#### Abstract

The homotopy analysis method (HAM) with two auxiliary parameters is employed to examine heat and mass transfer in a steady two-dimensional magneto hydrodynamic viscoelastic fluid flow over a stretching vertical surface by considering Soret and Dufour effects. The two-dimensional boundary-layer governing partial differential equations are derived by considering the Boussinesq approximation. The highly nonlinear ordinary differential forms of momentum, energy, and concentration equations are obtained by similarity transformation. These equations are solved analytically in the presence of buoyancy force. The effects of different involved parameters such as magnetic field parameter, Prandtl number, buoyancy parameter, Soret number, Dufour number, and Lewis number on velocity, temperature, and concentration profiles are plotted and discussed. The effect of the second auxiliary parameter is also illustrated. Results show that the effect of increasing Soret number or decreasing Dufour number tends to decrease the velocity and temperature profiles (increase in Sr cools the fluid and reduces the temperature) while enhancing the concentration distribution.

#### 1. Introduction

The analysis of the flow field in a boundary-layer near a stretching sheet is an important part in fluid dynamics and heat transfer. This type of flow occurs in a number of engineering processes such as extrusion of plastic sheets, polymer processing, and metallurgy [1, 2].

Some researchers neglect the Dufour and Soret effects on heat and mass transfer according to Fourier’s and Fick’s laws [3]; however, when density differences exist in the flow regime, these effects are important and cannot be neglected [4]. Afify [5] has shown that when heat and mass transfer occurred in a moving fluid, the energy flux can be generated by a composition gradient, namely, the Dufour or diffusion-thermo effect, and the mass fluxes developed by the temperature gradients are called the Soret or thermal-diffusion effect. In their numerical study they have used the Soret and Dufour effects of a steady flow due to a rotating disk in the presence of viscous dissipation and ohmic heating. Heat and mass transfer with hydrodynamic slip over a moving plate in porous media was investigated by Hamad et al. [6] via Runge-Kutta-Fehlberg fourth-fifth order method. The heat transfer of mixed convection of vertically moving surface in an ambient stagnant fluid was reported by Ali and Al-Yousef [7, 8] and the effect of variable viscosity of mixed convection was studied by Ali [9].

Das et al. [10] considered the effect of heat and mass transfer on a free convective flow of an incompressible electrically conducting fluid past a vertical porous plate. Chen [11] employed finite difference method in order to study the heat and mass transfer in MHD free convective flow with ohmic heating and viscous dissipation. Noor et al. [12] examined the MHD flow over an inclined surface with heat source/sink effects by shooting method. Abreu et al. [13] solved the boundary-layer flow with Dufour and Soret effects in both forced and natural convection. The effects of thermal radiation and first order chemical reaction on unsteady MHD convective flow past a semiinfinite vertical plate under oscillatory suction and heat source in slip-flow regime were taken into account by Pal and Talukdar [14]. Gbadeyan et al. [15] studied heat and mass transfer of a mixed convection boundary-layer flow considering porous medium over a stretching vertical surface. A vertical plate in a non-Darcy porous medium was selected to investigate the thermodiffusion and diffusion-thermo effects numerically using the Keller-box method by Prasad et al. [16]. Pal and Mondal [17–19] analyzed the effects of thermal diffusion and diffusion thermo on steady and unsteady MHD non-Darcy flow over a stretching sheet in a porous medium considering thermal radiation, nonuniform heat source/sink, variable viscosity, viscous dissipation, and first order chemical reaction using Runge-Kutta-Fehlberg integration method. Mansour et al. [20] analyzed the effects of chemical reaction and thermal stratification over a vertical stretching surface in a porous medium by Runge-Kutta scheme with considering Soret and Dufour numbers. Bég et al. [21] used Keller-box implicit method to analyze the heat and mass transfer micropolar fluid flow from an isothermal sphere with Soret and Dufour effects. Furthermore, Alam et al. [22], Tai and Char [23], Mahdy [24, 25], Pal and Sewli [26], and Tsai and Huang [27] have studied the effect of Soret and Dufour effects in their analyses for different aspects of heat and mass transfer flows.

One of the most effective and reliable methods in order to solve the high nonlinear problems is the homotopy analysis method. Homotopy analysis method (HAM) was firstly employed by Liao to offer a general analytic method for nonlinear problems [28, 29]. Rashidi et al. [30] analyzed the effect of partial slip, diffusion thermo, and thermal diffusion on MHD fluid flow in a rotating disk via HAM and discussed the effect of various slip parameters, magnetic field parameter, Schmidt number, and other important variables. Mustafa et al. [31] considered the effects of Brownian motion and thermophoresis in stagnation point flow of a nanofluid towards a stretching sheet. Rashidi and Pour [32] employed HAM for unsteady boundary-layer flow and heat transfer on a stretching sheet. Abbas et al. [33] studied the mixed convection of an incompressible Maxwell fluid flow over a vertical stretching surface by HAM. Dinarvand et al. [34] employed HAM to investigate unsteady laminar MHD flow near forward stagnation point of a rotating and translating sphere. Hayat et al. [35] illustrated the thermal-diffusion and diffusion-thermo effects on two-dimensional MHD axisymmetric flow of a second grade fluid in the presence of Joule heating and first order chemical reaction. The non-linear Brinkman equation for the stagnation-point flow was studied via HAM by Ziabakhsh et al. [36]. Analytical and numerical solutions of a radial stagnation flow over a stretching cylinder have been recently reported by Weidman and Ali [37] where aligned and nonaligned flow were studied. Rashidi et al. [38, 39] employed HAM to obtain the analytical solutions over stretching and shrinking sheets in the presence of buoyancy parameter.

The objective of this paper is to study the steady two-dimensional MHD viscoelastic fluid flow over a vertical stretching surface in the presence of the Soret and Dufour effects analytically via HAM. The effects of different involved parameters such as magnetic field parameter, Prandtl number, buoyancy parameter, Soret number, Dufour number, and Lewis number on the fluid velocity, temperature, and concentration distributions are plotted and discussed.

#### 2. Flow Analysis

Consider a steady two-dimensional heat and mass transfer flow of an incompressible electrically conducting viscoelastic fluid over a stretching vertical surface with a variable magnetic field normally applied to the surface. Keeping the origin fixed, two equal and opposite forces are applied along the -axis. It is assumed that the stretching velocity is in the form of , where and are constants. The induced magnetic field is neglected in comparison to the applied magnetic field and the viscous dissipation is small. The governing equations subject to Boussinesq approximation, the boundary-layer assumptions, and the above assumptions can be written as (for more details see [41])where and are velocity components in the directions of and along and normal to the surface, respectively (as shown in Figure 1). is the kinematic viscosity, is the viscoelasticity parameter, is the electrical conductivity, is the fluid density, is the acceleration due to gravity, is the coefficient of thermal expansion, is the coefficient of thermal expansion with concentration, is the thermal diffusivity, is the thermal diffusion ratio, is the concentration susceptibility, is the specific heat at constant pressure, is the coefficient of mass diffusivity, is the fluid temperature, is the fluid concentration, and is the mean fluid temperature.

The corresponding boundary conditions are as follows:

We assume that and , where and are constants. Introducing stream function , and similarity variable , [42] the continuity equation is satisfied and the momentum, energy, and concentration equations are transformed into ordinary differential equations as follows:where superscript denotes the derivative with respect to , is the viscoelasticity parameter (when the viscoelastic parameter takes the form of similar to the viscoelastic parameter obtained by Hayat et al. [41]), is the magnetic field parameter, is the buoyancy parameter, where is the Grashof number, is the Reynolds number, is the constant dimensionless concentration buoyancy parameter, is the Prandtl number, is the Lewis number, is the Soret number, and is the Dufour number. The corresponding boundary conditions are as follows:where is the suction/injection parameter ( for suction and for injection). In this paper the suction parameter has been considered because the primary assumption in boundary-layer definition says that the boundary-layer thickness is supposed to be very thin and we are not allowed to increase it, so we do not present the injection parameters that may lead to enlarging the boundary-layer thickness and contravening the boundary-layer assumption presented by Prandtl in 1904.

#### 3. HAM Solution

We choose the initial approximations to satisfy the boundary conditions. The appropriate initial approximations are as follows:where is the second auxiliary parameter. The linear operators , and arewith the following properties:where are arbitrary constants and the nonlinear operators areThe auxiliary functions are introduced asThe th order deformation equations (see (10)) can be solved by the symbolic software MATHEMATICAwhere is the auxiliary nonzero parameterFor more information about the HAM solution, see [28, 29].

In Figure 2, -curve is figured, obtained via 20th order of HAM solution. The averaged residual errors are defined as (12) to acquire optimal values of auxiliary parameters:

To check the accuracy of the method, the residual errors of (12) are illustrated in Figures 3 and 4. The residual errors are reduced when we use the second auxiliary parameter and this justifies why we use the second auxiliary parameter. In Figure 3, the effect of considering is to decrease the order of residual errors than at (without the second auxiliary parameter) in Figure 4 which improves the accuracy of the HAM method. The velocity profiles presented in Figure 5 show an excellent agreement between our results and [40].

#### 4. Results and Discussion

In this paper the MHD two-dimensional steady heat and mass transfer flow of an incompressible viscoelastic fluid over a stretching vertical surface with considering the effects of Soret and Dufour numbers is investigated. Applying numerical values to the problem parameters, we can discuss their effects on the velocity , temperature , and concentration distributions. Graphical illustration of the results is very useful and practical to discuss the effect of different parameters. In this analysis, it has been considered that [41]. Negative (thermal and concentration buoyancy forces oppose each other) induces a slight increase in both fluid temperature and concentration [43]. In this paper the value of is considered to be 0.5. The effect of magnetic parameter on the velocity is plotted in Figure 6. Transverse magnetic field parameter creates a drag force, namely, Lorentz force that resists the flow and slows down the flow and causes to decrease the velocity. In Figure 7 the effect of magnetic field parameter on temperature profiles is illustrated. Magnetic field parameter causes skin-frictional heating and so the wall temperature increases and the thickness of thermal boundary-layer increases. The effect of is to increase the concentration profile (Figure 8). The governing equations are coupled together only with the buoyancy parameters. When increases, the Grashof number accelerates the fluid so the velocity and the boundary-layer thickness increases, as shown in Figure 9. The effect of on temperature and concentration profiles is shown in Figures 10 and 11. Both the thermal and concentration boundary-layer thicknesses decrease with the increase in the value of buoyancy parameter. The effects of Prandtl number on velocity, temperature, and concentration distributions are illustrated in Figures 12–14, respectively. Increase in leads to increase in kinematic viscosity and velocity decreases. It is clearly shown that with the increase in the velocity profiles descends (Figure 12). With the increase in Prandtl number the thermal diffusion decreases, so the thermal boundary-layer becomes thinner and temperature decreases. A fluid with larger and higher heat capacity increases the heat transfer [40] (Figure 13). The Pr reduces the concentration distribution just the same as its effect on temperature profile (Figure 14). The Soret effect is a mass flux due to a temperature gradient and the Dufour effect is enthalpy flux due to a concentration gradient and appears in the energy equation. The effects of Soret and Dufour numbers on velocity, temperature, and concentration profiles are plotted in Figures 15, 16, and 17, respectively. We considered the effects of and so that their product remains constant at 0.05. As one can see the increase in the value of or decrease in descends the velocity and temperature profiles and enhances the concentration distribution. Increase in Soret number cools the fluid and reduces the temperature [43]. Lewis number is the ratio of thermal diffusivity to mass diffusivity. The Lewis number can also be expressed as the ratio of the Schmidt number to the Prandtl number (), where is the Schmidt number. Figure 18 displays the effect of Lewis number on the velocity profile. The effect of increasing the value of on the velocity is as the same as the effect of decreasing the value of and it can be easily understood that with the enhancement of the velocity distribution increases. The effect of on temperature profile is presented in Figure 19. The temperature decreases with the increase in Lewis number similar to the results presented by Hayat et al. [41]. With the increase in the mass diffusivity decreases and the concentration descends (Figure 20). It should be noticed that permits complete similarity solutions of the equations, where and are constants and not . However in this problem must be constant and is selected equal to 0.5 in order to reach the local similarity solution.

#### 5. Conclusion

In the present investigation, an analysis is carried out in order to study the steady magneto hydrodynamic incompressible viscoelastic fluid flow over a stretching surface in the presence of the Soret and Dufour effects analytically via HAM with two auxiliary parameters. Analytical solutions are obtained using the homotopy analysis method and its residual was reduced by using the second auxiliary parameter. These analytical solutions show excellent agreement with the data available in the literature (Figures 3–5). The effect of Mn is to decrease the velocity while increasing the thermal boundary-layer. The effect of increasing the buoyancy parameter is to reduce both the thermal and concentration boundary-layer thicknesses. The effect of increasing Sr or decreasing Du tends to decreases the velocity and temperature profiles while enhancing the concentration distribution. The temperature profiles are not sensitive to increasing Le however the concentration profiles are very sensitive.

#### Nomenclature

: | Constant values [–] |

: | Magnetic field [kg s^{−2} A^{−1}] |

: | Arbitrary constant [–] |

: | Concentration [kg m^{−2}] |

: | Specific heat at constant pressure [J kg^{−1} K^{−1}] |

: | Coefficient of mass diffusivity [m^{2} s^{−1}] |

: | Dufour number ( [–] |

: | Auxiliary nonzero parameter |

: | Auxiliary function |

: | Auxiliary linear operator |

: | Lewis number [–] |

: | Magnetic field parameter [–] |

: | Nonlinear operator |

: | Constant dimensionless concentration buoyancy parameter |

: | Prandtl number [–] |

: | Reynolds number [–] |

: | Soret number () [–] |

: | Mean fluid temperature [K]. |

*Greek Letters*

: | Thermal diffusivity [m^{2} s^{−1}] |

: | Coefficient of thermal expansion [K^{−1}] |

: | Coefficient of thermal expansion with concentration [kg^{−1} m^{3}] |

: | Dimensionless fluid concentration ( [–] |

: | The second auxiliary parameter |

: | Similarity variable ( [–] |

: | Dimensionless fluid temperature ( [–] |

: | Density [kg m^{−3}] |

: | Fluid electrical conductivity [S m^{−1}] |

: | Buoyancy parameter () [–] |

: | Fluid kinematic viscosity [m^{2} s^{−1}] |

: | Stream function. |

*Subscripts*

: | Wall condition |

: | Infinity condition. |

*Superscript*

′: | Differentiation with respect to . |

#### Conflict of Interests

All the authors have no conflict of interests to report.

#### Acknowledgments

The authors express their gratitude to the anonymous referees for their constructive reviews of the paper and for helpful comments. The authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through the research group Project no. RGP-VPP-080.