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^{1}Department of Mathematics, Annamacharya Institute of Technology and Sciences (Autonomous), Rajampet, A.P, India

^{2}Department of Mathematics, JNTUA College of Engineering, Pulivendula, A.P, India

- *Corresponding Author:
- Raju MC

Department of Mathematics

Annamacharya Institute of Technology and Sciences (Autonomous)

Rajampet-516126, A.P, India

**Tel:**9440355318

**E-mail:**[email protected]

**Received** August 12, 2015; **Accepted** August 24, 2015; **Published** August 30, 2015

**Citation:** Chandra RP, Raju MC, Raju GSS (2015) Thermal and Solutal Buoyancy Effect on MHD Boundary Layer Flow of a Visco-Elastic Fluid Past a Porous Plate with Varying Suction and Heat Source in the Presence of Thermal Diffusion. J Appl Computat Math 4:249. doi:10.4172/2168-9679.1000249

**Copyright:** © 2015 Chandra RP, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Applied & Computational Mathematics

An analytical solution is investigated for a fully developed free convective flow of a visco-elastic incompressible electrically conducting fluid past a vertical porous plate bounded by a porous medium in the presence of thermal diffusion, variable suction and variable permeability. A magnetic field of uniform strength is applied perpendicular to the plate and the presence of heat source is also considered. The novelty of the study is to investigate the effect of thermal diffusion on a visco-elastic fluid in the presence of time dependent variable suction. The importance is due to the applications of this kind of visco-elastic fluids in many industries. The coupled dimensionless nonlinear partial differential equations are transformed into a set of ordinary differential equations by using multiple parameter perturbation on velocity whereas simple perturbation method on temperature and concentration. With corresponds to these, the expressions for skin friction, Nusselt number and Sherwood number are derived. The numerical computations have been studied through figures and tables. The presence of thermal diffusion increases fluid velocity, whereas the influence of the magnetic field reduces it. In the case of heavier species, it is noticed that concentration increases with an increase in Soret number.

MHD; Visco-elastic fluid; Thermal diffusion; Variable suction; Variable permeability; Vertical porous plate; Heat and mass transfer

C^{l} Species concentration

D Molecular diffusivity

Gr Grashof number of heat transfer

K^{1} Permeability of the medium

k **Thermal** diffusivity

Rc Elastic parameter

Nu Nusselt number

S Heat source parameter

Sh Sherwood number

T Non-dimensional temperature

t Non-dimensional time

u Non-dimensional velocity

v_{o} Constant suction velocity

y Non-dimensional distance along y-axis

? A small positive constant

β Volumetric coefficient of expansion for heat transfer

τ Density of the fluid

β^{1} Volumetric coefficient of expansion with species concentration

S_{o} Frequency of oscillation

υ Kinematic coefficient of viscosity

σ Electrical conductivity

ω Non-dimensional frequency of oscillation

C Non-dimensional Species concentration

Gc Grashof number for mass transfer

g Acceleration due to gravity

Kp Porosity parameter

M Magnetic parameter

B_{0} **Magnetic field** of uniform strength

Pr Prandtl number

Sc Schmidt number

T^{1} Temperature of the field

t^{l} Time

u^{l} Velocity component along x-axis

v Suction velocity

y^{l} Distance along y-axis

ρ Density of the fluid

τ Skin friction

ω^{1} Condition on porous plate

S_{o} Soret number

w Condition on porous plate

The knowledge of visco-elastic fluids past a porous media plays vital role in many scientific and engineering applications. Most frequently this flow was basically utilized in the fields of petroleum engineering concerned with the oil, gas and water through reservoir to the hydrologist in the analysis of the migration of underground water. This process in chemical engineering is used for both purification and filtration. Its application is also found in the process of drug permeation through human skin. To recover the water for drinking and irrigation purposes the principles of this flow are followed. Natural convection flow over a vertical surface embedded in porous media also comes across in many engineering areas like the design of nuclear reactors, catalytic reactors, and compact heat exchangers, geothermal energy conversion, the use of fibrous materials, thermal insulation of buildings, heat transfer from storage of agricultural products which generate heat as a result of metabolism, petroleum reservoirs, nuclear wastes, etc. Many researchers [1-6] contributed in studying the application of visco-elastic fluid flow of several types past porous medium in channels of different cross-sections. Mohamad [7] studied the Soret effect on the unsteady magneto hydrodynamics (MHD) free convection **heat and mass** transfer flow past a vertical moving porous plate in a Darcy medium along with the chemical reaction and generation of heat. Appropriate solutions for a natural convection flow in porous media have been attained and presented by Combarnous and Bories [8], Catton [9], Bejan [10,11], and Tien and Vafai [12]. Bejan and Khair [13] considered heat and mass transfer by natural convection in a porous medium. Nakayama and Koyama [14] employed an integral method for free convection from a vertical heated surface in a thermally stratified porous medium. The natural convection flow along a vertical heated surface through a porous medium has been analysed by Kaviany and Mittal [15]. Gupta and Sharma [16] attended to study MHD flow of viscous fluid past a porous medium enclosed by an oscillating porous plate in slip flow regime. Magneto hydro dynamic flow of a dusty viscoelastic fluid past a porous medium under the influence of an oscillating porous plate in slip flow regime is analyzed by Singh and Singh [17]. Ali et al. [18] paid attention towards the study of radiation effect on natural convection flow along a vertical surface in a gray gas. Chen [19] gave some conclusions on heat and mass transfer in MHD flow with natural convection along a permeable, inclined surface under the influence of variable wall temperature and concentration. Ravi kumar et al. [20] investigated on MHD double diffusive and chemically reactive flow through porous medium bounded by two vertical plates. Reddy et al. [21] studied heat transfer in hydro magnetic rotating flow of viscous fluid through non-homogeneous porous medium with constant heat source/sink. Raju et al. [22] contributed his efforts to study MHD thermal diffusion natural convection flow between heated inclined plates in porous medium. Raju et al. [23] put their attention towards Soret effects due to Natural convection between heated inclined plates with magnetic field. Reddy et al. [24] studied thermo diffusion and chemical effects with simultaneous thermal and mass diffusion in MHD mixed convection flow with Ohmic heating. Seshaiah et al. [25] analyzed the effects of chemical reaction and radiation on unsteady MHD free convective fluid flow embedded in a porous medium with time-dependent suction with temperature gradient heat source. Ibrahim and Makinde [26] attended a study on chemically reacting MHD boundary layer flow of heat and mass transfer over a moving vertical plate with suction. Kim [27] has considered an unsteady MHD convective heat transfer along a semi-infinite vertical porous moving plate with variable suction. Chemical reaction and radiation absorption effects on free convection flow through porous medium with variable suction in the presence of uniform magnetic field was studied by Sudheer Babu and Satyanarayana [28]. Das et al. [29] have considered and made a review on the mass transfer effects on MHD flow and heat transfer past a vertical porous plate through a porous medium under the influence of oscillatory suction and heat source. Mishra et al. [30] considered and pointed out the mass and heat transfer effect on MHD flow of a visco-elastic fluid through porous medium with oscillatory suction and heat source. Uma et al. [31] discussed Unsteady MHD free convective visco-elastic fluid flow bounded by an infinite inclined porous plate in the presence of heat source, viscous dissipation and Ohmic heating. Recently, Ravi et al. [32] studied combined effects of heat absorption and MHD on convective Rivlin-Ericksen fluid flow past a semi-infinite vertical porous plate with variable temperature and suction. Uma et al. [33] analyzed combined radiation and Ohmic heating effects on MHD free convective visco-elastic fluid flow past a porous plate with viscous dissipation. Chatterjee [34] investigated on heat transfer enhancement in laminar impinging flows with a non- Newtonian inelastic fluid. Tarun et al. [35] studied Laminar natural convection of power-law fluids in a square enclosure submitted from below to a uniform heat flux density. Mokarizadeh et al. [36] considered the influence of heat transfer in Couette-Poiseuille flow between parallel plates of the Giesekus viscoelastic fluid. Ferras et al. [37] found analytical solutions for Newtonian and inelastic non-Newtonian flows with wall slip. Ben Khelifa et al. [38] investigated natural convection in a horizontal porous cavity filled with a non-Newtonian binary fluid of power-law type. Motivate by the above studies, in this paper, a fully developed free convective flow of a visco-elastic incompressible electrically conducting fluid past a vertical porous plate bounded by a porous medium in the presence of thermal diffusion, variable suction and variable permeability is investigated.

The unsteady free convection heat and mass transfer flow of a wellknown non-Newtonian fluid, namely Walters B visco-elastic fluid past an infinite vertical porous plate, embedded in a porous medium in the presence of thermal diffusion, oscillatory suction as well as variable permeability is considered. A uniform magnetic field of strength B_{0} is applied perpendicular to the plate. Let x^{1} axis be taken along with the plate in the direction of the flow and y^{1} axis is normal to it. Let us consider the magnetic Reynolds number is much less than unity so that the induced magnetic field is neglected in comparison with the applied transverse magnetic field. The basic flow in the medium is, therefore, entirely due to the **buoyancy** force caused by the temperature difference between the wall and the medium. It is assumed that initially, at tl ≤ 0, the plate as fluids are at the same temperature and concentration. When t^{l}>0, the temperature of the plate is instantaneously raised to T^{1}_{w} and the concentration of the species is set to C^{1}_{w} . Under the above assumption with usual Boussinesq’s approximation, the governing equations and boundary conditions are given in **Figure 1**.

(1)

(2)

(3)

With the boundary conditions

(4)

(5)

Let the permeability of the porous medium and the suction velocity be of the form

(6)

(7)

Where v_{0} ? 0 and ? « 1 are positive constants.

Introducing the non-dimensional quantities

(7)

The Equations (3), (4), (5) reduce to the following non-dimensional form:

(8)

(9)

(10)

With the boundary conditions

(11)

In view of the periodic suction, temperature and concentration at the plate let us assume the velocity, temperature, concentration at the neighborhood of the plate be

(12)

(13)

(14)

Inserting Equations (12) – (14) into the Equations (8) – (10) and by equating the harmonic and non-harmonic terms, the following set of equations are obtained.

(15)

(16)

(17)

(18)

(19)

(20)

The boundary conditions now reduce to the following form

(21)

Equations (15) and (16), are of third order differential equations with two boundary conditions only. Hence the perturbation method has been employed using Rc (Rc<<1), the elastic parameter as the perturbation parameter.

(22)

(23)

Zeroth order equations

(24)

(25)

First order equations

(26)

(27)

The corresponding boundary conditions are

(28)

Solving these differential equations with the help of boundary conditions, we get

(29)

(30)

(31)

The skin friction at the plate in non-dimensional form is given by

(32)

The rate of heat transfer in terms of Nusselt number is given by

(33)

Another important physical quantity is the mass transfer coefficient, i.e. the Sherwood number which is in non-dimensional form is given by

(34)

The present analysis is focused on the effect of heat and mass transfer on magnetohydro dynamic flow of a visco-elastic fluid through a porous medium with oscillatory suction and heat source to bring out the Soret effect due to natural convection. The general nature of the velocity profile is parabolic with picks near the plate. It is noticed that oscillatory behavior of the profile cannot be seen with the dominating effect of heat source. The outcomes of the present study are verified by comparing with that of Mishra et al. [30] and found to be in good agreement in the absence of **thermal diffusion**. **Figures 2 and 3** exhibit the velocity profiles with the effect of Grashof number for heat and mass transfer. It is noticed that the fluid velocity increases and reaches its maximum over a short distance from the plate and then gradually reduce to zero under the increment of both the cases of Grashof number and modified Grashof number. This is due to the presence of thermal and solutal buoyancy which has the tendency of increase in velocity. The effect of Schmidt number on velocity is presented in **Figure 4**, from this figure it is noticed that velocity decreases with the increasing values of Sc. Physically this is true as Schmidt number increases, viscosity of the fluid also increases that results a decrease in velocity. A similar effect is observed in the case of Prandtl number through **Figure 5**. The effect of magnetic parameter on the fluid velocity is illustrated in the **Figure 6**. This is due to the application of transverse magnetic field, which has the tendency of reducing the velocity. This drag force is called as Lorentz force. The effect of porosity parameter and heat source parameter and Soret number on velocity is displayed in **Figures 7-9 **respectively. It is seen that the velocity increases as the values of Kp, S and So increases.

The effects of prandtl number and heat source/sink parameter on the temperature are presented in the **Figures 10 and 11** respectively. It is observed that there is a reduction of temperature with the increasing values of Pr where as the profiles of temperature rise for the enhancement values of S. Higher Prandtl number fluid causes lower thermal diffusivity which reduces the temperature at all points and hence the thickness of the thermal **boundary layer**. This result is in good agreement with Mishra et al. [30]. The presence of the heat source increases the fluid temperature. In **Figures 12 and 13** the influence of Schmidt number and Soret number on the species concentration is presented. It is noticed that concentration field become thinner under the effect of Schmidt number where as it has enriched due to the Soret effect. Effect of various physical parameters on Skin friction, Nusselt number and Sherwood number are shown in **Table 1**. It is observed that Skin friction increases with an increase of Gr, Gc and S_{o} but a reverse effect is noticed in the case of Sc, Pr and M.

Pr | Gr | Gc | Sc | M | Kp | S | S0 | τ | Nu | Sh |
---|---|---|---|---|---|---|---|---|---|---|

0.71 | 5 | 5 | 0.22 | 5 | 5 | 0.05 | 0.1 | 2.0363 | 0.6742 | 0.2119 |

1 | 5 | 5 | 0.22 | 5 | 5 | 0.05 | 0.1 | 1.9857 | 0.9726 | 0.2052 |

7.1 | 5 | 5 | 0.22 | 5 | 5 | 0.05 | 0.1 | 1.5217 | 7.2233 | 0.0422 |

0.71 | 5 | 5 | 0.22 | 5 | 5 | 0.05 | 0.1 | 2.0363 | 0.6742 | 0.2119 |

0.71 | 10 | 5 | 0.22 | 5 | 5 | 0.05 | 0.1 | 3.0067 | 0.6742 | 0.2119 |

0.71 | 15 | 5 | 0.22 | 5 | 5 | 0.05 | 0.1 | 3.9743 | 0.6742 | 0.2119 |

0.71 | 20 | 5 | 0.22 | 5 | 5 | 0.05 | 0.1 | 4.9520 | 0.6742 | 0.2119 |

0.71 | 5 | 5 | 0.22 | 5 | 5 | 0.05 | 0.1 | 2.0363 | 0.6742 | 0.2119 |

0.71 | 5 | 10 | 0.22 | 5 | 5 | 0.05 | 0.1 | 3.0999 | 0.6742 | 0.2119 |

0.71 | 5 | 15 | 0.22 | 5 | 5 | 0.05 | 0.1 | 4.1657 | 0.6742 | 0.2119 |

0.71 | 5 | 20 | 0.22 | 5 | 5 | 0.05 | 0.1 | 5.2316 | 0.6742 | 0.2119 |

0.71 | 5 | 5 | 0.60 | 5 | 5 | 0.05 | 0.1 | 1.9614 | 0.6742 | 0.5755 |

0.71 | 5 | 5 | 0.78 | 5 | 5 | 0.05 | 0.1 | 1.9299 | 0.6742 | 0.7472 |

0.71 | 5 | 5 | 0.96 | 5 | 5 | 0.05 | 0.1 | 1.9006 | 0.6742 | 0.9194 |

0.71 | 5 | 5 | 0.22 | 1 | 5 | 0.05 | 0.1 | 9.2443 | 0.6742 | 0.2119 |

0.71 | 5 | 5 | 0.22 | 1.2 | 5 | 0.05 | 0.1 | 7.9651 | 0.6742 | 0.2119 |

0.71 | 5 | 5 | 0.22 | 1.4 | 5 | 0.05 | 0.1 | 6.9676 | 0.6742 | 0.2119 |

0.71 | 5 | 5 | 0.22 | 1.6 | 5 | 0.05 | 0.1 | 6.1760 | 0.6742 | 0.2119 |

0.71 | 5 | 5 | 0.22 | 1 | 4 | 0.05 | 0.1 | 9.0692 | 0.6742 | 0.2119 |

0.71 | 5 | 5 | 0.22 | 1 | 8 | 0.05 | 0.1 | 9.5267 | 0.6742 | 0.2119 |

0.71 | 5 | 5 | 0.22 | 1 | 16 | 0.05 | 0.1 | 9.7824 | 0.6742 | 0.2119 |

0.71 | 5 | 5 | 0.22 | 1 | 5 | 0.05 | 0.1 | 9.2443 | 0.6742 | 0.2119 |

0.71 | 5 | 5 | 0.22 | 1 | 5 | 0.1 | 0.1 | 9.4304 | 0.6076 | 0.2134 |

0.71 | 5 | 5 | 0.22 | 1 | 5 | 0.15 | 0.1 | 9.7326 | 0.5122 | 0.2156 |

0.71 | 5 | 5 | 0.22 | 1 | 5 | 0.2 | 0.1 | 10.2111 | 0.3719 | 0.2187 |

0.71 | 5 | 5 | 0.22 | 1 | 5 | 0.05 | 0.05 | 9.2149 | 0.6742 | 0.2194 |

0.71 | 5 | 5 | 0.22 | 1 | 5 | 0.05 | 0.1 | 9.2443 | 0.6742 | 0.2119 |

0.71 | 5 | 5 | 0.22 | 1 | 5 | 0.05 | 0.15 | 9.2737 | 0.6742 | 0.2045 |

0.71 | 5 | 5 | 0.22 | 1 | 5 | 0.05 | 0.2 | 9.3031 | 0.6742 | 0.1970 |

**Table 1:** Effect of various physical parameters on Skin friction, Nusselt number and Sherwood number.

In the present study the effect of thermal diffusion due to natural convection on MHD flow of a visco-elastic fluid past a porous plate with variable suction and heat source/sink is analyzed. The governing equations for the velocity field, temperature and concentration by perturbation technique in terms of dimensionless parameters. The findings of this study are as follows.

• Velocity increases as Gr, Gc and So increase whereas there is a reverse effect in case of Sc, Pr and M.

• The concentration reduces with an increase in Sc but in the case of S_{o} it enhances.

• Skin friction increases with an increase of Gr, Gc and So but a reverse effect is noticed in the case of Sc, Pr and M.

• Nusselt number increases as Pr increases but in the case of S it decreases.

Sherwood number increases with an increase in Sc and S but a reverse effect is noticed in the case of Pr and S_{o} (**Table 2**).

Results of Mishra et al. [30] | Results of the present study | ||
---|---|---|---|

Sc | Sh | Sc | Sh |

0.22 | 0.219095 | 0.22 | 0.2201 |

0.3 | 0.298966 | 0.3 | 0.3001 |

0.66 | 0.658814 | 0.66 | 0.6601 |

0.78 | 0.776574 | 0.78 | 0.7802 |

**Table 2:** Comparison of our results with the results of Mishra [30] in the absence of thermal diffusion.

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