## Abstract

The viscosity of ferrofluid has a vital role in liquid sealing of the hard disk drives, biomedical applications as drug delivery, hyperthermia, and magnetic resonance imaging. The theoretical expressions for ferrofluid viscosity and its limitation are presented in detail in this article. A qualitative comparison of the theoretical and experimental viscosity results is also shown. In the absence of a magnetic field, the viscosity of ferrofluid depends on the volume concentration of magnetic nanoparticles, including surfactant layers. However, the viscosity of ferrofluid due to stationary magnetic field depends on the angle between the applied magnetic field and vorticity in the flow. If this angle is 90 deg, then there is a maximum increase in the viscosity. When field frequency matches with the relaxation time, known as resonance condition, then there is no impact of an alternating magnetic field in the viscosity of ferrofluid. If the frequency of an alternating magnetic field is less than resonance frequency, then an alternating magnetic field increases the viscosity of ferrofluid. For diluted ferrofluid, present theoretical results of viscosity have good agreement with the experimental data.

## 1 Introduction to Ferrofluids

### 1.1 Formation of Ferrofluid.

Ferrofluid is also known as magnetic fluid. Ferrofluids are not directly available in nature [1]. These fluids are artificially synthesized of colloidal mixtures of carrier liquid, typically water or oil, and magnetic nanoparticles [1,2]. Surfactants are used in the colloidal mixtures to ensure the stability of the ferrofluid [1,3]. Surfactants prevent the aggregation of magnetic particles [4,5]. Ferrofluids, which work at zero gravity region, are recognized as colloidal suspension of superparamagnetic materials [1,4,5]. In ferrofluid preparation, we consider the size of the magnetic particles 5–15 nm in diameter and volume fraction up to approximately 10% [1,3]. We select water-based carrier liquid for medical purposes, mineral oil and silicon organic-based carrier liquid for lubrication and sealing system, and hydrocarbon-based carrier liquids for printing devices [4,5]. Properties of the ferrofluid depend on the size of the magnetic particles and their magnetization [1,3]. The stability of ferrofluid is ensured by the thermal motion, which prevents aggregation and precipitation [4,5]. Thermal motion increases with decreasing the size of the particles. The magnetic properties disappear if the size of magnetic particles is less than 1–2 nm [4,5]. Long-chain molecules (e.g., OOH, H2OH, H2NH2, and so on) are used for surfactants to ensure the stability of ferrofluids [4,5]. Surfactants produce a chemical reaction in the colloidal mixture, and this reaction reduces the size of the magnetic particles [5]. Reduction of the size of magnetic particles loses magnetic properties [5].

For the application of ferrofluid, it is essentially required that ferrofluid should be very stable concerning the temperature in the presence of a magnetic field [6]. Therefore, agglomeration of the magnetic particles must be avoided for proper commercial use [6]. Kikuchi et al. described experimentally that the reaction temperature from 200 °C to 250 °C, the size of the magnetic particles increases from 5 nm to 11 nm [7]. For the nonhomogeneous distribution of pH and dielectric constant, the microemulsion method is useful to prepare stable ferrofluids [8]. Imran et al. synthesized highly stable ferrofluid using motor oil as base fluid and found 13 nm average particle size of γ-Fe2O3 [9]. The stability of ferrofluid for small particle size, concentrations, and dipolar coupling energies are too low for field-induced dipolar structure formation [10]. Lalas and Carmi investigated the stability of motionless ferrofluid using the concept of Rayleigh number [11]. Tari et al. have investigated the role of magnetization and temperature for the stability of diester-based Fe3O4 ferrofluid [12]. In the transition from the laminar to turbulence motion of ferrofluid, we need to check the accuracy of the numerical solution along with the stability and uniqueness [13]. Thermal and magnetic stress is an important class of surface interactions of magnetic particles near Curie temperature [14].

Bolshakova et al. investigated the stability of ferrofluid in the presence of a non-uniform magnetic field using the method of magnetic particles redistribution process [15]. Internal structures and macroscopic physical properties [16] and arrangements of the magnetic particles in special structures [17] can serve in the development of the applications of magnetic fluid. There is no suitable procedure to explain the thermodynamical and dynamical properties of magnetic fluids with developed microstructure [18]. There is a critical chain number for phase transition in ferrofluid [19], and the magnetic field exhilarates the formation of chains from ferro-particles [20]. Wiedenmann investigated the stability of nanoparticles in ferrofluid against electrostatic repulsion or surfactants [21]. Gazeau et al. demonstrated the Brownian motion nanoparticles in ferrofluid under an applied magnetic field [22]. Sousa et al. investigated the surface magnetic properties of NiFe2O4 nanoparticles [23]. Raikher et al. demonstrated a magneto-optical way to analyze the internal and external magnetic relaxations in magnetic fluids [24]. Raikher et al. explained the particle orientation dynamics using the general Fokker–Planck equation [25]. The synthesis of ferrofluid, stability, and characteristics with magnetic properties has been investigated for different types of magnetic nanoparticles [28]. The theoretical results of viscosity do not include the role of types of surfactant used in the formation of ferrofluid. However, Borin et al. used the mixed surfactants in the formation of ferrofluid and observed that the nature of surfactant has a higher impact on the zero-field viscosity [29]. However, if the particle size is greater than 15 nm, then the shapes of the nanoparticles may influence the agglomeration [30,31].

### 1.2 Real-Life Applications of Ferrofluids.

In the mid-1960s, ferrofluid was developed by NASA as a method for controlling fluid in space since the flow of ferrofluid can be controlled by the external magnetic field. Conventional ferrofluids are useful in liquid seals, shock absorbers, controlling heat in loudspeakers, printing for paper money, and lubrication bearings [5]. The commercial use of ferrofluids has been available in the literature [32,33]. Ferrofluids can be used in heat transfer, sealing, damping, and lubrication [38]. The researchers investigated the performance of ferrofluids for lubrication [41]. The viscosity of magnetic fluid influences the strength and performance of the lubricant film [42,43]. Ferrofluids have an important role in biomedical applications for diagnostic and therapy, drug delivery, hyperthermia treatments, and magnetic resonance imaging [46]. Fundamental and applied research in the ferrofluid, nowadays, researchers are trying to develop magneto-optical devices and ferrofluidic sensors [47,48]. Even researchers have shown the application of ferrofluid in environmental engineering [49]. Ferrofluid actuation can control the flow of ferrofluid for applications in microfluidic pumping and targeting drug delivery [50]. Kole and Khandekar described the existing and emerging applications of ferrofluid based on element designing, biomedical, and thermal engineering [51].

### 1.3 Magnetic Field and Viscosity of Ferrofluid.

The main aim of this work is to demonstrate the influence of the magnetic field on the rotational viscosity of ferrofluid. Figure 1 shows the influence of magnetic field intensity on the rotational viscosity. If the vorticity and the applied magnetic field are collinear, then there is no impact of the magnetic field on the viscosity since the particle can rotate around the magnetic moment direction [52]. In this case, the viscosity of ferrofluid will be the same as zero-field viscosity. A finite angle between vorticity and magnetic field creates a hindrance of free rotation of the particles [53]. If the field is perpendicular to the vorticity, then there would be a maximum change in the rotational viscosity [54,55].

Fig. 1
Fig. 1
Close modal

In a quiescent ferrofluid (Ω = 0), the applied magnetic field does not rotate the magnetic particles in any preferable direction. Any vortex flow generates the rotational viscosity in the presence of a magnetic field. In vortex flow, a constant magnetic field rotates the particle slower than fluid, therefore rotational viscosity becomes positive [52,53,56]. However, in the presence of an alternating magnetic field, the rotational viscosity depends not only on the field strength but also on the frequency of the alternating magnetic field [57]. A slow oscillating magnetic field decreases the angular velocities of the magnetic particles and it becomes less than the angular velocity of the fluid [58]. Therefore, in a slow oscillating magnetic field, the rotational viscosity is positive. In a fast oscillating magnetic field, the magnetic particles rotate faster than the angular velocity of the fluid [57]. In this case, the rotational viscosity becomes negative. It is proved experimentally that for H ≈ 2000 Oe, the viscosity decreases from 220 cP to 58 cP at 700 Hz frequency [57]. Considering the above facts of rotational viscosity, we describe the mathematical procedure to understand these facts. Finlayson theoretically investigated the influence of spin viscosity and the Langevin function in a rotational flow of ferrofluid [59].

## 2 Governing Equations in Ferrohydrodynamic Flow

To describe the behavior of ferrofluid in different flow domains, the researchers use the following set of equations [58,60,61]:

The equation of continuity [1,57,62]
$∇⋅v=0$
(1)

In the above equation, v denotes the velocity of the fluid. Equation (1) represents the continuity equation of fluid mechanics for incompressible liquid. Ferrofluid can usually be assumed as an incompressible liquid in which density ρ is constant.

The equation of motion [57,58,63,64]
$ρnfdvdt=−∇p+μnf∇2v+μ0(M⋅∇)H+I2τs∇×(ωp−Ω)$
(2)

In the above equation, ρnf denotes the density of ferrofluid, μnf denotes the viscosity of ferrofluid, μ0 denotes the permeability of free space, M denotes the magnetization, H denotes the magnetic field intensity, τs denotes the rotational relaxation time, ωp denotes the angular velocity of magnetic particles in the flow, and $Ω=12(∇×v)$ denotes the vorticity in the flow.

In Eq. (2), $ddt=∂∂t+v⋅∇$ represents the convective derivative, and this expression measures the rate of change of the mass motion v. On the right-hand side of the momentum equation, $−∇p$ denotes the pressure gradient, $μ0(M⋅∇)H$ denotes the magnetization force in the presence of an applied magnetic field. The expression $I2τs∇×(ωp−Ω)$ denotes the viscous torque due to the difference between the average angular velocities of the magnetic particles and the angular velocity of the fluid.

The equation of magnetization [3,60,65]
$dMdt=ωp×M−1τB(M−M0)$
(3)
The equation of instantaneous magnetization [5]
$M0=nmL(ξ)HH,ξ=mH(t)kBT,L(ξ)=cothξ−ξ−1$
(4)
where M0 denotes the instantaneous equilibrium magnetization, H denotes the magnitude of magnetic field intensity, τB denotes Brownian relaxation time, n denotes the number of particles, m denotes the magnetic moment, L(ξ) denotes the Langevin function for paramagnetism, kB denotes the Boltzmann constant, and H(t) denotes the time-dependent magnetic field. Instantaneous equilibrium magnetization M0 in a given H(t) exists at τB = 0. The real magnetization M is different from M0 due to non-zero Brownian relaxation time τB.
The equation of rotational motion [57,58,66]
$Idωpdt=M×H−Iτs(ωp−Ω)$
(5)
where I denotes the sum of particles moment of inertia over unit volume. The rotational relaxation time τs is very small. Therefore, the inertial term is negligible in comparison to the relaxation term, i.e., $Idωpdt≪Iωpτs$ which reduces the above equations as [57,58,65]
$ωp=Ω+τsI(M×H)$
(6)
Using Eq. (6), the momentum equation can be written as [55,57,58,65,67]
$ρnfdvdt=−∇p+μnf∇2v+μ0(M⋅∇)H+12∇×(M×H)$
(7)
The energy equation [4,5,64,68]
$ρnf(cp)nf[∂T∂t+(v⋅∇)T]=knf∇2T−μ0T∂M∂Tv⋅∇H+μnfΦ$
(8)
where T denotes the temperature, (cp)nf denotes the specific heat at constant pressure, knf denotes the thermal conductivity of ferrofluid, and Φ denotes the viscous dissipation term.

The energy equation (Eq. (8)) and the momentum (Eq. (7)) are important to study the coupling of new ferrohydrodynamic models. The magnetization in the momentum equation depends on the magnetic field intensity and temperature [64]. Therefore, the ferrohydrodynamic flow should be investigated with the energy equation. The velocity and magnetization in the energy equation represent the heating due to adiabatic magnetization.

Two mechanisms of ferrofluid Neel relaxation and Brownian relaxation time have an important role in the study of ferrofluid [69]. This mechanism shows that the magnetization in ferrofluid can relax after changing the strength of the magnetic field [70]. Brownian relaxation time occurs due to nanoparticle rotation of the colloidal mixture and Neel relaxation time occurs due to the rotation of the magnetic vector within the particle [1,5].

A Brownian relaxation time τB is given by [1]
$τB=3μnfVkBT$
(9)
where $V=π(dm+2s)36$ denotes the hydrodynamic volume of the particle including surfactant layers. Here, dm denotes the diameter of the magnetic core and s denotes the thickness of the surfactant layer.
Under certain material conditions, the magnetic moment may rotate inside the particle relative to the crystal structure [58,71]. This kind of relaxation of the colloidal particles can occur if the thermal energy is high enough to overcome the energy barrier provided by the crystallographic anisotropy of the magnetic material [4,5]. The height of this energy barrier is given by KV, where K is the anisotropy constant of the material [3,5]. For the case KVkBT, the thermal energy is large enough to induce fluctuations of the magnetization inside the grain with a characteristic time τN
$τN=1f0exp(KVkBT)$
(10)
where f0 is a frequency having the approximate value 109 s−1.
When τNτB, relaxation occurs by the Neel mechanism, and the material is called to possess intrinsic superparamagnetism [2,5]. When τBτN, the Brownian mechanism is determined and the material exhibits extrinsic superparamagnetism [3,5]. However, if the smaller time constant is much greater in comparison to the time scale of the experiment, then the same may be regarded as ferromagnetic [2,71]. An effective relaxation time combined from Neel and Brown times for the relevant particle diameter can be calculated as [1,3,58]
$τeff=τBτN(τB+τN)$
(11)
For the specific types of ferrofluid, the researchers are using the thermophysical properties of ferrofluid. The following mathematical equations are being used by the researchers [7276]:
$ρnf=ρf[(1−φ)+φ(ρsρf)]$
(12)
$(ρcp)nf=(ρcp)f[(1−φ)+φ(ρcp)s(ρcp)f]$
(13)
$μnf=μf(1−φ)2.5$
(14)
$knfkf=ks+2kf−2φ(kf−ks)ks+2kf+φ(kf−ks)$
(15)
where ρf the density of the base fluid, φ denotes the volume concentration of nanoparticles, (ρs, ρf) denotes the density of nanoparticles and base fluid, respectively, ((ρcp)s, (ρcp)f ) denotes the heat capacitance of solid and base fluid, respectively, μf the dynamic viscosity of the base fluid, and (ks, kf) denotes the thermal conductivity of nanoparticles and base fluid, respectively.

## 3 Viscosity of Ferrofluid

### 3.1 Viscosity in the Absence of the Magnetic Field.

In the absence of a magnetic field, the viscosity of ferrofluid depends on the volume concentration. The mathematical expression for the viscosity of ferrofluid is given as [3,7779]
$μnf=μf(1+52φ~),φ~=φ(dm+2sdm)3$
(16)
μnf denotes the viscosity of ferrofluid in the absence of the magnetic fluid, μf denotes the viscosity of the base fluid, $φ~$ denotes the volume concentration of magnetic nanoparticles including the surfactant layer, where φ denotes the volume concentration of the magnetic nanoparticles, dm denotes the diameter of the magnetic core, and s denotes the thickness of the surfactant layers.

Equation (16) represents the linear relation between the viscosity of magnetic fluid and volume concentration. This expression of viscosity is valid for $φ≤2%$ [8082].

The volume concentration $φ~$ of the suspended material in the colloidal suspension can be expressed as:

In 1970, the first improvement Eq. (14) was given as [3,83]
$μnf=μf(1+52φ~+315φ~2)$
(17)

The above expression considers the interaction between particles within very low volume concentration [80]. However, the ferrofluids are synthesized in 5–30% volume concentrations [61].

In 1985, Rosensweig modified the expression for the viscosity of ferrofluid as [1]
$μnf=μf(1−52φ~+bφ~2)$
(18)
where $b=(52φ~c−1)φ~c2$ and $φ~c$ denotes the critical volume fraction of the suspended material.

Equation (18) measures the viscosity of ferrofluids and considers volume concentration up to the quadratic term [1]. The linear terms of volume concentration in Eqs. (18) and (16) are the same.

Recently for the viscosity of the solution, the researchers use the following expressions [8487]:
$μnf=μf(1−φ~)2.5$
(19)

Expanding the expression in Eq. (18) using binomial theorem, the higher-order term of volume concentration can also be considered in calculating the viscosity. This expression gives good results up to the 40% volume concentration [80,81]. However, the diameter of the magnetic core and surfactant layers are also the key factors in the viscosity of ferrofluids. In the absence of a magnetic field, the viscosity variation in ferrofluid follows Newton’s law. It can be assumed as Newtonian viscosity of ferrofluid.

### 3.2 Viscosity of Ferrofluid in the Presence of Magnetic Field.

In the presence of a magnetic field, the rotation of the magnetic particle was also considered in the viscosity of ferrofluid. In 1969, researchers had introduced the theoretical expressions for the viscosity of ferrofluid under the influence of an external magnetic field [88]. This expression of viscosity depends on the strength and direction of the magnetic field. This expression is [5,89]
$Δμ=μnf(1+52φ~+32φ~sin2ε1)$
(20)
where Δμ denotes the additional rotational viscosity in the presence of the magnetic field.
The term $sin2ε1$ includes the magnetic part in the following form [5]:
$sin2ε1=12(1+1ξr2)−[14(1+1ξr2)2−1ξr2sin2β]12$
(21)
where β denotes the angle between the vorticity of the flow and the magnetic field direction and ξr denotes the ratio of the magnetic torque and the viscous torque acting on a particle. The relation between ξr and magnetic field can be written as [4,5]
$1ξr=μ0mH4πμnfd3γr$
(22)
where γr denotes the shear rate, d denotes the mean diameter of the magnetic particles.
In a planer Couette flow, the additional viscosity due to magnetic field is given as [3,52,56,90,91]
$Δμ=μ0τBM0H4(1+μ0τsτBM0HI)$
(23)
where τs, τB, and I are defined as
$τs=d2ρnf15μnf,τB=πd3μnf2kBT,I=25d2ρnfφ~$
(24)
Using Eqs. (4) and (24), the expression for the rotational viscosity in Eq. (23) becomes [3,56,57,66,92]
$Δμ=32φ~μnfξ−tanhξξ+tanhξ$
(25)
where $ξ=mHkBT$ denotes the Langevin parameter (the ratio of the energy of particle’s magnetic moment to thermal energy).
Equation (25) shows the expression for the rotational viscosity due to the magnetic field when the magnetic field is perpendicular to the vorticity in the flow. For arbitrary, angle between magnetic field and vorticity, Eq. (25) can be written as [4,57,58]
$Δμ=32φ~μnfξ−tanhξξ+tanhξsin2β$
(26)
Using Eq. (25), Bacari et al. demonstrated the theoretical expression for the viscosity [57,58,93]
$Δμ=32μnfφ~Ω−ωpΩ$
(27)

The main physical reason for the appearance of rotational viscosity is the difference between vorticity and angular velocity of the magnetic particles. In the absence of the magnetic field, the angular velocity of magnetic particles is equal to the vorticity in the flow, i.e., Ω = ωp. In this case, the rotational viscosity Δμ becomes zero. A constant magnetic field impedes the free particle rotations so that the angular velocity of the particles is always less than the vorticity in the flow, i.e., ωp < Ω. In this case, rotational viscosity is always positive. However, under the influence of an alternating magnetic field, a fast oscillating magnetic field rotates the magnetic particle faster than fluid, i.e., ωp > Ω. In this situation, the rotational viscosity becomes negative.

Magnetic torque M × H and viscous torque (ωpΩ) in ferrofluid flow generate the rotational viscosity in ferrofluids [1,56,57]. In the presence of the magnetic field, the fluid and particles in the colloidal suspensions rotate with different angular velocities, and this difference of angular velocities creates an additional resistance in the flow. The equilibrium of these two torques gives [54,57,94]
$μ0M×H=6μnfφ~(ωp−Ω)$
(28)
For mean magnetic torque, we get [57,58,94]
$μ0M×H¯=−6μnfφ~ξ−tanhξξ+tanhξΩ$
(29)
Using Eq. (27), the expression $12∇×(M×H)$ in the momentum equation (Eq. (7)) can be expressed as
$12∇×(M×H)=−3μnfφ~ξ−tanhξξ+tanhξ(∇×Ω)=−32μnfφ~ξ−tanhξξ+tanhξ(∇×∇×v)=32μnfφ~ξ−tanhξξ+tanhξ∇2v$
(30)

Thus, in ferrohydrodynamic flow, the expression $12∇×(M×H)$ creates an additional viscosity due to the applied magnetic field. In the presence of a stationary magnetic field, the total viscosity is μnf + Δμ, where Δμ denotes the rotational viscosity. The additional viscosity depends on the strength of the magnetic field, which creates a difference between angular velocities between fluid and magnetic particles. The volume concentration of nanoparticles, diameter of the magnetic core, and thickness of the surfactant layer also affect the rotational viscosity.

The relative viscosity can be presented as [3,61,95]
$R=Δμμnf$
(31)
In the presence of strong the magnetic field [5]
$R(H→∞)=32φ~sin2β$
(32)
If the magnetic field is perpendicular to the vorticity in the flow, then the maximum relative viscosity is [3,96]
$Rmax=32φ~$
(33)
If the particle size is 10 nm including the surfactant layer, the maximum increase in the viscosity of ferrofluid is approximately 40%. In a weak magnetic field, we consider $tanhξ=ξ−13ξ3+0(ξ5)$, therefore
$ξ−tanhξξ+tanhξ≈16ξ2$
(34)
A weak magnetic field (ξ ≪ 1) represents the following expression of relative viscosity of ferrofluid
$R≈14φ~ξ2$
(35)

Researchers have used these expressions of viscosity in ferrohydrodynamic flow in different regimes [63,97101].

### 3.3 Negative Viscosity Effects in Ferrofluid.

A stationary magnetic field always increases the viscosity of ferrofluid and it depends on the strength of the magnetic field. In an alternating magnetic field, the viscosity of ferrofluid depends on the strength and frequency of an alternating magnetic field. The angular velocity of the magnetic particle can be written in terms of vorticity and field frequency [58,102,103]
$ωp=Ω(1−ξ26(1−ω02τB2)(1+ω02τB2)2)andξ=μ0mHkBT$
(36)
where ω0 denotes the frequency of an alternating magnetic field.
For the weak field (mHkBT), the viscosity of ferrofluid is [58,104]
$Δμ=14μnfφ~ξ2(1−ω02τB2)(1+ω02τB2)2$
(37)
Here, ω0τB denotes the dimensionless field frequency. The condition ω0τB = 1 is known as the resonance condition. This condition can be achieved when the frequency of alternating magnetic fields matches with relaxation time. In this case, there is no impact of rotational viscosity due to the magnetic field [57]. A case ω0τB < 1, the expressions in Eq. (37) remain positive. This case always enhances the viscosity of ferrofluid due to applied magnetic field. For the case, ω0τB > 1, the expressions in Eq. (37) become negative [57,58]. In other words, after applying a magnetic field, the viscosity of ferrofluid becomes less than without a magnetic field. This viscosity reduction is known as the negative viscosity effect. If we take, ω0τB → ∞, the impact of Eq. (37) in the viscosity of ferrofluid becomes negligible [58].
The viscosity of ferrofluid for arbitrary amplitude is [58]
$Δμ=14μnfφ~ξ2(2−tanhε−2tanhεε);ε=π2ω0τB$
(38)

Considering limit $ε→∞$, Eq. (38) becomes $Δμ=14μφ~ξ2$ (viscosity due to stationary magnetic field). It is to notice that for a strong magnetic field (ξ ≫ 1), the expression $14μnfφ~ξ2$ can be replaced by $32μnfφ~$.

The researcher validated the theoretical and experimental results of the negative viscosity, and it is found that at 130 Hz the viscosity of ferrofluids becomes negative [57]. These theoretical expressions show the experimental evidence of negative viscosity [57,58,105]. Zeuner et al. demonstrated the qualitative agreement with the theoretical results of rotational viscosity [106]. The quantitative data of the theoretical viscosity could not match exactly due to the variation in the diameter of the magnetic core and the thickness of the surfactant layer.

To avoid the discrepancies between theoretical and experimental results of the rotational viscosity, there is a need to improve the experiments and theoretical expressions in the rotational viscosity. From a theoretical point of view, distribution of particle size, the influence of Brownian and rotational relaxation time, variation in the thickness of the surfactant layer, and variation in angular velocities of the particles should also be included in the theoretical expressions of the rotational viscosity. The preparation of stable ferrofluids and machine error should be considered during experimental results of rotational viscosity of ferrofluids.

## 4 Comparison of Theoretical and Experimental Results of Rotational Viscosity

In the theoretical expressions for rotational viscosity, it is assumed that the ferrofluid contains spherical particles of uniform size which do not interact [53,54]. Further, in theory, we assume that the magnetic moment is fixed within the particles and the particles cannot rotate relative to the crystal structure [54]. However, this assumption is valid for magnetically weak particles since these particles allow internal rotation of the magnetic moment. The experimental results show that in highly concentrated ferrofluids or the magnetic particles having strong magnetic dipole-dipole interaction, the formation of chains or clusters of magnetic particles has been observed. According to McTague experiment [89], theoretical results of rotational viscosity have good agreement with the experimental results with 10 nm size of magnetic particles with a volume concentration of 0.05%. This agreement between results was obtained since the particle size was small enough to prevent the interaction of the particles [107]. If the ferrofluid is highly concentrated or the mean diameter of the magnetic particle is greater than 15.8 nm then interparticle interaction should be considered in the rotational viscosity in the presence of the magnetic field [53,108,109]. Odenbach experimented to investigate the chain formation and its influence on the rotational viscosity [108,110]. In the first case, ferrofluid is synthesized with a small volume of larger particles, and in the second case; it is synthesized with a larger volume of small particles. It was demonstrated that larger particles form chain in the presence of a magnetic field and influence the rotational viscosity [53]. To include the influence of chain formation in the theoretical model, the following expressions are given [53,108]:
$nc≈μ018μnfγrM0d3(d+2s)3$
(39)
where nc denotes a maximum length of the chains which is stable in a shear flow with shear rate γr, M0 denotes the spontaneous magnetization of magnetic particles.

The theoretical expressions of the rotational viscosity do not depend on the shear rate. However, most of the experimental results of rotational viscosity are published with different shear rates [111113]. Rosa and Chuna demonstrated that the dipolar interaction of the magnetic nanoparticles increases the rotational viscosity and magnetization in ferrofluid [114]. Odenbach suggested that considering $γ˙=0.05s−1$, the theoretical results of magnetoviscosity can be compared [53]. Odenbach compared the theoretical and experimental results of rotational viscosity of ferrofluid and found that experimental rotational viscosity was higher than the theoretical results [115]. It was concluded that the theoretical results of the rotational viscosity do not depend on the shear rate and the experimental results are obtained for different shear rates [115,116]. Increasing the shear rate reduces the agglomeration of the particles which is the reason for the viscosity reduction at a high shear rate [117]. However, Ambacher et al. already proved that the theoretical results of the rotational viscosity have a good agreement with the experimental results for diluted ferrofluid [61]. In the case of concentrated ferrofluid, theoretical results do not match with the experimental results [61]. Bacri et al. investigated the experimental evidence of negative rotational viscosity and obtained Δμ = −19 cP in the presence of an alternating magnetic field using a frequency greater than 7000 Hz [57]. The quantitative comparison between the theoretical and experimental results is difficult since in the presence of a magnetic field it is difficult to ensure the stability of ferrofluid and theoretical analysis of experimental magnetic field [118]. Odenbach et al. considered the role of interparticle interaction in the rotational viscosity and used Eq. (38) to validate the theoretical results with experimental findings [119]. Rosa and Cunha demonstrated that the theoretical results, simulated results, and experimental results of rotational viscosity are identical at low magnetic field intensity [120]. When the results were obtained for higher magnetic field intensity, the theoretical results of the viscosity are 30% lower than experimental results [120].

Figure 2 demonstrates the effect of the angle between magnetic field intensity and vorticity on the relative viscosity. This result is obtained from Eq. (26) for different values of β. In this graph, we consider 10 nm diameter of the magnetic core, 2 nm thickness of the surfactant layer, the dimensionless magnetic field intensity ξ = 2, and volume concentration up to 10%. Variation in the direction between the magnetic field and vorticity from $30deg$ to $45deg$ enhances the rotational viscosity by 41.41% approximately. The variation in the direction from $45deg$ to $60deg$ and $60deg$ to $90deg$, the enhancement in the viscosity is approximately 22.48% and 12.71%, respectively. Whenever the magnetic field and vorticity are parallel to each other, the expression for rotational viscosity in Eq. (26) becomes zero and the magnetic field does not create any additional viscosity in the flow.

Fig. 2
Fig. 2
Close modal

Figure 3 demonstrates the theoretical dependence of the rotational viscosity for different values of the dimensionless magnetic field. In this case, we consider the dilute ferrofluid with a 10 nm diameter of the magnetic core and 2 nm thickness of the surfactant layer. At 7% volume concentration, if we enhance the dimensionless magnetic field intensity from ξ = 2 to ξ = 5, the relative rotational viscosity enhances approximately 86.24%. At the same concentration, the enhancement in ξ from 5 to 8 and 8 to 11 increases the relative viscosity by approximately 16.66% and 5.98%, respectively. In the presence of a stationary field, when all particles are aligned in the direction of the magnetic field then enhancement in the magnetic field does not make any more any enhancement in the viscosity of ferrofluid [121123]. Figure 4 demonstrates the reduced rotational viscosity profile with the variation of dimensionless magnetic field intensity. This graph is plotted using Eqs. (26) and (32) and the applied magnetic field is considered perpendicular to the vorticity in the flow. At the initial stage, increasing the magnetic field intensity increases the rotational viscosity and for high magnetic field intensity, it tends to saturate. This data represents a good agreement with the experimental results published in previous research papers [122,124,125]. Figure 4 in Ref. [125] shows the experimental results for reduced rotational viscosity. The results, however, are validated by the theoretical calculations shown in Fig. 4. Table 1 displays the reduced viscosity values for various values ξ. It is derived from Fig. 4. It would be simpler to compare the reduced viscosity values given in Ref. [125].

Fig. 3
Fig. 3
Close modal
Fig. 4
Fig. 4
Close modal
Table 1

Reduced rotational viscosity at different dimensionless magnetic field intensity

ξΔμ(ξ)/μ(∞)
00
10.13534380
20.35878878
30.50783578
40.60415112
50.66944306
60.71631416
70.75155276
80.77777779
90.79999999
100.81818181
110.83333332
120.84615376
ξΔμ(ξ)/μ(∞)
00
10.13534380
20.35878878
30.50783578
40.60415112
50.66944306
60.71631416
70.75155276
80.77777779
90.79999999
100.81818181
110.83333332
120.84615376

## 5 Conclusions

This review on the viscosity of ferrofluid has presented the recent fundamental theoretical development of the viscosity of ferrofluid. The viscosity of ferrofluid has a significant role in the application of ferrofluid in sealing, biomedical engineering, and heat transfer analysis. The rotational viscosity of ferrofluid depends on the difference between the vorticity in the flow and rotation of the magnetic nanoparticles in the presence of a magnetic field. The magnetic field can be directed perpendicular to the vorticity in the flow to enhance the maximum viscosity. On the other hand, an alternating magnetic field with a higher frequency than resonance frequency can reduce the viscosity of ferrofluid. Recently, researchers have been publishing research papers on different types of magnetic nanofluids. Still, some of them did not consider the impact of viscosity due to the applied magnetic field. For more realistic results for which theoretical and experimental results can coincide, the theoretical expressions of the ferrofluid viscosity should be considered in the computational work.

The experimental results on the rotational viscosity match the theoretical results for dilute ferrofluids. In concentrated ferrofluids, there are discrepancies between theoretical and experimental results. If the size of the magnetic particles is greater than 15.8 nm, then the particle interaction is strong and it allows the chain formation. The size distributions of the magnetic nanoparticles are not uniform; however, the theoretical results of viscosity consider the uniform size distribution of the particles. Theoretical results of the viscosity do not depend on the shear rate and experimental results are obtained with different shear rates. Therefore, the exact comparison of the data is not possible. Experimentally it is difficult to measure the exact angle between vorticity and magnetic field. The theoretical expressions of the rotational viscosity represent the ideal situation and the experimental results show the real situation. Therefore, we need to address the above issues in the theoretical expressions of the viscosity of ferrofluid for quantitative comparison. The following issues must be addressed in theoretical expressions of ferrofluid viscosity:

• Particle size distributions should be considered non-uniform when calculating viscosity. However, it may make computation more difficult.

• The chain formation effect should be considered in theoretical expressions at higher magnetic particle concentrations.

• Shear rate must be included in the main expression of rotational viscosity in order to provide a more accurate experimental comparison.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

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