1 Introduction

A number of studies have been reported in the literature focusing on the stagnation-point flow towards a stretching sheet because of its industrial and engineering applications such as extrusion,paper production,insulating materials,glass drawing,and continuous casting. Hiemenz^[1] initiated the two-dimensional stagnation-point flow. He discovered that the Navier- Stokes equations governing the flow can be transformed into an ordinary differential equation (ODE) of third-order by using the similarity transformation. The boundary layer flow over a stretching surface was studied by Sakiadis^[2]. He modeled the laminar boundary-layer behavior on a moving continuous flat surface and presented the numerical solutions for the boundary- layer equations. Later on,this idea was extended by Crane^[3] for both linear and exponentially stretching sheets. Chiam^[4] discussed the steady two-dimensional stagnation-point flow of a viscous fluid towards a stretching sheet. He made the analysis with the assumption that the sheet was stretched in its own plane with a velocity proportional to the distance from the stag- nation point. Wang^[5] studied the free convection on a vertical stretching surface. Nazar et al.^[6]analyzed the unsteady two-dimensional stagnation-point flow of an incompressible viscous fluid over a flat deformable sheet. The magnetohydrodynamic (MHD) stagnation-point flow towards a stretching vertical sheet was discussed by Ishak et al.^[7]. The two-dimensional stagnation- point flow of a viscoelastic fluid was studied by Sadeghy et al.^[8] assuming that the fluid obeyed the upper-convected Maxwell (UCM) model. The boundary-layer hypothesis was used to sim- plify the equations of motion which were further reduced to a single nonlinear third-order ODE with the idea of the stream function coupled with the technique of the similarity solution. The resulting equations were solved using the Chebyshev pseudo-spectral collocation-point method. Attia^[9] made an analysis of the steady hydromagnetic laminar three-dimensional stagnation- point flow of an incompressible viscous fluid impinging on a permeable stretching surface with heat generation or absorption.

MHD flows play an important role in the motion of fluids. Normally,a uniform magnetic field is applied normal to the plate which is maintained at a constant temperature. The steady MHD mixed convection flow of a viscoelastic fluid in the neighborhood of two-dimensional stagnation- points with a magnetic field has been investigated by Kumari and Nath^[10] considering the UCM model. The boundary layer theory could simplify the equations of motion,the induced magnetic field,and the energy into three coupled nonlinear ODEs. These equations were finally solved by using the finite difference method. The results indicated that the increase in the elasticity number causes the reduction in the surface velocity gradient and the surface heat transfer.

The dual solutions of the boundary layer flow over moving surfaces are of practical impor- tance in engineering analysis. It gives the possibility to determine the most realistic and phys- ically meaningful solutions. Extensive studies have been carried out to investigate the multiple solutions of the boundary layer flows driven by moving surfaces. Kemp and Acrivos^[11] found the dual solutions for a moving-wall boundary layer reverse flow. Riley and Weidman^[12] pre- sented multiple solutions for the Falker-Skan equation with a stretching boundary. Ingham^[13] presented the dual solutions of a steady mixed convection boundary layer assisting flow over a moving vertical flat plate. Ridha and Curie^[14] found the dual solutions for both assisting and opposing flows. Recently,Mahapatra et al.^[15] showed that the upper branch solution was al- ways stable whereas the lower branch was unstable. Moreover,the stable branch was physically meaningful. Makinde et al.^[16] showed that the dual solutions existed for the stagnation-point flow of an electrically conducting nanofluid towards a vertically stretching sheet.

In the present article, we discuss the stagnation-point flow of a Prandtl fluid towards a shrinking sheet with a magnetic field. To the best of the authors’ knowledge, no such investiga- tion has been carried out yet. The main objective of the article is to discuss the dual solutions for both stagnation points and shrinking flows.

2 Mathematical formulation

We consider a two-dimensional stagnation-point flow of an incompressible Prandtl fluid over a wall coinciding with the plane y = 0. The flow is confined to the plane y > 0,and the sheet shrinks linearly along the x-axis. Moreover,we consider the effects of stagnation away from the x-axis,and the uniform magnetic field is applied normal to the fluid flow (see Fig. 1).

The extra stress tensor for the Prandtl fluid is defined as^[17]

where A and C are material constants of the Prandtl fluid model,respectively. The flow equations for the Prandtl fluid model after applying the boundary layer approximations can be defined as follows: where u and v are the velocity components along the x- and y-axes,respectively. ν is the kinematic viscosity,B0 is the magnetic field,σ is the permeability of the fluid,and ρ is the density of the fluid. The flow velocity outside the boundary layer (inviscid fluid) is ue(x) = ax, where a > 0. From Eq. (3),we can see that most of the liquid metals possess low conductivity which may lead to small electrical currents generated by the fluid flow in the presence of a magnetic field. In other words,the magnetic Reynolds number is much smaller than one^[18,19].

The corresponding boundary conditions are

where a > 0 is the constant,and uw is the velocity at wall.

Introduce the following similarity transformations:

By transformations (5),Eqs. (2)-4) take the forms In the above equations,α = A/C is the Prandtl parameter,β = a³x²A/2Cν is the elastic parameter, λ = b a is the stretching (λ > 0) or shrinking (λ < 0) parameter,M² = σB₀ ²/ρa is the magnetic field parameter,and s is the mass transfer parameter with s > 0 for suction and s < 0 for injection.

After using the boundary layer approximations,the wall shear stress λ_w can be given by

The coefficient of the skin friction is defined by In the dimensionless form,the skin friction is defined as

3 Numerical method for solutions

Numerical solutions to the governing ordinary differential equation (6) subject to the bound- ary conditions (7) are obtained by the shooting method. First,we convert the boundary value problem (BVP) into the initial value problem (IVP) and assume a suitable finite value for the far field boundary condition,i.e.,η→∞,saying η∞. Then,we set the following first-order system:

with the boundary conditions To solve (11) with (12) as an IVP,the values for q(0),i.e.,f′′(0),are needed. However,no such a value is given prior to the computation. The initial guess values of f′′(0), are chosen,and the fourth-order Runge-Kutta method is used to obtain a solution. We compare the calculated values of f′(η) at the far field boundary condition η∞(= 20) with the given boundary condition f′(η) ! 1,,and the values of f′′(0) are adjusted by the Secant method for better approximation. The step-size is taken as ▽η = 0.01,and the accuracy to the fifth decimal place is regarded as the criterion of convergence. It is important to note that the dual solutions are obtained by setting two different initial guesses for the values of f′′(0),where both profiles (first and second solutions) satisfy the far field boundary condition (7) asymptotically but with different shapes.

To validate the accuracy of the proposed numerical scheme,the obtained results correspond- ing to the skin-friction coefficient f′′(0) (for the Newtonian fluid case,i.e.,β = 0) are compared with those available in the literatures^[20,21,22] in Table 1 and Table 2 for the stretching case (λ > 0) and the shrinking case (λ < 0).

4 Results and discussion

In the present section,we discuss the effects of the physical parameters such as the suc- tion/injection parameter s,the Prandtl parameter α,the elastic parameter β,and the magnetic field parameter M on both the velocity profile and the skin friction coefficient. The effects of the various fluid flow parameters on the skin friction coefficient are presented in Fig. 2,Fig. 3,Fig. 4. From Fig. 2,it can be observed that for fixed suction and MHD parameters, the local skin friction increases as the elastic parameter (β) decreases. Meanwhile, the existence of the dual solution for the shrinking sheet case is found when λc < λ < 0, where the critical values of λ are found to be λc ≈ −4.69, −4.78, and −5.47 for β = 0.3, 0.2, and 0.1, respectively. Moreover, as the elastic parameter increases, the curve for the local skin friction first decreases gradually, and then decreases to zero at λc = −4. It is also observed that for decreasing values of the elas- tic parameter β, both the branches of the skin friction coefficient exhibit the same increasing tendency.

It is clear from Fig. 2,Fig. 3,Fig. 4 that dual solutions exist beyond the critical value (turning point) λc. Large imposition of suction is required so that dual solutions are possible for the flow with large magnetic parameters. In reality,between these two solutions,only one solution is stable. The first solution is assumed to be physically stable because its solution is the continuation of the case of injection (s < 0). The second has negative values of the skin friction coefficient. This solution shows the occurrence of flow separation and reverse,which cause the difficulties in the numerical computation. Merkin^[23],Weidman et al.^[24],Paullet and Weidman^[25],and Harris et al.^[26] have presented the mathematical proof of the conjecture of dual numericalsolutions. They performed stability analyses and revealed that the solutions along the upper branch (first solution) are linearly stable while those on the lower branch (second solutions) are linearly unstable.

The effects of the magnetic field parameter M and the mass transfer parameter s are pre- sented in Fig. 3 and Fig. 4,respectively,while the rest parameters are fixed. It is observed from Fig. 3 that in the present region for the values of M,we can have dual solutions of the skin friction coefficient against λ. Figure 3 shows that in both cases,lower and upper branches give the same increasing behavior for higher values of M. It is observed from Fig. 4 that both curves have the same increasing behavior for higher values of s. Finally,we show that for the stretching/shrinking parameter λ,dual solutions exist.

The variations of the velocity profile f′(η) with various fluid flow parameters are plotted in Figs. 5−7. The effects of the Prandtl fluid parameter α and the elastic parameter β are plotted in Figs. 5−6. The dual velocity profiles show that the velocity decreases with the increases in α and β while conversely increases for the second solution. It is also noted that the boundary layer is thicker for the second solution in comparison with that for the first solution. The effects of the suction/injection parameter s on the dual velocity profiles are shown inFig. 7. As the mass transfer parameter s increases in the magnitude,both the profiles increase. For large values of the far field boundary η∞, the second solution velocity profiles decrease with the increase in s.

5 Conclusions

We theoretically study the boundary layer stagnation-point flow of the Prandtl fluid model towards a shrinking sheet in the presence of a magnetic field. We discuss the effects of the suction/injection parameter s,the Prandtl parameter α,the elastic parameter β,and the magnetic parameter M on the fluid flow. The key conclusions of this study are as follows.

(i) We show that dual solutions exist for the proposed Prandtl fluid flow model.

(ii) The skin friction increases with the increases in the suction/injection parameter s and the magnetic parameter M,whereas with the increase in the elastic parameter β,the skin friction decreases.

(iii) It has been observed that the boundary layer thickness for the second solution is thicker than that for the first solution.

(iv) For the increasing suction/injection parameter s,the boundary layer thickness for both solutions decreases.