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MHD Mixed Convective Heat and Mass Transfer in a Visco-Elastic Boundary Layer Slip Flow past a Vertical Permeable Plate with Thermal Radiation and Chemical Reaction

Rita Choudhury¹ and Utpal Jyoti Das^2*

¹Department of Mathematics, Gauhati University, Guwahati-781014, Assam, INDIA.

^*Corresponding Address:

 [email protected]

Research Article

Abstract: An analytical study for the problem of unsteady mixed convection with thermal radiation and first-order chemical reaction on magneto hydrodynamic boundary layer flow of an electrically conducting visco-elastic fluid past a vertical permeable plate has been investigated. Slip boundary layer condition is applied at the porous interface. The classical model is used for studying the effect of radiation for optically thin media. The perturbation scheme has been used to solve the problem. Analytical expressions for dimensionless velocity, temperature, concentration fields, skin friction co-efficient, rate of heat transfer and Sherwood number have been obtained. The profiles of the velocity and skin friction have been presented graphically for different values of the visco-elastic parameters with the combination of the other flow parameters encountered in the problem under investigation.

2000 Mathematics Subject Classification: 76A05, 76A10.

Keywords: MHD, visco-elastic, thermal radiation, heat transfer, porous medium, boundary layer flow, slip flow, first order chemical reaction.

Mixed convection flows with simultaneous heat and mass transfer in porous media under the influence of a magnetic field and chemical reaction are frequently encountered in many transport processes in nature. Its application is found in many industries viz. in the chemical industry, power and cooling industry for drying, chemical vapour deposition on surfaces, cooling of nuclear reactors and magnetohyrodynamic power generators. Simultaneous heat transfer and evaporation of crude oil in different stages of refining process are physical examples of heat and mass transfer. Many transport processes exist in nature and in industrial applications in which the simultaneous heat and mass transfer occurs as a result of combined buoyancy effects of diffusion of chemical species. A comprehensive description of the theoretical work for both laminar and turbulent mixed convection boundary layer flows has been given in a review paper by Chen and Armaly [1] and in the book by Pop and Ingham [2]. The problem of mixed convection under the influence of magnetic field has attracted numerous researchers viz. Soundalgekar et al. [3], Elbasheshy [4], Abel et al. [5, 6] in view of its applications in geophysics and astrophysics. In the above mentioned

studies the radiation effect is ignored. But the effects of thermal radiation heat transfer cannot be neglected when technological processes take place at high temperature (Siegel and Howell [7], Modest [8]). Recent developments in hypersonic flights, missile re-entry rocket combustion chambers, gas cooled nuclear reactors and power plants for inter planetary flight have focussed the attention of many researchers  [9]-[17].

Convection in porous media has gained significant attention in recent years because of its importance in engineering applications. Reviews of the applications related to convective flows in porous media can be found in the book by Nield and Bejan [18]. The fundamental problem of flow through and past porous media has been discussed by Cheng [19] and Rudraiah [20] on thermal radiation as a mode of energy transfer and emphasize the need for inclusion of radiative transfer in these processes. The inadequacy of the no-slip condition is quite evident in polymer melts which often exhibit microscopic wall slip. The boundary conditions to be satisfied at the interface between a porous medium and fluid layer are the matching of velocity and stresses. Several authors [21]-[26] have studied in this line.

The combined effect of heat and mass transfer with chemical reaction in porous medium has important engineering applications e.g. tubular reactors, oxidation of solid materials and synthesis of ceramic materials. Chemical reaction can be codified as either a heterogeneous or homogeneous process. This depends on whether it occurs at an interface or a single-phase volume reaction. In most chemical reactions, the reaction rate depends on the concentration of the species itself. A chemical reaction is said to be of first order, if the rate of reaction is directly proportional to concentration itself. During chemical reaction between two species concentration heat is also generated. Chemical reaction effects on heat and mass transfer laminar boundary layer flow have been discussed by various authors [27, 28, 29, 30, 31] in various situations. For the problem of coupled heat and mass transfer in MHD, the effect of Ohmic heating are not studied in the above investigations. However, it is more realistic to include Ohmic effect in order to explore the impact of the magnetic field on the thermal transport in the boundary layer. The effect of Ohmic heating on the MHD free convective heat transfer has been examined by Hossain [32]. Chaudhary et al. [33] have analyzed the effect of radiation on heat transfer in MHD mixed convection flow with simultaneous thermal and mass diffusion from an infinite vertical plate with viscous dissipation. Pal and Mondal [34] analyzed the effect of variable viscosity on MHD non-Darcy mixed convective heat transfer over a stretching sheet embedded in a porous medium with non-uniform heat source/sink and Ohmic dissipation. Recently, Pal and Talukdar [35] studied the combined effect of MHD and Ohmic heating in unsteady two-dimensional boundary layer slip flow, heat and mass transfer of a viscous incompressible fluid past a vertical permeable plate with the diffusion of species in the presence of thermal radiation incorporating the first-order chemical reaction. They used the classical model introduced by Cogley et al. [36] for the radiation effect. In the present study, an attempt has been made to extend the problem studied by Pal and Talukdar to the case of visco-elastic fluid characterised by second-order fluid.

The constitutive equation for the second-order fluid is of the form

   (1)

where  is the stress tensor, is hydrostatic pressure, is unit tensor, are the kinematic Rivlin –Ericksen tensors, are the material co-efficients describing the viscosity, visco-elasticity and cross-viscosity respectively. The material coefficients are taken constants withandas positive and as negative (Coleman and Markovitz [37]). The equation (1) was derived by Coleman and Noll [38] from that of simple fluids by assuming that the stress is more sensitive to the recent deformation than to the deformation that occurred in the distant past.

We consider unsteady two-dimensional visco-elastic flow of an incompressible, electrically conducting and heat-absorbing fluid past a semi-infinite vertical permeable plate embedded in a uniform porous medium which is subject to slip boundary condition at the interface of porous and fluid layers. A uniform transverse magnetic field of magnitude  is applied in the presence of radiation and concentration buoyancy effects in the direction of - axis. The transversely applied magnetic field and magnetic Reynolds number are assumed to be very small so that the induced magnetic field and the Hall effect are negligible. It is assumed that there is no applied voltage which implies the absence of an electric field. Since the motion is two dimensional and length of the plate is large enough so all the physical variables are independent of. The wall is maintained at constant temperature  and concentration  higher than the ambient temperature and concentration respectively. Also, it is assumed that there exists a homogeneous first-order chemical reaction with rate constant R between the diffusing species and the fluid. It is assumed that the porous medium is homogeneous and present everywhere in local thermodynamic equilibrium. Rest of properties of the fluid and the porous medium are assumed to be constant. The governing equations for this investigation are based on the balances of mass, linear momentum, energy and concentration species. Taking into consideration these assumptions, the equations that describe the physical situation can be written as follows:

Continuity Equation:

   (2)

Momentum Equation:

       (3)

Energy Equation:

       (4)

Mass Diffusion Equation:

    (5)

where  are the dimensional distances along and perpendicular to the plate, respectively. are the components of dimensional velocities along and directions, respectively. is the gravitational acceleration, is the dimensional temperature of the fluid near the plate, is the free stream dimensional temperature, is the dimensional concentration, is the free stream dimensional concentration.  and are the thermal and concentration expansion coefficients, respectively. is the pressure, is the specific heat of constant pressure, is the magnetic field coefficient, is the radiative heat flux, is the density, is the thermal conductivity, is the magnetic permeability of the fluid, is the molecular diffusivity, is the dimensional heat absorption coefficient,  is the coefficient of proportionality of the absorption of the radiation,  is the chemical reaction parameter and The fifth and the sixth terms on R.H.S. of the momentum equation (3) denote the thermal and the concentration buoyancy effects, respectively. The second and the third term on the R.H.S. of equation (4) denote the inclusion of the effect of the thermal radiation and heat absorption effects, respectively.

The radiative heat flux is given by (Cogley et al. [36]) as

    (6)

where  is the absorption coefficient at the wall and is Planck’s function.

Under these assumptions, the appropriate boundary conditions for velocity involving slip flow, temperature and concentration fields are defined as 

 (7)

 (8)

   (9)

where  and  are the wall dimensional concentration and temperature, respectively. is the permeability of the porous medium, is the frequency and is the porous parameter. From equation (3), it is clear that the suction velocity at the plate surface is a function of time only. Hence the suction velocity normal to the plate is assumed in the form

    (10)

where  is a real positive constant, and are small less than unity, and is a scale of suction velocity which is non-zero positive constant.

Outside the boundary layer, equation (3) gives

   (11)

In order to write the governing equations and boundary conditions in dimensionless form, the following non-dimensional quantities are introduced

    (12)

In view of the above non-dimensional variables, the basic field of equations (3)-(5) can be expressed in non-dimensional form as

  (13)

  (14)

    (15)

where  is the thermal Grashof number, is the solutal Grashof number, is the Prandtl number, is the magnetic field parameter, is the radiation parameter,  the Schmidt number, is the heat source parameter, is the visco-elastic parameter and is the chemical reaction parameter and is the absorption of radiation parameter which are defined as follows:

   (16)

The corresponding boundary conditions are

 (17)  

  (18)

where  is the porous permeability parameter.

The set of partial differential equations (13)-(15) cannot be solved in closed-form. However, it can be solved analytically after these equations are reduced to a set of ordinary differential equations in dimensionless form which can be done by representing the velocity temperature  and concentration as

 (19)

 (20)

  (21)

Substituting (19)-(21) into equations (13)-(15) and equating the harmonic and non-harrnonic terms, and neglecting the higher order of we obtain                    

 (22)

  (23)                        

 (24)

 (25)

 (26)                          

  (27)

where the prime denotes ordinary differentiation with respect to  The corresponding boundary conditions are

 (28)   

 (29)

The solutions of the equations (24)-(27) consistent with the boundary conditions (28) and (29) are obtained but not presented here for the sake of brevity.

For small shear rate we note that.  Substituting

   (30)

into equations (22), (23) and boundary conditions (28), (29) up to the first order of and comparing the coefficients of like powers of we obtain

  (31)

   (32)

     (33)

   (34)

The corresponding boundary conditions are

 (35)    

 (36)

Solutions of the equations (31)-(34) consistent with the boundary conditions (35) and (36) are obtained but not presented here for the sake of brevity.

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

  (37)

The rate of heat transfer coefficient, which in non-dimensional form, in terms of Nusselt number, is given by

Sherwood number is given by

where  is the local Reynolds number.

In the present investigation, we have analyzed the heat and mass transfer on mixed convection flow of a visco-elastic incompressible, electrically conducting fluid over an infinite vertical porous plate in the presence of magnetic field and thermal radiation using the classical model for the radiating heat flux. We have obtained the results for varieties of physical parameters, which are illustrated by means of graphs. The purpose of this study is to bring out the effects of the visco-elastic parameter  on the governing flow with the combination of the other flow parameters. All the corresponding results for Newtonian fluid are obtained by setting

Figures 1-10 depict the velocity profile in the boundary layer against to observe the visco-elastic effects for various sets of values (Table.1) of the   thermal  Grashof  number   solutal Grashof number  Schmidt number absorption radiation parameter chemical reaction parameter radiation parameter heat source parameter magnetic field parameter Porous permeability parameter  with fixed values of  and It is evident from the figures that the velocity in the boundary layer increases with the increasing values of the visco-elastic parameter in comparison with the Newtonian cases when               (i) increases (Figs. 1 and 2), (ii)  increases (Figs. 2 and 3), (iii) increases (Figs. 3 and 4), (iv) increases (Figs. 4 and 5), (v)  increases (Figs. 5 and 6), (vi) increases (Figs. 6 and 7), (vii)  increases (Figs. 7 and 8), (viii) increases (Figs 8 and 9), (ix)  increases (Figs. 9 and 10) ; by keeping other flow parameters fixed.

Also, we note from the figures that the velocity in the boundary layer increases for Newtonian as well as non-Newtonian cases when (Figs. 1 and 2), (Figs 2 and 3), (Figs. 9 and 10) increase whereas reverse trend is seen for (Figs. 3 and 4), and (Figs. 7 and 8).

Table1: Description of various cases:

Fig. 1: Variation of u against  for case I.

Fig. 2: Variation of u against  for case II.

Fig. 3: Variation of u against for case III.

Fig. 4: Variation of u against  for case IV.

Fig. 5: Variation of u against  for case V.

Fig. 6: Variation of u against  for case VI.

Fig. 7: Variation of u against  for case VII.

Fig. 8: Variation of u against  for case VIII.

Fig. 9: Variation of u against  for case IX.

Fig. 10: Variation of u against  for case X.

Fig. 11: Variation of against t for case I.

Fig. 12: Variation of against t for case II.

Fig. 13: Variation of against t for case III.

Fig. 14: Variation of against t for case IV.

Fig. 15: Variation of against t for case V.

Fig. 16: Variation of  against t for case VI.

Fig. 17: Variation of against t for case VII.

Fig. 18: Variation of against t for case VIII.

Fig. 19: Variation of  against t for case IX.

Fig. 20: Variation of against t for case X.

Figures 11-20 represent the non-dimensional shearing stress against  to observe the visco-elastic effects for various sets of values given in table 1. Figures 11-20 reveal that the skin friction coefficient decrease with the increasing values of the visco-elastic parameter in compared to Newtonian fluid. It is also noted that the behaviours of the profiles of skin friction coefficient remain the same when with the increasing values of the visco-elastic parameter  (i) increases (Figs. 11 and 12), (ii)   increases (Figs. 12 and 13), (iii) increases (Figs. 13 and 14), (iv) increases (Figs. 14 and 15), (v)  increases (Figs. 15 and 16), (vi) increases (Figs. 16 and 17 ), (vii) increases (Figs. 17 and 18), (viii) increases (Figs. 18 and 19), and (ix) increases (Figs. 19 and 20) keeping other flow parameters constant.

It is seen from the figures 11-20 that the skin friction coefficient increase with increasing time  for Newtonian as well as non-Newtonian fluid in all the cases of table 1. Further it is found that skin friction coefficient   increase for both Newtonian and non-Newtonian cases with the increasing values of (Fig. 11 and 12),  (Figs. 12 and 13),  (Figs.13 and 14), (Figs. 14 and 15), (Figs. 18 and 19), whereas reverse trend is seen for  (Figs. 17 and 18), (Figs. 19 and 20).

It has also been observed that the temperature field, concentration field, Nusselt number and Sherwood number are not significantly affected by the visco-elastic parameter.

4. Conclusion

The above study brings out the flowing results of physical interest:

1. The velocity in the boundary layer increase with the increasing values of

the visco-elastic parameter  in comparison with the Newtonian cases. Also, velocity increase as any one values of the Grashof number for heat transfer, Solutal Grashof number, and porous permeability parameter () increases but it decrease as any of the values of the Schmidt number and heat source parameter increases.

1.  Skin friction increase with the increasing time for both Newtonian and non-Newtonian fluid. Further, it is found that skin friction co-efficient increase for both Newtonian and non-Newtonian cases with the increasing values of Grashof number, Schimdt number, absorption radiation parameter, or Hartmann number whereas skin friction decrease for heat source parameter or porous permeability parameter increasing.

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