Free Convective Fluctuating Flow and Mass Transfer Through A Porous Medium Bounded By A Vertical Plate in the Presence of Hall Current and Variable Permeability

Suresh Rana

Department of Mathematics, Govt. College of Teacher Education, Dharamshala (H.P) India

*Corresponding Author E-mail: sureshrana 428@gmail.com.

ABSTRACT:

In this paper the effect of hall current, chemical reaction, radiation on a free convective flow bounded by a vertical plate embedded in a porous medium under the influence of uniform magnetic field, which is applied normal to the surface , is studied .The permeability of the porous medium fluctuates in time about constant mean. The problem is solved analytically and expressions for velocity, temperature and concentration have been obtained. The effect of permeability, magnetic number ,hall parameter , radiation parameter ,Grashof number ,modified Grashof number ,chemical reaction on velocity ,temperature and skin friction and heat transfer amplitude and phase are obtained and shown respectively with the help of figures and tables and are discussed in detail.

KEYWORDS: Chemical reaction; Radiation; Permeability; Amplitude; Concentration.

I. INTRODUCTION:

The phenomenon of free convection arises in the fluid when temperature and concentration changes cause density variation leading to buoyancy forces acting on the fluid elements.The mass transfer differences effect the rate of heat transfer. In industries, many transport processes exist in which heat and mass transfer take place simultaneously as a result of combined buoyancy effect of thermal diffusion and diffusion thermo chemical species. The phenomenon of heat and mass transfer frequently exists in chemically processed industries such as food processing and polymer production. Free convection flows are also of great interest in a number of industrial applications such as fiber and granular insulation geothermal system etc.Convection in porous media has applications in geothermal energy recovery, oil extraction, thermal energy storage and flow through filtering devices.Megneto-hydrodynamics is attracting the attention of many authors due to its application in geophysics. In engineering in MHD pumps, MHD bearing etc. at high temperature attained in some engineering devics.Since some fluids can 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 when it is capable of emitting and absorbing thermal radiation. This is of interest because heat transfer by thermal radiation is becoming of great importance when we are concerned with space application and higher operating temperatures.

The growing need for chemical reactions in chemical and hydrometallurgical industries require the study of heat and mass transfer with chemical reaction. Chemical reactions occur in air or water due to the presence of foreign mass. It may be present by itself or as mixtures 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 such as polymer production, manufacturing of ceramics or glassware and food processing.

Soundalgekar and Takhar [1] studied the effects of radiation on the natural convection flow of a gas past a semi-infinite plate using the Cogly-Vincentine-Gillas equilibrium model. Takhar et.al.[2] also investigated the effect of radiation on MHD free convection flow past a semi-infinite vertical plate for same gas.Muthucumarswamy and Kumar[3] studied the thermal radiation effects on moving infinite vertical plate in presence of variable temperature and mass diffusion.Hussain et.al.[4] studied the effect of radiation on free convection on porous vertical plate.Chamkha et.al.[5] studied the effect of hydro-magnetic combined heat and mass transfer by natural convection from a permeable surface embedded in fluid saturated porous medium.

The effect of chemical reaction on heat and mass transfer in a laminar boundary layer flow has been studied under different conditions by several authors. The effect of a chemical reaction on moving isothermal vertical surface with suction has been studied by Muthucumarswamy [6].Manivannan et.al. [7] has investigated radiation and chemical reaction effects on isothermal vertical oscillatory plate with variable mass diffusion.

Sharma et.al. [8] studied the influence of chemical reaction and radiation on unsteady MHD free convective flow and mass transfer through viscous incompressible fluid past a heated vertical plate immersed in porous medium in presence of heat source. Mahapatra et.al.[9] studied the effects of chemical reaction on free convection flow through a porous medium bounded by a vertical surface.Rajsekhar et.al.[10],Kishan and Srinivas[11] ,Anjalidevi and David[12],Kishan and Deepa[13] and Gaikwad and Rahuldev[14] studied the effects of various parameters on fluid flow. Recently Tavva Sudhakar Reddy et.al. [15] studied the MHD free convection heat and mass transfer flow through a medium bounded by a vertical surface in presence of hall current. The purpose of this paper is to study Heat and Mass Transfer on Free Convective unsteady Fluctuating flow through a Porous medium bounded by a vertical plate in the presence of Hall current,radiation,chemical reaction and permeability.

II. FORMULATION OF THE PROBLEM:

Consider an electrically conducting, radiating, viscous, incompressing fluid through a porous medium bounded by an infinite vertical porous plate with constant suction .The -axis is taken along the surface in an upward direction and -axis is taken normal to it. A uniform magnetic field B₀ is assumed to be applied in a direction perpendicular to the surface .The fluid properties are assumed to be constant except that the influence of density in the body term. A chemically reactive species are emitted from the vertical surface in to a hydrodynamic flow field .It diffuses into the fluid when it undergoes a homogenous chemical reaction. The reaction is assumed to take place entirely in the stream. Under these conditions, the problem is governed by the following equations;

Table-1

K_r

K₀

Tan

0.5

1.0

2.0

0.5

0.04

-.04

0.04

0.1

2.2805

2.3091

2.2005

2.3472

2.0971

2.4355

2.4906

2.5768

2.2545

2.1208

2.1803

2.3043

2.1085

2.3968

2.4451

2.4681

2.3312

2.361

2.2491

2.3095

2.209

2.4579

2.500

2.520

0.21169

0.2076

0.21022

0.14942

0.24564

0.19899

0.19530

0.18089

0.31423

0.33158

0.34288

0.27166

0.31131

0.28312

0.27556

0.272

0.41911

0.40693

0.44072

0.38972

0.43164

0.38113

0.37055

0.36557

Table-2

K_r

K₀

0.5

1.0

2.0

0.5

0.04

0.00

-.04

0.04

0.1

0.7

1.0

2.0

0.1

2.4794

2.2862

2.116

2.5862

3.1612

2.8859

2.1968

41.799

62.317

112.86

2.6140

2.6726

2.7004

2.4184

2.2302

2.0653

2.5108

3.038

2.8363

2.1617

38.926

55.844

96.913

2.5496

2.6069

2.6339

2.3861

2.2002

2.0367

2.4792

2.991

2.8048

2.1376

36.065

51.341

86.367

2.5153

2.5716

2.5984

0.10538

0.10151

0.099577

0.10181

0.10174

0.047814

0.69587

0.16597

0.16038

0.13993

0.10499

0.10557

0.10579

0.12565

0.12296

0.12453

0.12179

0.12193

0.091793

0.11020

0.20858

0.10035

0.22097

0.12769

0.12855

0.12896

0.14934

0.14620

0.14319

0.14454

0.14053

0.11795

0.13516

0.27402

0.29424

0.33085

0.15192

0.15301

0.15352

Table 3

K_r

Sherwood Number

0.01

0.22

0.60

0.78

0.22

0.04

0.00

-0.04

0.25457

0.63764

0.81814

0.22

0.16745

V. RESULTS AND DISCUSSION:

In order to point out the effects of various parameters on flow characteristic, the following discussion is set out. The value of Prandtl number is chosen as 0.71 to represent air. The value of the Schmidt number (0.22) is chosen to represent the presence of species hydrogen.Figure1 depicts the effect of radiation F on the velocity profile. It is clear from the figure that velocity decreases with increase of radiation . In figure (2) and (3) the effect of Grashof number Gr and modified Grashof number Gm on velocity is observed and it is found that velocity increases with increase of both Gr and Gm but increase in velocity is more in case of increasing values of Gr than Gm .Figure (4) depicts the effect of permeability in the absence of magnetic field and radiation. It is observed that with increase of permeability the velocity increases . In figure (5) the effect of Hall parameter m has been depicted when Pr=0.71 in the presence of magnetic parameter and radiation. It is observed that velocity increases with increase of m. Figure (6) depicts the effect of chemical reaction. It has been observed that for (generative reaction) the velocity increases but for (destructive reaction) the velocity decreases. Figure (7) depicts the effect of magnetic field M on the velocity field. It has been observed that velocity decreases with increasing magnetic parameter M.

The effect of magnetic parameter M on temperature profile has been presented in figure (8) .It has been observed that temperature decreases with increase of magnetic parameter M. In figure (9) the effect of radiation parameter F on temperature profile has been observed. It has been found that temperature decreases as F is increased. Figure (10) depicts effect of permeability on temperature profile. It is observed that temperature increases with increase of permeability. The effect of Schmidt number on temperature field has been depicted in figure (11) and it has been observed that temperature decreases with increase of .

The effect of chemical reaction parameter on concentration has been depicted in figure (12).It is clear from the figure that for concentration decreases and for the concentration increases. The effect of on concentration has been observed in figure (13).It is clear from the figure that concentration decreases with increasing. The numerical values of the amplitudes and the phases of the skin friction and the rate of heat transfer are presented in Table 1 and Table 2.From Table 1 the amplitude of the skin friction , , has been observed to be decreasing for to and increasing between and except for and for for which the amplitude of the skin friction , , has been observed to be decreasing with increasing value of An increase in the value m leads to the increase in the value .But the value decreases between F=0.5 and F=1 and increases between F=1 and F=2.The values in this table exhibits that there is always a phase lead. Table 2 shows that the amplitude of the rate of heat transfer,, decreases with increasing values of The values increase sharply for increasing of and the value increases for and decreases for .The values of,, increase for increasing values of F and decrease for increasing values M.The values of,,increases moderately for increasing values of hall parameter m. From the Table 2 it is obvious that there is always a phase lead .Table 3 presents the values of Sherwood number. It is clear from the table that the value of Sherwood increases for increasing value of Sc and value of Sherwood number increases for and decreases for and is independent of Eckert number E.

VI. ACKNOWLEDGEMENTS:

The author will be highly thankful to Referee for his valuable suggestions.

VII. NOMENCLATURE:

C - dimensionless concentration U₀ -------- mean stream velocity

- Concentration; u --- dimensionless velocity of the fluid at the direction

---Species concentration at the plate; -- velocity of the fluid at the direction

---Species concentration far away from the plate; - velocity of the fluid at the direction

C_p--- specific heat at constant pressure; v₀ - suction velocity

D --- Chemical diffusivity; -- co-ordinate axis along the plate

--acceleration due to gravity; -- co-ordinate axis along the plate

G_r --- Grashof number; --- suction parameter

G_m --- modified Grashof number; ---- coefficient of thermal expansion

k --- thermal conductivity; ---- coefficient of concentration expansion

P_r --- Prandtl number; ----- kinematic viscosity

P^’ --- pressure; ----- density

q^’ ---- heat flux at the plate; ----- Stefan-Boltzman constant

F ---- radiation parameter; ----- dimensionless frequency of vibration of fluid

S_c ----schmidt number ----- frequency of vibration of fluid

T ---- dimensionless fluid temperature ; M ---- magnetic parameter

---- temperature of fluid away from the plate ; m --- Hall current parameter

---- temperature of the fluid; k_r ---- chemical reaction parameter

t --- dimensionless time; k_o ----- permeability parameter

t^’ --- time; R ----- Reynolds number.

----------- temperature of fluid at the plate ;

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Received on 16.01.2014 Accepted on 01.02.2014

Int. J. Tech. 4(1): Jan.-June. 2014; Page 47-56