- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents

International Journal of Engineering Mathematics

Volume 2013 (2013), Article ID 692827, 7 pages

http://dx.doi.org/10.1155/2013/692827

## Falkner-Skan Flow of a Maxwell Fluid with Heat Transfer and Magnetic Field

^{1}Department of Mathematics, COMSATS Institute of Information Technology, Park Road, Chak Shahzad, Islamabad 44000, Pakistan^{2}Department of Mathematics, COMSATS Institute of Information Technology, Attock 43600, Pakistan

Received 11 February 2013; Accepted 19 May 2013

Academic Editor: George S. Dulikravich

Copyright © 2013 M. Qasim and S. Noreen. 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

This investigation deals with the Falkner-Skan flow of a Maxwell fluid in the presence of nonuniform applied magnetic fi*
*eld with heat transfer. Governing problems of flow and heat transfer are solved analytically by employing the homotopy analysis method (HAM). Effects of the involved parameters, namely, the Deborah number, Hartman number, and the Prandtl number, are examined carefully. A comparative study is made with the known numerical solution in a limiting sense and an excellent agreement is noted.

#### 1. Introduction

The Falkner-Skan problem under various aspects has attracted the attention of several researchers [1]. This problem under various aspects has been discussed extensively for viscous fluid. The interested readers may consult the studies in [2–11] for detailed information in viscous fluids. There are several materials which do not obey the Newton's law of viscosity, for example, biological products like blood and vaccines, foodstuffs like honey, ketchup, butter, and mayonnaise, certain paints, cosmetic products, pharmaceutical chemicals and so forth. These fluids are characterized as the non-Newtonian fluids. Investigation of such fluids is very useful in industrial, engineering, and biological applications. However, such fluids cannot be studied by employing a single constitutive relationship. This is due to diverse properties of non-Newtonian fluids in nature. These non-Newtonian fluid models are discussed in view of three main categories, namely, the differential, the rate, and the integral types. The simplest subclass of rate type fluids is called Maxwell. The Maxwell fluid allows for the relaxation effects which cannot be predicted in differential type fluids, namely, second, third, and fourth grades. Recently, there has been an increasing interest in the theory of rate type fluids and, in particular, a Maxwell fluid model has been accorded much attention. The Falkner-Skan wedge flow of a non-Newtonian fluid was firstly investigated by Rajagopal et al. [12]. Massoudi and Ramezan [13] discussed the effect of injection or suction on the Falkner-Skan flows of second grade fluids. The Falkner-Skan wedge flow of power-law fluids embedded in a porous medium is investigated by Kim [14]. Olagunju [15] studied this flow problem for viscoelastic fluid. In [10–15], the attention has been given to the differential type fluids. To the best of our knowledge, no one investigated the Falkner-Skan flow problem for rate type fluids.

In [10], Yao has examined the Falkner-Skan wedge flow. He established series solution for the velocity and temperature by using homotopy analysis method [16–25]. The purpose of the present contribution is to extend the flow analysis of study [10] in two directions. The first generalization is concerned with the consideration of electrically conducting fluid. Such analysis has inserted in power generators the cooling of reactors, MHD accelerators, the design of heat exchanges and electrostatic filters. Secondly, we consider the Maxwell fluid instead of viscous fluid. The rest of the paper is arranged as follows. The description of the problem is presented in Section 2. Section 3 develops the homotopy solutions for velocity and temperature. Convergence of the derived solution is examined in Section 4. Further, the variations of embedded parameters have been discussed in this section.

#### 2. Problem Development

We study the steady two-dimensional flow of a Maxwell fluid in the presence of a magnetic field. The magnetic Reynolds number is small so that induced magnetic field is neglected. The stream velocity varies according to . The constant temperature of surface and free stream is and , respectively. The boundary layer equations for the considered problem are where and are the velocity components in the - and -directions, respectively, is kinematic viscosity, is the electrical conductivity, is the thermal conductivity, is the fluid density, is the fluid temperature, is relaxation time, is the magnetic field, and is specific heat.

The relevant boundary conditions are prescribed as follows: with [1] and [7, 8] Putting (1) is satisfied identically and (2)–(4) give

Here is the stream function, prime denotes the differentiation with respect to , is the local Deborah number, is the constant parameter, is Hartmann number, and is the Prandtl number. The values of , , and are

The local Nusselt number and heat transfer from the plate are given by Invoking (7) one obtains in which the local Reynolds number . For , from (9), we have where .

#### 3. Homotopy Analysis Solutions

For an interest in homotopy solutions, we express and by a set of base functions as follows: where and are the coefficients. The initial approximations of and and the auxiliary linear operators and are in which are the arbitrary constants. If is the embedding parameter and , and , are the nonzero auxiliary parameters, then the problems at zeroth order give For and , we have When increases from to , and vary from and to the exact solutions and . In view of Taylors theorem and (20) and (21), one arrives at The auxiliary parameters are so properly chosen that the series (28) and (29) converge at , and hence The problems at th-order deformation satisfy the following equations and boundary conditions:

#### 4. Convergence of the Homotopy Solutions

Obviously the auxiliary parameters and in the series solutions (31) and (32) have a definite role in adjusting and controlling the convergence. For the admissible values of and , the and curves are portrayed for 15th-order of approximations. It is noticed that the ranges for the admissible values of and are and (Figure 1). Moreover, the series given by (31) and (32) converge in the whole region of when and .

#### 5. Results and Discussion

The purpose of this section is to investigate the variations of parameters including Deborah number , the constant , Hartman number , and the Prandtl number on the velocity and the temperature . The variation of the Deborah number , Hartman number , and the parameter on the velocity can be seen in Figures 2–5. It is found from Figure 2 that the boundary layer thickness decreases with an increase in . Figure 3 is plotted for the effects of Hartman number on the velocity profile . The effects of on are qualitatively similar to those of . Figure 4 depicts the velocity field for different values of parameter . Clearly is an increasing function of while the boundary layer thickness decreases. Figure 5 shows the effects of Prandtl number on the temperature profile . The thermal boundary layer thickness decreases when increases. The variation of Hartman number on the temperature profile is shown in Figure 6. It is noticed that the temperature profile increases when is increased. The effects of parameter are shown in Figure 7. It is observed that an increase in the value of decreases the thermal boundary layer thickness. Figures 8 and 9 are displayed to analyze the comparison of the present analytical results with the existing numerical solutions. An excellent agreement is found between the two solutions for different values of . In Table 1, we have computed the numerical values of , , and for the comparison of the present analytical results with the numerical solution [26]. Table 2 is also a comparison between homotopy solution and numerical solution [26] in a special case. An excellent agreement is found between the two solutions.

#### References

- H. Schlichting,
*Boundary Layer Theory*, McGraw Hill, New York, NY, USA, 6th edition, 1979. - V. M. Falkner and S. W. Skan, “Some approximate solutions of the boundary layer equations,”
*Philosophical Magazine*, vol. 12, pp. 865–896, 1931. - T. Cebeci and H. B. Keller, “Shooting and parallel shooting methods for solving the Falkner-Skan boundary-layer equation,”
*Journal of Computational Physics*, vol. 7, no. 2, pp. 289–300, 1971. View at Scopus - S. P. Hastings and W. Troy, “Oscillatory solutions of the Falkner-Skan equation,”
*Proceedings of the Royal Society of London A*, vol. 397, no. 1813, pp. 415–418, 1985. View at Scopus - E. F. F. Botta, F. J. Hut, and A. E. P. Veldman, “The role of periodic solutions in the Falkner-Skan problem for
*λ*> 0,”*Journal of Engineering Mathematics*, vol. 20, no. 1, pp. 81–93, 1986. View at Publisher · View at Google Scholar · View at Scopus - E. Alizadeh, M. Farhadi, K. Sedighi, H. R. Ebrahimi-Kebria, and A. Ghafourian, “Solution of the Falkner-Skan equation for wedge by Adomian Decomposition Method,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 3, pp. 724–733, 2009. View at Publisher · View at Google Scholar · View at Scopus - S. Abbasbandy and T. Hayat, “Solution of the MHD Falkner-Skan flow by Hankel-Padé method,”
*Physics Letters A*, vol. 373, no. 7, pp. 731–734, 2009. View at Publisher · View at Google Scholar · View at Scopus - S. Abbasbandy and T. Hayat, “Solution of the MHD Falkner-Skan flow by homotopy analysis method,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 9-10, pp. 3591–3598, 2009. View at Publisher · View at Google Scholar · View at Scopus - E. Magyari, “Falkner-Skan flows past moving boundaries: an exactly solvable case,”
*Acta Mechanica*, vol. 203, no. 1-2, pp. 13–21, 2009. View at Publisher · View at Google Scholar · View at Scopus - B. Yao, “Series solution of the temperature distribution in the Falkner-Skan wedge flow by the homotopy analysis method,”
*European Journal of Mechanics B*, vol. 28, no. 5, pp. 689–693, 2009. View at Publisher · View at Google Scholar · View at Scopus - B. Yao, “Approximate analytical solution to the Falkner-Skan wedge flow with the permeable wall of uniform suction,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 8, pp. 3320–3326, 2009. View at Publisher · View at Google Scholar · View at Scopus - K. R. Rajagopal, A. S. Gupta, and T. Y. Na, “A note on the falkner-skan flows of a non-newtonian fluid,”
*International Journal of Non-Linear Mechanics*, vol. 18, no. 4, pp. 313–320, 1983. View at Scopus - M. Massoudi and M. Ramezan, “Effect of injection or suction on the Falkner-Skan flows of second grade fluids,”
*International Journal of Non-Linear Mechanics*, vol. 24, no. 3, pp. 221–227, 1989. View at Scopus - Y. J. Kim, “The Falkner-Skan wedge flows of power-law fluids embedded in a porous medium,”
*Transport in Porous Media*, vol. 44, no. 2, pp. 267–279, 2001. View at Publisher · View at Google Scholar · View at Scopus - D. O. Olagunju, “The Falkner-Skan flow of a viscoelastic fluid,”
*International Journal of Non-Linear Mechanics*, vol. 41, pp. 825–829, 2006. - S. J. Liao,
*Beyond Perturbation: Introduction to homotopy Analysis Method*, Chapman and Hall, CRC Press, Boca Raton, Fla, USA, 2003. - S. J. Liao, “Notes on the homotopy analysis method: some definitions and theorems,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, pp. 983–997, 2009. - S.-J. Liao, “A general approach to get series solution of non-similarity boundary-layer flows,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 5, pp. 2144–2159, 2009. View at Publisher · View at Google Scholar · View at Scopus - S. J. Liao, “On the homotopy analysis method for nonlinear problems,”
*Applied Mathematics and Computation*, vol. 147, no. 2, pp. 499–513, 2004. View at Publisher · View at Google Scholar · View at Scopus - T. Hayat, M. Qasim, and Z. Abbas, “Homotopy solution for the unsteady three-dimensional MHD flow and mass transfer in a porous space,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 15, no. 9, pp. 2375–2387, 2010. View at Publisher · View at Google Scholar · View at Scopus - I. Hashim, O. Abdulaziz, and S. Momani, “The homotopy analysis method for Cauchy reaction diffusion problems,”
*Physics Letters A*, vol. 372, pp. 613–618, 2008. - I. Hashim, O. Abdulaziz, and S. Momani, “Homotopy analysis method for fractional IVPs,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 3, pp. 674–684, 2009. View at Publisher · View at Google Scholar · View at Scopus - A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Solutions of time-dependent Emden-Fowler type equations by homotopy analysis method,”
*Physics Letters A*, vol. 371, no. 1-2, pp. 72–82, 2007. View at Publisher · View at Google Scholar · View at Scopus - A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “On a new reliable modification of homotopy analysis method,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 2, pp. 409–423, 2009. View at Publisher · View at Google Scholar · View at Scopus - M. Dehghan and R. Salehi, “The use of homotopy analysis method to solve the time-dependent nonlinear Eikonal partial differential equation,”
*Zeitschrift fur Naturforschung A*, vol. 66, no. 5, pp. 259–271, 2011. View at Scopus - F. M. White,
*Viscous Fluid Flow*, McGraw-Hill, New York, NY, USA, 2nd edition, 1991.