Lie group analysis for MHD effects on the convectively heated stretching porous surface with the heat source/sink
 Limei Cao^{1},
 Xinhui Si^{1}Email author,
 Liancun Zheng^{1} and
 Huihui Pang^{2}
https://doi.org/10.1186/s1366101503264
© Cao et al.; licensee Springer. 2015
Received: 19 December 2014
Accepted: 7 April 2015
Published: 22 April 2015
Abstract
This paper investigates magnetohydrodynamic (MHD) boundary layer flow and convective heat transfer of a fluid with variable viscosity through a porous medium towards a stretching sheet by considering the effects of viscous dissipation in presence of heat source/sink. The fluid viscosity is assumed to be a linear function of the temperature. The application of scaling group of transformations on the generalized stretching surface with injection velocity leads to two possible surface conditions. The governing equations with two types of boundary conditions are solved numerically using Bvp4c with MATLAB, respectively. Furthermore, more attention is paid to the effects of some physical parameters on the velocity and the temperature distribution with considering the permeability and the heat sink or the heat source.
Keywords
1 Introduction
The interest in MHD fluid flows stems from the fact that liquid metals that occur in nature and industry are electrically conducting, which is attractive both from a mathematical and a physical standpoint. This type of flow has received much attention of many researchers due to its applications in technological and engineering problems such as MHD generators, plasma studies, nuclear reactors, geothermal energy extractions. By the application of a magnetic field, hydromagnetic techniques are used for the purification of molten metals from nonmetallic inclusions. Then the type of problem that we are dealing with is very useful to polymer technology and metallurgy [1–4]. In addition, some theoretical work also has been done. For example, Rasmussen [5] numerically studied the problem of the steady viscous symmetric flow between two parallel porous coaxial disks. Hayat et al. [6] studied the MHD flow of an upperconvected Maxwell fluid over a porous stretching plate with the homotopy analysis method. Noor [7] presented an analysis of the MHD flow of a Maxwell fluid past a vertical stretching sheet in the presence of thermophoresis and chemical reactions. Hayat et al. [8] constructed an analytic solution for unsteady MHD flow in a rotating Maxwell fluid through a porous medium. Ibrahim and Shankar [9] investigated MHD boundary layer flow and heat transfer of a nanofluid past a permeable stretching sheet with slip boundary conditions.
Lie group analysis, also called symmetry analysis, was developed by Sophus Lie to find point transformations which map a given differential equation to itself. This method has been used by many researchers to solve some nonlinear problems in fluid mechanics [10–17]. Furthermore, the scaling group techniques, a special form of Lie group transformations, have been applied by many researchers [18–26] to study different flow phenomena over different nonlinear dynamical systems, aerodynamics, and some other engineering branches. For example, Kanadasamy and Muhaimin [21] discussed steady twodimensional flow of incompressible fluid over a vertical stretching sheet by scaling group of transformations. Hamad [23] used this method to study the effect of a magnetic field on the free convection flow of a nanofluid over a linear stretching. Das [24] analyzed the MHD boundary layer flow of an electrically conducting nanofluid past a vertical convectively heated permeable stretching surface with variable stream conditions in presence of chemical reaction.
Motivated by the above works, the aim of this paper is to investigate the MHD effects on the convectively heated stretching porous surface with the heat source/sink. With the assistance of scaling group of transformations, two types of boundary conditions satisfying the similarity transformations are obtained, and then the coupled differential equations are deduced according to different boundary conditions, respectively. The effects of different parameters on the velocity and temperature distribution for these two cases are plotted in graphs and discussed in detail.
2 Preliminaries
Consider a steady twodimensional forced convection flow of a viscous incompressible laminar dissipating fluid past a convectively heated stretching sheet immersed in a porous medium in the region \(y>0\). Keeping the origin fixed, a force is applied along the xaxis which results stretching of the sheet. All body forces except by magnetic field are neglected etc. and the uniform magnetic field of strength \(B_{0}\) is assumed to be perpendicular to the xaxis. We assume the temperature of the sheet to be different from that of the ambient medium and the fluid viscosity to vary with temperature, while the other fluid properties are assumed to be constants. u and v are the components of velocity in x and y directions and T is the temperature of the fluid, respectively.
3 Application of group transformations
Two results are presented subject to the two different boundary conditions in the following section.
3.1 Case 1: equations with the first type of boundary conditions
The relation \(3\alpha_{2}\alpha_{3}=\alpha_{2}\alpha_{3}\) can result in \(\alpha_{2}=0\). Hence, \(\alpha_{1}+2\alpha_{2}2\alpha_{3}=3\alpha_{2}\alpha_{3}\alpha_{6}\) gives \(\alpha_{6}=0\), \(\alpha_{2}=\frac{1}{4}\alpha_{1}=\frac{1}{3}\alpha_{3}\). Also the boundary conditions yield \(\alpha_{4}=m\alpha_{1}=\frac{1}{2}\alpha_{1}\), \(\alpha_{5}=\frac{m1}{2}\alpha_{1}=\frac{1}{4}\alpha_{1}\), and then \(m=\frac{1}{2}\).
3.2 Case 2: equations with the second type of boundary conditions
4 Numerical solutions and discussion
Since (3.16)(3.18) and (3.27)(3.29) are coupled nonlinear boundary value problems, these equations are solved numerically by Bvp4c with MATLAB, which is a collocation method equivalent to the fourth order monoimplicit RungeKutta method. Since the velocity changes sharply in the boundary layer near the plate, this region with a sharp change makes this boundary value problem a relatively difficult one. In order to resolve better the boundary layer and obtain a more accurate solution, the relative error tolerance on the residuals is defined to be 10^{−6} (i.e. \(\operatorname{RelTol}=10^{6}\)) during the process of numerical computation. The results are presented graphically and in tables.
Comparison of \(\pmb{f''(0)}\) for \(\pmb{Pr=10}\) , \(\pmb{Le=1}\) , and \(\pmb{Nb=0.1}\)
Computed values of skin friction coefficient \(\pmb{f''(0)}\) for various values of M , S , and \(\pmb{k_{1}}\)
\(\boldsymbol {M^{2}}\)  S  \(\boldsymbol {k_{1}}\)  Ibrahim and Shankar [ 9 ]  Present result 

1  0  0  1.4142  1.414214 
1  0.2  0  1.5177  1.517745 
1  0.2  1  0.5656  0.565566 
1  0.2  1.2  0.5055  0.505457 
1.5  0.5  0  1.8508  1.850781 
2  0.5  0  2.0000  2.000000 
5 Conclusion

The two types of boundary conditions satisfying the similarity solution can be obtained with the assistance of scaling group of transformations.

As there exists a slip parameter, the influence of various physical parameters, such as the viscosity parameter, the Eckert number, the Prandtl number, the permeability parameter, and the Hartmann number on the velocity and temperature distribution is significantly different from the case of no slip velocity.

Because of the existence of the slip parameter, the velocity at the plate have different values influenced by Hartmann number, permeability parameter, and viscosity parameter.

Whether there exists slip parameter or not, the temperature distribution is an increasing function of the Eckert number, the convection heat transfer parameter, and the heat source/sink parameter, respectively. However, it is the decreasing function of increasing Prandtl number.
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundations of China (No. 11302024, No. 61440058), the Fundamental Research Funds for the Central Universities (No. FRFTP14071A2, No. FRFBR13023) and Beijing Higher Education Young Elite Teacher Project (No. YETP0387, No. YETP0322), Research Foundation of Engineering Research Institute of USTB (No. Yj2011015).
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
References
 Crane, LJ: Flow past a stretching plate. Z. Angew. Math. Phys. 21, 645647 (1970) View ArticleGoogle Scholar
 Hashizume, H: Numerical and experimental research to solve MHD problem in liquid blanket system. Fusion Eng. Des. 81, 14311438 (2006) View ArticleGoogle Scholar
 Aoyagi, M, Ito, S, Hashizume, H, Muroga, T: MHD pressure drop characteristics in a threesurfacemultilayered channel under a strong magnetic field. Fusion Eng. Des. 85, 11811184 (2010) View ArticleGoogle Scholar
 Zhang, C, Zheng, L, Zhang, X, Chen, G: MHD flow and radiation heat transfer of nanofluids in porous media with variable surface heat flux and chemical reaction. Appl. Math. Model. 39, 165181 (2015) View ArticleMathSciNetGoogle Scholar
 Rasmussen, H: Steady viscous flow between two porous disks. Z. Angew. Math. Phys. 21, 187195 (1970) View ArticleGoogle Scholar
 Hayat, T, Abbas, Z, Sajid, M: Series solution for the upperconvected Maxwell fluid over a porous stretching plate. Phys. Lett. A 358, 396403 (2006) View ArticleMATHGoogle Scholar
 Noor, NFM: Analysis for MHD flow of a Maxwell fluid past a vertical stretching sheet in the presence of thermophoresis and chemical reaction. World Acad. Sci., Eng. Technol. 64, 10201023 (2012) Google Scholar
 Hayat, T, Fetecau, C, Sajid, M: On MHD transient flow of a Maxwell fluid in a porous medium and rotating frame. Phys. Lett. A 372, 16391644 (2008) View ArticleMATHGoogle Scholar
 Ibrahim, W, Shankar, B: MHD boundary layer flow and heat transfer of a nanofluid past a permeable stretching sheet with velocity, thermal and solutal slip boundary conditions. Comput. Fluids 75, 110 (2013) View ArticleMATHMathSciNetGoogle Scholar
 Ovsiannikov, LV: Group Analysis of Differential Equations. Academic Press, New York (1982) MATHGoogle Scholar
 Bluman, GW, Kumei, S: Symmetries and Differential Equations. Springer, New York (1989) View ArticleMATHGoogle Scholar
 Olver, J: Application of Lie Groups to Differential Equations. Springer, New York (1989) MATHGoogle Scholar
 AbdelMalek, MB, Badrana, NA, Hassanb, HS: Using group theoretic method to solve multidimensional diffusion equation. J. Comput. Appl. Math. 147, 385395 (2002) View ArticleMATHMathSciNetGoogle Scholar
 Boutros, YZ, AbdelMalek, MB, Badran, NA, Hassan, HS: Liegroup method for unsteady flows in a semiinfinite expanding or contracting pipe with injection or suction through a porous wall. J. Comput. Appl. Math. 197, 465494 (2006) View ArticleMATHMathSciNetGoogle Scholar
 Boutros, YZ, AbdelMalek, MB, Badran, NA, Hassan, HS: Liegroup method solution for twodimensional viscous flow between slowly expanding or contracting walls with weak permeability. Appl. Math. Model. 31, 10921108 (2007) View ArticleMATHGoogle Scholar
 AbdelMalek, MB, Aminb, AM: Lie group analysis of nonlinear inviscid flows with a free surface under gravity. J. Comput. Appl. Math. 258, 1729 (2014) View ArticleMATHMathSciNetGoogle Scholar
 Mekheimer, KS, ElSabbagh, MF, AboElkhair, RE: Lie group analysis and similarity solutions for hydromagnetic Maxwell fluid through a porous medium. Bound. Value Probl. 2012, Article ID 15 (2012) View ArticleMathSciNetMATHGoogle Scholar
 Pakdemirli, M, Yurusoy, M: Similarity transformations for partial differential equations. SIAM Rev. 40, 96101 (1998) View ArticleMATHMathSciNetGoogle Scholar
 Yurusoy, M, Pakdemirli, M, Noyan, OF: Lie group analysis of creeping flow of a second grade fluid. Int. J. NonLinear Mech. 36, 955960 (2001) View ArticleMathSciNetMATHGoogle Scholar
 Mukhopadhyay, S, Layek, GC, Samad, SA: Study of MHD boundary layer flow over a heated stretching sheet with variable viscosity. Int. J. Heat Mass Transf. 48, 44604466 (2005) View ArticleMATHGoogle Scholar
 Kanadasamy, R, Muhaimin, I: Lie group analysis for the effect of temperature dependent fluid viscosity and thermophoresis particle deposition on free convective heat and mass transfer under variable stream conditions. Appl. Math. Mech. 31, 317328 (2010) View ArticleMathSciNetMATHGoogle Scholar
 Zhu, J, Zheng, L, Zhang, Z: Effects of slip condition on MHD stagnationpoint flow over a powerlaw stretching sheet. Appl. Math. Mech. 31, 439448 (2010) View ArticleMATHMathSciNetGoogle Scholar
 Hamad, MAA: Analytical solution of natural convection flow of a nanofluid over a linearly stretching sheet in the presence of magnetic field. Int. Commun. Heat Mass Transf. 38, 487492 (2011) View ArticleGoogle Scholar
 Das, K: Lie group analysis for nanofluid flow past a convectively heated stretching surface. Appl. Math. Comput. 221, 547557 (2013) View ArticleMathSciNetMATHGoogle Scholar
 Kundu, PK, Das, K, Jana, S: Nanofluid flow towards a convectively heated stretching surface with heat source/sink: a lie group analysis. Afr. Math. 25, 363377 (2014) View ArticleMATHMathSciNetGoogle Scholar
 Asghar, S, Jalil, M, Hussan, M, Turkyilmazoglu, M: Lie group analysis of flow and heat transfer over a stretching rotating disk. Int. J. Heat Mass Transf. 69, 140146 (2014) View ArticleGoogle Scholar
 Dessie, H, Kishan, N: MHD effects on heat transfer over stretching sheet embedded in porous medium with variable viscosity. viscous dissipation and heat source/sink. Ain Shams Eng. J. 5, 967977 (2014) View ArticleMATHGoogle Scholar
 Batchelor, GK: An Introduction to Fluid Dynamics. Cambridge University Press, London (1987) MATHGoogle Scholar
 Cortell, R: Flow and heat transfer of a fluid through a porous medium over a stretching surface with internal heat generation/ absorption and suction/blowing. Fluid Dyn. Res. 37, 231245 (2005) View ArticleMATHGoogle Scholar
 Mukhopadhyay, S, Layek, GC, Samad, SA: Effects of variable fluid viscosity on flow past a heated stretching sheet embedded in porous medium in presence of heat source/sink. Meccanica 47, 863876 (2011) View ArticleMathSciNetMATHGoogle Scholar