Magnetohydrodynamics Stagnation-Point Flow of a Nanoﬂuid Past a Stretching/Shrinking Sheet with Induced Magnetic Field: A Revised Model

: The revised Buongiorno’s nanoﬂuid model with the effect of induced magnetic ﬁeld on steady magnetohydrodynamics (MHD) stagnation-point ﬂow of nanoﬂuid over a stretching or shrinking sheet is investigated. The effects of zero mass ﬂux and suction are taken into account. A similarity transformation with symmetry variables are introduced in order to alter from the governing nonlinear partial differential equations into a nonlinear ordinary differential equations. These governing equations are numerically solved using the bvp4c function in Matlab solver, a very adequate ﬁnite difference method. The inﬂuences of considered parameters ( Pr , M , χ , Le , Nb , Nt , S , and λ ) on velocity, induced magnetic, temperature, and concentration proﬁles together with the reduced skin friction and heat transfer rate are discussed. Results from these criterion exposed the existence of dual solutions when magnetic ﬁeld and suction are applied for a speciﬁc range of λ . The stability of the solutions obtained is carried out by performing a stability analysis.


Introduction
In preceding decades, a very active topic of studies continues to be the study of fluid flow and heat transfer past a stretching/shrinking surface. Although a large sum of works have made remarkable contributions to the development of the theory, a proportionally good amount of efforts have also been dedicated to daily engineering applications that include electronic equipment, thermal energy storage systems, glass-fiber manufacturing, wire drawing, paper milling, and extraction of polymer sheets (see Fisher [1]). As a matter of fact, stretching confers a unidirectional orientation to the extrudate, thereby improving its mechanical properties as the features of the end product are substantially influenced by the rate of cooling. Correspondingly, it is crucial to meticulously control the fluid and heat transfer mechanism as the desired quality of the final product relies heavily on the cooling procedure. Following the famous works of Tsou et al. [2], Sakiadis [3], and that of Crane [4] in spearheading the stretching sheet problem with boundary layer approximation, several similar studies have arose, emphasizing on the shrinking sheets aspect (see Miklavčič and Wang [5] and Wang [6]). The general deduction in previously reported manuscripts suggested that the shrinking/stretching is linearly proportional to the axial distance. The solution approach somehow leaves behind uncertainties pertaining the uniqueness of the solution.
The concept of magnetohydrodynamics (MHD), which analyses the performance of magnetically induced nanofluids in various boundary-layer flow control systems can be integral in the theoretical where suction is an existent and unsteady case. For those interested in engineering analysis, the idea of discovering dual solutions and defining stability is of practical significance, as it offers a way to determine if a steady state solution is physically relevant [45]. This kind of research is therefore very useful in determining the dual solutions of fluid flow problems. Here, the work of Hussaini and Lakin [46] is worth mentioning among the pioneering researchers who discovered the uniqueness of solutions in the boundary layer problems. Magyari et al. [47] has presented dual solutions in homogenous boundary layer flows caused by continuous surface stretching with rapidly decreasing power-law and exponential velocities. Subsequently, in unsteady cases, Merrill et al. [48] studied the mixed stagnation-point convection flow on a vertical surface in a fluid-saturated porous medium and discovered that there exist dual solutions for certain buoyancy parameter values. By conducting a stability analysis by Merkin [49], Merrill et al. [48] demonstrated that the solutions of the upper branch are stable and therefore asymptotically available solutions. Very recently, the effect of induced magnetic field on MHD stagnation-point flow of nanofluids towards stretching/shrinking sheet was analyzed by Junoh et al. [50]. Since the existence of dual solutions for a certain range of shrinking rate, a stability analysis has been performed and the result showed that only the upper branch solution is stable and physically acceptable.
The purpose of this present paper is to extend the work by Junoh et al. [50] by considering the presence of suction on MHD stagnation-point flow of heat and mass transfer over a permeable stretching/shrinking sheet in a nanofluid. The effect of induced magnetic field is also taken into account. In this study, the new boundary condition proposed by Kuznetsov and Nield [51] will be implemented. This work also highlights the stability of the solutions obtained. Therefore, we presume the findings are new and original, which all those interested in stretching/shrinking sheet problems in nanofluids can use with strong confidence.

Problem Formulation
We consider a steady two-dimensional MHD stagnation-point flow of heat and mass transfer over a permeable stretching/shrinking sheet in a nanofluid with induced magnetic field. The coordinate system is selected, in which the x-axis is in horizontal direction and the y-axis is in vertical direction. u and v represents the velocity and H 1 and H 2 are the magnetic components along the xand y-axes, respectively. u e is the velocity at the edge of the boundary layer while u w and v w are, respectively, the velocity and mass flux velocity at the surface. At the plate, y = 0, and the temperature T and concentration C take constant values T w and C w . The ambient values as y → ∞, temperature and concentration, respectively, are T ∞ and C ∞ . The physical flow model and coordinate system are shown in Figure 1. Following the mathematical nanofluid model proposed by Davies [52] and Kuznetsov and Nield [53], the governing equations of this problem can be derived as follows: ∂u ∂x along with their boundary conditions: where = (ρC p ) p (ρC p ) f is defined as the ratio of nanoparticle heat capacity to the base fluid heat capacity.
The boundary conditions D B ∂C ∂y (7) states that the normal flux of nanoparticles is zero at the boundary when the thermophoresis is considered (Kuznetsov and Nield [51]). Following Kuznetsov and Nield [53], we introduce symmetry variables where primes denote differentiation with respect to η. After substitution Equation (8) into Equations (1)-(6), we get the following reduced form: subjected to the new boundary conditions: In the above equations, Pr is the Prandtl number, M is the magnetic parameter, χ is the reciprocal magnetic parameter, Nb is the Brownian motion parameter, Nt is the thermophoresis parameter, Le is the Lewis number, S is the suction/injection parameter which is defined as and λ stands for stretching/shrinking parameter. Here, λ > 0 indicates that the plate is stretched; while when λ < 0 the plate is shrunk. It should be highlighted that the use of the variables (8) will change the position of the Pr from the energy Equation (11) to the momentum Equation (9), which is a new fact for stretching/shrinking sheet problems.
The physical quantities of interest in this study are the skin friction or shear stress coefficient C f and the local Nusselt number Nu x are Here τ w is the surface shear stress and q w is the surface heat flux given by Then, applying the symmetry variables (8), we obtain where Re x = u e x/ν is the local Reynolds number. However, the local Sherwood number Sh x can be derived from the boundary conditions (13) as φ (0) = − Nt Nb θ (0).

Stability Analysis
To conduct the stability analysis of the solutions, we consider this problem in unsteady form. The continuity in Equations (1) and (2) where t denotes the time. The following new dimensionless variables are introduced:

Results and Discussion
The governing Equations . We fix Pr = 0.72, M = 0.5, χ = 1, Le = 1, Nb = 0.01, Nt = 0.01, and S = 3 into our computation procedure, modifying one parameter at a time. We take η ∞ = 10 as our far-field boundary condition. All the profiles presented in the form of figures (Figures 2-10) satisfy the boundary condition (13) and produce asymptotic graph. A comparison of the results for some values of f (0) has been carried out with the results in the existing literature [55,56] in order to validate this study. This comparison data is shown in Table 1 and shows a positive agreement. Consequently, we are assured that the numerical values obtained in this study are veritable.  Figures 2 and 3 show the velocity profiles with the effects of M and Pr, respectively. It is seen that as the value of M and Pr increases, the boundary layer thickness of these profiles also increase. Figures 4-6 portray the impact of M, χ, and Pr on the induced magnetic profiles. As the value of these parameters increase, the boundary layer thickness become thicker. Figure 7 illustrates the influence of the Pr on temperature profile. It is noticed that the thermal boundary layer thickness increases as the Pr increase.        Figure 8 exhibits the influence of Le on concentration graph profile. The concentration boundary layer thickness decreases as Lewis number increases. This is likely due to the fact that mass transfer rate increases parallel to Le. Likewise divulges that the concentration gradient at the plate surface is also increasing. As can be seen in Figure 9, increasing values of Nt leads to an increase in the thickness of concentration graph profile. A different behavior is shown in Figure 10, the concentration graph profile is decreasing when the value of Nb increasing. Thus, the concentration boundary layer also decreases. From these figures it may very well be seen that the boundary layer thickness for the upper solution is slightly slimmer contrasted with that of the lower branch solution.     (12) can be obtained both for shrinking (λ < 0) and stretching (λ > 0) cases. Figure 11 exhibits the variation of the f (0) increasing as the value of S increases. The presence of suction will escalate the skin friction at the surface of sheet. Thereupon, the flow speed will decrease and as a result increase the velocity gradient at the surface of the sheet. The variation of the −θ (0), as shown in Figure 12, indicates that when S increases, the heat transfer rate at the surface also increases. The heat flux becomes larger when suction is applied in the flow. Hence, it will also increase the magnitude of the temperature gradient at the surface of the sheet. Based on the critical value λ c in these figures, the influence of suction S also causes a delaying of the boundary layer separation. From Figures 13 and 14, we can observe that as the values of M increase, the f (0) and −θ (0) decrease. In magnetohydrodynamics problems, it is a fact that the presence of transverse magnetic field sets in Lorentz force, which exudes a retarding force on the velocity field. The existing of this force has the urgency to decelerate the fluid motion in the boundary layer. As a result, the boundary layer separation occurs faster when the magnetic field is adapted.
Based on all these figures, the existence of nonunique solutions (dual solutions) are clearly shown. The stability of these solutions are tested to analyze which solutions are stable and physically realizable in practice. It is also intended to find out which solutions are not suitable as well. Therefore, the eigenvalue problem (34)- (38) is solved for the smallest eigenvalues γ 1 regarding the upper and lower solution branches. These calculations are presented in Table 2 for some values of λ when S = 1, 2, 3. From the table, it can be noticed that the smallest eigenvalue γ 1 gives a positive value for the first solution and a negative value for the second solution. Hence, a conclusion is drawn that only the upper branch solutions are physically significant; whilst the lower branch solutions are not. Furthermore, as the value of λ approaches the critical point λ c , the smallest eigenvalue γ 1 converges to 0 for both upper and lower branches, as discovered by Merkin [49].

Conclusions
A numerical study is carried out for the revised model of the steady MHD stagnation-point flow of nanofluid over a stretching or shrinking sheet with induced magnetic field. The effect of all parameters are studied numerically and graphically. The suction detains the boundary layer separation, however, the magnetic parameter enhances the boundary layer separation. Non-unique (dual) solutions are found to exist for both stretching and shrinking cases. Therefore, a stability analysis is done via bvp4c function in MATLAB software, and their results found that the first solution (upper branch) is stable and valid physically; while the second solution (lower branch) is not stable.

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

Abbreviations
The following abbreviations are used in this manuscript: