Stability Analysis of Boundary Layer Flow and Heat Transfer of Fe2O3 and Fe-Water Base Nanofluid οver a Stretching/Shrinking Sheet with Radiation Effect

In this paper, the radiation and slip effects are investigated on the boundary layer flow and heat transfer of Fe2O3 and Fe-water base nanofluids over a porous stretching/shrinking sheet. A similarity transformation is used to convert the system of governing partial differential equations into ordinary differential equations, which are then numerically solved in Maple software with the help of the shooting technique. At different ranges of the applied parameters, dual solutions are found. The effects of the different physical factors such as radiation, nanoparticle volumetric fractions, suction, and slip parameters are determined and discussed. The skin-friction coefficient and local Nusselt number are influenced significantly by the applied parameters. In the boundary layer regime, the increase in nanoparticle volume fractions and radiation parameters enhance the temperature and boundary-layer thicknesses, while increasing Prandtl number, suction, and thermal slip parameters decrease the temperature and reduce thermal boundary-layer thicknesses. The suspension of iron nanoparticles shows more enhancement in skin friction and Nusselt number than the iron oxide nanoparticles in base fluid water. Keywords-boundary layer; dual solutions; shooting method; rediation; nanofluid

INTRODUCTION Nanofluids are made up of solid nanoparticles and common base fluids such as water, oil, glycol, polymer fluids, and so on [1,[3][4][5]. The nanoparticles are made from metals, metal oxides, and carbides, and have far greater heat conductivity. Their insertion considerably enhances thermo-physical properties such as thermal conductivity, viscosity, thermal diffusivity, and the convective heat transfer coefficient. Nanofluids may be employed in many industrial and technological applications [2], including microelectronics, coolants, fuel cells, pharmaceutical operations, household freezers, chillers, and lubricants. Several methods have been developed to improve the thermal conductivity of typical common liquids, including suspending macro/micro particles of solid materials in them.
Two types of nanofluid modelshave been created and effectively employed as a result of recent developments in nanotechnology. Buongiorno [6] proposed the initial model, which incorporated 7 slip factors as critical components in generating a relative velocity between the nanoparticles and the base fluid. Only thermophoresis and Brownian motion were considered essential among the 7 slip components when creating a model for convective transport in nanofluids. The second nanofluid model was proposed by Tiwari and Das [7] to study the effect of nanoparticle volume fractions on thermal characteristics enhancement. Researchers used and modified these two well-known nanofluid models by using different physical flow parameters [8][9][10][11].
Many engineering and practical science applications, such as paper manufacturing, wire drawing, extrusion, metal spinning, and hot rolling, rely on the boundary layer flow through a stretching/shrinking sheet. Crane [12] was the first to work on the stretched sheet. Since then, a broad range of issues related to these topics was examined. Authors in [13] examined the heat and mass transfer on a stretched sheet with suction and blowing. Many researchers were also interested to study the flow over a stretching sheet. The study of viscous flow past on shrinking sheet along the suction action was pioneered in [14]. It is worth mentioning that mass suction is required to keep the flow inside boundary layer flowing. The analytic solution of the magnetohydrodynamic flow of a second grade fluid on a shrinking sheet was investigated in [15]. Authors in [16] explored the MHD rotational flow of a viscous fluid over a shrinking surface. Authors in [17] were the first to investigate the nanofluid flow through the boundary layer of a stretching sheet. The boundary layer slip flow problem is quite relevant in practice. Wang [18] analyzed the impact of the partial slip on flow of a viscous fluid through a stretching sheet using numerical simulations. Authors in [19] examined the natural convection of magnetohydrodynamics nanofluid flow on a vertically flat plate. Authors in [20] examined the impact of the particle sizes and volume fractions on the Al 2 O 3 -water base nanofluid. Authors in [21] studied the steady boundary-layer flow through a continuously moving plate with different nanoparticles immersed in water in a 2D (two-dimensional) flow. Author in [22] considered 2D nanofluid flow past a moving plate in a continuous free-stream. Authors in [23] examined 2D laminar stagnant-point fluid flow with heat absorption and generation. Authors in [24] investigated the behavior of slip factors using single and multiphase nanomaterial models. Authors in [25] inspected the inclination of the magnetic field in a convective radiative heat transfer micro-nanofluid using a porous medium.
Various scholars have studied the occurrence of multiple solutions to the boundary value problems. These solutions are caused by the non-linearity of the equations at various ranges of the physical parameters that can be seen in the literature. Several academics assessed the stability of a range of solutions to find which is the stable and physically realizable. Many scholars, including the authors in [26][27][28] have come up with different solutions and they proved that only one solution was found stable through stability analysis.
The primary aim of this study is to determine how various flow parameters impact water-based nanofluids created by stretching and shrinking sheets. The Tiwari and Das model is used to investigate the heat radiation and slip effects of iron (Fe) and iron oxide (Fe 2 O 3 ) nanoparticles. The similarity transformations are used to convert the partial differential equations into ordinary differential equations,. The shooting approach is applied for solution of the resulting system of equations. According to the findings of this study, dual solutions exist for certain ranges of the physical parameters that are illustrated in the graphs. Due to the occurrence of dual solutions, the bvp4c Matlab program is used to perform the stability analysis. The stability tests reveal that the first solution is the stable and the physically realizable, while the second is not stable.

II. PROBLEM FORMULATION
Let's consider a two-dimensional boundary layer flow and heat transfer in a water-based nanofluid containing iron (Fe) and iron oxide (Fe 2 O 3 ) nanoparticles over a stretching/ shrinking sheet. The sheet is considered to be parallel to the plane ‫ݕ‬ = 0, and the flow is contained at ‫ݕ‬ > 0. The sheet is stretched and shrunk with velocity ‫ݑ‬ ௪ = ‫,ݔܿ‬ where ܿ > 0 is the stretching rate, with two opposed and equal forces applied in the direction of the x-axis. The base fluid (water) and nanoparticles are considered to be in thermal equilibrium. Table I lists the thermo physical properties of water as well as the nanoparticles that are used in the present study. The governing equations of the problem by using the Tiwari and Das model [7] are written as: The Roseland approximation, ‫ݍ‬ is defined as: where ߪ * is Boltzmann's constant and k* is the absorption coefficient. It is assumed that the temperature difference is very small. As a result, when the Taylor series is applied to the temperature of the free stream fluid flow which is indicated by ܶ ஶ and ignoring higher order terms, the ܶ ସ can be defined as: The specified boundary conditions are: where v, u are the velocity components in the x and y directions and ܷ ஶ and ‫ݑ‬ ௪ are the velocities of free stream fluid and sheet respectively, A and B are slip factors, T, ߤ , ߙ and ߩ are the temperature, viscosity, thermal diffusivity and density of the nanofluid respectively, as defined in [32]: where ߩ ௦ and ߩ are the densities of solid nanoparticles volume fractions and fluid (water) respectively, ݇ and ݇ ௦ are the thermal conductivities of the base fluid and solid nanoparticles volume fractions respectively. Furthermore, the ߤ (the viscosity of the nanofluid) was estimated in [33] and the similarity transformation is taken according to [34]: Transformation (8) instantly fulfills the differentiation in terms of (1), however (2) and (3) are reduced to the following non-linear ordinary differential equations: and the boundary conditions take the form: at ߟ = 0, ݂ ᇱ ሺߟሻ → 0; ߠሺߟሻ → 0; as ߟ → ∞ሻ (11) where prime stands for the derivative with respect to ߟ, ൗ is the thermal slip parameter and ߣ is the stretching/shrinking parameter. When 0 < ߣ < 1, the fluid and the plate are moved in the similar direction. Likewise, for ߣ < 0 and ߣ > 1, these are in opposite direction of the motion. When ߣ < 0 shows that the flow of the ambient fluid is in the positive x-direction and the movement of sheet is in negative x-direction, ߣ > 1 shows that the flow of ambient fluid is in the negative x-direction and the movement of the sheet in the positive x-direction. In the present paper, both cases are considered but main focus was given on ߣ < 0.
The local Nusselt number ‫ݑܰ‬ ௫ and the skin friction coefficient ‫ܥ‬ are key physical parameters that are described as: Shear stress ߬ ௪ and surface heat diffusion ‫ݍ‬ ௪ are defined as: By the use of (8) and (13) in (12), we get: (14) where ܴ݁ ௫ = ௫௨ ೢ ణ reports the Reynolds number.
III. STABILITY ANALYSIS As this problem has dual solutions, stability analysis is required to find a stable and physically reliable solution. To evaluate the stability of the solutions, at first the governing system of (2) and (3) will be written into an unsteady form like: where t denotes the time. The transformation (8) (17) By implementing (17) into the (15) and (16) we get: The boundary conditions are: To obtain stability for the solutions ݂ሺߟሻ = ݂ ሺߟሻ and ߠሺߟሻ = ߠ ሺߟሻ that satisfy the specified boundary value problem given in equations (2) ‫ܩ‬ሺߟሻ and ‫ܨ‬ሺߟሻ are the related functions to ߠ ሺߟሻ and ݂ ሺߟሻ respectively, and ߛ is the unknown minimum eigenvalue.
The following linear eigenvalue problem system is generated by using (19) into (18)-(21): The set ‫ܨ‬ሺߟሻ = ‫ܨ‬ ሺߟሻ and ‫ܩ‬ሺߟሻ = ‫ܩ‬ ሺߟሻ is used in (22) and (23) to acquire the initial decay or growth of solutions of (21). Therefore, the following system of eigenvalues has been solved: with boundary conditions: To get the smallest eigenvalue, the linearized equations (24) and (25) with boundary conditions (26) are solved in MATLAB using the bvp4c solver techniques. As mentioned in [35], we must relax one boundary condition into the initial condition. In this case, ‫ܨ‬ ′ሺߟሻ → 0 as ߟ → ∞ into ‫′ܨ‬ ᇱ ሺ0ሻ ൌ 1.
The smallest negative eigenvalues indicate the disturbance's initial growth. As a result, the solutions related to the fluid flow are said to be unstable. The fluid flow is said to be stable in case obtained smallest positive eigenvalues.
IV. NUMERICAL METHOD The boundary value problem given in (9) and (10) is solved using the shooting method in Maple software, using the initial and boundary conditions (11). On the other hand, this method turns the boundary value problem into an initial value problem. Therefore, we get: The boundary conditions take the form: where ߙ ଵ , ߙ ଶ are taken as the unknown initial conditions. So, the shooting values for the missing initial values of ߙ ଵ , ߙ ଶ are important. The solution must satisfy the boundary conditions ‫ܨ‬ ሺߟሻ → 0, ߠሺߟሻ → 0 as ߟ → ∞ of the specified boundary value problem. The computation is done in Maple by using the shootlib function and the shooting method. In this paper, the thickness of the boundary layer ߟ ஶ is taken between 4 and 7, at which the convergence shown in the graphs is obtained. For fixed values of the used parameters, 2 velocity and temperature profiles are achieved, each totally satisfying the problem's required boundary conditions. As a result, the quantities ݂ ᇱ ′ሺ0ሻ and െߠ′ሺ0ሻ, have two distinct values.

V. RESULTS AND DISCUSSION
The shooting method in Maple is used to obtain the numerical solutions to the system of ordinary differential equations (9) and (10) that are subjects to the boundary conditions of (11). Figures 2-17 show the results of the impact of the non-dimensional physical parameters on the velocity and temperatures profiles as well as the skin friction coefficient and Nusselt numbers. Table I shows the thermo physical properties of the base fluid and nanoparticles used in the present study. Table II shows the comparison result of the 2 numerical methods, the shooting method used in Maple and the bvp4c used in Matlab. The comparison shows much symmetry in the results that encourage us to consider that our obtained results are correct and trustworthy. In addition, in the case of dual solutions, stability analysis has been performed with bvp4c in Matlab to determine which solution is reliable and stable by obtaining the smallest eigenvalue using (24)- (26). The solution which corresponds to a positive least eigenvalue is considered as a reliable and stable solution. The solution which corresponds to a negative least eigenvalue is considered as an unstable solution. Table III Table IV shows the critical points concerned to Figures 2-5, where two solutions are merged. Furthermore, the impact of the various used parameters of this study are presented and discussed below through graphs.  Figure 2 shows that when the suction rate is increased, the skin friction coefficient reduces for values ߣ 0 and rises for ߣ ൏ 0 in the first (stable) solution. Furthermore, the rate of skin friction (drag force) is examined to be larger in Fe 2 O 3 -water base nanofluid than in Fe-water base nanofluid for ߣ 0 (for stretching case of the sheet) and vice versa for ߣ < 0 (for shrinking case of the sheet) for the same values of the suction parameter ሺܵሻ. The Nusselt number (rate of heat transfer) rises as the suction parameter ሺܵሻ is increased in both solutions ( Figure 3).
In addition, the variations of skin friction coefficient ሺ݂ ′′ሺ0ሻሻ and local Nusselt number ሺ−ߠ ′ሺ0ሻሻ along the variation of the suction parameter at various values of nanoparticles volume fraction ሺ߶ሻ are shown in  For ܵ > ܵ , dual solutions also exist. At the critical point ܵ , both solutions merge, while no solution is observed for ܵ < ܵ . In both Fe 2 O 3 -water base and Fe-water base nanofluids, the skin friction coefficient rises as the nanoparticle volume fraction increases, while it decreases in the second solution ( Figure 4). The rate of skin friction for the Fe-water base nanofluid is observed to be greater than that of the Fe 2 O 3 -water base nanofluid. It may be noted that the viscous fluid flow in Figures 4 and 5 is shown by black colored lines. Figure 5 shows that the Nusselt number decreases with an increase in the rate of nanoparticles volume fraction in both solutions in both types of nanofluids. Heat transfer rate is observed greater in Fe-water base nanofluid than in Fe 2 O 3 -water base nanofluid. The comparative results of the variations of the skin friction coefficient and local Nusselt number with the variation of nanoparticles volume fraction regarding to Fe 2 O 3 -water base nanofluid and Fe-water base nanofluid are presented in Figures  6 and 7. It is seen that the Fe-nanoparticles show the greater resistance in flow, which means greater rate of skin friction coefficient is observed as compared to the Fe 2 O 3 in water base nanofluid. Due to the greater resistance of suspending Fe nanoparticles in water, the rate of heat transfer of Fe-water base nanofluid remains greater than that of the Fe 2 O 3-water base nanofluid as shown in Figure 7. Furthermore, due to the repeated trend of the result, only graphs of Fe-water base nanofluid are presented here for further study.
Furthermore, Figures 8-17 show the velocity and temperature profiles for various values of the applied parameters. Figures 8 and 9 show the velocity and the temperature profiles of the boundary layer flow of the Fe-water base nanofluid at different values of nanoparticle volume fraction ሺ߶ሻ. The influence of ߶ shows that with any increment in the concentration of nanoparticles the velocity of the fluid decreases throughout the boundary layer region in the first (stable) solution. Actually, the nanoparticles develop a friction in fluid molecules which retards the flow. The nanoparticles also increase the thermal conductivity of the base fluid, so the heat is transferred from the hot regime to the cold one in faster rate and in result, warms the thermal boundary layer regime. Hence, the concentration of nanoparticles increases the temperature and the thermal boundary layer thickness of the base fluid as shown in Figure 9. The effects of the suction parameter ሺܵሻ on ݂ ′ሺߟሻ and ߠሺߟሻ profiles are shown in Figures  10 and 11. It can be clearly observed that any increment in S raises the suction rate of the fluid at sheet and in result, momentum boundary layer thickness and fluid velocity decrease in the first solution throughout the nanofluid flow. In the second solution, the opposite result can be observed. Any increment in ܵ decreases the thermal boundary layer thickness and the temperature of the nanofluid throughout the flow in both solutions. Figures 12 and 13 show the impact of ߣ (for shrinking case) on ݂ ′ሺߟሻ and ߠሺߟሻ profiles respectively. Both the profiles of velocity ݂ ′ሺߟሻ and temperature ߠሺߟሻ are decreasing for increasing ߣ (in the first solution) and increasing for increasing ߣ (in the second solution) for the shrinking case.
The influence of the Prandtl number ሺܲ‫ݎ‬ሻ on the temperature profile ߠሺߟሻ of the Fe-water base nanofluid is shown in Figure 14. When the parameter ‫ݎܲ‬ is raised, the temperature of the nanofluid decreases. Because fluids with a high ‫ݎܲ‬ value have a low thermal diffusivity, the temperature of the moving fluid drops. The temperature and thickness of the boundary layer decrease as ‫ݎܲ‬ increases. The influence of radiation parameter ሺܴ݀ሻ on the temperature profile is drawn in Figure 15. The temperature profile increases as the radiation parameter increases. As the value of ܴ݀ increases, the divergence of radiating heat flux increases as ݇ * decreases. As a consequence, the rate of radiating heat transfer to the fluid's boundary layer flow rises which increases the temperature of the fluid. The velocity profile ݂′ሺߟሻ of the Fe-water base nanofluid in the first solution decreases when the velocity slip parameter (ߜ) increases, as can be seen in Figure 16. Basically, any increment in ߜ shows more fluid particles slipping on sheet, so the fluid flow decelerates near the sheet. In the second solution, any change in parameter values causes the nanofluid velocity to rise. The influence of thermal slip parameter (ߜ ் ) on the temperature profile ߠሺߟሻ is shown in Figure 17. When the thermal slip increases, the thickness of the thermal boundary layer and the temperature profile both decrease in both solutions. The fluid velocity is initially decreased when the thermal slip parameter is raised, resulting in a decrease in net molecular mobility. Consequently, less molecular momentum decreases the thermal boundary layer thickness and the temperature profile.

VI. CONCLUSION
The steady two-dimensional boundary layer flow and heat transfer of Fe 2 O 3 and Fe-water base nanofluid over a flat stretching/shrinking sheet with radiation and slip effects were studied numerically in this paper. The nanofluid model in [7] is used with a porous medium. The similarity transformation is used to convert the system of partial differential equations into ordinary differential equations. The shooting method is used to solve these equations numerically. Due to the existence of dual solutions, stability analysis is performed using bvp4c in Matlab. The following conclusions are drawn based on the numerical investigations: • Dual solutions are observed for certain ranges of stretching/shrinking and suction parameters.
• Stability analysis shows that one solution is stable and physically reliable while the other is unstable.
• The skin friction coefficient decreases for ߣ 0 and increases for ߣ < 0 as the rate of suction is increased.
• The heat transfer rate and the skin friction coefficient are decreasing in the second solution when the velocity slip parameter ߜ increases. • Fe-water nanofluid shows a greater rate of skin friction and Nusselt number as compared to Fe 2 O 3 -water nanofluid.
• An increase in nanoparticles' volume fraction decreases the skin friction coefficient and increases the Nusselt number.
• Increase in velocity slip, suction, nanoparticles volume fraction, and shrinking parameters decreases the velocity profiles and the associated boundary layers thicknesses.
• The rise in nanoparticle volumetric fraction and radiation increases the temperature profiles and the boundary layer thicknesses.