Effect of suction on the MHD fluid flow past a non-linearly stretching/shrinking sheet: Dual solutions

This paper investigated the effects of suction for incompressible magnetohydrody-namics (MHD) fluid over a non-linearly stretching/shrinking sheet. A proper similarity trans-formation has been used to reduce the system of partial differential equations to a system of ordinary differential equations, which was then solved numerically by using the bvp4c function in the MATLAB software. Dual solutions are found for certain values of the suction parameter when the sheet is shrinking. The generated results were presented and discussed in the relevance of the governing parameters.


Introduction
The boundary layer flow past a continuous moving surface has many industrial and engineering applications, such as the extrusion of a polymer sheet from a dye, hot rolling, metallic plates cooling and continuous casting. The study was initiated by Sakiadis [1] in 1961, then followed by Crane [2] where he discovered the closed analytic solution of twodimensional Navier-Stokes equations by considering the linear relationship between the flow velocity and the distances from a fixed point. Erickson et al. [3] extended [1] by considering the case of non-zero transverse velocity. Further, the three-dimensional unsteady fluid flow along a continuously stretching surface has been presented by Lakshmisha et al. [4] by considering the effects of the magnetic field, rate of heat and mass transfer.
Several years later, Miklavcic and Wang [5] studied the steady three-dimensional boundary layer flow d ue t o a s hrinking s urface, a nd t hey d iscovered t hat a n a dequate s uction i s required to sustain the flow p ast t he s hrinking s heet. T his t ype o f fl ow is es sentially a ba ckward flow, as discussed by Goldstein [6] and it shows physical phenomena quite distinct from the forward stretching flow. Besides, they have also discovered the non-unique/multiple solutions at a specific rate of suction. This discovery has been discussed by Davey [7] where he mentioned that a unique solution indicates the nodal point flow, non-uniqueness solutions reflect a saddle point flow, and no solution means the equations become insoluble. The analysis done in [5] was also extended in various directions for different fluids by many researchers (see [8][9][10][11][12][13][14][15]).
The papers discussed in all the above are related to the case of linearly stretching/shrinking surface. There exists another physical phenomenon in which the surface is stretched or shrunk in a nonlinear fashion. In 2001, Vajravelu [16] obtained the numerical solution of flow and heat transfer in a viscous fluid over a nonlinearly stretching sheet by using a fourth-order Runge-Kutta integration scheme and he showed that the heat flow is always from the sheet to the fluid. Later, some works on the boundary layer flow along with the nonlinearly stretching or shrinking sheet. Shit and Haldar [17] investigated the thermal radiation effects on the magnetohydrodynamic (MHD) flow and heat transfer over a nonlinear shrinking porous sheet. Recently, Khan et al. [18] studied the nonlinear radiation effects on MHD flow of nanofluid over a nonlinearly stretching/shrinking wedge, while Jusoh and Nazar [19] discussed the magnetohydrodynamic (MHD) stagnation point flow and heat transfer of an electrically conducting nanofluid over a nonlinear stretching/shrinking sheet.
The present study considers the numerical solutions of the magnetohydrodynamic twodimensional boundary layer flow past a non-linearly stretching/shrinking sheet with suction effect, which is an extension from [16]. The present work also attempted to obtain the second solution, which has not been done in [16]. This study is essential in industrial applications such as automotive plastic fuel tanks, heating pipe and rigid packing for entrees [20]. The numerical results presented in Section 5 are generated by using the bvp4c function in the MATLAB software.

Problem Formulation
We consider the steady two dimensional and incompressible free conventional boundary layer flow of a Newtonian fluid over a stretching or shrinking sheet, as shown in figures 1 and 2. The pressure gradient and external force are assumed to be neglected in this problem. The flow is generated by the nonlinear stretching/shrinking sheet along the x-axis where x is the coordinate measured along the sheet. We also assume that the variable magnetic field β is applied normal to the sheet and the induced magnetic field is neglected. Using boundary layer approximations, the governing equations of this problem can be expressed as the following while the corresponding boundary conditions for the current problem can be written as where u and v are the velocity components in x and y directions, respectively, while ν is the kinematic viscosity, ρ is the density, β is the strength of the applied magnetic field, σ i s the electrical conductivity of the fluid, v w is the suction ( v w < 0) and injection ( v w > 0) parameter, and λ is the stretching or shrinking parameter with λ > 0 is for stretching surface and λ < 0 is for shrinking surface. Furthermore, m is the nonlinear parameter with m = 1 for the linear case and m ̸ = 1 is for the nonlinear case.

Similarity Transformation
Following [16], we look now for a similarity solution of equations (1) and (2) subjected to the initial and boundary condition (3) of the following form where primes denote derivatives with respect to η. Furthermore, we assume where s > 0 corresponds to suction while s < 0 corresponds to the fluid withdrawal, respectively. Using (4), equation (1) is satisfied, while equation (2) is transformed into the following ordinary differential equation along with the corresponding boundary conditions where M = 2σβ 2 ρU (m+1) is the magnetic parameter, while s is the mass transfer parameter with s > 0 denotes suction, while s < 0 refers to injection and s = 0 is for nonpermeable surface.
The main physical quantity of interest is the reduced skin friction coefficient in x−direction which is defined as where Re x = Uwx ν .

Application of the bvp4c function Method
As mentioned in the previous section, we implement the bvp4c function to solve the resulting non-linear ordinary differential e quation ( 6 w ith b oundary c ondition ( 7. T he MATLAB solver bvp4c function uses the collocation method and effective in solving the boundary value problem [21]. To achieve this, we first write equation (6) as a system of first order equation in the following form while the boundary condition (7) is transformed to Aside from equations (9) and (10) , a good initial guess is necessary to obtain the dual solutions. Dual solutions comprise of upper branch solution and lower branch solution. The upper branch solution is a solution which converged asymptotically with a thin boundary layer, whereas the lower branch solution converged asymptotically with the thicker boundary layer. The relative tolerance has been fixed to 1 × 10 −10 throughout the computation process.

Results and Discussion
The nonlinear ordinary differential equation (6) and its boundary condition (7) are solved numerically using the bvp4c function in MATLAB, as stated in the previous section. The validity of the method is measured by comparing the present numerical results with the previous study.
The following table 1 shows an excellent comparison of the present study with the previous results obtained in [16], which proves that the present method is accurate and the results are correct.

Conclusions
The present study was devoted to solving boundary layer flow problem over a permeable stretching/shrinking sheet under the influence of the magnetohydrodynamics (MHD). Interestingly, this paper identified the dual solutions, namely upper branch solution and lower branch solution and explained its behaviour physically as the governing parameters vary. When the magnetic parameter M and the mass transfer parameter s increases, the upper branch solution shows increment while the lower branch solution conveys decrement. The upper branch solution decreases while the lower branch solution increases when the velocity power index m increases.