Effects of Velocity and Thermal Slip Conditions with Radiation on Heat Transfer Flow of Ferrofluids

To analyze the thermal convection of ferrofluid along a flat plate is the per-sistence of this study. The two-dimensional laminar, steady, incompressible flow past a flat plate subject to convective surface boundary condition, slip velocity in the presence of radiation has been studied where the magnetic field is applied in the transverse direction to the plate. Two different kinds of magnetic nanoparticles, magnetite Fe 3 O 4 and cobalt ferrite CoFe 2 O 4 are amal-gamated within the base fluids water and kerosene. The effects of various physical aspects such as magnetic field, volume fraction, radiation and slip conditions on the flow and heat transfer characteristics are presented graphically and discussed. The effect of various dimensionless parameters on the skin friction coefficient and heat transfer rate are also tabulated. To investigate this particular problem, numerical computations are done using the implicit finite difference method based Keller-Box Method.


Introduction
Owing to low thermal conductivity of the customary heat transfer fluids such as water, oil and ethylene glycol, the act of engineering apparatus such as heat exchangers and electronic devices face complications. To recover this problem, fluids with higher thermal conductivity like nanofluids are used as additional to these fluids [1]. A nanofluid is a class of fluids comprising nanoparticles with the size range under 100 nm that are uniformly and stably adjourned in a liquid. The nanoparticles used in nanofluids are made of metals, oxides, carbides, or carbon nanotubes. Communal base fluids include water, ethylene glycol and oil [2].
Among various kinds of explores on nanofluids, some of the studies have been absorbed on the nanofluids prepared by diffusing magnetic nanoparticles in a transporter liquid. These are called "ferrofluids". This is a liquid that becomes highly magnetized in the presence of a magnetic field. Ferrofluids are colloidal liquids made of nanoscale particles (diameter usually 10 nanometers or less) of magnetite, hematite or some other ferromagnetic (metallic) such as iron (Fe), cobalt (Co), and nickel (Ni) as well as their oxides such as magnetite (Fe 3 O 4 ) and ferrites (MnZn, Co ferrites) particles suspended in a carrier fluid (usually an organic solvent or water) [3] [4] [5]. A crunching process for ferrofluid was designed in 1963 by NASA's Steve Papell as a liquid rocket fuel that could be drawn toward a pump inlet in a weightless environment by applying a magnetic field [6]. The name "ferrofluid" was introduced, the process enhanced, more highly magnetic liquids manufactured, supplementary carrier liquids discovered, and the physical chemistry elucidated by R.E. Rosensweig and colleagues [6]; which evolved as a new branch of fluid mechanics termed ferrohydro dynamics. The applications of ferrofluids are enormous in real life sectors like technological applications and materials research. Ferrofluids have vast applications in biomedical sectors such as cancer treatment and Magnetic Resonance Imaging (MRI).
The technology for production for nanoparticles and suspensions, synthesis of nanofluids, thermal transport in stationary fluids, thermal conductivity in nanofluids, convective heat transfer under both natural and forced flow and future developments in nanotechnology have been thoroughly studied by many researchers [7] [8] [9]. Magnetohydrodynamic (MHD) flow and heat transfer from fluids along flat plates have many engineering submissions in different industries. Most of the former studies explore MHD convection heat transfer from surfaces under no slip condition [10] [11] [12] [13] [14]. But in many situations, fluids demonstrate boundary slip for example, micro-scale-fluid dynamics in Micro-Electro-Mechanical System (MEMS). The studies interrelated to slip boundary can be originated by Cao and Baker [15]. Different aspects of the effect of thermal slip condition over a permeable stretching sheet, shrinking sheet and more recently the effects of thermal radiation and ohmic dissipation are also investigated [16] [17] [18]. The MHD boundary layer flow and heat transfer of ferrofluids along a flat plate or an oscillatory infinite sheet with various slip conditions have been studied by different authors [19]- [24]. Ferrofluid properties are derived using the volume fraction of solid nanoparticles in combination with the properties of base fluid and nanoparticles. This model formerly industrialized by Choi [8] has been used among others. The studies related to the use of ferrofluids have been led by to enhance heat transfer in the boundary layer and the convective stability of ferrofluids in magnetic field [16] could be helpful in understanding the physics behind the flow of ferrofluids along plates. However, no attempts so far have been made to analyze heat transfer flow of ferrofluids along a flat plate with velocity and thermal slip conditions with radiation.

Governing Equations
Consider the convective heat transfer of selected ferrofluids along a stationary flat plate in a constant magnetic field. The plate is embedded in a medium saturated with water-or kerosene-based ferrofluids. The flow is assumed to be laminar, steady, incompressible and two-dimensional. The base fluids and the selected nanoparticles are assumed to be in thermal equilibrium. In the presence of magnetic field, the ferroparticles moments almost instantly orient along the magnetic field lines and when the magnetic field is removed, the particles moments are quickly randomized. This orientation along the magnetic-field lines shows a certain precise positioning of the ferroparticles depending upon the position of the magnetic field ( Figure 1).
The hydrodynamic slip is assumed at the fluid-solid interface with convective surface boundary condition. Here, we consider a situation where the work done by a fluid on adjacent layers due to action of shear forces that is transformed into heat is negligible compared to radiation. The viscous dissipation is insignificant for low viscous flow such as laminar flow. So, we have neglected the viscous dissipation in this study. The constant temperature w T is assumed to be greater than the ambient temperature T ∞ . Using an order-of-magnitude analysis, the standard boundary layer equations for this problem can be written as follows: The transverse magnetic field is assumed to be a function of the distance from B is the magnetic field strength [20].
Using the Rosseland approximation [25] as in Cortell [26], the radiative heat flux is simplified as: We assume that the temperature differences within the flow region are sufficiently small, so that the term 4 T can be expressed as a linear function of temperature. The best linear approximation of 4 T is obtained by expanding it in a Taylor series about T ∞ and neglecting higher order terms, i.e. 4 3 4 The effective properties of ferrofluids may be expressed in terms of the properties of base fluid, ferroparticles and the volume fraction of solid nanoparticles as follows [20]: , The boundary conditions for the problem are given by where γ is the slip parameter. The bottom of the surface is heated due to the convective heat transfer from a hot fluid at a temperature w T , yielding a heat transfer coefficient f h as a function of x, with its strength given as We look for a similarity solution of Equation (1), Equation (2) and Equation (5) of the following form: where η is the similarity variable, is the local Reynolds number based on the free stream velocity and f ν is the kinematic viscosity of the base fluid. The stream function ψ is defined as Employing the similarity variables (8), Equation (1), Equation (2) and Equation (5) reduce to a nonlinear system of ordinary differential equations: The associated boundary conditions are Here, primes denote differentiation with respect to η , is the Prandtl number and is the dimensionless thermal slip parameter and β is the dimensionless slip velocity. We have to take where c is a constant of dimension 1 2 L . Thus we take β as the dimensionless slip parameter ranging from zero (total adhesion) to infinity (full slip) defined .
Here, wx τ is the surface shear stress along the x direction and w q is the heat flux given by Reducing dimensionless form we get,

Numerical Keller Box Method
The coupled non-linear two-point boundary value problem (9) and (10) combined with the boundary conditions (11a) and (11b) is solved numerically using the implicit finite difference scheme, the Keller-Box method.

The Finite Difference Method
As described in Cebeci and Bradshaw [28] [29] and Na [30] Equation (9) to Eq-uation (10) subject to the boundary conditions (11a) and (11b) are first written as a system of five first-order equations.
In terms of the new dependent variables, the boundary conditions become We approximate the quantities ( ) Equation (14) are imposed for 1, 2, , j J =  and the transformed boundary layer thickness, j η , is sufficiently large so that it is beyond the edge of the boundary layer [31]. At n x x = the boundary conditions (13a) and (13b) become

Newton's Method
The nonlinear system of Equation (14) is linearized using Newton's method and the following iterates are introduced Then we obtain the following tridiagonal system, To complete the system (17) we recall the boundary conditions (15) which can be satisfied exactly with no iteration [28] [29]. So, to maintain these correct values in all the iterates, we take

Block-Elimination Method
The linear system (17) can now be solved numerically by the block-elimination method [30]. The linearized difference equations of the system (17) have a block-tridiagonal structure. In a matrix-vector form, this can be written as The coefficient matrix A in (20) is known as a tridiagonal matrix due to the facts that all elements of A are zero except those on the three main diagonals. To apply the block elimination method, we assume that A is nonsingular and The solution of equation (21) by the block-elimination method consists of two sweeps: forward sweep, backward sweep. Once the elements of δ are found, Equation (17) can be used to find the (i + 1)th iteration in Equation (16). Calculations are stopped when ε is a small prescribed value.

Results and Discussions
The proposed numerical method is programmed in MATLAB with a step size  Table 1.
In order to validate the accuracy of our numerical procedure, the skin friction coefficient is computed for pure fluid, magnetite and cobalt ferrite for specific values of velocity slip, magnetic parameters (Table 2) and the heat transfer rate   (Table  3). It is well established that our results are well matched with the previous studies.
From Table 4, we see that for both water-based and kerosene-based ferroflu- The variation of the dimensionless velocity and temperature profiles with magnetic parameter M is shown in Figure 3 for magnetite and cobalt ferrite in base fluids, water and kerosene. In each case, the dimensionless velocity profile (Figure 3(a) and Figure 3(c)) increases at the surface and within the boundary layer with the increase of magnetic parameter. In the absence of magnetic field, the dimensionless velocity is found to be smaller. As the magnetic field is applied, it arranges the ferroparticle in order and enhances the velocity. The effect of magnetic parameter on dimensionless temperature is shown in Figure 3(b) (magnetite) and Figure 3(d) (cobalt ferrite) respectively. It is evident that the      increase of the magnetic parameter M results in the decrease of temperature profiles. Figure 4(a), Figure 4(c) show the effect of volume fraction ϕ of ferrofluids on velocity profile. In both cases, the dimensionless velocity profile decreases with increasing volume fraction ϕ . In the absence of volume fraction, the velocity is found to be highest. From Figure 4(b) and Figure 4(d) we observe that, increase in the volume fraction increases the dimensionless temperature for both ferrofluids. In fact, addition of ϕ increases the thermal conductivity of the pure fluid and thus results in increasing the thermal diffusion in the boundary layer.   number has no effect on the velocity profiles as radiation parameter regardless of the base fluid or ferroparticles. For each case, it is observed that due to the higher Prandtl number of kerosene, the thermal boundary layer thickness as well as the dimensionless surface temperature is smaller for kerosene-based ferrofluids.

Conclusions
The present study investigates the flow and heat transfer of ferrofluids over a flat plate with slip conditions and radiation using the Keller-Box method. For each ferrofluid, it is concluded that: • The skin friction coefficient increases with the increase of magnetic field, volume fraction of solid ferroparticles and decreases with increasing velocity slip parameter, but remains the same for variation of radiation parameter and Biot number. • The heat transfer rate or Nusselt number increases with increasing magnetic parameter, volume fraction, velocity slip parameter, Biot number and decreases with increasing radiation parameter.
• Kerosene-based ferrofluids have higher skin friction and heat transfer rate than the water-based ferrofluids.
• Kerosene-based magnetite (Fe 3 O 4 ) provides the higher skin friction coefficient and heat transfer rate at the wall compared to the kerosene-based cobalt ferrite (CoFe 2 O 4 ). • In the slip flow regime, under low Biot number conditions, the permeability effects on heat transfer rate tend to be negligible.
• The dimensionless velocity profiles increase with the increase of magnetic parameter and velocity slip parameter, decrease with the increase of volume fraction of solid nanoparticles and there is no effect with the variation of radiation parameter and Biot number.
• The dimensionless temperature profile increases with the increase of volume fraction of solid nanoparticles, radiation parameter, Biot number and decreases with the increase of magnetic and velocity slip parameter. For future work, investigation can be carried out to study the effects of magnetic field, radiation, slip conditions, viscous dissipation, internal heating, suction, injection of ferrofluids for unsteady case and turbulent flow.