thermodynamic variables such as density, 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 $$\lambda_2$$ 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
dynamic viscosity $$\tilde{\eta}$$ [Pa s] 0.5 1 1.5 2 0.128 0.5
density $$\rho$$ [103 kg/m3] 1.06 - 1.07 1.07 1.28 -
kinematic viscosity ν=$$\tilde{\eta}/\rho$$ [10-3 m2/s] 0.47 - 1.4 1.87 0.1 -
initial susceptibility $$\chi_0$$ 1.09 0.73 0.58 0.85 1.57 0.44
saturation magnetization Msat [kA/m] 18.15 16.4 17.89 19.01 32 6.51
volume fraction Φ [%] 4.1 3.7 4.0 4.3 7.2 1.5
average diameter (log-normal) $$D_0$$ [nm] 7 7.1 6.3 7 - 11
average diameter (volume) <D3>1/3 [nm] 8.7 8.5 7.6 8.4 10 12
standard deviation (log-normal) σD 1.47 1.42 1.44 1.42 - 1.27
Brownian relaxation time (experimental) τB [ms] 0.599 0.485 0.355 0.48 0.78 -

## FERROHYDRODYNAMIC EQUATIONS

### 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. $\tag{2}(\partial_t + {\bf u \cdot \nabla}) {\bf u} - \nabla^2 {\bf u}+ \nabla p = \alpha_1({\bf M}\cdot \nabla) {\bf H} + \frac{1}{2} \alpha_2 \nabla \times ({\bf M}\times {\bf H}),$ $\nabla \cdot {\bf u} = 0,$ with $${\bf u} = (u_1,u_2,u_3)$$ the velocity vector, H the magnetic field, and M the magnetization. Using characteristic time and length scales, the coefficients $$\alpha_1,\alpha_2$$ can be scaled to 1 in Equation (2) representing the non-dimensional ferrohydrodynamic equation 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 ${\bf M}_{eq} = M_{eq}(H){\bf H}/H$ or to use the relaxation of an effective field ${\bf H}_{eff} = M^{-1}_{eq}(H){\bf H}/H$ 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 $\tag{3}({\bf \Omega} + \kappa {\bf M} \times {\bf H}) \times {\bf M} = \gamma_\tau ({\bf M} - \gamma_\chi {\bf H}),$ with $$2 {\bf \Omega} = \nabla \times {\bf u}$$ the local vorticity and $$\kappa=\mu_0/(6\Phi\tilde{\eta})$$ . Both coefficients $$\gamma_\tau$$ and $$\gamma_\chi$$ are functions of H, M, and of some other material properties of the ferrofluid and model dependent.

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) $\tag{4}{\bf \partial}_t {\bf M}_j = {\bf \Omega} \times {\bf M}_j - \frac{1}{\tau_j}({\bf M}_j - {\bf M}^{eq}_j),$ with $${\bf M}^{eq}_j$$ 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 direction of the magnetic field, $${\bf M}^{eq}=\chi{\bf H}$$, where $$\chi$$ is the magnetic susceptibility of the ferrofluid, determined using Langevin’s formula (Langevin 1905). This is an approximation of Equation (1) considering small values α for weak magnetic fields $M\approx M_{sat}\frac{1}{3}\frac{\mu_0mH}{k_B T}= \frac{M_{sat}}{3} \frac{\mu_0 \pi \overline{d}^3 M_0}{6k_B T}H = \chi H.$ Here $$\overline{d}$$ is the mean diameter of the particles, $$M_0$$ is the spontaneous magnetization of the magnetic material, and χ is the initial susceptibility of the fluid. Typically, for particles with mean diameter $$\overline{d}$$ of about 10 nm and saturation magnetization of about 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 stationary magnetization near equilibrium with small $$||{\bf M}-{\bf M}_{eq}||$$ and small relaxation times $$\tau >> 1$$, and τ is the magnetic 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 ${\bf M} - {\bf M}^{eq} = c_N {\bf \Omega}\times {\bf H},$ where cN is the Niklas coefficient $\tag{5}c_N^2 = \frac{\tau}{\left(\frac{1}{\chi}+\frac{\tau \mu_0 H^2}{6 \mu \Phi}\right)} \qquad \left[\frac{\tau_j}{\left(\frac{1}{\chi_j}+\frac{\tau_j \mu_0 H^2}{6 \mu \Phi}\right)} \,\, \text{for polydispersity}\right],$ where μ is the dynamic viscosity, and $$\Phi$$ 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 $(\partial_t + {\bf u} \cdot \nabla) {\bf u} = (1+s_N^2) \nabla^2 {\bf u} -\nabla p_M - {\bf s}_N \times \nabla [(\nabla \times {\bf u}) \cdot {\bf s}_N].$ In this approach, the magnetic field and all the magnetic properties of the ferrofluid influence the velocity field only via the magnetic field parameter $\tag{6}{\bf s}_N = \sqrt{\frac{c_N}{2}} {\bf H}.$

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 ferrofluides APG933, APG935, APG936, and APG513A for $$\kappa=\mu_0/(6\Phi\tilde{\eta})$$ (S72, straight line) and $$\kappa=0$$ (DEBYE, dashed) (with relaxation time $$\tau=\tau_B(<D^3>^{1/3})$$)

Figure 5 illustrates the dependence of the parameter $${\bf s}_N(H)$$ on the magnetic field as well as the influence of the used magnetization model. Here, the absolute value sN(H) is presented for the different models and used ferrofuids, respectively. (a) shows (i) a simple Debye-model (DEBYE); $$\gamma_\chi=\chi, \gamma_\tau=1/\tau, \kappa=0$$ (ii) a polydisperse Debye-model (POLY) [Equation (4)], and a model introduced by Shliomis 1972 (S72); $$\gamma_\chi=\chi, \gamma_\tau=1/\tau, \kappa=\mu_0/(6\Phi\tilde{\eta})$$. Further, material parameters of the 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

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.

### Couette flow as prototypical system to study ferrofluids

Figure 6: Schematic of the Taylor-Couette system illustrating different external applied homogeneous magnetic fields; radial $$H_r {\bf e}_r$$, axial $$H_z {\bf e}_z$$, azimuthal $$H_\theta {\bf e}_\theta$$, and transverse $$H_x {\bf e}_x = H_x (\cos(\theta){\bf e}_r - \sin(\theta){\bf e}_\theta)$$. Basic control parameters in TCS are the inner (index $$_i$$) and outer (index $$_o$$) Reynolds numbers (Taylor-Couette flow, Taylor 1923), $$Re_i=\omega_i r_i d/\nu$$ and $$Re_o=\omega_o r_o d/\nu$$, respectively $$r_i=R_i/(R_o-R_i)$$ and $$r_o=R_o/(R_o-R_i)$$ are the non-dimensionalized inner and outer cylinder radii).

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.

#### 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. However, the amount of up-shift, i.e. strength in stabilization depends on the field orientation.

Figure 7: Stabilization effect of basic state in TCS for a ferrofluid in axial magnetic field. (a) Influence of an axial magnetic field on the location of the bifurcation thresholds of Taylor vortex flow and spirals out of basic state (Circular Couette flow, CCF) in the $$Re_i-Re_o$$ diagram. Blue (full) lines refer to Taylor vortices and orange (dashed) ones to spirals. Magenta line with points indicates the moving of the (bi-critical) co-dimension two point, $$\gamma$$, with sz towards smaller $$Re_o$$. Further shown are the bifurcation thresholds $$Re_{i,c}$$, with increasing sz2 for fixed (b) $$Re_o=-100$$ and (c) $$Re_o=0$$, respectively, obtained by full non-linear simulations. (© Altmeyer 2018)

Figure 7 provides an example illustrating the stabilization effect in the Taylor-Couette system for a pure axial magnetic field configuration. Increasing the magnetic field strength sz as (Niklas 1987) control parameter, the bifurcation thresholds move towards larger inner Reynolds number $$Re_i$$ (see also caption in Figure 6). This holds for both primary bifurcating solutions, toroidally closed Taylor Vortex flow (TVF) and helical spiral vortex flow (spiral, SPI), while the first is stable and the second is unstable close to onset. However, as the the strength of the stabilization is different for both of these structures, the bi-critical, co-dimension two point, $$\gamma$$, at which the primary bifurcating solution and thus the stability changes depends on sz (Altmeyer et al. 2010, Altmeyer 2018, Reindl and Odenbach 2011a).

#### Mode interaction and flow structure modification

The orientation of the applied magnetic field plays a crucial role when it comes to the flow structures itself. For such magnetic fields that also conserve the basic system symmetries (radial, azimuthal, axial) the flow structures remain unchanged, except in strength due to the former mentioned stabilization effect. In contrast, a symmetry breaking magnetic field, i.e. a transverse magnetic field, changes the scenario completely. Here the flow profiles are altered and all flow states (including the basic state) become inherently three-dimensional (Altmeyer et al. 2010, Altmeyer 2018, Reindl and Odenbach 2011b) including further modification in bifurcation thresholds and flow dynamics in general.

Figure 8: Schematic sketch of modes in the presence of different externally imposed (homogeneous) magnetic fields in Taylor-Couette geometry. Here, m represents the azimuthal, and n the axial mode index in a Fourier expansion. (a) Modes either in the absence of any magnetic field or in a pure axial magnetic field. (b) Pure transversal magnetic field with additional stimulation of $$m=\pm2$$ modes. (c) Superposition of axial and transversal fields (oblique) with even further finite $$m=\pm1$$ modes. Strength of mode amplitudes, the magnitudes are characterized from dark (large amplitudes) to bright (small amplitudes) colors (legend right). Red and blue lines indicate the dominant mode amplitudes of the flow structures. (© Altmeyer 2018)

While first theoretically/numerically predicted (Altmeyer et al. 2010) these flow modifications were confirmed in experimental studies (Reindl and Odenbach 2011b). Regarding the basic equations, the reason is based on a non-linear interaction of modes, which become stimulated in the presence of a symmetry breaking magnetic field. Figure 8 illustrates schematics for such non-linear mode stimulation in the presence of different magnetic fields.

Here the modes are obtained by solving the ferrohydrodynamic equations of motion by combining a standard, second-order finite-difference scheme in r with a Fourier spectral decomposition in θ and z together with (explicit) time splitting. $f(r,\theta,z,t)=\sum_{m=-m_{\max}}^{m_{\max}} \sum_{n=-n_{\max}}^{n_{\max}} f_{m,n}(r,t)e^{inkz} \,e^{im\theta},$ where k is the axial wavenumber. Here the more interested reader is referred to the corresponding literature, e.g. Partial differential equation in Scholarpedia.

While pure Taylor vortices and spirals live in the m-n-plane in a straight line with m=0 and m=1, respectively, additional modes may appear in the presence of a magnetic field. Which modes become additionally stimulated depends on the orientation of the applied magnetic field. The resulting effect onto the flow structures due to the additional stimulated modes is illustrated in Figure 9. While a pure axial magnetic field does not stimulate any further modes and thus leaves the flow structures unchanged (except of smaller amplitudes/strength due to stabilization effect) a finite symmetry breaking transversal component renders the flow to be inherently three-dimensional (cf. mode spectra in bottom row in Figure 9). The wavy-like characteristics are obvious. However, in the case of toroidally closed wavy vortices, their solutions are crucially different from the other classical higher order instability of wavy states in the Taylor-Couette flow (Akonura and Lueptow 2003). While classical wavy vortices are azimuthally rotating structures, wavy vortices introduced by a symmetry breaking magnetic field do not rotate and are pinned in phase. This is a very good example of how magnetically controllable fluids allow for the variation in dynamics and structure.

Figure 9: Visualization of flow structures in Taylor-Couette system for different field configurations. Top row: azimuthal vorticity isosurfaces η over two axial wavelengths (for visualization purposes) with pure axial magnetic field (same as in the absence of any magnetic field): (a) Taylor vortices at $$Re_o=0, Re_i=125$$, (b) spirals at $$Re_o=-150, Re_i=160$$, with pure transversal applied magnetic field: (c) wavy vortices at $$Re_o=0, Re_i=125$$, (d) wavy spirals at $$Re_o=-150, Re_i=160$$; and oblique applied magnetic field: (e) wavy vortices at $$Re_o=0, Re_i=150$$, and (f) wavy spirals at $$Re_o=-150, Re_i=190$$. Red (yellow) isosurfaces correspond to positive (negative) values as indicated. Bottom row: Mode amplitudes $$|u_{m,n}|$$ of the radial velocity field u corresponding to structures over the m-n-plane. Values are scaled with the maximum mode amplitude to be 1 (Taylor vortex flow $$|u_{m,n}| = 1$$). (© Altmeyer 2018)
Units and dimensionless parameters for some ferrofluids APG933, APG935, APG936 and APG513A from FerroTec used in Tayolor-Couette system.
Ferrofluid APG933 APG935 APG936 APG513A
diffusion time τD = d2 [s] 212.8 71.4 53.5 1000
velocity d/τD = ν/d [10-2 m/s] 4.7 14 18.7 1
pressure ρν2/d2 [Pa] 2.34 20.97 37.42 0.13
external magnetic field $$H_{ext}=(2\rho/\mu_0)^{1/2}(\nu/d)$$ [kA/m] 1.93 5.78 7.72 0.45
nondim. relaxation time $$10^3 \tau_B (<D^3>^{1/3})\tau_D$$ 1.8 12.5 27.1 0.13
saturation magnetization Msat/Hext 9.4 3.1 2.5 71.1
$$\mu_0(6\Phi\eta H_{ext}\tau)$$ 8.09 8.35 7.65 4.6

Table II gives an overview of units and dimensionless parameters (FerroTec) for different magnetite-based ferrofluids as used in the Taylor-Couette flow. As already seen for the basic ferrofluid data presented in table I, there is a relatively large range in the given numbers. The rheological configuration of the ferrofluid itself has significant influence and thus strong effects on dynamics, flow structures and properties in the system.

### Rayleigh-Bénard system to study ferrofluids

Another well known and extensively investigated hydrodynamic pattern forming system is the Rayleigh-Bénard system (Bodenschatz et al. 2000, Rayleigh-Benard convection), which consists of a horizontal fluid layer between two plates that are (classical) heated from below and cooled from above. The control parameter is the dimensionless Rayleigh number $$Ra$$ as a measure of the buoyancy force (Fb ∝ gΔT, with gravitational acceleration g and applied temperature difference ΔT across the layer). Becoming supercritical, i.e. $$Ra$$ is above the critical value $$Ra_c$$, the motionless fluid layer looses its stability against small perturbations and convection starts in the form of straight parallel rolls as they occur (in narrow channels with roll axes perpendicular to the long side walls).

Considering magnetic fluids (as ferrofluids) offers an alternative method to investigate periodic forcing into the system. Using a time-periodic and spatially homogeneous magnetic field Matura and Lücke 2009 observed a competition between stabilizing thermal and viscous diffusion against the destabilizing buoyancy and Kelvin forces. Studying the system stability they found the conductive state and particularly the type of response, which can be harmonic or sub-harmonic, is in general determined by the system parameters. The high-frequency limit coincides with the stationary stability boundary (using a mean magnetic Ra) while the stability boundary for low-frequency modulation is shifted. The conductive state gets stabilized. They concluded, that sub-harmonic response is not typical for ferrofluids because of their high Prandtl numbers.

## APPLICATIONS

There are various examples of scientific studies of ferrofluids that eventually found their way from the engineering field into industrial applications (Berkovsky & Bashtovoy 1996). Some of them even gained high commercial value. Some of these applications are now widely used and probably in all-day use by the the reader without knowing it. Here to mention are:

• Sealing of rotary shafts: a rotating shaft is surrounded by a magnet creating a strong magnetic field in the small gap between the shaft and the magnet. Thus the shaft is sealed by the ferrofluid placed in this region with low friction, and the magnetic forces can easily be strong enough to keep the fluid in place against pressure differences of about 1 bar. One prominent example that uses such sealing by ferrofuids are computer hard drives.
• Loudspeakers: for the loudspeaker system the generally present magnetic field is used to fix the ferrofluid in a way that helps to cool the voice coil.
• Damping technologies: the most prominent example for the industrial application of the magnetorotational viscosity effect of ferrofluids are found in the automotive industry. Due to fast and accurate modification of a magnetic field, a simple press on a button allows a car driver to switch between differnt damping setups and thus experience completely different driving characteristics.
• Turbulence control: the potential of turbulence control via use of magnetic fields and ferrofluids (Altmeyer et al. 2015) offers promising aspects for future applications. Thus, the additional option of applying a magnetic field provides further features for separation. For instance, using ferrofluids which are able to stick to a specific type of particles in fluid suspension, it is possible to separate such particles which do not differ in their specific weight.

## INTERDISCIPLINARY FEATURES

Based on the fact that suspensions of magnetic nanoparticles exhibit the possibility to modify their properties and to control their flow under the influence of external magnetic fields creates a research field which is strongly developing.

Moreover the overall field of ferrofluid research has a high interdisciplinary character, bringing engineers, experimental and theoretical physicists, chemists, applied mathematicians, and physicians together. Potential applications range from magneto-mechanical and magneto-optical devices, as well as fundamental insights in structures of such fluids and internal dynamic mechanisms. For instance the determination of a connection between the internal microstructure forced by interparticle interaction on the one side and their macroscopic properties on the other side make such magnetic fluids interesting model substances to understand the rheological behaviour of suspensions.

Besides pure scientific interest, deeper understanding of magnetic suspensions can also lead to the design of future applications which use the magnetic control of the properties of the fluid. First steps in this direction have already been successfully done (see above) with promising outlook into future developments.

A highly important branch of recent and especially future research on magnetic fluids is their use in biomedical applications. Here special focus lies on cancer treatment. First studies suggest that the magnetic hyperthermia or magnetic drug targeting seem to be promising new therapeutic techniques. For more information on these recent developments and applications as well as on ferrofluids in general, the interested reader is referred to the available literature (Rosensweig 1985, Blumes et al. 1997, Odenbach 2003).

The fact that the utilization of ferrofluid systems is not only restricted to technical applications but also useful for various medical treatments makes the field very attractive for future studies and challenging from a scientist’s point of view.

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## Internal references

• R. Benzi and U. Frisch (2010), Turbulence. Scholarpedia, 5(3):3439.
• P. Holmes and E. T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.
• G. Nicolis and C. Rouvas-Nicolis (2007), Complex systems. Scholarpedia, 2(11):1473.