Advances in Mathematical Physics

Volume 2016, Article ID 7036728, 14 pages

http://dx.doi.org/10.1155/2016/7036728

## Homotopy Analysis of the Radiation Effect on MHD Flow with Heat and Mass Transfer due to a Point Sink

Department of Mathematics, Dr. Babasaheb Ambedkar Technological University, Lonere, Raigad District 402103, India

Received 4 June 2016; Revised 8 September 2016; Accepted 27 September 2016

Academic Editor: Alkesh Punjabi

Copyright © 2016 C. N. Guled and B. B. Singh. 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

An analytical solution of the magnetohydrodynamic, steady, and incompressible laminar boundary layer flow in the presence of heat and mass transfer as well as magnetic field on a cone due to a point sink by using the homotopy analysis method (HAM) has been studied under the radiative fluid properties. The HAM produces an analytical solution of the governing self-similar nonlinear two-point boundary layer equations. The effects of the suction/injection, magnetic, and radiation parameters over the obtained solution have been discussed. The effects of Prandtl number on temperature and Schmidt number on concentration profiles have also been studied. It has been observed that the temperature profiles exhibit an increasing trend with radiation in case of injection while an opposite trend is observed in case of suction. The results obtained in the present study have also been compared numerically as well as graphically with the corresponding results obtained by using other methods. An excellent agreement has been found between them. The analytical solution obtained by the HAM is very near to the exact solution for a properly selected initial guess, auxiliary, and convergence control parameters and for higher orders of deformations.

#### 1. Introduction

In many areas of applied engineering and industry, we often come across the boundary layer flow of a steady incompressible laminar fluid flow in the presence of mass transfer and applied magnetic field. This kind of study is of interest for the fields of different branches of innovation, for example, in vortex chambers, magnetohydrodynamic (MHD) power generators, atomic reactors, and geophysical liquid flow. The examination of the boundary layer flow of an electrically conducting fluid on a cone because of a point sink with an applied magnetic field is significant in the investigation of conical nozzle or diffuser-flow problems and it was first of all concentrated on by Choi and Wilhelm [1]. Prior to this problem, Rosenhead [2] studied the same problem in the absence of magnetic field, mass flux diffusion, and heat transfer. Ackerberg [3] presented the series solution for the converging motion of the viscous fluid inside a cone. Takhar et al. [4] extended the same problem for electrically conducting fluid and discussed the heat and mass transfer effects. Eswara et al. [5] investigated the problem for the transient case. Eswara and Bommaiah [6] revisited the problem by taking into account temperature dependent viscosity. Turkyilmazoglu [7] considered the Falkner-Skan flows past stretching boundaries when the momentum and thermal slip boundary conditions are allowed at the boundary. Turkyilmazoglu extended the flow model set-up in a moving convergent channel by Magyari [8] by taking into consideration the momentum slip condition at the wall and found exact analytical solutions for the converging channel, for example, wedge nozzle.

In the context of space technology and processes involving temperatures, the effects of radiation are of vital significance. Recent developments in hypersonic flights, missile reentry, rocket combustion chambers, power plants for interplanetary flight, and gas cooled thermal reactors have focused their attention on thermal radiation as a mode of energy transfer. As a consequence of this, Vyas and Rai [9] made an elaborate analysis of the radiative flow inside a circular cone due to a point sink at the vertex of the cone. It is here worth mentioning that, unlike convection/conduction, the radiative heat transfer mechanism is rather more complex. However, some reasonable approximations have been found satisfactory to make the radiative systems solvable. The works of Sparrow and Cess [10] and Howell [11] describe the essentials of the radiative heat transfer. Many other pertinent radiative heat transfer studies for different configurations have also been reported by authors like Plumb et al. [12], Hossain and Takhar [13], Raptis [14], Sedeek and Salem [15], Al-Odat et al. [16], Prasad et al. [17], Mukhopadhyay [18], Vyas and Srivastava [19], Vyas and Ranjan [20], Chauhan and Kumar [21], Baoku et al. [22], Babu et al. [23], and so forth.

The objective of the present paper is to extend the work of Takhar et al. [4] by taking into account the radiative properties of the fluid at the wall. The flow problem which is governed by nonlinear equations with two-point boundary conditions has been solved by using HAM. Using the recursive method derived by Liao [24, 25], rigorous recursive formulae have been developed. Symbolic computation software and high performance computers have been used to derive the analytic solutions. The flow characteristics have been analyzed, and the results have been compared with those of [4] by setting the radiation parameter as zero.

In 1992, Liao [26] investigated the homotopy analysis method (HAM). The strength of HAM is that it leads to convergent analytic series solutions of strongly nonlinear problems faster than any other existing methods, independent of small or large physical parameter/s involved in the problem [27]. This behaviour of HAM makes it a superior technique to the conventional perturbation methods. The methods such as Adomian decomposition method [28–30], -expansion method [31], and Lyapunov artificial small parameter method [32] may not be valid for strong nonlinear problems due to the divergent nature of their obtained solution series. Liao [33], indeed, showed that HAM is the general case and the Adomian decomposition method, -expansion method and Lyapunov artificial small parameter method are the special cases of HAM. Moreover, He’s homotopy perturbation method (HPM) [34, 35] is also a special case of the HAM (cf. Liao [36]). Actually, for some auxiliary linear operators, the traditional HPM turns out to be the Taylor series expansion (cf. Turkyilmazoglu [37]).

The HAM has been applied to an extensive variety of nonlinear problems in science and engineering ever since it was first introduced in 1992. The problems of viscous flows of non-Newtonian fluids that have been mainly tackled by Hayat and his coworkers [38–40] and problems of heat transfer [41, 42] are some of the examples of applications of HAM. Even a much wider range of applications of HAM can be found in [27].

#### 2. Governing Equations

We consider here the boundary layer flow of a fluid which is electrically conducting. The flow is assumed to be steady, laminar, incompressible, and axisymmetric in a circular cone having three-dimensional sink at the vertex (Figure 1).