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Unsteady MHD Free Convection Flow of a Viscous Dissipative Kuvshinski Fluid Past an Infinite Vertical Porous Plate in the Presence of Radiation, Thermal Diffusion and Chemical Effects
ISSN: 2168-9679
Journal of Applied & Computational Mathematics
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Unsteady MHD Free Convection Flow of a Viscous Dissipative Kuvshinski Fluid Past an Infinite Vertical Porous Plate in the Presence of Radiation, Thermal Diffusion and Chemical Effects

Vidyasagar B1, Raju MC2* and Varma SVK1

1Department of Mathematics, Sri Venkateswara University, Tirupati, AP, India

2Department of Humanities and Sciences, Annamacharya Institute of Technology and Sciences, AP, India

*Corresponding Author:
Raju MC
Department of Humanities and Sciences
Annamacharya Institute of Technology and Sciences
Rajampet (Autonomous) Cuddapah, AP, India
Tel: +919848998649
E-mail: [email protected]

Received date: August 07, 2015; Accepted date: September 04, 2015; Published date: September 09, 2015

Citation: Vidyasagar B, Raju MC, Varma SVK (2015) Unsteady MHD Free Convection Flow of a Viscous Dissipative Kuvshinski Fluid Past an Infinite Vertical Porous Plate in the Presence of Radiation, Thermal Diffusion and Chemical Effects. J Appl Computat Math 4:255. doi:10.4172/2168-9679.1000255

Copyright: © 2015 Vidyasagar B, 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.

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Abstract

The objective of present problem is to investigate the effects of thermal diffusion, viscous dissipation, radiation and chemical reaction on a well-known non Newtonian fluid namely Kuvshinski fluid interaction on unsteady MHD flow over a vertical moving porous plate. The fluid is considered to be a gray, absorbing emitting but non scattering medium, and the Rosseland approximation is used to describe the radiative heat flux energy equation. The plate moves with constant velocity in the direction of fluid flow while the free stream velocity is assumed to follow the exponentially increasing small perturbation law. A uniform magnetic field acts perpendicular to the porous surface. The dimensionless governing equations are solved by using a simple perturbation law. The expressions for velocity, temperature and concentration are derived. With the aid of these the expressions for Skin friction, Nusselt number and Sherwood number are also derived. The effects of various material parameters on the above flow quantities are studied numerically with the help of figures and tables. It is observed that an increases in the Prandtl number results in a decreasing in temperature. An increase in Kr leads to decrease in both of concentration and velocity.

Keywords

Kuvshinski fluid; MHD; Porous medium; Thermal radiation; Chemical reaction; Thermal diffusion and vertical plate.

Introduction

The phenomenon of heat and mass transfer frequently exists in chemically processed industries such as food processing and polymer production. Free convection flows are of great interest in a number of industrial applications such as fiber and granular insulation, geothermal systems etc. convection in porous media has applications in geothermal energy recovery, oil extraction, thermal energy storage and flow through filtering devices. Magneto hydrodynamics is attracting the attention of the many authors due to its applications in geophysics; it is applied to study the stellar and solar structures, interstellar matter, radio propagation through the ionosphere etc. in engineering in MHD pumps, MHD bearings etc. Nasser [1] examined numerically the problem of unsteady free convection with heat and transfer from an isothermal vertical flat plate to a non-Newtonian fluid saturated porous medium. The flow in the porous medium was described via-the Darcy– Brinkman for Chheimer model. Anwar and Wilson [2] investigated natural convection flow in a fluid saturated porous medium enclosed by non-isothermal walls with heat generations. Combined effect of magnetic field and viscous dissipation on a power law fluid over plate with variable surface heat flux embedded in a porous medium was studied by Amin [3]. Isreal-Cookey et al. [4] studied the influence of viscous dissipation and radiation on unsteady MHD free convection flow past an infinite heated vertical plate in a porous medium with time–dependent suction, Kinyanjui et al. [5] considered magneto hydrodynamic free convection heat and mass transfer of a heat generating fluid past an impulsively saturated infinite vertical porous plate with Hall current and radiation absorption. Shvets and Vishevskiy [6] discussed the effect of dissipation on convective heat transfer in flow of non-Newtonian fluids. Cheng and Lin [7] considered an unsteady forced heat transfer on a plate embedded in the fluid-saturated porous medium with inertia effect and thermal dispersion. An analysis is carried out by Salama [8] to study the effect of heat and mass transfer on a non-Newtonian fluid between two infinite parallel walls, one of them moving with a uniform velocity under the action of a transverse magnetic field. The moving wall moves with constant velocity in the direction of fluid flow while the free stream velocity is assumed to follow the exponentially increasing small perturbation Law, Time-dependent wall suction is assumed to occur at permeable surface. Eldabe and Hassan [9] have considered non-Newtonian flow formation in Couette motion in magneto hydrodynamics with time varying suction. Char [10] investigated heat and mass transfer in a hydro magnetic flow of the viscoelastic fluid over stretching sheet. Eldabe et al. [11] consider mixed convective heat and mass transfer in a non- Newtonian fluid at a peristaltic surface with temperature-dependent viscosity. Chamka [12] investigated an unsteady MHD convective heat and mass transfer past a semi-infinite vertical permeable moving plate with heat absorption. In his study Salam [13] addressed a coupled heat and mass transfer flow in Darcy-Forchheimer mixed convection from a vertical flat plate embedded in fluid saturated porous medium under the influence of radiation and viscous dissipation. Chamka [14] addressed a heat and mass transfer problem of a non-Newtonian fluid flow over permeable wedge in porous medium with variable wall temperature and concentration and source or sink. In his investigation Chamka and Humoud [15] discussed mixed convection heat and mass transfer of a non-Newtonian fluids from a permeable surface embedded in a porous medium.

At high temperatures attained in some engineering devices, gas, for example, can be ionized and so becomes an electrical conductor. The ionized gas or plasma can be made to interact with the magnetic and alter heat transfer and friction characteristic. Since some fluids can also emit and absorb thermal radiation, it is of interest to study the effect of magnetic field on the temperature distribution and heat transfer when the fluid is not only an electrical conductor but also it is capable of emitting and absorbing thermal radiation. This is of interest because heat transfer by thermal radiation has become of greater importance when we are concerned with space applications and higher operating temperatures. In their study Sondulgekar and Takhar et al. [16] addressed radiative convective flow past a semiinfinite vertical plate. Thakar et al. [17] considered radiation effects on MHD free convection flow of a radiating gas past semi-infinite vertical plate. Hossain et al. [18] studied the effect of radiation on free convection from a porous vertical plate. Muthucumarswamy and Kumar [19] studied heat mass transfer effects on moving vertical plate in the presence of thermal radiation. MHD flow past an impulsively started infinite vertical plate in the presence of thermal radiation was discussed by Mzumder and Deka [20]. The growing need for chemical reactions in chemical and hydrometallurgical industries require the study of heat and mass transfer with chemical reaction. The presence of a foreign mass in water or air causes some kind of chemical reaction. This may be present either by itself or as mix with air or water. In many chemical engineering processes, a chemical reaction occurs between a foreign mass and the fluid in which the plate is moving. These processes take place in numerous industrial applications, for example, polymer production, manufacturing of ceramics or glassware and food processing. A chemical reaction can be codified as either a homogenous or heterogeneous process. This depends on whether it occurs on an interface or a single phase volume reaction. A reaction is said to be of first order if its rate is directly proportional to the concentration itself. Thakar et al. [21], discussed the study of flow and mass transfer on a stretching sheet with a magnetic field and chemically reactive species. Muthucumarswamy and Ganesan [22], considered the effect of chemical reaction and radiation on flow characteristics in a unsteady upward motion of an isothermal plate. MHD Flow of a uniformly stretched vertical permeable surface in the presence of heat generation/absorption and a chemical reaction has been discussed by Chamkha [23]. Muthucumarswamy [24], considered the effect of chemical reaction on a moving isothermal vertical surface with suction. Manivanna et al. [25] studied the radiation and chemical reaction effects on isothermal vertical oscillating plate with variable mass diffusion. Raju et al. [26], analyzed the unsteady MHD radioactive and chemically reactive free convection flow near a moving vertical plate in porous medium. Again Raju et al. [26,27] worked on a problem of radiation and mass transfer effects on a free convection flow through a porous medium bounded by vertical surface. Kim [28] investigated on MHD convective heat transfer flow past a moving vertical porous plate. Chamkha, continued this in the presence of mass transfer and heat absorption. Barik and Dash [29] considered the radiation effect on MHD flow past inclined porous plate. Raju et al. [30], Chamkha et al. [31], Umamaheswar et al. [32], Ravikumar et al. [33,34], have reported relevant studies on MHD convective flows in the presence of heat and mass transfer and thermal radiation past a plate. Motivated by the above studies in this paper we made an attempt to study the effects of thermal diffusion, viscous dissipation, radiation and chemical reaction on a well-known non Newtonian fluid namely Kuvshinski fluid interaction on unsteady MHD flow over a vertical moving porous plate. The fluid is considered to be a gray, absorbing emitting but non scattering medium, and the Rosseland approximation is used to describe the radiative heat flux energy equation. The plate moves with constant velocity in the direction of fluid flow while the free stream velocity is assumed to follow the exponentially increasing small perturbation law. A uniform magnetic field acts perpendicular to the porous surface.

Problem Formulation

An unsteady two dimensional free convection flow of viscous, incompressible, conducting radiating and chemically reacting Kuvshinski fluid through porous medium occupying a semi–infinite region of the space bounded by an infinite vertical plate is considered the x*-axis is taken along the vertical plate in the up-ward direction and y*-axis is taken normal to it. A uniform magnetic field of strength B0 is applied perpendicular to the fluid flow direction. Initially it is assumed that the plate and the fluid are the same temperature T*and concentration level C* everywhere in the fluid. The level of foreign mass is assumed to below, so that the Dofour effect is neglected. The fluid is assumed to be gray emitting and absorbing radiation but non-scattering medium. The radioactive heat flux in the x*- direction is negligible in comparison to their in y*- direction. All the fluid properties are assumed to be constant except the influence of the density variation with temperature in body force term. Electric field and induced magnetic field effects are neglected. Energy dissipation is neglected under these assumptions the equation describing the flow field are given by

Continuity Equation: (1)

image (1)

Momentum Equation:

image (2)

Energy Equation:

image (3)

Diffusion Equation:

image (4)

In the momentum equation right hand side first two terms represent the effect of buoyancy force, third term represents the presence of non- Newtonian fluid, fourth and fifth terms represent the application of transverse magnetic field and porous medium respectively. The last two terms of RHS energy equation represent the presence of viscous dissipation and thermal radiation. Similarly the last two terms of diffusion equation represent the presence of homogeneous chemical reaction and thermal diffusion respectively.

The initial and boundary conditions are:

image (5)

The Equation (1) gives

image (6)

Where ν0 is the constant suction velocity using the Rosseland approximation for optically thick fluids and the Taylor’s series about T neglecting the second and higher order terms we have Equation (7)

image (7)

Where σ*is the Stefan-bottzmann constant and is k* the mean absorption coefficient

In view of (6) and (7) the Equations (2)-(4) reduce to

image (8)

image (9)

image (10)

On introducing the following non- dimensional quantities

image

image

image

image

image (11)

In view of (11) the Equations (8)-(10), reduce to the following nondimensional form

image (12)

image (13)

image (14)

Where Gr is the thermal Grashof number, Gm is the modified Grashof number, Pr is the fluid Prandle number, Sc is the Schmidt number S0 is the Soret number and Kr is the chemical reaction Parameter.

The corresponding boundary conditions reduce to

image (15)

Problem Solution

The governing Equations (12), (13) and (14) of the flow, temperature and concentration are coupled non- linear differential equations. Assuming E to be small, we write

image(16)

Substituting Equations (16) into (12), (13) and (14) and equating the powers of E, we obtain equations to the zeroth order as

image(17)

image(18)

image(19)

The first order equations are

image(20)

image(21)

image(22)

The corresponding boundary conditions (15) now become

image (23)

Where image

Here a prime denotes the differentiation with respect to y. solving Equations (19) and (20) under the corresponding boundary conditions (23) we obtain

image (24)

image (25)

image (26)

image

image (27)

image (28)

image (29)

Where the constants are given by

image

image

image

image

image

image

image

image

image

image

image

image

image

image

image

image

image

image

Results and Discussion

The problem of an unsteady free convection flow of viscous, incompressible, radiating Kuvshinski fluid flow past through porous medium was formulated and solved by means perturbation method. The expressions obtained in previous section are studied with the help of figures from 1-14. The effects of various physical parameters viz., Schmidt number (Sc), thermal Grashof number (Gr), mass Grashof number (Gm), magnetic parameter (M), Prandtl number (Pr), chemical reaction parameter (Kr), thermal radiation parameter (N), Soret number (S0) are studied numerically by choosing arbitrary values. The effect of Schmidt number Sc on velocity is presented in Figure 1, as the Schmidt number increases, the velocity decreases. It is seen that velocity attains maximum near the plate and it reaches the free stream near the vicinity of the either side of the plate. This is due to the concentration buoyancy effect that decreases the fluid velocity. Figure 2 depicts the effect of Soret number on velocity field. From this figure it is observed that velocity increases with an increase in Soret number. The effect of magnetic parameter on velocity field is present in Figure 3, where velocity is observed to be decreasing with an increase in magnetic parameter. Physically this is true as an increase in magnetic field results a Lorentz’s force which has the tendency of retarding the flow. Therefore velocity obviously decreases with an increase in magnetic parameter.

computational-mathematics-effect-sc-velocity

Figure 1: Effect of Sc on velocity.

computational-mathematics-effect-s0-velocity

Figure 2: Effect of S0 on velocity.

computational-mathematics-effect-m-velocity

Figure 3: Effect of M on velocity.

In Figures 4 and 5, effects of thermal and solute buoyancy on velocity is presented, in which it is noticed that velocity increases in both the cases as both the parameters namely Grashof number and modified Grashof number increase. Effect of Prandtl number on velocity is presented in Figure 6. The numerical results show that the effect of increasing values of Prandtl number results a decrease in velocity. From Figure 7 it is observed that an increases in the thermal radiation parameter results a decrease in velocity. Effect of porosity parameter on velocity is shown in Figure 8, from this figure it is seen that velocity increases with the increasing values of porosity parameter. Chemical reaction effect on velocity is presented in Figure 9. From this numerical study it is noticed that velocity decreases with an increase in chemical reaction parameter. Effect of Schmidt number on concentration is displayed in Figure 10. From this figure it is noticed that concentration decreases with an increase in Schmidt number. Schmidt number is defined as the ratio of kinematic viscosity to the thermal diffusivity. As Schmidt number increases, usually viscosity increases and subsequently thermal diffusion decreases, therefore concentration decreases. Similar effect is noticed in the presence of chemical reaction parameter, which is shown in Figure 11 But, Figure 12 witnesses the reverse action on concentration in the presence of Soret number. In Figures 13 and 14, effect of radiation parameter and Prandtl number are presented. From these figure it is noticed that temperature decreases with the increasing in both radiation parameter and Prandtl number. Effect of Schmidt number, Soret number, magnetic number, Grashof number and modified Grashof number on Skin friction, Nusselt number and Sherwood number is presented in Table 1, from this table it is observed that, Skin friction and Nusselt number increase with an increase in Soret number, Grashof number and modified Grashof number whereas reverse phenomenon is noticed in the case of Schmidt number and magnetic parameter. But Sherwood number increases with an increase in Schmidt number, Grashof number and modified Grashof number and it decreases in the case of magnetic parameter and Soret number. Effect of chemical reaction parameter, Prandtl number, porosity parameter and radiation parameter on Skin friction, Nusselt number and Sherwood number. This table witnesses that when an increase in chemical reaction parameter results a decrease in Skin friction and Nusselt number whereas Sherwood number increases. The influence of Prandtl number decreases the shear force whereas it increases the rate of heat transfer the rate of mass transfer. The presence of porosity parameter increases the Skin friction, Nusselt number and Sherwood number. Radiation parameter increases the Nusselt number whereas it decreases the Skin friction and Sherwood number (Tables 1 and 2).

computational-mathematics-effect-gm-velocity

Figure 4: Effect of Gm on velocity.

computational-mathematics-effect-gr-velocity

Figure 5: Effect of Gr on velocity.

computational-mathematics-effect-pr-velocity

Figure 6: Effect of Pr on velocity.

computational-mathematics-effect-n-velocity

Figure 7: Effect of N on velocity.

computational-mathematics-effect-kp-velocity

Figure 8: Effect of kp on velocity.

computational-mathematics-effect-kr-velocity

Figure 9: Effect of Kr on velocity.

computational-mathematics-effect-sc-velocity

Figure 10: Effect of Sc on concentration.

computational-mathematics-effect-kr-concentration

Figure 11: Effect of Kr on Concentration.

computational-mathematics-effect-s0-concentration

Figure 12: Effect of S0 on concentration.

computational-mathematics-effect-n-concentration

Figure 13: Effect of N on Temperature.

computational-mathematics-effect-pr-temperature

Figure 14: Effect of Pr on Temperature.

Sc S0 M Gr Gm τ Nu Sh
0.22 2 5 5 5 3.977 0.5552 0.2211
0.6 2 5 5 5 3.8308 0.5042 0.4109
0.78 2 5 5 5 3.7789 0.485 0.5001
0.96 2 5 5 5 3.7339 0.4685 0.5872
0.22 2 5 5 5 3.977 0.5552 0.2211
0.22 3 5 5 5 4.0307 0.5766 0.1627
0.22 4 5 5 5 4.0846 0.5987 0.1074
0.22 5 5 5 5 4.1387 0.6217 0.0553
0.22 2 5 5 5 3.997 0.5552 0.2211
0.22 2 6 5 5 3.7133 0.4578 0.2169
0.22 2 7 5 5 3.495 0.386 0.2139
0.22 2 8 5 5 3.3104 0.3312 0.2118
0.22 2 5 5 5 3.977 0.5552 0.2211
0.22 2 5 6 5 4.3548 0.6674 0.2259
0.22 2 5 7 5 4.7316 0.7906 0.2315
0.22 2 5 8 5 5.1074 0.9252 0.238
0.22 2 5 5 5 3.977 0.5552 0.2211
0.22 2 5 5 6 4.3901 0.6818 0.228
0.22 2 5 5 7 4.8028 0.8222 0.2366
0.22 2 5 5 8 5.2152 0.977 0.2471

Table 1: Effect of Schmidt number, Soret number, magnetic number, Grashof number and modified Grashof number on Skin friction, Nusselt number and Sherwood number.

Kr Pr Kp N τ Nu Sh
0.2 0.71 0.5 2 3.977 0.5552 0.2211
0.3 0.71 0.5 2 3.9355 0.5402 0.2791
0.4 0.71 0.5 2 3.8291 0.5274 0.5932
0.5 0.71 0.5 2 3.8701 0.517 0.3424
0.2 0.71 0.5 2 3.977 0.5552 0.2211
0.2 1 0.5 2 3.8978 1.0746 0.1518
0.2 7.1 0.5 2 3.5124 32.3181 1.5026
0.2 0.71 0.5 2 3.977 0.5552 0.2211
0.2 0.71 0.6 2 4.0779 0.5957 0.223
0.2 0.71 0.7 2 4.1546 0.6277 0.2246
0.2 0.71 0.8 2 4.215 0.6537 0.2258
0.2 0.71 0.5 2 3.977 0.5552 0.2211
0.2 0.71 0.5 3 3.9459 0.7293 0.1946
0.2 0.71 0.5 4 3.9273 0.8509 0.1784
0.2 0.71 0.5 5 3.9149 0.94 0.1673

Table 2: Effect of chemical reaction parameter, Prandtl number, porosity parameter and radiation parameter on Skin friction, Nusselt number and Sherwood number.

Conclusion

The problem of an unsteady free convection flow of viscous, incompressible, radiating Kuvshinski fluid flow past through porous medium was formulated and solved by means perturbation method. The following conclusions are made.

1) Velocity increases with an increase in Grashof number, modified Grashof number, Soret number, and porosity parameter but it decreases with an increase in Schmidt number, magnetic parameter, Prandtl number, radiation parameter, and chemical reaction parameter.

2) Concentration increases with an increase in Soret number but it shows reverse phenomenon in the case of Schmidt number and chemical reaction parameter.

3) Temperature decreases with an increasing in both radiation parameter and Prandtl number.

4) Skin friction increases with an increase in Soret number, Grashof number, modified Grashof number and porosity parameter, whereas it decreases in the presence of Schmidt number, magnetic parameter, chemical reaction parameter, Prandtl number and radiation parameter.

5) Nusselt number increases with an increase in Soret number, Grasof number, modified Grashof number, Prandtl number, porosity parameter and radiation parameter, where as it decreases in the presence of Schmidt number, magnetic parameter and chemical reaction parameter.

6) Sherwood number increases with an increase in Schmidt number, Grashof number, modified Grashof number, Prandtl number and porosity parameter, whereas it has different approach in the case of Soret number, magnetic parameter and radiation parameter.

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