Numerical Simulation of Water Based Ferroﬂuid Flows along Moving Surfaces

: The steady two-dimensional boundary layer flow past a stretching flat sheet in a water-based ferrofluid is investigated. The spatially varying magnetic field is created by two line currents. The similarity method is applied to transform the governing equations into a system of coupled ordinary differential equations. Numerical investigations are performed for ferrofluids, the suspensions of water, and three types of ferroparticles (magnetite, cobalt ferrite, and Mn-Zn ferrite). The impact of the solid volume fraction, the surface stretching parameter, and the ferromagnetic coefficient on the dimensionless velocity and temperature profiles, the skin friction coefficient, and the local Nusselt number are analysed for the three types of ferrofluid.


Introduction
The motivation for studying the flow of nanofluids is their significance in widespread industrial applications. The industrial intensification has a demand on the development of more efficient heat transfer systems.
Nanofluids are used as energy saving cooling liquids instead of traditional coolants. First, Choi [1] introduced the term nanofluid for a liquid when nanosized (1-100 nm) solid particles are dispersed in a base solution. Nanoparticles can be, e.g., metals, metal oxides, and carbide ceramics, which increase heat conduction and convection. Due to their thermophysical properties and heat transfer performance, these fluids can improve the heating or cooling efficiency [2][3][4].
Ferrofluids are nanofluids in which magnetic particles (magnetite, cobalt or iron) are suspended in the base fluid. The studies on ferrofluids have shown that the addition of these particles improves significantly the heat transfer in the fluid flow [5,6]. Ferrites (magnetite (Fe 3 O 4 ), cobalt ferrite (CoFe 2 O 4 ) and Mn-Zn ferrite (Mn-ZnFe 2 O 4 )) incorporated within the base fluid (water) have a wide range of industrial applications in biomedical applications (for example, therapeutic applications in hyperthermia, in cancer treatment, drug delivery) and in diagnostic applications (nuclear magnetic resonance imaging) [7][8][9][10]. Ferroparticles have a high capacity to remove high concentrations of organic compounds. Its industrial application is the removal of dyes that appear in wastewater as a result of manufacturing processes (e.g., paint manufacturing, textile manufacturing, or tanning) [11][12][13][14][15][16].
Boundary layer theory was introduced by Prandtl [17] in order to understand the flow behaviour of a viscous fluid above the boundary. One of the classic applications of this theory is the flow of the boundary layer over a moving plate. Sakiadis [18,19] and Tsou et al. [20] examined the flow behaviour when the plate moves at a constant speed. This problem can be applied during extrusion processes. In the case of linear stretching, Crane [21] provided a solution to the Sakiadis problem for heat and mass

Formulation of the Problem
We investigate the steady boundary layer flow of a viscous, electrically nonconducting, and incompressible nanofluid over an impermeable flat surface placed in the horizontal direction.
Cartesian coordinates are chosen so that the x-axis is along the flow direction while y-axis is perpendicular to the surface. The nanofluid is confined above the surface. The ambient fluid has a constant temperature T ∞ , the temperature of the surface is a decreasing function of x and is given by T w = T C − Ax m+1 , where A and m are real parameters and T c denotes the Curie temperature. The horizontally stretching surface is placed at y = 0 and the nanofluid flow is above the surface. The sheet is being stretched at speed u w = U w x m with a power function of distance x. The ambient fluid temperature is T ∞ = T c , and the exterior streaming speed is zero.
The magnetic field is generated by currents perpendicular to the flow plane and generated by two wires, which are equidistant from the flat surface [40]. The direction of magnetization of the suspension's element is taken into account in the direction of the local magnetic field, and the scalar potential function can be described by where I 0 is the dipole moment per unit length and a denotes the distance of the line current from the origin. The ferrofluid flow is affected by the magnetic field created by the magnetic dipole. In this case, the magnetic field can be expressed as a linear function of the temperature where K is the pyromagnetic coefficient [5]. The governing equations in two-dimensions can be formulated as follows [41] ∂u ∂x where u and v are the parallel and normal velocity components to the plate, respectively, µ n f is the dynamic viscosity, ρ n f denotes the density, α n f is the thermal diffusivity of the nanofluid, which will be assumed constant, and µ 0 is the magnetic permeability. Equations (3)-(5) are subjected to the boundary conditions on the sheet (y = 0) with u w = U w x m , T w = T C − Ax m+1 and far from the surface: The parameter m controls the stretching of the surface and the behaviour of boundary layers. Three types of nanofluids will be investigated. Different volume fractions of the nanoparticles Fe 3 O 4 , CoFe 2 O 4 , and Mn-ZnFe 2 O 4 dispersed in water are considered. In this work, nanofluids are considered as single-phase fluids. The thermophysical properties of the nanofluids can be expressed in term of the properties of the base fluid, nanoparticles and nanoparticle concentration φ in the base fluid as follows: where µ n f is the viscosity of the nanofluid and µ b is the viscosity of the base fluid water, where ρ b and ρ p denote the density of the base fluid and nanoparticles, respectively, and the heat capacity of the nanofluid (C p ) n f is assumed as The thermal conductivity k n f of the nanofluid is calculated with the thermal conductivity k b of base fluid and the thermal conductivity k p of the particles with the formula  The similarity functions and similarity variable are introduced as follows: Equation (3) is automatically satisfied with stream function ψ(x, y) choosing such as u = ∂ψ/∂y and v = −∂ψ/∂x . Applying the similarity transformations (12), the governing Equations (3)-(5) can be rewritten as follows: where the Prandtl number for the nanofluid is defined by Pr = (µρ/α) n f and β = I 0 µ 0 KA/πρ n f U 2 w is the ferromagnetic parameter. The primes denote differentiation with respect to η. Also, boundary conditions (6) and (7) are converted to Based on the above quantities, the skin friction coefficient C f and the local Nusselt number Nu x can be given as follows where the wall shear stress τ w and the wall heat flux q w are Using the above equations, we get where the local Reynolds number Re x is defined by We remark that the exact solution to Equation (13) with boundary conditions f (0) = 0, f (0) = 1, f (∞) = 0 for m = 1 and β = 0 can be given in the closed form Processes 2020, 8, 830 5 of 12 then, the corresponding stream function is and the velocity components are These solutions are equivalent to those obtained by Crane [11].

Results and Discussion
The coupled transformed system of differential Equations (13)- (14) subjected to boundary conditions (15) and (16) are solved numerically by employing the fourth-order method (bvp4c) in MATLAB. The influence of several physical parameters, namely the stretching exponent m, the ferromagnetic interaction parameter β, the Prandtl number Pr of the three types of nanofluids on the boundary layer flow and heat transfer in ferromagnetic viscoelastic fluid are investigated numerically. The system of Equations (13)- (14) are solved in the interval 0 ≤ η ≤ 10, the solutions at infinity converge to the boundary conditions. The physical parameters are taken from Table 1 for the nanoparticles Fe 3 O 4 , CoFe 2 O 4 , and Mn-ZnFe 2 O 4 and for the water base fluid. The two differential equations are converted into a system of first order differential equations and solved.
The influence of the nanoparticle concentration on the Prandtl number for Fe 3 O 4 -water ferrofluid is depicted in Figure 1. It can be noticed that the addition of nanoparticles will cause a significant decrease in the Prandtl number of the ferrofluid.
Processes 2018, 6, x FOR PEER REVIEW 5 of 12 and the velocity components are These solutions are equivalent to those obtained by Crane [11].

Results and Discussion
The coupled transformed system of differential Equations (13)-(14) subjected to boundary conditions (15) and (16) are solved numerically by employing the fourth-order method (bvp4c) in MATLAB. The influence of several physical parameters, namely the stretching exponent , the ferromagnetic interaction parameter , the Prandtl number Pr of the three types of nanofluids on the boundary layer flow and heat transfer in ferromagnetic viscoelastic fluid are investigated numerically. The system of Equations (13)- (14) are solved in the interval 0 10, the solutions at infinity converge to the boundary conditions. The physical parameters are taken from Table 1 for the nanoparticles Fe3O4, CoFe2O4, and Mn-ZnFe2O4 and for the water base fluid. The two differential equations are converted into a system of first order differential equations and solved. The influence of the nanoparticle concentration on the Prandtl number for Fe3O4-water ferrofluid is depicted in Figure 1. It can be noticed that the addition of nanoparticles will cause a significant decrease in the Prandtl number of the ferrofluid. Figure 2 demonstrates the impact of the ferromagnetic solid particles on the nondimensional velocity distribution when the volume fraction is = 0.02, the ferromagnetic parameter is fixed as = 0.05 and the stretching parameter is = 0.05. This plot shows that the velocity is the smallest and the boundary layer thickness is the shortest one for Fe3O4-water solution. For the same parameters, the thermal distribution is shown in Figure 3. It can be observed that the temperature gradient is also the largest for CoFe2O4. Based on the figures, the thickness of the momentum and the  Figure 2 demonstrates the impact of the ferromagnetic solid particles on the nondimensional velocity distribution when the volume fraction is φ = 0.02, the ferromagnetic parameter is fixed as β = 0.05 and the stretching parameter is m = 0.05. This plot shows that the velocity is the smallest and the boundary layer thickness is the shortest one for Fe 3 O 4 -water solution. For the same parameters, the thermal distribution is shown in Figure 3. It can be observed that the temperature gradient is also the largest for CoFe 2 O 4 . Based on the figures, the thickness of the momentum and the thermal boundary layer for the variable η are around 8 and 2, respectively.           Figure 6 shows that, for a negative value of , the velocity initially shows an increasing trend and then decreases monotonically to zero. For = 0 or 0, this phenomenon cannot be observed. In addition, for negative values, the boundary layer is thicker.    Figure 6 shows that, for a negative value of , the velocity initially shows an increasing trend and then decreases monotonically to zero. For = 0 or 0, this phenomenon cannot be observed. In addition, for negative values, the boundary layer is thicker.    Figure 6 shows that, for a negative value of , the velocity initially shows an increasing trend and then decreases monotonically to zero. For = 0 or 0, this phenomenon cannot be observed. In addition, for negative values, the boundary layer is thicker.   For the temperature distribution we see a decrease of the temperature with increasing for both positive and negative values of .
The effect of nanoparticle's volume fraction on dimensionless velocity and temperature are shown in Figures 8 and 9 for the ferrofluid water-Fe3O4. From these figures, it is noted that velocity decreases for increasing values of volume fraction . The influence of increasing nanoparticle concentration is the opposite in the temperature profiles.   both positive and negative values of . The effect of nanoparticle's volume fraction on dimensionless velocity and temperature are shown in Figures 8 and 9 for the ferrofluid water-Fe3O4. From these figures, it is noted that velocity decreases for increasing values of volume fraction . The influence of increasing nanoparticle concentration is the opposite in the temperature profiles.       Figure 6 shows that, for a negative value of m, the velocity initially shows an increasing trend and then decreases monotonically to zero. For m = 0 or m > 0, this phenomenon cannot be observed. In addition, for m negative values, the boundary layer is thicker.
For the temperature distribution we see a decrease of the temperature with increasing m for both positive and negative values of m.
The effect of nanoparticle's volume fraction on dimensionless velocity and temperature are shown in Figures 8 and 9 for the ferrofluid water-Fe 3 O 4 . From these figures, it is noted that velocity decreases for increasing values of volume fraction φ. The influence of increasing nanoparticle concentration is the opposite in the temperature profiles.
Quantities important for technical practice are the skin coefficient of friction C f and the local Nusselt number Nu given by (17) and (18). For the ferrofluid with different nanoparticle concentration, the variation of the Re x Nu x x − m 2 are depicted on Figures 10 and 11 for Fe 3 O 4 with β = 0.05 , m = 0.05. It is observed that the variation of the skin friction with the volume fraction of solid particles is decreasing in the interval 0 ≤ φ ≤ 0.2. However, the local Nusselt number is increasing with values of the volume fraction φ.
Tables 2-4 give the variation of − f (0) and −θ (0) depending on the volume fraction φ, the ferromagnetic parameter β, and the stretching parameter m. These values have a great influence on the wall shear stress and on the wall heat flux.

Conclusions
The effects of volume fraction of solid ferroparticles and the non-uniform magnetic field on the dimensionless velocity and temperature and on the skin friction coefficient and local Nusselt number are investigated in ferrofluid flows along nonlinearly stretched sheets for three selected ferroparticles (magnetite, cobalt ferrite and Mn-Zn ferrite) in water base fluid. It has been shown that a very small amount of ferro particle affects the velocity and temperature of a ferrofluid in a non-constant magnetic field. The effect of the concentration of solid particles in the base fluid is examined in the range 0 ≤ φ ≤ 0.2, φ = 0 means the case when only the base fluid is considered. The thermophysical properties of the base fluid and the applied nanoparticles are given in Table 1 Figure 1.
We considered the impacts provided by the varying values of the pertinent parameters (stretching parameter, volume fraction, and ferromagnetic parameter) on the dimensional velocity, dimensional temperature, the skin friction, and local Nusselt number. Figures 2-9 illustrate the variation in the velocity and temperature. We observed that the increase in the volume fraction slows down the flow and increases the temperature. The increase in the magnetic effect has the same impact on the velocity and temperature profiles. An increase in the stretching parameter will decrease both the velocity and the temperature. The Nusselt number and the skin friction for different values of the volume fraction are also analysed in Figures 10 and 11. The Nusselt number is increasing and the skin friction is decreasing with an increasing value of φ.