Journal of Applied Mathematics

Volume 2014 (2014), Article ID 256761, 8 pages

http://dx.doi.org/10.1155/2014/256761

## MHD Flow of a Viscous Fluid over an Exponentially Stretching Sheet in a Porous Medium

^{1}Department of Mathematics, University of Azad Jammu and Kashmir Muzaffarabad, Azad Kashmir 13100, Pakistan^{2}Theoretical Physics Division, PINSTECH, P.O. Nilore, Islamabad 44000, Pakistan^{3}Department of Physics, University of Azad Jammu and Kashmir Muzaffarabad, Azad Kashmir 13100, Pakistan^{4}Department of Computer Sciences and Information Technology, University of Azad Kashmir, Azad Kashmir 13100, Pakistan

Received 18 January 2014; Revised 2 April 2014; Accepted 8 April 2014; Published 29 April 2014

Academic Editor: Ning Hu

Copyright © 2014 Iftikhar Ahmad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Radiation effects on magnetohydrodynamic (MHD) boundary-layer flow and heat transfer characteristic through a porous medium due to an exponentially stretching sheet have been studied. Formulation of the problem is based upon the variable thermal conductivity. The heat transfer analysis is carried out for both prescribed surface temperature (PST) and prescribed heat flux (PHF) cases. The developed system of nonlinear coupled partial differential equations is transformed to nonlinear coupled ordinary differential equations by using similarity transformations. The series solutions for the transformed of the transformed flow and heat transfer problem were constructed by homotopy analysis method (HAM). The obtained results are analyzed under the influence of various physical parameters.

#### 1. Introduction

Since the revolutionary work of Sakiadis [1, 2] on boundary-layer flow past a moving surface a great deal of research work has been carried out for the two-dimensional boundary-layer flows. Crane [3] extended the Sakiadis [1, 2] flow problem by assuming a stretching boundary. Crane’s problem is one of the flow problems in boundary layer theory that possesses an exact solution. The stretching velocity in Crane’s problem is linearly proportional to the distance from origin. The heat and mass transfer on the flow past a porous stretching surface is discussed by P. S. Gupta and A. S. Gupta [4]. Brady and Acrivos [5] proved the existence and uniqueness of the solution for the stretching flow. Three-dimensional flow due to a stretching surface was analyzed by McLeod and Rajagopal [6]. In another paper Wang [7] extended Crane’s problem for a stretching cylinder. Flow problems due to a stretching surface have important applications in technology, geothermal energy recovery, and manufacturing process such as oil recovery, artificial fibers, metal extrusion, and metal spinning. Extensive literature is available regarding the steady flows in Newtonian and non-Newtonian fluid flows over a stretching sheet. The readers are referred to the studies [8–14] and references therein. The transportation of heat in a porous medium has applications in geothermal systems, rough oil mining, soil-water contamination, and biomechanical problems. Vajravelu [15] reported steady flow and heat transfer of viscous fluids by considering different heating process in a porous medium.

The above mentioned problems deal with linear stretching of the surface. Magyari and Keller [16] initiated the work by assuming an exponentially stretching surface. The heat transfer analysis for flow past an exponentially stretching surface was carried out by Elbashbeshy [17]. The same problem by considering the constitutive equation of a viscoelastic fluid was investigated by Khan [18]. Due to the applications in electrical power generator, astrophysical flows, solar power technology, space vehicle reentry, and so forth, the heat transfer analysis in the presence of radiation is another important area of research. Raptis [19] investigated the radiation effects for the flow past a semi-infinite flat plate. The influence of thermal radiation in a viscoelastic fluid past a stretching sheet is illustrated by Chen [20]. Sajid and Hayat [21] discussed the homotopy series solution for the influence of radiation on the flow past an exponentially stretching sheet. Chiam [22] investigated the heat transfer analysis with variable thermal conductivity in a stagnation point flow towards a stretching sheet. In another paper, the analysis of effects of variable thermal conductivity was discussed by Chiam [23].

The purpose of the present paper is to demonstrate an analysis for the MHD flow and heat transfer analysis of a viscous fluid towards an exponentially stretching sheet in the presence of radiation effects and Darcy’s resistance. The analytic series solution is presented by using homotopy analysis method [24–30].

#### 2. Mathematical Formulation of the Problem

Consider a steady, incompressible, two-dimensional MHD flow of an electrically conducting viscous fluid towards an exponentially stretching sheet in a porous medium. The fluid occupies the space and is flowing in the direction and -axis is normal to the flow. A constant magnetic field of strength is applied in the normal direction and the induced magnetic field is neglected which is a valid assumption when the magnetic Reynolds number is small. The boundary-layer equations that govern the present flow and heat transfer problem are where and are the components of velocity in and directions, respectively, is the pressure, is the density, represents kinematic viscosity, is the permeability of porous medium, is the electrical conductivity, is the fluid temperature, is the specific heat, is radiative heat flux, and is the variable thermal conductivity defined in the following way: in which is the small parameter ( for gases and for solids and liquids) and and are dimensionless temperature distributions for PST and PHF cases, respectively. The relevant boundary conditions for the present problem are where is a characteristic velocity, is the characteristic length scale, and , , , and are the parameters of temperature profiles depending on the properties of the liquid. Equation (7) suggests that far away from the plate. Since there is no variation in the pressure in direction as evident from (3), therefore, is zero throughout the flow field and we have from (2) The radiative heat flux is defined as in which is the Stefan-Boltzmann constant and is the coefficient of mean absorption. Assuming as a linear combination of temperature, then Substituting (9) and (10) in (4) one gets Introducing the dimensionless variables In terms of above variables flow and heat transfer problems take the following form: where the dimensionless parameters are given as

#### 3. HAM Solution

To find the series solutions for velocity and temperature profiles given in (14), we use homotopy analysis method (HAM). The velocity and temperature distributions can be expressed in terms of the following base functions: in the form where , , and are the coefficients. The following initial guesses are selected for solution expressions of , , and : and auxiliary linear operators are which satisfy where () are arbitrary constants. The zero-order deformation problems are The nonlinear operators are given by in which is an embedding parameter and , , and are nonzero auxiliary parameters. For and , we, respectively, have As varies from 0 to 1, , , and vary from , , and to , , and , respectively. Using Taylor’s series one can write where Note that the zero-order deformation equations contain three auxiliary parameters , , and . The convergence of series solutions strongly depends upon these three parameters. Assuming , , and are chosen in such a way that the above series are convergent at , then Taking the derivative of (21)–(24) -times with respect to , then selecting and dividing by the following expressions for th order deformation problems are obtained: in which The system of linear nonhomogeneous equations (29)–(32) is solved using Mathematica in the order .

#### 4. Results and Discussion

The governing nonlinear equations for the MHD flow and heat transfer analysis for a viscous fluid in a porous medium with radiation effects and variable thermal conductivity are solved analytically using homotopy analysis method for both PST and PHF cases. The solutions are given in the form of analytic expressions in (28). These analytic expressions contain the auxiliary parameters , , and . The convergence and rate of approximation of the obtained series solutions strongly depend upon these parameters. To examine the range of acceptable values of these parameters, we draw the -curves in Figure 1 for the functions , , and . This figure depicts that the admissible values of , , and are in the interval .

To see the influence of porosity parameter , Prandtl number , thermal conductivity , temperature parameters and , and thermal radiation on the temperature distributions for both PST and PHF cases, Figures 2–6 have been plotted. Figures (a) are for the PST case and Figures (b) are for PHF case. In Figures 2(a) and 2(b), variation of temperature distributions with variation in thermal conductivity parameter is shown. Figure 2(a) illustrates that temperature and thermal boundary-layer thickness in the PST case increases by increasing thermal conductivity. The results in the PHF case are opposite to that of PST case and are shown in Figure 2(b). Figure 3 elucidates that the temperature and thermal boundary-layer thickness increases by an increase in the porosity parameter. Since in the governing equation magnetic parameter and porosity parameter appear in the same way, therefore, the effects of magnetic parameter are similar to those of porosity parameter. The behavior of temperature with changing Prandtl number is quite opposite to that of porosity parameter and is given in Figure 4. Figure 5 is made to show the influence of parameter in the PST case and parameter in the PHF case on the temperature distributions. It is evident from Figure 5 that effects of these parameters are the same as that of Prandtl number. The influence of thermal radiation on the temperature profiles for both PST and PHF cases is depicted in Figure 6. The plotted profiles show that the temperature and thermal boundary-layer thickness is an increasing function of the thermal radiation.

#### 5. Conclusions

Magnetohydrodynamic boundary-layer flow and heat transfer characteristic due to exponentially stretching sheet in the presence of thermal radiation, Darcy resistance, and variable thermal conductivity have been studied in the paper. The similarity solutions are developed using homotopy analysis method. Following are the findings of current study.(1)The variable thermal conductivity has an impact in enhancing the temperatures profile in PST case and reduction in temperature in PHF case.(2)The effect of porosity parameter increases the temperature in both PST and PHF cases.(3)The effect of increasing the values of Prandtl number is to decrease the thermal boundary-layer thickness in both of the cases.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgment

Muhammad Sajid acknowledges the support from AS-ICTP.

#### References

- B. C. Sakiadis, “Boundary layer behaviour on continuous solid surfaces,”
*AIChE Journal*, vol. 7, no. 1, pp. 26–28, 1961. View at Publisher · View at Google Scholar - B. C. Sakiadis, “Boundary layer behaviour on continuous solid surfaces: II, the boundary layer on a continuous flat surface,”
*AIChE Journal*, vol. 17, pp. 221–225, 1961. View at Google Scholar - L. J. Crane, “Flow past a stretching plate,”
*Zeitschrift für Angewandte Mathematik und Physik*, vol. 21, no. 4, pp. 645–647, 1970. View at Publisher · View at Google Scholar · View at Scopus - P. S. Gupta and A. S. Gupta, “Heat and mass transfer on a stretching sheet with suction and blowing,”
*The Canadian Journal of Chemical Engineering*, vol. 55, no. 6, pp. 744–746, 1977. View at Publisher · View at Google Scholar - J. F. Brady and A. Acrivos, “Steady flow in a channel or tube with an accelerating surface velocity. An exact solution to the Navier-Stokes equations with reverse flow,”
*Journal of Fluid Mechanics*, vol. 112, pp. 127–150, 1981. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. B. McLeod and K. R. Rajagopal, “On the uniqueness of flow of a Navier-Stokes fluid due to a stretching boundary,”
*Archive for Rational Mechanics and Analysis*, vol. 98, no. 4, pp. 385–393, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Y. Wang, “The three-dimensional flow due to a stretching flat surface,”
*Physics of Fluids*, vol. 27, no. 8, pp. 1915–1917, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Y. Wang, “Fluid flow due to a stretching cylinder,”
*Physics of Fluids*, vol. 31, pp. 466–468, 1988. View at Publisher · View at Google Scholar - R. Cortell, “Flow and heat transfer of a fluid through a porous medium over a stretching surface with internal heat generation/absorption and suction/blowing,”
*Fluid Dynamics Research*, vol. 37, no. 4, pp. 231–245, 2005. View at Publisher · View at Google Scholar · View at Scopus - R. Cortell, “A note on magnetohydrodynamic flow of a power-law fluid over a stretching sheet,”
*Applied Mathematics and Computation*, vol. 168, no. 1, pp. 557–566, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S.-J. Liao, “On the analytic solution of magnetohydrodynamic flows of Non-Newtonian fluids over a stretching sheet,”
*Journal of Fluid Mechanics*, no. 488, pp. 189–212, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. D. Ariel, T. Hayat, and S. Asghar, “The flow of an elastico-viscous fluid past a stretching sheet with partial slip,”
*Acta Mechanica*, vol. 187, no. 1–4, pp. 29–35, 2006. View at Publisher · View at Google Scholar · View at Scopus - P. D. Ariel, “Axisymmetric flow of a second grade fluid past a stretching sheet,”
*International Journal of Engineering Science*, vol. 39, no. 5, pp. 529–553, 2001. View at Publisher · View at Google Scholar · View at Scopus - T. Hayat and M. Sajid, “Analytic solution for axisymmetric flow and heat transfer of a second grade fluid past a stretching sheet,”
*International Journal of Heat and Mass Transfer*, vol. 50, no. 1-2, pp. 75–84, 2007. View at Publisher · View at Google Scholar · View at Scopus - K. Vajravelu, “Flow and heat transfer in a saturated porous medium,”
*Zeitschrift für angewandte Mathematik und Mechanik*, vol. 74, pp. 605–614, 1994. View at Publisher · View at Google Scholar - E. Magyari and B. Keller, “Heat and mass transfer in the boundary layers on an exponentially stretching continuous surface,”
*Journal of Physics D: Applied Physics*, vol. 32, no. 5, pp. 577–585, 1999. View at Publisher · View at Google Scholar · View at Scopus - E. M. A. Elbashbeshy, “Heat transfer over an exponentially stretching continuous surface with suction,”
*Archives of Mechanics*, vol. 53, no. 6, pp. 643–651, 2001. View at Google Scholar · View at Scopus - S. K. Khan, “Boundary layer vicoelastic flow over an exponential stretching sheet,”
*International Journal of Applied Mechanics and Engineering*, vol. 11, pp. 321–335, 2006. View at Google Scholar - A. Raptis, C. Perdikis, and H. S. Takhar, “Effect of thermal radiation on MHD flow,”
*Applied Mathematics and Computation*, vol. 153, no. 3, pp. 645–649, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C.-H. Chen, “On the analytic solution of MHD flow and heat transfer for two types of viscoelastic fluid over a stretching sheet with energy dissipation, internal heat source and thermal radiation,”
*International Journal of Heat and Mass Transfer*, vol. 53, no. 19-20, pp. 4264–4273, 2010. View at Publisher · View at Google Scholar · View at Scopus - M. Sajid and T. Hayat, “Influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet,”
*International Communications in Heat and Mass Transfer*, vol. 35, no. 3, pp. 347–356, 2008. View at Publisher · View at Google Scholar · View at Scopus - T. C. Chiam, “Heat transfer with variable conductivity in a stagnation-point flow towards a stretching sheet,”
*International Communications in Heat and Mass Transfer*, vol. 23, no. 2, pp. 239–248, 1996. View at Publisher · View at Google Scholar · View at Scopus - T. C. Chiam, “Heat transfer in a fluid with variable thermal conductivity over a linearly stretching sheet,”
*Acta Mechanica*, vol. 129, no. 1-2, pp. 63–72, 1998. View at Google Scholar · View at Scopus - S. J. Liao,
*The proposed homotopy anslysis technique for the solution of non-linear problems [Ph.D. thesis]*, Shanghai Jiao Tong University, 1992. - S. J. Liao,
*Beyond Perturbation: Introduction to Homotopy Analysis Method*, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2003. - S. Liao, “A new branch of solutions of boundary-layer flows over an impermeable stretched plate,”
*International Journal of Heat and Mass Transfer*, vol. 48, no. 12, pp. 2529–2539, 2005. View at Publisher · View at Google Scholar · View at Scopus - S. J. Liao, “An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 11, no. 3, pp. 326–339, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - M. Sajid, T. Hayat, and S. Asghar, “On the analytic solution of the steady flow of a fourth grade fluid,”
*Physics Letters A: General, Atomic and Solid State Physics*, vol. 355, no. 1, pp. 18–26, 2006. View at Publisher · View at Google Scholar · View at Scopus - Z. Abbas, M. Sajid, and T. Hayat, “MHD boundary-layer flow of an upper-convected Maxwell fluid in a porous channel,”
*Theoretical and Computational Fluid Dynamics*, vol. 20, no. 4, pp. 229–238, 2006. View at Publisher · View at Google Scholar · View at Scopus - S. Abbasbandy, “The application of homotopy analysis method to nonlinear equations arising in heat transfer,”
*Physics Letters A*, vol. 360, no. 1, pp. 109–113, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet