Theoretical Development in the Viscosity 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].

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].

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].

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].

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

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.

In the above equation, ρ_nf denotes the density of ferrofluid, μ_nf denotes the viscosity of ferrofluid, μ₀ denotes the permeability of free space, M denotes the magnetization, H denotes the magnetic field intensity, τ_s denotes the rotational relaxation time, ω_p 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 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].

Equation (16) represents the linear relation between the viscosity of magnetic fluid and volume concentration. This expression of viscosity is valid for $φ≤2%$ [80–82].

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

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].

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.

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.

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 = ω_p. 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., ω_p < 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., ω_p > Ω. In this situation, the rotational viscosity becomes negative.

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 μ_nf + 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.

Researchers have used these expressions of viscosity in ferrohydrodynamic flow in different regimes [63,97–101].

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.

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 [111–113]. 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.

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 [121–123]. 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].

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.

There are no conflicts of interest.

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