Mathematical Analysis of Two Phase Saturated Nanoﬂuid Inﬂuenced by Magnetic Field Gradient

: Nanoﬂuids are composed of nano-sized particles dispersed in a carrier liquid. The present investigation’s aim is to examine theoretically the magneto-thermomechanical coupling phenomena of a heated nanoﬂuid on a stretched surface in the presence of magnetic dipole impact. Fourier’s law of heat conduction is used to evaluate the heat transmission rate of the carrier ﬂuids ethylene glycol and water along with suspended nanoparticles of a cobalt–chromium–tungsten–nickel alloy and magnetite ferrite. A set of partial differential equations is transformed into a set of non-linear ordinary differential equations via a similarity approach. The computation is performed in Matlab by employing the shooting technique. The effect of the magneto-thermomechanical interaction on the velocity and temperature boundary layer proﬁles with the attendant effect on the skin friction and heat transfer is analyzed. The maximum and minimum thermal energy transfer rates are computed for the H 2 O-Fe 3 O 4 and C 2 H 6 O 2 -CoCr 20 W 15 Ni magnetic nanoﬂuids. Finally, the study’s results are compared with the previously available data and are found to be in good agreement.


Introduction
The problems of fluids' flow and heat transfer for the rate of cooling and heating is very important in various industrial processes, such as polymer extrusion, drawing of plastic and sheets, fiberglass, the production of paper and many more. The transfer of heating or cooling in an object or between two distinct surfaces takes place due to temperature differences. A higher temperature difference causes higher thermal energy transmission [1][2][3]. The transmission of thermal energy can be computed theoretically using the well known law, i.e., Fourier's law. This law uses the correlation between thermal conductivity and heat capacity to evaluate the transmission rate of thermal energy. The law also states that the transmission of thermal energy occurs with an inertial rate [4][5][6]. Later on, Cattaneo-Christov [7] added the thermal relaxation time into Fourier's law. It plays an important role when experiments are difficult to conduct for heat transfer simulation. These laws are used for the theoretical computation of heat transfer [8][9][10]. The above discussed laws are used for evaluation of heat flux in two phase fluids as well as regular fluids.
The boundary layer flows on the stretching sheet has been examined by many researchers for the rate of heat transfer and cooling as it has promising applications in many industries. The stretching sheet velocity of the flow field has gained considerable attention as it has a certain effect on the excellence of the final products in the fabric and polymer industry. This stretching velocity can be linear, polynomial, hyperbolic or exponential. The pioneering work of Crane [11] has laid the foundation for theoretical investigations on the flow field over a stretching surface. He presented an exact analytical solution of the flow over a linearly stretching sheet. The boundary layer heat transport flow of multiphase magnetic fluids past a stretching sheet under the impact of a circular magnetic field was described in [12]. The general fluid model for the magnetohydrodynamic fluid flow and heat transfer was analyzed by Hatzikonstantinou and Vafeas [13]. The authors computed heat transfer along with the skin friction coefficient in the flow of a micropolar fluid. In this direction, Papadopoulos et al. [14] demonstrated the theoretical investigation of pipe flow in the presence of a magnetic field in a ferrofluid along with a cylindrical coil. The available literature in this direction includes the study of ferrite nanoparticles [15][16][17][18][19].
The literature consists of study of ferrite and regular particles, which are studied in different base fluids for heat transfer along with skin friction. These particles have been studied multiple times with different effects to compute which mathematical model better suits the physical situation. Thus, in the field of fluid dynamics the same problem is normally studied with different effects and parameters to predict the heat transfer that better suits the physical situation modeled. The fluid flow computed for heat transfer and skin friction is available in [20][21][22][23].
Ferromagnetic fluids were first synthesized by Rosensweig [24] in his pioneering work, in which he also first used the term ferrohydrodynamics. After that, the work has received increasing attention over the last few decades. In his later detailed work [25], he explored the basic equations for ferromagnetic fluids with internal rotation. The heat transfer is efficient in solids instead of liquids or gasses. Therefore, smart liquids called ferrofluids are synthesized by introducing ferromagnetic nanoparticles into a carrier fluid. The suspension of solid nanoparticles (1-100 nm) in a base fluid enhances the transport properties of the considered nanofluid and hence its heat transfer rate. The ferrofluids are famous due to the suspension of nano-sized ferrite particles instead of regular particles. These ferrite particles have the extra property of Curie temperature; thus, ferrofluids are of great interest and have been studied extensively by researchers after the work of Rosensweig [24]. To make electronic devices long-lasting, ferrofluids can evacuate heat from these devices, such as in speakers and Lenovo laptops [26]. Heat transfer in these devices takes place through the magnetic fluid, made up of a base fluid (water, oil, ethylene glycol) and ferrite nanoparticles. Later on, the flow problem discussed by Crane [11] was extended by Andreson and Valnes [27] with the addition of a magnetic field gradient on the stretched surface of an incompressible non-conducting ferrofluid. Odenbach [28] pointed out the behavior of magnet field impact over magnetite ferrofluid experimentally and discussed the practical applications in technical and medical fields.
In this study, the dynamics of heat transfer processes associated with the motion of viscous , incompressible fluids in the presence of a magnetic field and temperature gradient are analyzed. One of the striking features of the considered nanofluid is the dependence of the magnetization upon the temperature gradient, and this thermomagnetic coupling makes the study of these fluids demanding. Since the suspension of nano-sized particles in a viscous fluid enhances the transport properties of the flowing fluid, the transmission rate of the thermal energy is also enhanced. Therefore, the aim of the article is to describe the interaction between a magnetic field gradient and solid nano-sized particles and its influence on the convection, heat transfer and friction drag.
A cobalt-chromium-tungsten (wolfram)-nickel (CoCr 20 W 15 Ni) alloy and magnetite ferrite (Fe 3 O 4 ) nano-sized particles were added into a base fluid of ethylene glycol (C 2 H 6 O 2 ) and water (H 2 O). The graphical results are computed in the discussion section. The particles and the base fluid properties are taken under the assumption of isothermal equilibrium. The analysis is made in such a way that a single solid particle is suspended in the base fluid. The suspension is then computed for thermal energy transmission. The comparison is made for the ferrite (Fe 3 O 4 ) and alloy (CoCr 20 W 15 Ni) two phase nanofluid flow. The results are discussed and presented for the base fluids ethylene glycol (C 2 H 6 O 2 ) and water (H 2 O).

Mathematical Modeling
The steady state two phase incompressible laminar flow in the presence of an external magnetic dipole that is placed near the x surface at distance l is incorporated. The disturbance to the laminar flow is induced via the stretching x−surface sheet, and the stretching velocity is defined to be u = Sx. The temperature of the fluid at the surface is T w , whereas the temperature of fluid far away from the sheet is taken to be a Curie temperature, i.e., T = T c . The ferrite Fe 3 O 4 and alloy CoCr 20 W 15 Ni solid nanoparticles satisfy the Curie temperature T c . By contrast, the ferrite Fe 3 O 4 and alloy CoCr 20 W 15 Ni along with the base fluids ethylene glycol C 2 H 6 O 2 and water H 2 O are taken under the assumption of isothermal equilibrium and no slip occurs between them.
The vector form of the equation for the flow of the ferrofluid is given in [25]. In vector form the equations for the present problem are: Physically, Equation (1) represents the conservation of mass. The left-hand side of Equation (2) represents momentum due to convection or inertial forces; the first term in Equation (2) on the right-hand side represents the pressure, whereas the second term appears due to surface forces or viscous forces, and the third term denotes the body forces due to the magnetic dipole moment. The first term on the left-hand side of Equation (3) represents heat transfer due to convection, and the second term represents the internal energy due to magnetic dipole. The right-hand side of Equation (3) shows the heat transfer due to conduction.
The mathematical equation for the steady state two-dimensional flow of an incompressible ferrofluid and boundary layer flow described by the dynamical equations governing convective flow and the heat transfer of the nanofluid can be written as: The thermo-mechanical coupling at the wall and far from the wall is defined as: The term µ 0 M ∂H ∂x in Equation (5) represents the component of magnetic force per unit volume. This term depends on the existence of the magnetic field gradient along the corresponding x direction, called the Kelvin force, and is very well known in ferrohydrody- in Equation (6) is due to the magneto caloric effect and represents the thermal power per unit of volume. The relation of the ferrofluid between some of the physical characteristics of the solid nanoparticle and the considered carrier fluid were explained by Islam et al. [29] and is given in Table 1: Table 1. Mathematical expressions of thermo-physical properties.

Properties
Islam et al. [29] Thermal Conductivity k

Magnetic Dipole
Artificially synthesized nanofluids behave like normal fluids except that they experience a force due to magnetization. One of the striking features of nanofluids is the dependence of magnetization upon the temperature gradient, and this thermomagnetic coupling makes these fluids demanding in various applications. An external magnetic dipole is placed in the flowing fluid to magnetize the two phase C 2 15 Ni nanofluids. The introduction of magnetic forces into the magnetizable liquid gives rise to the effect known as ferrohydrodynamics. Now, since the nanoparticles are mechanically free to align with the fields of lines, the expression of the body force could be a good approximation for the flows of nanofluids. The magnetic scalar potential Ω is stated below: here γ 1 symbolizes the dipole moment per unit length, while the x and y components for the magnetic field are: Since it is generally known that the magnetic body force is relative to the magnetic field gradient of H, we have: Using Equations (10)- (12), and then differentiating the resultant equation with respect to x and y, we obtain the following equations: ∂H ∂y The variation of the magnetization M with the magnetic field intensity H and temperature T can be fairly approximated by the linear relation shown below. The detailed explanation for the derivation of H is presented in [27]. The magnetization is defined as:  Figure 1.

Similarity Analysis
For the numerical solution of the governing boundary layer model, Equations (5) and (6), along with boundary conditions, need to be transformed into non-dimensional ordinary differential equations. The transformations is performed by introducing the similarity analysis. The similarity transformation for the two phase nanofluid introduced by Andersson and Valnes [27] is: The stream function ψ(x, y) defined by the velocity components is u = ∂ψ ∂y , v = − ∂ψ ∂x , which satisfies the conservation Equation (4) identically, while f and θ are the dimensionless functions.
By using Equations (16) and (17) in the momentum and energy Equations (5) and (6), we obtain the transformed nonlinear ordinary differential equations of the momentum and temperature as follows: The boundary conditions in (6) and (7) now become: while A 1 and A 2 are: The dimensionless parameters calculated in the problem include β (ferrohydrodynamic interaction), σ 2 (viscous dissipation), Pr (Prandtl number), and σ 1 (dimensionless Curie temperature). Mathematical expressions are presented below: The physical interest parameters for the boundary layer flow problems are the skin friction coefficient, which is the rate of shear stress, and the Nusselt number, which is the rate of heat transfer. The wall shear stress characterized by the skin friction coefficient C f , along with the Nusselt number, is: where q w = −k n f ∂T ∂y y=0 is the surface heat flux. The substitution of the non-dimensional similarity transformation considered in Equations (16) and (17) into Equation (24) transforms the skin friction and local Nusselt number into dimensionless form. Hence, Equation (24) takes the following form: where the local Reynolds number Re x is defined as Re x = xS/ν f , which is based on the stretching sheet velocity S (x) .

Discussion
The  W 15 Ni. Therefore, we conclude that the maximum velocity field is determined for the H 2 O-Fe 3 O 4 nanofluid and the minimum distribution of velocity is determined for the magnetic C 2 H 6 O 2 -CoCr 20 W 15 Ni nanofluid. This variation occurs due to the thermophysical values of the transport properties of the solid particles and base fluids. The comparison is shown in Figure 2. The relation is compared for the temperature field in Figure 3. The variation in temperature distribution is noticed in the order C 2 15 Ni magnetic nanofluid exhibits the higher temperature field. The ferrite nanoparticles considered in the problem have higher thermal properties, which help in the enhancement of heat transfer; as a result, the lower temperature field occurs for the magnetite ferrite base magnetic nanofluid. Cobalt particles, however, have the property of Curie temperature, but that is not sufficient for a higher temperature field. Thus, the lower temperature field is observed for the alloy-based magnetic nanofluid.  The influence of ferrohydrodynamic interaction parameter β on the temperature and the axial velocity is displayed in Figures 4 and 5. The magnetic property of the solid particles, i.e., alloys and ferrite, makes the two phase nanofluid magnetize in the presence of a magnetic dipole placed near the surface. The dipole attracts the alloy and the ferrite particles until the alloy and ferrite reach their Curie temperature. Thus, the axial velocity for varying β declines, whereas the temperature profile for β shows the reverse nature for all of the magnetic nanofluids C 2 Figure 4, whereas its influence on the temperature field is evident in Figure 5. The interaction between the nanoparticles of the ferrite and alloys with the flowing fluid leads to resisting the flow and the internal energy of the fluid. The higher resistance to the flowing fluid is induced by the alloy as well as the ferrite nanoparticles; as a result, the velocity field declines, as presented in Figure 4. On the other hand, the internal energy in the presence of the alloy and ferrite nanoparticles arises when β is enhanced; thus, the temperature field enhancement is evident, as shown in Figure 5.  Generally, thermal engineers are interested in the reduction of wall shear stresses, whilst they need to enhance the heat transfer rate of the system. For this purpose, the mathematician constructs different mathematical models that interpret the heat transfer and the fluid flow to examine which model better predicts the friction drag and Nusselt number. Regular fluids have low thermal properties; thus, they are not fast in transmission of thermal energy, whereas solids have higher transport properties, and hence their dispersion in base fluids make them more efficient for heat transfer. Therefore, C 2 Figure 6. The higher resistance has a better impact on the corresponding velocity. This means that the corresponding velocity will be higher for the considered solid particle and the base fluid. Figure 7 [30]. The comparison is made for the Nusselt number as presented in Table 3.

Concluding Remarks
The phenomena of heat transfer analysis in ferromagnetic nanofluids on a stretching surface was explored with nanoparticles of magnetite ferrite and alloy in water and ethylene glycol (EG) as a carrier fluid. The main objective of the present theoretical study was to reduce the friction drag in the boundary layer flow and to intensify the efficiency of the considered nanofluid. The characteristics of nanoparticles together with the magnetic dipole impact enhanced the thermal conductivity of the engineered nanofluid and hence their influence on the heat flow. An appropriate transformation was employed on the mathematical model to convert the described system of partial differential equations into nonlinear ordinary differential equations and then computations were performed using the shooting method.
Combinations of nanoparticles of ferrite magnetite with a base fluid of water and ethylene glycol, and then of a cobalt-chromium-tungsten-nickel alloy in water and EG were examined. Moreover, the impact of various physical parameters involved in the flow model, such as ferrohydrodynamics interaction and others, were also analyzed on the velocity and temperature profiles to predict heat transfer rate. Finally, a comparison of different values of the Prandtl number indicated excellent agreement with previous data available in the literature. In conclusion, the nanofluid H 2 O-Fe 3 O 4 was found to exhibit a maximum heat transfer rate and C 2 H 6 O 2 -CoCr 20 W 15 Ni with higher resistance as compared to the rest of the nanofluid combinations considered.

Conflicts of Interest:
The authors declare no conflict of interest.