Radiation and Soret Effects to MHD Flow in Vertical Surface with Chemical reaction and Heat generation through a Porous Medium

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1 ISSN (e): 5 35 Vol, 4 Issue, 7 Jul 4 International Journal of Computational Engineering Research (IJCER) Radiation and Soret Effects to MHD Flow in Vertical Surface with Chemical reaction and Heat generation through a Porous Medium Srinathuni Lavana, D Chenna Kesavaiah Department of H&S, Naraana Engineering & Technical Campus, Jafarguda, Haatnagar, R.R.Dist, T.S, India. Department of H & BS, Visvesvaraa College of Engg & Tech, MP Patelguda, Ibrahimpatnam, R.R.Dist, T.S, India ABSTRACT The present paper is the effects of radiation, chemical reactions and thermo Diffusion (Soret effect) to MHD flow of an electricall conducting, incompressible, viscous fluid past an impulsivel moving isothermal vertical plate through porous medium in the presence of uniform suction. Thermo diffusion is one of the mechanisms in transport phenomena in which molecules are transported in a multi component mixture driven b temperature gradients. A flow of this tpe represents a new class of boundar laer flow at a surface of finite length. The equations governing the flow field are solved b analtical method. The velocit, temperature, concentration and skin friction have been evaluated for variation in the different governing parameters. KEYWORDS: Chemical reaction, Heat generation/absorption, Radiation, Soret number, MHD, Porous medium, Radiation I. INTRODCTION Thermo diffusion along with molecular diffusion occurs in man engineering sstems and in nature. Thermo diffusion has a great effect on the concentration distribution in a multi component mixture. The variations of composition and temperature ma either lessen or enhance the separation in mixtures. The thermo diffusion phenomenon also plas a major role in the hdrodnamic instabilit analsis in mixtures, investigations of mineral migration, the mass transport modeling in living matters, and composition variation studies in hdrocarbon reservoirs. Diffusion is one of the major mechanisms of transport phenomena. Molecular diffusion is the movement of molecules from a higher to lower chemical potential. Thermo diffusion and pressure diffusion are additional was in which molecules are transported in a multi component mixture driven b temperature and pressure gradients, respectivel. The thermo diffusion phenomenon was discovered b Ludwig [] and Soret [], and named as the Soret effect. The Soret coefficient is the ratio of the thermo diffusion coefficient to the molecular diffusion coefficient. Furthermore, the stud of thermo haline instabilit with thermo diffusion in a fluid saturated porous medium is of importance in geophsics, ground water hdrolog, soil science, oil extraction. The reason is that the earth s crust is a porous medium saturated b a mixture of different tpes of fluids such as oil, water, gases and molten form of ores dissolved in fluids. Thermal gradients present between the interior and exterior of the earth s crust ma help convection to set in. The thermal gradient in crude oil can have a strange effect on the distribution of petroleum components in an oil deposit. The thermal gradient causes the Soret effect which makes the larger molecule components to have a tendenc to rise, while smaller molecule components go down to the bottom of the well. However, gravit causes the heav components in a fluid to fall and the lighter ones to rise. This means that the distribution of components in a given well is neither consistent nor readil predictable. Pal and Talukdar [3] analzed the combined effect of mixed convection with thermal radiation and chemical reaction on MHD flow of viscous and electricall conducting fluid past a vertical permeable surface embedded in a porous medium is analzed. Mukhopadha [4] performed an analsis to investigate the effects of thermal radiation on unstead mixed convection flow and heat transfer over a porous stretching surface in porous medium. Haat et al. [5] analzed a mathematical model in order to stud the heat and mass transfer characteristics in mixed convection boundar laer flow about a linearl stretching vertical surface in a porous Open Access Journal Page 6

2 Radiation and Soret Effects to MHD Flow in Vertical Surface medium filled with a viscoelastic fluid, b taking into account the diffusion thermo (Dufour) and thermaldiffusion (Soret) effects. Bhavana et.al [6]. The Soret effect on free convective unstead MHD flow over a vertical plate with heat source. Chenna Kesavaiah et.al [7] Radiation and Thermo - Diffusion effects on mixed convective heat and mass transfer Flow of a viscous dissipated fluid over a vertical surface in the presence of chemical reaction with heat source. MHD flows has received the attention of several researchers due to its diverse applications in man branches of sciences and technolog such as in planetar fluid dnamics, power generation sstems, nuclear reactor thermal dnamics and electromagnetic materials processing. On the other hand, MHD flows with combined heat and mass transfer has numerous applications in chemical, mechanical and biological sciences. Some important application are stellar and solar structures cooling of nuclear reactors, interstellar matter, liquid metals fluid, power generation sstem, radio propagation and aerodnamics. The stud of heat generation or absorption in moving fluid is important in problems dealing with dissociating fluids. Possible heat generation effect ma alter the temperature distribution; consequentl, the particle deposition rate in nuclear reactors, electronic chips and semi conductor wafers. Rahman et al. [8] studied thermo-micropolar fluid flow along a vertical permeable plate with uniform surface heat flux in the presence of heat generation. Ibrahim et al. [9] found the analtic solution of MHD mixed convection heat and mass transfer over an isothermal, inclined permeable stretching plate immersed in a uniform porous medium in the presence of chemical reaction, internal heating, Dufour effect and Hall effects. Pal and Chatterjee [] studied heat and mass transfer in MHD non-darcian flow of a micropolar fluid over a stretching sheet embedded in a porous media with non-uniform heat source and thermal radiation. Cortell [] investigated theoreticall the effect of viscous dissipation as well as radiation on the thermal boundar laer flows over non-linearl stretching sheets considering prescribed surface temperature and heat flux. Raptis and Perdikis [] studied numericall the stead two dimensional flow of an incompressible viscous and electricall conducting fluid over a non-linear semi-infinite stretching in the Presence of a chemical reaction and under the influence of magnetic field. Rajagopal et al. [3] studied a Falkner-Skan flow field of a second-grade viscoelastic fluid. Shit and Haldar [4] investigated the effects of thermal radiation and Hall current on MHD free-convective flow and mass transfer over a stretching sheet with variable viscosit in the Presence of heat generation/absorption. Therefore the objective of the present paper is to stud the MHD heat transfer flow of a Newtonian fluid moving vertical surface with heat generation or absorption b considering the radiation effect along with Soret effect, uniform heat and mass flux and chemical reaction. II. FORMLATION OF THE PROBLEM We consider the stead, two-dimensional laminar, incompressible flow of a chemicall reacting, viscous fluid on a continuousl moving vertical surface in the presence of a uniform magnetic field, radiation absorption with heat generation, uniform heat and mass flux effects issuing a slot and moving with uniform velocit in a fluid at rest. Let the x-axis be taken along the direction of motion of the surface in the upward direction and -axis is normal to the surface. The temperature and concentration levels near the surface are raised uniforml. The induced magnetic field, viscous dissipation is assumed to be neglected. Now, under the usual Boussinesq s approximation, the flow field is governed b the following equations. Flow configuration and coordinate sstem Open Access Journal Page 63

3 Radiation and Soret Effects to MHD Flow in Vertical Surface Continuit equation u x v () Momentum equation u u u B u v g T T g C C u u x K p () Energ equation T T T q r C p u v k Q T T x C C C T u v D K r M C C D T x (3) (4) The relevant boundar conditions are u u, v v c o n st w T q, C j a t k D (5) u, T T, C C a s The radiative heat flux q is given b equation (5) in the spirit of Cogl et.al [5] r q r 4 ( T T ) I where e b I K d, w T K w is the absorption coefficient at the wall and e b is Planck s function, I is absorption coefficient In order to write the governing equations and the boundar conditions the following non dimensional quantities are introduced. u v T T C C K v p B Q u, Y, T, C, k, M, Q u v C v w q j k v k v p q j g g C k v k v p Kr Q j l P r, K r, Q, G r, S c, G c l k v q v C u v D u v s p p p w M w D K j 4 q I qd M T Du, R, S T c c q C k v j (6) Where u, v are the velocit along the x, -axis, is constant obtained after integration conservation of mass in pre-non dimensional form not mentioned, is the kinematic viscosit, g is the acceleration due to gravit, T is the temperature of the fluid, is the coefficient of volume expansion, pressure, is a constant, is the Stefan-Boltzmann constant, C is the specific heat at constant p T exceeds the free steam temperature T w, C is Open Access Journal Page 64

4 the species concentration, is the wall temperature, Radiation and Soret Effects to MHD Flow in Vertical Surface C is the concentration at the plate, T w is the free steam temperature far awa from the plate, C is the free steam concentration in fluid far awa from the plate, Pr is the Prandtl number, Gr is the Grashoff number, k is the thermal conductivit of the fluid, B is uniform magnetic field strength, M is the magnetic field parameter which is the ratio of magnetic force to the inertial force. It is a measure of the effect of flow on the magnetic field. The term Q T T is assumed to be the amount of heat generated or absorbed per unit volume. Q is a constant, which ma take on either positive or negative values. III. SOLTION OF THE PROBLEM In view of (6) the equations (), (3) and (4) are reduced to the following non-dimensional form d d M G r T G c C d Y d Y K (7) d T d Y d T P r Q R P r T (8) d Y d C d C d T S c K rs cc S cs (9) o d Y d Y d Y The corresponding boundar conditions can be written as, T, C a t Y Y Y, T, C a s Y () Where Gr is the thermal Grashof number, Gc is the solutal Grashof number, Pr is the fluid Prandtl number, Sc is the Schmidt number Kr is the chemical reaction, Q is Heat source parameter, M is the Magnetic parameter, S is the Soret number and R is the Radiation Parameter. The stud of ordinar differential equations (7), (8) and (9) along with their initial and boundar conditions () have been solved b using the method of ordinar linear differential equations with constant coefficients. We get the following analtical solutions for the velocit, temperature and concentration. m m m J e J e J e J e T B e m m m 4 C Z e Z e m6 The computed solution for the velocit is valid at some distance from the slot, even though suction is applied from the slot onward. This is due to the assumption that velocit field is independent of the distance parallel to the plate. The fluids considered in this stud are air (Pr =.7) and water (Pr = 7.). Skin friction J m J m J m J m Nusselt number T N u B m Open Access Journal Page 65

5 Radiation and Soret Effects to MHD Flow in Vertical Surface Sherwood number C S h Z m Z m 4 APPENDIX m S c S c 4 K rs c m 4 P r P r 4 Q R P r m 4 M K Q R Pr B B m S cs o, Z m m S cm K rs c Z Z m m 4 J G rb m m M J J J J 4 3 K J G c Z m m M K J 3 G c Z m m M 4 4 K IV. RESLTS AND DISCSSION The variation of non-dimensional velocit for different values of Grashof number Gr, Solutal Grashof number Gc, Porosit permeabilit K, Chemical reaction parameter Kr, Magnetic parameter M, Prandtl number Pr, Heat source parameter Q, Radiation parameter R Soret number S and Schmidt number Sc are shown in figures () - (). Figure () shows the influence of thermal buoanc force parameter Gr on the velocit. As can be seen from this figure, the velocit profile increases with increases in the values of the thermal buoanc. We actuall observe that the velocit overshoot in the boundar laer region. Buoanc force acts like a favourable pressure gradient which accelerates the fluid within the boundar laer therefore the solutal buoanc force parameter Gc has the same effect on the velocit shown in figure () asg r. Figure (3) shows that the axial velocit increases with a rise in permeabilit parameter K. Moreover, for small permeabilit, the axial velocit reduces faster than when the permeabilit is higher. Figure (4) depict effect of chemical reaction on the flow variables is shown. In figure (4) velocit is clearl boosted with stronger chemical reaction i.e. as the chemical reaction parameter Kr increases profiles are lifted continuousl throughout the boundar laer, transverse to the plate. A distinct velocit escalation occurs near the wall after which profiles deca smoothl to the stationar value in the free stream. Chemical reaction therefore boosts momentum transfer i.e. accelerates the flow. Figure (5) reveals that axial velocit decreases as Magnetic parameter M increases. This ma be attributed to the fact that an increase in M signifies an enhancement of Lorentz force, thereb reducing the magnitude of the velocit. This figure further indicates that blood velocit in the capillar decreases with increase in distance from the lower wall of the capillar. Figure (6) illustrates the effect of Prandtl number Pr on the velocit. It is noticed that as the Prandtl number Pr increases, the velocit decreases. As seen in the earlier cases, far awa from the plate, the effect is much significant. Figure (7) illustrate the variation of velocit with the effects of heat absorption parameter Q. The velocit values are clearl reduced with increasing Q ; again an overshoot is computed close to the plate both in the presence and absence of heat source. Heat source Open Access Journal Page 66

6 Radiation and Soret Effects to MHD Flow in Vertical Surface however suppresses the overshoot. Figure (8) show the effect of radiation parameter R on the velocit it is observed that the velocit decrease as the radiation parameter R increases. This result qualitativel agrees with expectations. Figure (9) shows the variation of the velocit boundar-laer with the Soret number Sr. It is found that the velocit boundar laer thickness increases with an increase in the Soret number. Figure () illustrates the effect of the Schmidt number (Sc) on the velocit. The Schmidt number Sc embodies the ratio of the momentum diffusivit to the mass (species) diffusivit. It phsicall relates the relative thickness of the hdrodnamic boundar laer and mass-transfer (concentration) boundar laer. It is noticed that as Schmidt number Sc increases the velocit field decreases. Figures () (3) give some characteristic temperature profiles for different values of Prandtl number Pr, heat source parameter Q and radiation parameter R. Figure () presents the change in the temperature distribution in the boundar laer, when the Prandtl number Pr changes graduall. It shows that as the Prandtl number increases, the temperature of the boundar laer diminishes. This ma be attributed to the fact that the thermal boundar laer thickness reduces with an increase in Prandtl number. Further, this figure indicates that the temperature gradient at the surface increases with a rise in Prandtl number. This implies that an increase in Prandtl number is accompanied b an enhancement of the heat transfer rate at the wall of the blood vessel. The underling phsics behind this can be described as follows. When blood attains a higher Prandtl number, its thermal conductivit is lowered down and so its heat conduction capacit diminishes. Thereb the thermal boundar laer thickness gets reduced. As a consequence, the heat transfer rate at the vessel wall is increased. Figure () illustrates the effect of the heat source parameter Q on the temperature. It is noticed that as the heat source parameter increases, the temperature decreases. Figure (3) illuminates a ver important effect of thermal radiation on the temperature profile. This figure emphasizes that as thermal radiation increases during blood flow in capillaries; there is a significant rise in the thickness of boundar laer. Thereb the temperature of the boundar laer in enhanced b an appreciable extent. The effects of order of chemical reaction on concentration distributions are displaed in figure (4) respectivel. It is observed from these figures that an increase in the chemical reaction order n leads to decrease the concentration profiles. Figure (5) illustrate the effect of the Prandtl number on the concentration profiles respectivel. It is evident that increases in the Prandtl number leads to increases on the concentration profiles. From figure (6) Soret number S increases with increasing values on the concentration profiles. Figure (7) illustrate the concentration field distributions with transverse coordinate for different Schmidt number Sc. An increase in Sc causes a considerable figure (7). A much greater reduction is observed in concentration values in figure (7). An increase in Sc will suppress concentration in the boundar laer regime. Higher Sc will impl a decrease of molecular diffusivit D causing a reduction in concentration boundar laer thickness. Lower Sc will result in higher concentrations i.e. greater molecular (species) diffusivit causing an increase in concentration boundar laer thickness. For the highest value of Sc, the concentration boundar laer thicknesses are of the same value approximatel i.e. both species and momentum will diffuse at the same rate in the boundar laer. The effects of Soret number S versus Grashof number Gr on the local skin-friction coefficient shown in figure (8). From these figure we observe that the local skin-friction coefficient increase as Soret number S Open Access Journal Page 67

7 Radiation and Soret Effects to MHD Flow in Vertical Surface CONCLSIONS The problem of stead laminar, hdromagnetic two-dimensional mixed convection flow due to stretching sheet in the presence of Soret effects was investigated. The resulting partial differential equations, describing the problem, are transformed into ordinar differential equations b using similarit transformations. These equations are more convenientl solved analticall b perturbation technique for the computation of flow, heat and mass transfer characteristics, skin-friction coefficient, for various values of the magnetic field parameter, heat source, Soret numbers, etc. Comparisons of the present results with previousl published work on some limiting cases were performed and results were found to be in excellent agreement. From the present investigation following conclusions are drawn: Increasing the magnetic field parameter decreases the flow velocit near the stretching sheet in the momentum boundar laer, whereas its effect is reversed as the space variable of porous permeabilit parameter. Temperature deceases where as increases in radiation parameter. Skin friction increases with increase in the value of Soret number. Temperature decreases with increase in non-uniform heat source parameter Increase in the value of Schmidt number the concentration profiles decreases. It is hoped that the present investigation is useful in the stud of MHD flow over a vertical plate embedded in non-darcian porous medium which can be utilized as the basis for man scientific and industrial applications and studing more complex problems involving Soret effects. REFERENCES [] Ludwing (856): The thermo diffusion phenomenon was discovered, Akad. Wiss. Wien Math.-naturw. Kl, : 539, pp [] Soret (879): and named as the Soret effect. The Soret coefficient is the ratio of the thermo diffusion coefficient to the molecular diffusion coefficient, Arch. Geneve, 3:48. [3] Pal D and Talukdar B (). Buoanc and chemical reaction effects on MHD mixed convection heat and mass transfer in a porous medium with thermal radiation and ohmic heating. Communications in Nonlinear Science and Numerical Simulation, 5(), pp [4] Mukhopadha S (9): Effect of thermal radiation on unstead mixed convection flow and heat transfer over a porous stretching surface in porous medium. International Journal of Heat and Mass Transfer, 5(3), pp [5] Haat T, Mustafa M and Pop I (): Heat and mass transfer for Soret and Dufour's effect on mixed convection boundar laer flow over a stretching vertical surface in a porous medium filled with a viscoelastic fluid. Communications in Nonlinear Science and Numerical Simulation, 5(5), pp [6] M Bhavana, D Chenna Kesavaiah, A Sudhakaraiah (3): The Soret effect on free convective unstead MHD flow over a vertical plate with heat source, International Journal of Innovative Research in Science, Engineering and Technolog Vol. (5), pp [7] D Chenna Kesavaiah, P V Satanaraana, S Venkataramana (3): Radiation and Thermo - Diffusion effects on mixed Convective heat and mass Transfer flow of a viscous dissipated fluid over a vertical surface in the presence of chemical reaction with heat source, International Journal of Scientific Engineering and Technolog., Vo.(), pp : [8] Rahman M M, Eltaeb I A and Rahman S M (9): Thermo micropolar fluid flow along a vertical permeable plate with uniform surface heat flux in the presence of heat generation, Thermal Science, 3(), pp [9] Ibrahim A A (9): Analtic solution of heat and mass transfer over a permeable stretching plate affected b chemical reaction, internal heating, Dufour Soret effect and hall effect, Thermal Science, 3 (), pp [] Pal D and Chatterjee S (): Heat and mass transfer in MHD non-darcian flow of a micropolar fluid over a stretching sheet embedded in a porous media with non-uniform heat source and thermal radiation, Commun., Nonlinear Sci., Numer., Simulat., 5, pp [] Cortell (7): Viscous flow and heat transfer over a non-linear stretching sheet. Applied Mathematics Compute, Vol.84 (), pp [] Raptis, A., and Perdikis, C. (6): Viscous Flow over a non- linear stretching sheet in the Presence of a chemical reaction and magnetic field. Int. J. Non-Linear Machiner. Vol. 63, pp [3] Rajagopal K R (983): On Stokes problem for a non-newtonian fluid. Acta Mechanica, Vol.48, pp [4] Shit G C and Haldar R (9): Effect of thermal radiation on MHD viscoelastic fluid flow over a stretching sheet with variable viscosit. Communicated for publication in Int. J. Fluid Mech. [5] Cogl A C, Vincentr W C and Gilles S E (968): A Differential approximation for radiative transfer in a non-gra gas near equilibrium, AIAA Journal, 6 Open Access Journal Page 68

8 Radiation and Soret Effects to MHD Flow in Vertical Surface 3.5 Gc=., M=., K=., R=. Kr=.,Pr=.7,Q=.,Sc=.65,S =..5 Gr= 5.,.,5., Figure (): Velocit profiles for different values of Gr Gr=., M=., K=., R=. Kr=.,Pr=.7,Q=.,Sc=.65,S = Gc= 5.,.,5., Figure (): Velocit profiles for different values of Gc.5 Gr=., Gc=., M=., R=. Kr=.,Pr=.7,Q=.,Sc=.65,S o =..5 K=.,.,3., Figure (3): Velocit profiles for different values of K.5 Gr=.,Gc=.,M=.,K=.,R=. Pr=.7, Q=., Sc=.65, S o =..5 Kr=.,.,3., Figure (4): Velocit profiles for different values of Kr Open Access Journal Page 69

9 Radiation and Soret Effects to MHD Flow in Vertical Surface 3.5 Gr=., Gc=., K=., R=. Kr=.,Pr=.7,Q=.,Sc=.65,S o =..5 M=.,.,3., Figure (5): Velocit profiles for different values of M.5 Gr=., Gc=., M=., K=. R=.,Kr=.,Q=.,Sc=.65,S o =..5.5 Pr=.,.,3., Figure (6): Velocit profiles for different values of Pr.5 Gr=., Gc=., M=., K=. R=.,Kr=.,Pr=.7,Sc=.65,S o =..5.5 Q=.,.,3., Figure (7): Velocit profiles for different values of Q.5 Gr=., Gc=., M=., K=. Kr=.,Pr=.7,Q=.,Sc=.65,S o =..5 R=.,.,3., Figure (8): Velocit profiles for different values of R Open Access Journal Page 7

10 T T Radiation and Soret Effects to MHD Flow in Vertical Surface 5 4 Gr=., Gc=., M=., K=. R=.,Kr=.,Pr=.7,Q=.,Sc=.65 3 S =.,.,3., Figure (9): Velocit profiles for different values of S.5 Gr=., Gc=., M=., K=. R=.,Kr=.,Pr=.7,Q=.,S o =..5 Sc=.6,.65,.7, Figure (): Velocit profiles for different values of Sc.5.4 R=.,Q=..3. Pr=.,.,3., Figure (): Temperature profiles for different values of Pr Pr=.7,R=..3. Q=.,.,3., Figure (): Temperature profiles for different values of Q Open Access Journal Page 7

11 C C C T Radiation and Soret Effects to MHD Flow in Vertical Surface Pr=.7,Q= R=.,.,3., Figure (3): Temperature profiles for different values of R.4. Sc=.65,Pr=.7,Q=.,S o =.,R= Kr=.,.,3., Figure (4): Concemtration profiles for different values of Kr.9 Sc=.6,Q=5.,S o =.,Kr=.,R=..8.7 Pr=.,.,.3, Figure (5): Concemtration profiles for different values of Pr.5 Sc=.65,Pr=.7,Q=.,Kr=.,R=..5 S =.,.,3., Figure (6): Concemtration profiles for different values of S Open Access Journal Page 7

12 C Radiation and Soret Effects to MHD Flow in Vertical Surface.4. Pr=.7,Q=.,S o =.,Kr=.,R=..8.6 Sc=.6,.65,.7, Figure (7): Concemtration profiles for different values of S 4 S =.,.,3., Gc=., M=., K=.,R=. Kr=.,Pr=.7,Q=.,Sc= Gr Figure (8):Skin friction for different values of S versus Gr Open Access Journal Page 73