INTRODUCTION:

Viscoelastic flows and transport phenomena arise in numerous areas of chemical, industrial process, biosystems, food processing and biomedical engineering. These include the rheology of liquid crystal precursors employed in the manufacture of carbon super-fibers, crude oil emulsion processing, paper coating rheological processing, propulsive culinary transport of respiratory airway mucus, thermocapillary bubble dynamics in weakly elastic fluids, rheo-reactor phosphatation flows, flour rheology, mayonnaise elasticoviscous flows, xanthan gum hydrogel flows, polygalacturonase based food stuff. The heat transfer in the forced convection flow of a viscoelastic fluid of Walter model was investigated by Rajagopal [15]. Siddappa et al. [17] studied the flow of visoelastic fluids of the type Walter’s liquid B past a stretching sheet. Abel and Veena [1] studied the viscoelasticity on the flow and heat transfer in a porous medium over a stretching sheet.

Chowdhury and Islam [5] were studied the MHD free convection flow of visco-elastic fluid past an infinite vertical porous plate. Mittal et al. [13] discussed the vorticity of fluctuating flow of a visco-elastic fluid past an infinite plate with variable suction in slip flow regime.

The study of heat generation or absorption in moving fluids is important in problems dealing with chemical reactions and those concerned with dissociating fluids. Heat generation effects may alter the temperature distribution and this in turn can affect the particle deposition rate in nuclear reactors, electronic chips and semi conductor wafers. Although exact modeling of internal heat generation or absorption is quite difficult, some simple mathematical models can be used to express its general behavior for most physical situations. Heat generation or absorption can be assumed to be constant, space-dependent or temperature-dependent. Chambre and Young [6] have analyzed a first order chemical reaction in the neighbourhood of a horizontal plate. Applblat [4] studied analytical solution for mass with a chemical reaction of first order. Crepeau and Clarksean [7] have used a space-dependent exponentially decaying heat generation or absorption in their study on flow and heat transfer from a vertical plate. Das et al. [8] have studied the effect of homogeneous first order chemical reaction on the flow past an impulsively started infinite vertical plate with uniform heat flux and mass transfer. Several interesting computational studies of reactive MHD boundary layer flows with heat and mass transfer in the presence of heat generation or absorption have appeared in recent years Mahdy [12], Abo-Eldahab et al. [2], Das [9].

The Soret effect, for instance has been utilized for isotope separation, and in mixture between gases with very light molecular weight (H₂ ,He) and of medium molecular weight (N2 ,air)Soret effect [thermal-diffusion] refers to mass flux produced by a temperature gradient. Following Eckert and Drakes [10] work several investigators have analyzed the effects of Soret and Dufour effects in various types of heat and mass transfer problems. Kafoussias and Williams [11] examined the boundary layer flows in the presence of Soret and Dufour effects associated with the thermal-diffusion and diffusion-thermo for the mixed-forced-natural convection. Postelnicu [14] studied simultaneous heat and mass transfer by natural convection from a vertical plate embedded in an electrically conducting fluid-saturated porous medium in the presence of Soret and Dufour effects using Darcy-Boussinesq model. Srihari et al. [19] discussed Soret effect on unsteady MHD free convective mass transfer flow past an infinite vertical porous plate with oscillatory suction velocity and heat sink. Srenivas Reddy [18] has discussed the soret effect on mixed convective heat and mass transfer through a porous cylindrical annulus. Saritha and Satya Narayana [16] thermal diffusion and chemical reaction effects on unsteady MHD free convection flow past a semi infinite vertical permeable moving plate. Very recently SudarsanReddy.P, et al [20] has analyzed finite element analysis of thermo-Diffusion and Diffusion-Thermo effects on convective heat and mass transfer flow through a porous medium in a cylindrical annulus with constant heat source. Ahmed and Kalita [3] Soret and magnetic field effects on a transient free convection flow through a porous medium bounded by a uniformly moving infinite vertical porous plate in presence of heat source.

The aim of the present chapter is to study the combined influence of chemical reaction and soret effects on MHD convection heat and mass transfer flow of a chemically reacting visco-elastic fluid through porous medium bounded by an oscillating porous plate in the slip-flow regime with heat generation or absorption is investigated. Approximate solutions have been derived for the mean velocity, mean temperature and mean concentration using multi-parameter perturbation technique and these are presented in graphical form.

FORMULATION OF THE PROBLEM:

Physical Model:

An unsteady convective heat and mass transfer flow of a viscous electrically conducting viscoelastic fluid (Walter’s Liquid Model B) through a porous medium bounded by an oscillating porous plate in the slip-flow regime is considered. The x-axis taken along the vertical plate in the upward direction and y-axis is taken normal to it. Let u and v be the components of velocity in x and y directions respectively. A uniform magnetic field of strength is applied normal to the fluid flow direction. The effects of first order chemical reaction, Soret effect and heat generation/absorption sources are (taken into account) considered. The flow is assumed that

(i) The surface is maintained at constant temperature and concentration.

(ii) As the plates are long enough, all the physical variables are functions of y and t only

(iii) Magnetic Reynolds number is very small so that the induced magnetic field is neglected

(iv) Applied electric field is also neglected

(v) Dissipation effects are neglected

(vi) Pressure in flow field is assumed to be a constant

Under the above assumptions and the constitutive equation for the rheological equation of state for Visco-elastic fluid are characterized by Walter’s Liquid Model B. The governing boundary layer equations of the flow field are given by

(1)

(2)

(3)

(4)

Boundary conditions are

(5)

Where ; = Maxwells reflexion coefficient.

The equation (1) gives (6)

Where is the constant suction to the velocity normal to the plate.

Introducing the following non-dimensional quantities

M = Sc = , , (7)

In view of the above non-dimensional quantities and (6) the equations (2) to (4) reduce to the form

(8)

(9)

(10)

Where .

Corresponding boundary conditions are

(11)

SOLUTION OF THE PROBLEM:

For solving equations (8), (9) and (10) subject to the boundary conditions (11), we assume the following for the velocity, temperature and concentration distribution of the flow field

U= (12)

T = (13)

(14)

Substituting (12) to (14) in equations (8 ) – (10), and equating the coefficients of harmonic and non harmonic,we get

(15)

(16)

(17)

(18)

(19)

(20)

Corresponding boundary conditions are

as y (21)

Solving the equations (15) to (20) with the corresponding boundary conditions (21), we obtain

(22)

(23)

(24)

(25)

(26)

(27)

Using the above expressions (22) to (27) in equations (12) - (14) we obtain the velocity, temperature and concentration fields as

(28)

(29)

(30)

Skin friction:

The Skin friction at the wall is given by

(31)

Heat flux

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

(32)

Sherwood number

The rate of mass transfer on the wall in terms of Sherwood number is given by

(33)

RESULTS AND DISCUSSION:

Numerical caluculations are carried out for different values of dimensionless parameters and representative set of results is reported graphically in Figures 1-19. These results are obtained to illustrate the influence of the Chemical reaction parameter , the Heat source parameter S, the Soret number So, the Magnetic field parameter M , the Schmidt number Sc on the velocity, temperature and concentration profiles, while the values of the physical parameters are fixed at real constants.

The concentration profiles for different values of Soret number So are shown in figure.1. This shows that the concentration increases with increasing values of Soret number. Figure.2. illustrates the effect of Chemical reaction parameter on concentration profiles. It is noticed that as Chemical reaction parameter increases, concentration decreases. For various values of the Schmidt number Sc, the concentration profiles are plotted in figure 3. It can be seen that as Schmidt numberincreases, the concentration decreases.

The influence of the Suction parameter Voon the velocity profiles is presented in figure.4. This shows that the concentration decreases with increasing values of Suction parameter. Figure.5. shows the effect of Heat source parameter S on temperature profiles. It is observed that temperature decreases with increasing values of Heat source parameter. The temperature profiles for different values of Prandtl number Pr are shown in figure.6. This shows that the temperature decreases as Prandtl number increases. Figure.7. illustrates the effect of Suction parameter Vo on temperature profiles. It is seen that as Suction parameter increases, the temperature decreases.

For different values of the Grashof number Gr on velocity profiles are shown figure.8. it is obvious that as Grashof number increases, the velocity increases. Figure.9. shows the velocity profiles for different values of modified Grashof number Gm. It is seen that as modified Grashof number increases, the velocity increases. The effect of Soret number So on velocity profiles are shown in figure.10. It is observed that velocity increases with increasing values of Soret number.

Figure.11. illustrates the velocity profiles for different values of Chemical reaction parameter . It is observed that as Chemical reaction parameter increases, the velocity field decreases. The velocity profiles for different values of Viscoelastic parameter K₁ are shown in figure.12. It is seen that velocity decreases with increasing values of Viscoelastic parameter. Figure.13. illustrates the velocity profiles for different values of Prandtl number Pr. It is obvious that as Prandtl number increases, the velocity decreases.

Velocity profiles for different values of Porosity parameter Kp are shown in figure.14. it is observed that as Porosity parameter increases, the velocity increases. Figure.15. illustrates the velocity profiles for different values of Magnetic parameter M. it is obvious that as Magnetic parameter increases, the velocity decreases. The effect of Heat source parameter S on velocity profiles are shown in figure.16. It is seen that velocity decreases with increasing values of Heat source parameter.

The influence of the Rarefraction parameter Ron the velocity profiles is presented in figure.17. This shows that the velocity increases with increasing values of Rarefraction parameter R. Figure.18. illustrates the velocity profiles for different values of Schmidt number Sc. it is obvious that as Schmidt number increases, the velocity decreases. The velocity profiles for different values of Suction parameter Vo are shown in figure.19. It is seen that velocity decreases with increasing values of Suction parameter.

Table.1 shows the numerical values of Skinfriction coefficient for various values of Viscoelastic parameter K_1, Porosity parameter Kp, Prandtl number Pr, Soret number So, Chemical reaction parameter , Heat source parameter S and Schmidt number Sc. From table 1 we observed that, as Viscoelastic parameter K_1, Heat source parameter S,Prandtl number Pr, Schmidt number Sc,Chemical reaction parameter increases, the skinfriction coefficient increases while an increase the Soret number So and Porosity parameter Kp, thevalue of the skinfriction coefficient decreases.

Table.2 shows the numerical values of heat transfer coefficient in terms of Nusselt number Nu for various values of Heat source parameter S, Prandtl number Pr and suction parameter Vo, It is observed that, an increase in the Heat source parameter, Prandtl number and Suction parameter, increases the value of heat transfer coefficient.

Table.3 shows numerical values of mass transfer coefficient in terms of Sherwood number Sh for various values of Soret number So, Chemical reaction parameter, Heat source parameter S, Prandtl number Pr , Suction parameter Vo, Schmidt number Sc. From table – 1, we observed that, an increase in the Chemical reaction, Suction parameter and Schmidt number, increases the value of mass transfer coefficient and an increase in the Soret number, Heat source parameter and Prandtl number, decreases the value of mass transfer coefficient.

Fig. 1: Effect of Soret number So on concentration profiles with Sc=0.5, Kr=0.5, Pr=0.71, S=0.5, Vo=0.5

Fig. 2: Effect of Chemical reaction parameter on concentration profiles with Sc=0.5, Kr=0.5, Pr=0.71, S=0.5, Vo=0.5

Fig. 3: Effect of Schmidt number Sc on concentration profiles with =0.5, Kr=0.5, Pr=0.71, S=0.5, Vo=0.5

Fig. 4: Effect of Suction parameter Vo on concentration profiles with Sc=0.5, Kr=0.5, Pr=0.71, S=0.5, =0.5

Fig. 5: Effect of Heat source parameter S on temperature profiles with Pr=0.71, Vo=0.5

Fig. 6: Effect of Prandtl number Pr on temperature profiles with S=0.5, Vo=0.5

Fig. 7: Effect of Suction parameter Vo on temperature profiles with Pr=0.71, S=0.5

Fig. 8: Effect of Grashof number Gr on velocity profiles with Sc=0.5, =0.5, S=0.5, Pr=0.71, So=0.5, Gm=4, Vo=0.5, R=0.5, =0.5, Kp=0.5, t=0.5, M=0.5, =0.5

Fig. 9: Effect of modified Grashof number Gm on velocity profiles with Sc=0.5, =0.5, S=0.5, Pr=0.71,Gr=4, So=4, Vo=0.5, R=0.5, =0.5, Kp=0.5, t=0.5, M=0.5, =0.5

Fig. 10: Effect of Soret number So on velocity profiles with Sc=0.5, =0.5, S=0.5, Pr=0.71, Gr=4, Gm=4, Vo=0.5, R=0.5, =0.5, Kp=0.5, t=0.5, M=0.5, =0.5

Fig. 11: Effect of Chemical reaction parameter on velocity profiles with Sc=0.5, Pr=0.71, S=0.5, So=0.5, Gr=4, Gm=4, Vo=0.5, R=0.5, =0.5, Kp=0.5, t=0.5, M=0.5, =0.5

Fig. 12: Effect of Viscoelastic parameter on velocity profiles with Sc=0.5, =0.5, S=0.5, Pr=0.71

Fig. 13: Effect of Prandtl number Pr on velocity profiles with Sc=0.5, =0.5, S=0.5, So=0.5, Gr=4, Gm=4, Vo=0.5, R=0.5, =0.5, Kp=0.5, t=0.5, M=0.5, =0.5

Fig. 14: Effect of Porosity parameter Kp on velocity profiles with Sc=0.5, =0.5, S=0.5, Pr=0.71, Gr=4, Gm=4, Vo=0.5, R=0.5, =0.5, Pr=0.71, t=0.5, M=0.5, =0.5

Fig. 15: Effect of Magnetic field parameter M on velocity profiles with Sc=0.5, =0.5, So=0.5, Pr=0.71

Fig. 16: Effect of Heat source parameter S on velocity profiles with Sc=0.5, =0.5, So=0.5, Pr=0.71

Fig. 17: Effect of Rarefraction parameter R on velocity profiles with Sc=0.5, =0.5, So=0.5, Pr=0.71, Gr=4, Gm=4, Vo=0.5, S=0.5, =0.5, Kp=0.5, t=0.5, M=0.5, =0.5

Fig. 18: Effect of Schmidt number Sc on velocity profiles with So=0.5, =0.5, S=0.5, Pr=0.71, Gr=4, Gm=4, Vo=0.5, R=0.5, =0.5, Kp=0.5, t=0.5, M=0.5, =0.5

Fig. 19: Effect of Suction parameter Vo on velocity profiles with Sc=0.5, =0.5, S=0.5, Pr=0.71, Gr=4, Gm=4, So=0.5, R=0.5, =0.5, Kp=0.5, t=0.5, M=0.5, =0.5

Table 1: Values of Skinfriction with Gr=2, Gm=2, R=0.1, Kp=0.1, M=0.5,, t=0.5

	S		So	Kp	Pr	Sc
0.2	0.5	0.5	0.5	0.5	0.71	0.5	1.6798
0.4	0.5	0.5	0.5	0.5	0.71	0.5	1.7116
0.6	0.5	0.5	0.5	0.5	0.71	0.5	1.7486
0.5	1	0.5	0.5	0.5	0.71	0.5	1.7284
0.5	2	0.5	0.5	0.5	0.71	0.5	1.7578
0.5	3	0.5	0.5	0.5	0.71	0.5	1.7770
0.5	0.5	0.5	0.5	0.5	0.71	0.5	1.6798
0.5	0.5	1	0.5	0.5	0.71	0.5	1.7027
0.5	0.5	1.5	0.5	0.5	0.71	0.5	1.7188
0.5	0.5	0.5	1	0.5	0.71	0.5	1.6609
0.5	0.5	0.5	2	0.5	0.71	0.5	1.6374
0.5	0.5	0.5	3	0.5	0.71	0.5	1.6138
0.5	0.5	0.5	0.5	0.1	0.71	0.5	1.6798
0.5	0.5	0.5	0.5	0.2	0.71	0.5	0.7921
0.5	0.5	0.5	0.5	0.3	0.71	0.5	0.2859
0.5	0.5	0.5	0.5	0.5	0.71	0.5	1.6798
0.5	0.5	0.5	0.5	0.5	1	0.5	1.6904
0.5	0.5	0.5	0.5	0.5	3	0.5	1.7591
0.5	0.5	0.5	0.5	0.5	0.5	0.2	1.6487
0.5	0.5	0.5	0.5	0.5	0.5	0.4	1.6709
0.5	0.5	0.5	0.5	0.5	0.5	0.6	1.6878

Table 2: Values of Nusselt number

S	Pr	Vo	Nu
1	0.71	0.5	1.1931
2	0.71	0.5	1.6028
3	0.71	0.5	1.9186
0.5	0.71	0.5	0.9065
0.5	1	0.5	1
0.5	3	0.5	1.7808
0.5	0.71	1	1. 1462
0.5	0.71	2	1.7120
0.5	0.71	3	2.3434

Table 3: Values of Sherwood number

So	Kr	S	Pr	Vo	Sc	Sh
0.2	0.5	0.5	0.71	0.5	0.5	0.5607
0.5	0.5	0.5	0.71	0.5	0.5	0.4411
0.7	0.5	0.5	0.71	0.5	0.5	0.3641
0.5	1	0.5	0.71	0.5	0.5	0.6679
0.5	2	0.5	0.71	0.5	0.5	0.9834
0.5	3	0.5	0.71	0.5	0.5	1.2220
0.5	0.5	1	0.71	0.5	0.5	0.3737
0.5	0.5	2	0.71	0.5	0.5	0.2754
0.5	0.5	3	0.71	0.5	0.5	0.1986
0.5	0.5	0.5	0.71	0.5	0.5	0.4411
0.5	0.5	0.5	1	0.5	0.5	0.4023
0.5	0.5	0.5	3	0.5	0.5	0.0922
0.5	0.5	0.5	0.71	1	0.5	0.4978
0.5	0.5	0.5	0.71	2	0.5	0.6493
0.5	0.5	0.5	0.71	3	0.5	0.8355
0.5	0.5	0.5	0.71	0.5	0.2	0.2834
0.5	0.5	0.5	0.71	0.5	0.4	0.3953
0.5	0.5	0.5	0.71	0.5	0.6	0.4832

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