Study on MHD Cylindrical Couette Flow and Rheological Properties of some Magnetic Suspensions

The study of magnetic suspensions (MS) and magnetic field effects on their rheological properties is of evident practical importance due to its ability to orient and change their physical properties, especially their viscosity, by magnetic fields. This research presents the effect of a uniform magnetic field on the flow of MS in the annular region between two concentric cylinders. The motion of the fluid is due to the rotation of the inner cylinder with a constant angular velocity. An exact solution of the governing equations is obtained in the form of modified Bessel functions of the first and second kinds. The torque, which must be applied to the inner cylinder in order to maintain the rotation, is also calculated. The results show that as the magnetic parameter increases, the velocity profile decreases, while the torque increases due to the effect of magnetic force against the flow direction. 
In order to model the magnetoviscous effects, experiments were performed for different shear rates and different magnetic field strengths by using specially designed rheometers. The studied samples are iron oxide-water-glycerol system,  ferrofluid nanoparticles, MAG DX biocompatible ferrofluid. The theoretical analysis is based on Giesekus model for MS. This model gives more accurate results and takes into account the effects of viscoelastic shear thinning characteristics. It is found that a magnetic field increases the viscosity of all suspensions under consideration. Finally, new proposed correlation for the viscosity of MS as a function of both shear rate and magnetic field has been suggested.


INTRODUCTION
Couette flow is one in which the shearing surfaces are coaxial cylinders which rotate with an individual angular velocities around their common axis. An analytical solution to the flow of viscous Newtonian fluid through vertical annulus can be found in the classical textbooks of Bird et al. [1]. In real life, many fluids exhibit non-Newtonian nature. The study of the flow of such fluids is necessary due to its numerous technical and 122 industrial applications such as petroleum, oil refining, food industries etc. A single fluid model cannot describe the behavior of all non-Newtonian fluids. Therefore, several models have been proposed. One of these models is the Oldroyd-B fluid model. This fluid has been used quite widely in many applications and the results of simulations fit experimental data in a wide range.
One of the most important features of MS is the possibility to change their physical properties, especially their viscosities, by means of moderate magnetic fields. This capability makes MS very useful in the fields of engineering, medicine and fundamental researches. Under certain conditions, experimental results correlate the change of the viscosity of sheared MS under the influence of a magnetic field to the internal structure formation. Because of its specific characteristics, MS behaves as functional fluid and has found many useful applications such as dynamic sealing, dampers, heat conductors, temperature and magnetic field sensors, DNA detection, magnetic drug targeting and cancer hyperthermia treatment [2][3][4].
Magnetic fluids are a smart materials which are prepared by dispersing the magnetic particles into the carrying fluids, either aqueous or non-aqueous. Because of the magnetic dipolar-dipolar interaction among the magnetic particles, the magnetic fluids possessed typical magnetoviscous effects [5]. Experimental investigations have shown that an increase of the magnetic field strength yields an increase of the fluid's viscosity, the so-called magnetoviscous effect [6]. In the classical theory of rotational viscosity, this behavior is explained by the hindrance of the free rotation of particles in a shear flow by magnetic fields. This theory applies for highly diluted ferrofluids, assuming vanishing interactions between the particles. However, the commercial ferrofluids used for most applications are generally concentrated and, therefore, the interaction between the particles cannot be ignored [7].
Magnetic suspensions can be considered as liquid mixtures made of magnetic particles chains and small molecules of solvent. At rest, the chains of suspensions are randomly entangled and they do not set up the suspensions structure. When the fluid is in motion, the chains tend to align themselves parallel to the direction of flow. This tendency increases with increasing shear rate, so that the effective viscosity decreases [8]. In dilute suspensions, the rheology of suspension depends solely on the dynamics of an individual chain and the number of chains in the system. At high concentrations, interactions between suspension's particles impact the rheology in a significant way. As concentration increases, suspensions exhibit a change from fluid-like to more elastic-like behavior [9]. Viscosities of MS are known to be variable quantities which decrease with increasing shear rate. This shear thinning effect has been known for a long time, but its adequate explanation is still absent. In the case of concentrated suspensions, a varying viscosity is assumed to be due to the entanglement of magnetic particles chains. For a dilute solution, viscosity variations are associated with a behavior of separate particles in the flow, i.e., their deformation, orientation, etc.
The orientation of magnetic particles in solvent under the influence of an external magnetic field is of great importance owing to the possibility of changes in structure and the products formation. Under the effect of magnetic field these particles may rearrange themselves taking the same direction as the magnetic field lines (i.e. oriented parallel to the magnetic lines), a circumstance that lead to an increase in suspension viscosity. The effects of shear rate, magnetic field and concentration on the rheological properties of MS were investigated by Vshivkov et al. [10], Ghasemi et al. [11] and Hong et al. [12]. Many researchers have tried to explain the magnetoviscous effect [13,14]. Although the theories of chains have provided good explanations for the magnetoviscous effect, there are still many unexplained matters in this area. It is widely accepted that magnetoviscous effect is due to the complicity of the interaction originates from the disorder to order transition of the particles under applying an external magnetic field [15]. Therefore, the magnetic property, shape, size, and inner structure of the magnetic particles become the critical roles for affecting the rheological properties [16,17]. During the past decade, several groups performed intensive work in this area to investigate the magnetoviscous mechanism [18][19][20][21][22][23].
A ferrofluid is a suspension that consists of ferromagnetic nanoparticles dispersed in a carrier liquid such as water or organic solvents. Usually solid particles in these fluids are about 10 nm in size. In most engineering 123 applications of ferrofluids, the magnetic material can be one of . Hence, most studies have focused on the effects of magnetic field on the rheological properties of this ferrite [24]. A special character of ferrofluids is that, their flow and properties can be controlled by applying moderate magnetic fields. This has been a source of various technical and clinical applications [25,26]. Such fluids show non-Newtonian behaviour in presence of magnetic field. In different applications, the key parameters that dominate the process performance are viscosity and thermal conductivity of the ferrofluid. Recently, Fe 3 O 4 based magnetic fluids have attracted increasing interests since their better stability than the common carbonyl iron based magnetorheological fluid [27][28][29].
At present, the flow field of a viscoelastic fluid in the annular region between two concentric cylinders is investigated. The inner cylinder rotates with a uniform angular velocity about its own z-axis centered in the origin of the system and the outer cylinder is being at rest. To benefit from this theoretical study in the interpretation of magnetoviscous phenomenon, we reconsider the experimental data measured by Vshivkov et al. [27], Dhumal et al. [28], Nowak et al. [29] and Linke et al. [30] for iron oxide and biocompatible ferrofluids. We collect and reorder their experimental data and carry out its theoretical analysis based on Giesekus model for viscoelastic suspensions. As we have seen from our previous papers [31][32][33] this model takes into account the effects of shear thinning characteristics on the viscosity of the polymer solutions and gives more accurate results. In addition, the effects of magnetic field on the magnetorheological properties of the used MS are investigated. The chosen model fits adequately the experimental data.

GOVERNING EQUATIONS
To get a comprehensive idea about the behavior of a viscoelastic fluid, we consider an incompressible Oldroyd-B fluid for which the Cauchy stress tensor T is given as follows: where p is the pressure, s is the extra stress tensor,   is the rate of deformation tensor,  is the shear viscosity while where V is the velocity vector. The tensors L and   are defined as: Let us consider the motion of a viscoelastic fluid between two semi infinite concentric cylinders of circular cross sections having radii ). The inner cylinder rotating steadily around its axis by an angular velocity  while the outer cylinder is kept at rest. The fluid is assumed to be an incompressible and electrically conducting of constant density . The Cauchy dynamical equation of motion and the equation of continuity are given by: and where f is the body force per unit volume. The presence of a magnetic field requires that an additional force be included in the equations of fluid motion aside from the usual pressure and shear forces. The added force takes the form: where J is the current density and B is the magnetic induction vector. The current density is given as: in which the terms , respectively, represent the conduction and induction currents and  is the electrical conductivity.
We shall make use of the cylindrical coordinate system ( z , , r  ) for the solution of this boundary value problem. The z-axis is coincident with the two cylinders axis and 0   anticlockwise, see figure 1. We shall seek an axisymmetric two dimensional solution and this assume that the only nonzero component of velocity is then (r) (9) Furthermore, this velocity component as well as the stress components and the pressure will be functions of r only.
In the present study, we assume that the magnetic field is in the direction of z-axis. Neglecting the electric field E in equation 8 and replacing B by the externally applied field with the choice of the velocity given in equation 9, the continuity equation is satisfied automatically and Lorentz force takes the form: hence, the r-and -components of the equation of motion are written, respectively, as: r dr dp r The constitutive equation, equation 1, gives: (13c) Equations 12 and 13 represent the governing flow equations with the boundary conditions: Introducing the following change of variables: and defining the dimensionless constants with     / is the kinematic viscosity. After substitution from equations 15 into equations 12 and 13 we get (dropping the dimensionless mark "*" for simplicity): (16b) r dr dp r using equations 16a and 16d, it is easy to prove that the velocity component  satisfies the second-order boundary value problem: with the boundary conditions: Equations 18 and 19 admit the solution: also the dimensionless torque, which must be applied in order to maintain the rotation, is calculated as: The torques on the inner cylinder ( 1 r  ) and outer cylinder (   r ) are given as: Notice that, the velocity field   From the theoretical solution and the obtained curves, we can conclude the following remarks:


Increasing the magnetic parameter m, the velocity profile reduces monotonically due to the effect of magnetic force against the flow direction.


The shear stress and torque become larger for large magnetic parameter m.
 An increasing in  yields an effect opposite to that of the magnetic field.

TEST MATERIALS AND EXPERIMENTAL MEASUREMENTS
Since the viscosity of MS is very important factor, it is necessary to collect some of the viscometric data to get more precise formula, which describe the viscosity. In this work, four different MS were studied. Vshivkov et al. [27] employed a suspension of iron oxide (Fe 2.8 O 4 ) nanoparticles in glycerol-water solution. Iron oxide nanopowder colored black with average particle diameter of nm 22 was used. The viscosity of the suspension was measured using a modified Rheotest RN 4.1 rheometer a coaxial cylindrical work unit made of a weak magnetic material (brass). A magnetic field was generated using a equipped with magnetic system generating a permanent magnetic field with strength of 280 kA/m and field lines directed parallel to the rotor-rotation 129 axis. Vshivkov and his group investigated experimentally the changes in the viscosity of iron oxide over shear rates Figure 4 shows schematic representation of Rheotest RN 4.1 rheometer and its installation method in magnetic field. The working unit was exposed to a magnetic field and the viscosity was measured while shear rate was gradually increased from 0 to 30 s -1 . The materials and preparation of Fe 3 O 4 ferrofluid nanoparticles were employed by Dhumal et al. [28]. The rheological measurements were carried out using MCR300 rheometer, see     Different directions of the magnetic field also have a huge influence on magnetoviscous behavior. Linke and Odenbach [30] concluded that the highest magnetoviscous effect has been obtained in the magnetic field perpendicular to the flow field, table 4. In both low and high shear rate studies, the lowest magnetoviscous effect has been observed in a parallel orientation to the flow because the elongated microstructure oriented in the direction of the flow offers less hindrance to the flow.

CONSTITUTIVE EQUATION AND ELASTIC CHARACTERISTICS
The flow behavior of MS can be described by the flow curve which is usually presented as a function of the form where )(

Magnetic suspensions flow curves
In this section, fitting results of the experimental data given in [27][28][29][30] are overviewed to give a coherent picture about the rheological behaviors of the considered suspensions. Four sets of experimental data were used in order to describe their rheological behavior, see tables 1, 2, 3 and 4. A good fit for the experimental flow curves can be obtained starting from the understanding the effects of the Giesekus parameters on the flow curve under consideration and finding the proper values for these parameters. Then by inserting the proper values in Giesekus model explained above we can draw a theoretical curves. The proper parameter values must be able to the theoretical curve to catch the experimental points. The suitable Giesekus parameters required for the best fit of the four sets of experimental data are listed in table 5.  Figure 5a shows the shear dependence of viscosity for Fe 2.8 O 4 suspensions in the presence and in the absence of a magnetic field. It is obvious that this suspension has shear thinning behavior, this also reported for other ferrofluids such as Fe 3 O 4 and MAG DX in different base fluids, figures. 5b, 5c and 5d. The studied suspensions have a non-Newtonian fluid behaviors, where the viscosity is decreasing when the shear rate increases. As seen from the figures, samples exhibit shear thinning over the whole shear rate range investigated. When the ferrofluid exposed to the magnetic field, the magnetic particles tend to arrange themselves along the direction of the magnetic field (in this study, perpendicular to flow direction) and form new structures with higher resistance under the magnetic field. By increasing the field intensity, the interaction between the magnetic particles and their arrangement become stronger. Therefore, the viscosity of ferrofluid increases under an application of the field. By increasing shear rates, the viscosity drops significantly and its profile changes to one exhibiting three discrete regions as following:  Low shear rates, in which the velocity profile consists of a shear thinning. These situations indicate the breaking of the initial structures of magnetic particles.


Intermediate shear rates, where the suspension viscosity continues to decrease and the particles chains orient along the flow direction during deformation.


High shear rates region, where the shear stress is high enough to destroy the suspension domains into individual particle chains and orient these chains along the flow direction. Therefore, the viscosity tends to become a constant value.

Magnetoviscous effect and the suggested correlation
During analysis of the viscosity data, we must take into account that it is affected by shear rate and the applied magnetic field. As we have seen above, the viscosity of all samples decreases with increasing shear rate and increases with increasing the applied magnetic field. To study the effect of magnetic field on the rheological behavior of MS, data measured by Dhumal et al. [28] are considered. The range of shear rate is sufficient to cover the regions in which the viscosities at high shear rates arise to some constant values. In the range of low and intermediate magnetic fields, the viscosity is a function of magnetic field. Similar dependence were obtained from that data measured by Vshivkov et al. [27], Nowak et al. [29] and Linke et al. [30].
The relationship between magnetic field and viscosity of iron oxide Fe 3 O 4 suspensions at constant shear rates is shown in figure 6a. It is clear that, when shear rate is zero, applying a constant magnetic field causes the formation of large chains. Then, by applying a small shear rate the particles will find a chance to move and rearrange to form small chains. However, at higher shear rate, the shear force can break the chains and the magnetoviscous effect is weakened. It also notices that for all different magnetic fields in figure 6b, the behavior of Fe 3 O 4 suspension is shear thinning. The magnetic field leads to magnetization of magnetic 137 particles and then they adhere to each other. Therefore, aggregated structures are formed and these structures increase the viscosity. In spite of applying the shear rate, these structures don't break; in other words the shear force is not too strong to break down these structures. With more increasing of shear rate the applied shear stress overcome to the strength of these structures and then the viscosity decreases gradually.
For practical engineering, it is useful to get a simple equation describing the combined effect of magnetic field and shear rate on MS. The magnetoviscous effect shown in figure 6a reveals that formation of new structures is occurring as a function of the magnetic field. As we have seen in the previous sections, Giesekus model accurately predicts the viscosity as a function of shear rate equation 28. Referring to my previous works [31][32][33] we can relate the measured viscosity to shear rate and magnetic flux density. Therefore, we introduce a new proposed correlation which predicts the viscosity of Fe 3 O 4 ferrofluid in the form: Since Giesekus model accurately predicts the viscosity as a function of shear rate, the term   by correlating the experimental results using Mathematica program, Giesekus parameters

CONCLUSIONS
The main objective of the theoretical solution is to examine in detail the basic MHD flow of a magnetic suspension between two concentric cylinders. The momentum equation and the equation of state that characterize the suspension are solved in exact manner considering the Lorentz force to account the resistance offered by the magnetic field. The obtained velocity profiles possess modified Bessel functions of first and second kind. The effects of various parameters on the flow characteristics such as viscoelasticity, magnetic parameters have been studied and discussed graphically. Therefore, we conclude the following remarks:


An increase in magnetic parameter m reduces the velocity profile monotonically due to the effect of magnetic force against the flow direction (magnetoviscous effect).


An increase in the parameter  yields an effect opposite to that of the magnetic field.
To confirm these conclusions, some experimental results measured in more than one source have been studied and analyzed. The flow curves of all selected MS under different magnetic fields show that the viscosity decreases with increasing the shear rate. Therefore, Giesekus model is used to describe the viscoelasticity nature of the studied MS. The curves show also that this model fits the experimental data very well. Therefore, Giesekus model is currently the adequate one describing the rheological properties of MS with or without applied magnetic field. By experimental measurement and theoretical analysis, the following conclusions can be addressed:


The MS demonstrated the shear thinning behavior.
 Giesekus model could describe the magnetic suspension, whenever the suspension is subjected to applied magnetic field or not.


Magnetic particles exist as aggregates in the suspension and these aggregates further formed chaining structure under applied magnetic field.


The mechanical toque induced by the shear flow would divert the chain-like aggregates formed by the field.


The external magnetic field gives rise to a magnetic torque to contrast action the mechanical torque.
 When the shear rate was large enough such that the aggregate structures had been broken, the viscosity would not change any more with the increasing magnetic field.


A general constitutive equation was proposed to calculate the viscosity of MS subjected to applied magnetic field.