MHD mixed convection slip flow near a stagnationpoint on a nonlinearly vertical stretching sheet
 Ming Shen^{1}Email author,
 Fei Wang^{1} and
 Hui Chen^{2}
https://doi.org/10.1186/s1366101503406
© Shen et al.; licensee Springer. 2015
Received: 9 February 2015
Accepted: 29 April 2015
Published: 14 May 2015
Abstract
The problem of magnetohydrodynamic (MHD) mixed convection flow near a stagnationpoint region over a nonlinear stretching sheet with velocity slip and prescribed surface heat flux is investigated; this has not been studied before. Using a similarity transformation, the governing equations are transformed into a system of ordinary differential equations, and then are solved by employing a homotopy analysis method. The effects of the nonlinearity parameter, the magnetic field, mixed convection, suction/injection, and the boundary slip on the velocity and temperature profile are analyzed and discussed. The results reveal that the increasing exponent of the powerlaw stretching velocity increases the heat transfer rate at the surface. It is also found that the velocity slip and magnetic field increase the heat transfer rate when the free stream velocity exceeds the stretching velocity, i.e. \(\varepsilon< 1\), and they suppress the heat transfer rate for \(\varepsilon> 1\).
Keywords
1 Introduction
The problem of stagnationpoint flow and heat transfer on stretching sheet arises in an abundance of practical applications in industry and engineering, such as cooling of electronic devices and nuclear reactors, polymer extrusion, drawing of plastic sheets, etc.; and, moreover, in the magnetohydrodynamic (MHD) flow which has both liquid and magnetic properties and can exhibit particular characteristics in thermal conductivity. Thus the study of MHD stagnationpoint flow on stretching sheet has attracted many researchers in recent times, and many problems are discussed as regards different aspects, including the stretching sheet with variable surface temperature [1] or viscous dissipation [2, 3], the effect of slip [4, 5], and the analysis of the unsteady case [6].
Different from the flow induced by a stretching horizontal sheet, the effect of mixed convection due to a buoyancy force could not be neglected for the vertical sheet. There has been increasing interest in studying the problem of MHD with mixed convection boundary layer flow and heat transfer characteristics over a stretching vertical surface [7–15]. Very recently, Ali et al. [16] studied the MHD mixed convection stagnationpoint flow and heat transfer of an incompressible viscous fluid over a vertical stretching sheet, and the MHD boundary layer flow over a vertical stretching/shrinking sheet in a nanofluid was investigated by Makinde et al. [17] and Das et al. [18].
The above investigations considered the flow on the linearly stretching sheet or vertical sheet, but the real stretching velocity does not always need to be linear or uniform. Some work has been done in this field. The similarity solution of the boundary layer equations for a nonlinearly stretching sheet has been found by Akyildiz et al. [19]. The flow and heat transfer over a nonlinearly stretching sheet has been investigated by Akyildiz and Siginer [20] by using a Legendre spectral method. Recently, Dhanai et al. [21], Ashraf et al. [22], and Mabood et al. [23] analyzed the boundary layer flow and heat transfer on a nonlinearly shrinking/stretching sheet immersed in a nanofluid.
In the present paper, motivated by the above studies, the problem of MHD mixed convection stagnationpoint flow on the nonlinearly vertical stretching sheet is discussed in the presence of buoyancy force, suction/injection parameters, and boundary slip. The governing nonlinear coupled partial differential equations are reduced to a set of ordinary differential equations by means of similarity transformations. The reduced equations are solved by the homotopy analysis method (HAM) [24], which has been successfully applied to various interesting complicated fluid problems [25–30]. Graphs are plotted to gain physical insight toward the key embedding physical parameters. To the best of our knowledge, the series solutions for this model have not been presented before.
2 Mathematical formulation of the problem
3 Series solutions of HAM
4 Analysis of the results
4.1 Convergence of the solutions
Values of \(\pmb{f''(0)}\) and \(\pmb{1/\theta(0)}\) for different orders of approximations
Order of approximation  (a) ε = 0.1  (b) ε = 1  (c) ε = 2  

\(\boldsymbol{f''(0)}\)  1/ θ (0)  \(\boldsymbol{f''(0)}\)  1/ θ (0)  \(\boldsymbol{f''(0)}\)  1/ θ (0)  
1  0.5878  1.8038  −0.0556  2.1622  −0.7858  2.7746 
5  0.6511  2.1674  −0.0283  2.2484  −0.7940  2.3482 
10  0.6591  2.1700  −0.0252  2.2587  −0.7943  2.3501 
15  0.6597  2.1736  −0.0248  2.2635  −0.7944  2.3559 
20  0.6598  2.1758  −0.0248  2.2657  −0.7944  2.3580 
25  0.6599  2.1768  −0.0248  2.2666  −0.7944  2.3589 
30  0.6599  2.1771  −0.0248  2.2670  −0.7945  2.3593 
35  0.6599  2.1772  −0.0248  2.2670  −0.7945  2.3593 
40  0.6599  2.1772  −0.0248  2.2670  −0.7945  2.3593 
Values of \(\pmb{f''(0)}\) and \(\pmb{1/\theta(0)}\) for different orders of approximations
40th order of approximation  Yacob and Ishak [ 31 ]  Present results  

Pr  \(\boldsymbol{f''(0)}\)  1/ θ (0)  \(\boldsymbol{f''(0)}\)  1/ θ (0) 
0.7  1.8339  0.7776  1.8337  0.7771 
1  1.7338  0.8780  1.7337  0.8780 
4.2 Results and discussion
5 Conclusions

The increase of nonlinearity parameter m leads to an increases of the heat transfer rate at surface \(1/\theta(0)\), and to a decrease of both the thicknesses of the velocity and the thermal boundary layer.

The coefficient of the skin friction \(f''(0)\) and the heat transfer rate at the surface \(1/\theta(0)\) increase with increasing mixed convection parameter λ.

The heat transfer rate increases as the velocity slip parameter δ and magnetic parameter M increase for \(\varepsilon < 1\), but it decreases with these two parameters for \(\varepsilon> 1\).

Inside the velocity boundary layer, the velocity increases with the increasing S, δ, and M for \(\varepsilon<1\), and the opposite trend is displayed for \(\varepsilon>1\).

Inside the thermal boundary layer, the temperature always decreases with increasing λ, m, and S.
Declarations
Acknowledgements
The authors would like to thank the referees for their pertinent comments and valuable suggestions. This work is supported by the National Natural Science Foundation of China (Grant No. 51305080).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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