The viscous boundary layer flow caused by a stretching sheet moving with a certain velocity often occurs in many industrial and engineering processes. These flows have extensive applications in industries, for example in the drawing of plastic films or in the extrusion of polymer sheet from a die. During the production of the respective sheets, the melt issues from a slit are accordingly stretched to get the required thickness. The mechanical characteristic of the subsequent product depends on the cooling and stretching rates in the process. Crane^[1] proposed the steady two-dimensional flow of a viscous fluid past a stretching surface. An exact closed-form solution was obtained for Craneʼs problem in fluid dynamics.Further, the work of Crane was extended in several ways to add other physical features like blowing or suction, magnetic field, and heat transport phenomenon, etc. Mehmood et al.^[2] incorporated the heat transport phenomena in a flow of Jeffrey nanofluid past a stretched plate. Muhammad et al.^[3] investigated the heat transfer phenomenon in the presence of thermal stratification of ferrofluid over a stretching surface. Muhammad et al.^[4] utilized the Cattaneo-Christov heat flux model to discuss the squeezed flow in nanofluid over a stretching sheet. Zeeshan et al.^[5] demonstrated the heat transfer in the flow of ferrofluid via stretching surface. Maqbool et al.^[6] used a stretching surface to explore the Falkner-Skan boundary layer flow.

In fluid mechanics, porous medium is a substance containing voids (pores). The voids are filled usually with a fluid. The skeletal portion of the substance is called frame. The frame is typically solid, but structures like foams are analyzed using the concept of porous media. The porous medium is incorporated by its porosity. Porous medium has also the properties of permeability, electrical conducting, tortuosity, and tensile strength. The deviation of these properties of porous medium is complex due to the pores structure and media porosity. Certain natural materials and substances including rocks and soil (e.g. petroleum resources, aquifers), biological tissues (e.g. wood, bones, cork), zeolites and several other man made substances like cements and ceramics can be considered as porous media. In this direction, lots of work exists in the literature. Sheikholeslami et al.^[7] scrutinized the heat transfer flow past a permeable stretching wall embedded in porous medium. The Numerical study of nanofluid transportation in porous media with external magnetic source is established by Sheikholeslami and Ganji.^[8] Nadeem and Muhammad^[9] indicated the flow dripping with porous medium using Cattaneo-Christov heat flux model. Debasish^[10] symbolized the non-Newtonian effect on hydromagnetic stratified dusty fluid flow along a porous media. Devakar et al.^[11] examined fully developed flow in a straight uniform square duct of non-Newtonian fluids saturated in a porous medium. Ellahi et al.^[12] expressed the heat transfer phenomenon in a third grade fluid embedded in porous medium. Vajravelu^[13] taken stretching surface to discuss heat transfer flow embedded in porous medium.

With rapid progress of modern engineering technology, the curved stretching has got extensive attention due to its various applications in transportation, industrial, and electronics. Sanni et al.^[14] obtained the numerical solution of viscous fluid saturated in curved stretching sheet. In the presence of magnetic field, the viscous fluid flow over a curved sheet were established by Abbas et al.^[15] Nadeem et al.^[16] indicated the chemically reactive species in the flow of Maxwell fluid over a stretching channel. Hayat et al.^[17] utilized the curved stretching sheet to examine the convective flow with chemical reaction. Rosca and Pop^[18] explored the unsteady boundary layer flow along a permeable curved shrinking/stretching surface. Hayat et al.^[19] analyzed the homogeneous-heterogeneous reaction past a curved sheet with magnetohydrodynamic. In all these articles the stretching is curved, and to the best of our knowledge there is not a single article in the existing literature regarding the stretching flow in presence of porous medium and magnetohydrodynamic past a curved surface. In this article, we present a numerical study to obtain the solution of non-linearly ordinary differential equation by using bvp4c from MATLAB. The graphical behavior of sundry parameters is presented. Some application about stretching sheet, porous media, and magnetic field is given in Refs.[20–28].

Consider the steady two-dimensional and an incompressible viscous fluid past a curved stretching channel coiled in a circle of radius R about the curvilinear coordinates r and s, as shown in Fig. 1, so that large value of R compares to a slightly curved surface. The fluid is embedded in a porous medium and the surface is stretched along the s-direction with velocity u_w = cx and at r-direction the transfer magnetic field of strength B₀ is applied.

Fig. 1.

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	Fig. 1. Geometry of the flow.

Under the boundary layer approximation, the governing equations for the flow problem are given as:


where s and r are the curvilinear coordinates taken along the sheet and normal to it respectively, u and v are the velocity components respectively, p indicates pressure, υ demonstrates the kinematic viscosity, σ expresses the electrical conductivity of the fluid, and B₀ signifies the constant magnetic field.

For the velocity profile boundary conditions are
Here u_w is the stretching velocity along s-direction and c is positive stretching constant.

Here we introduce the following dimensionless variables as:
where prime denotes differentiation with respect to η. These transformations satisfy the continuity equation identically and Eqs. (2) and (3) are reduced to ordinary differential equations

Here indicates parameter of curvature and it should be constant, is the dimensionless Hartman number or magnetic parameter and characterizes the porosity parameter. It is worth mentioning that when , Pm=0, and in the absence of body forces and pressure gradient Eq. (7) reduces to classical problem of flat stretching surface studied by Crane.^[1]
Now eliminating pressure between Eqs. (6) and (7) first we differentiate Eq. (7) with respect to η and then put Eq. (6), we get finally
which the boundary conditions are
The pressure can be obtained from Eq. (7) of the form
Now the surface drag force C_f in s-direction is defined as
In above equation τ_rs simplifies the wall shear stress beside the s-direction can be given as
Using transformation Eqs. (5) and (14), Eq. (13) becomes
Here demonstrates the local Reynold number.

The numerical method bvp4c from MATLAB is utilized to solve the numerical solution for the non-linear ordinary differential Eq. (9) with boundary condition (10) for several values of magnetic parameter M, porosity parameter Pm, and dimensionless curvature parameter K. For this method first we transformed the fourth order differential equation to system of the first order differential equation by introducing new variables. The tolerance for the present problem is taken as 10⁻⁷ and the function bvp4c requires an initial guess for the solution. The guess we chose must fulfill the boundary condition (10) and keep the behavior of the solution. Table 1 signifies the comparison of skin friction coefficient when M = 0 and Pm=0 with those of Abbas et al.^[15] and Table 2 shows the numerical solution for skin friction coefficient with different values of M, Pm and K.

Table 1.

Table 1.

Table 1. The comparison of Skin friction with previous result Abbase et al.[15] . K Abbase et al.[15] Present results 5 1.157 63 1.157 630 10 1.073 49 1.073 490 30 1.023 53 1.023 530 50 1.014 05 1.014 050 200 1.003 56 1.003 560 1000 1.000 79 1.000 790

Table 1. The comparison of Skin friction with previous result Abbase et al.[15] .

Table 2.

Table 2.

Table 2. The value of skin friction for different values of parameters. . K M Pm 5 0.2 0.5 1.533 00 10 1.413 00 20 1.357 00 10 0 0.5 1.328 00 0.2 1.413 00 0.4 1.493 00 10 0.2 0.1 1.236 00 0.2 1.283 00 0.3 1.328 00

Table 2. The value of skin friction for different values of parameters. .

Figure 2 illustrates the influence of curvature parameter K with M = 0.3 and Pm = 0.5 is fixed, on the velocity profile. It can be seen that for rising values of K, result in the enhancement of fluid velocity. This is because the radius of the surface increases with enlarging the value of curvature parameter K, which demonstrates the increment of fluid velocity. The boundary layer thickness also increases for larger values of K. Figure 3 elucidates the change in the velocity profile for various estimation of magnetic parameter M with K = 0.1 and Pm = 0.5 are fixed. Here enhancement in M, leads to decay the magnitude of fluid velocity. It is due to fact that the resistive force (so-called Lorentz force) produced by magnetic field that is why the fluid velocity decreases. Figure 4 displays the characteristic of dimensionless porosity parameter Pm on fluid velocity field.

Fig. 2.

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	Fig. 2. Effect of K on .

Fig. 3.

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	Fig. 3. Effect of M on .

Fig. 4.

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	Fig. 4. Effect of Pm on .

It is found that the fluid velocity has inverse relation with porosity parameter, i.e. for enlarging values of porosity parameter the fluid velocity becomes thinner. In fact porous medium presence increases the resistance to fluid motion, which produces reduction in fluid velocity and related momentum layer thickness. The values assign to remaining parameters are K = 0.1 and M = 0.3.

The mathematical expression for skin friction coefficient is given in Eq. (14). The effect of parameters K on skin friction coefficient via magnetic parameter M and Pm=0.5 are fixed is indicated in Fig. 5. It is observed that the surface drag force decreases with enlarging the value of curvature parameter K. From Fig. 5 one can see that the absolute value of the skin friction increases by increasing the value of M, however it is decreasing by increasing the curvature parameter K. While in Fig. 6 the impact of porosity parameter Pm verses M on skin friction coefficient is demonstrated and it is found that the surface drag force increases with increment in porosity parameter Pm and K = 10 are fixed.

Fig. 5.

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	Fig. 5. Effect of K on wall shear stress.

Fig. 6.

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	Fig. 6. Effect of Pm on wall shear stress.

The property of the magnetic field in a viscous fluid over a steady curved stretching surface embedded in the porous medium is investigated. The following conclusion is made.