Chemical Reaction and Soret effect on MHD free Convective Flow past an Infinite Vertical Porous Plate with Variable Suction

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1 International Journal of Chemical Engineering Research. ISSN Volume 9, Number (7), pp. 5-6 Research India Publications Chemical Reaction and Soret effect on MHD free Conectie Flow past an Infinite Vertical Porous Plate with Variable Suction A. Mthree * and K.S. Balamurugan Department of Mathematics, Stanle College of Engineering and Technolog for women, Abids, Hderabad, Telengana, India Department of Mathematics, RVR & JC College of Engineering, Guntur, Andhra Pradesh, India Abstract This paper deals with the stud of chemical reaction and soret effects on unstead MHD free conectie mass transfer flow past an infinite ertical porous plate with ariable suction, where the plate temperature oscillates with the same frequenc as that of ariable suction elocit. The goerning boundar laer equations are soled analticall using two-term harmonic and non-harmonic functions. The effects of arious thermal phsical parameters on the elocit, temperature, concentration and skin-friction coefficient are discussed in detail through graphs. Ke ords: MHD, porous plate, chemical reaction, soret, ariable suction. INTRODUCTION: In recent ears, the problems of free conectie heat and mass transfer flows through a porous medium under the influence of a magnetic field hae been attracted the attention of a number of researchers because of their possible applications in man branches of science and technolog, such as its applications in transportation cooling of reentr ehicles and rocket boosters, cross-hatching on ablatie surfaces and film

2 5 A. Mthree and K.S. Balamurugan aporization in combustion chambers. On the other hand, flow through a porous medium hae numerous engineering and geophsical applications, for example, in chemical engineering for filtration and purification process; in agriculture engineering to stud the underground water resources; in petroleum technolog to stud the moement of natural gas, oil and water through the oil reseroirs. In iew of these applications, man researchers hae studied MHD free conectie heat and mass transfer flow in a porous medium; some of them are Raptis and Kafoussias [], Sattar [] and Kim [3]. Due to the importance of Soret (thermal-diffusion) and Dufour (diffusion-thermo) effects for the fluids with er light molecular weight as well as medium molecular weight man inestigators hae studied and reported results for these flows of whom the names are Eckert and Drake [4], Dursunkaa and orek [5], Anghel et al. [6], Postelnicu [7] are worth mentioning. Alam and Rahman [8] studied the Dufour and Soret effects on stead MHD free conectie heat and mass transfer flow past a semi-infinite ertical porous plate embedded in a porous medium. Man practical diffusie operations, the molecular diffusion of a species are inoled in the presence of chemical reaction within or at the boundar. There are two tpes of reactions i.e. homogeneous and heterogeneous reactions. A homogeneous reaction occurs uniforml throughout a gien phase. In such tpe of reaction the species generation is analogous to internal source of heat generation. In contrast, a heterogeneous reaction takes place in a restricted region or within the boundar of a phase. It is so treated as a boundar condition similar to the constant heat flux condition in heat transfer. All industrial chemical processes are so designed that the cheaper raw materials can be transformed to high alue products b chemical reaction. For a specific chemistr, the reactor performance is a complex function of the underling transport processes. An analsis of the transport processes and their interaction with chemical reactions are quite difficult and is directl connected to the underling fluid dnamics. Moreoer, the chemical reaction, heat and mass transfer on MHD flow oer a ertical stretching surface with heat source and thermal stratification hae been presented b Kandaswam et al. [9]. Al-Odat and Al-Azab [] studied the influence of magnetic field on unstead free conectie heat and mass transfer flow along an impulsiel started semi-infinite ertical plate taking into account a homogeneous chemical reaction of first order. The effect of radiation on the heat and fluid flow oer an unstead stretching surface has been analzed b El-Aziz []. J Prakash et.al [] has studied the thermo diffusion and chemical reaction effects on MHD three dimensional free conectie couette flow. Singh et. al. [3] inestigated MHD oblique stagnation-point flow towards a stretching sheet with heat transfer for stead and unstead cases. Elbashbesh et. al. [4] inestigated the effects of thermal radiation and magnetic field on unstead boundar laer mixed conection flow and heat transfer problem from a ertical porous stretching surface. The opposing buoanc effects on simultaneous heat and mass transfer b natural conection in a fluid saturated porous medium inestigated b Angirasa et.al. [5]. Ahmed [6] inestigates the effects of unstead free conectie MHD flow through a porous medium bounded b an infinite ertical porous plate. Mthree et al. [7] analzed the effect of chemical reaction on unstead MHD conectie heat and mass transfer past a semi-infinite ertical permeable moing plate with heat absorption.

3 Chemical Reaction and Soret effect on MHD free Conectie Flow 53 In this paper, an analtical stud was performed to inestigate the effects of chemical reaction and soret on the unstead MHD free conectie mass transfer flow past an infinite ertical porous plate with ariable suction. The goerning equations of motion are soled analticall b using a regular perturbation technique. The behaiour of elocit, temperature, concentration and skin-friction coefficient for different alues of thermo-phsical parameters has been computed and the results are presented graphicall and discussed in detail.. FORMATION OF THE PROBLEM: Unstead flow of an incompressible, electricall conducting iscous fluid past an infinite ertical porous plate embedded under the influence of a uniform transerse magnetic field is considered. Here the origin of the coordinate sstem is taken to be at an point of the plate. Let the components of elocit along x and axes are u, and which are chosen in the upward direction along the plate and normal to the plate respectiel. The polarization effects are assumed to be negligible. Hence the goerning equations of the problem are: () u u g ( T T ) g (C C ) u u u t k T T k T T t cp () (3) C C C T D D K ( ) r C C t The corresponding boundar conditions are: i t t> u, T T e, C C at (5) u, T T C C as ' From the continuit equation, it can be seen that is either a constant or a function of time. So, assuming suction elocit to be oscillator about a non-zero constant mean, one can write ( i t ) Ae (6) here is the mean suction elocit and ε, A are small such that ε A<<. The negatie sign indicates that the suction elocit is directed towards the plate. In order to write the goerning equations and the boundar conditions in dimensional following non-dimensional quantities are introduced. (4)

4 54 A. Mthree and K.S. Balamurugan, u t u, t, 4 T ( TT ) ( TT) w, C C C, C C Sc, D, g(t T ) g (C C ) 4 Gr, Gc 3,, 3 B M, K K, D (T ) T Cp So, Pr (7) ( C C ) KT Hence, using the aboe non - dimensional quantities, the equations ()-(5) in the nondimensional form can be written as u it u u ( Ae ) GrT GcC M u 4 t K 4 t Pr T it T T ( Ae ) C it C C T ( Ae ) So KrC 4 t Sc The corresponding boundar conditions are it t>, u, T T e, C at = u, T, C as () (8) (9) () Here t is the time, g the acceleration due to grait, the coefficient of olume * expansion, the coefficient of thermal expansion with concentration, T temperature of the fluid, T the temperature of the fluid far awa from the plate, C the species concentration, C the species concentration of the fluid far awa from the plate, T the plate temperature, C the species concentration near the plate, ν the kinematic iscosit, ρ the densit, Cp the specific heat at constant pressure, KT the thermal conductiit, D the chemical molecular diffusiit, µ the coefficient of iscosit, M the magnetic field parameter, So the soret number, Sc the Schmidt number and chemical reaction parameter Kr, all the phsical quantities hae their usual meaning.

5 Chemical Reaction and Soret effect on MHD free Conectie Flow SOLUTION OF THE PROBLEM: In order to reduce the aboe sstem of partial differential equation to a sstem of ordinar differential equations, the elocit, temperature and concentration in the neighbourhood of the porous plate are taken as u t u e u O it (, ) ( ) ( ) ( ) T t T e T O it (, ) ( ) ( ) ( ) C t C e C O it (, ) ( ) ( ) ( ) Substituting equation () in equations (8) to () and equating the harmonic and nonharmonic terms, and neglecting the higher order terms of O(ε ), we obtain u Lu M u GrT GcC K i u Lu M u GrT GcC 4 K T LPrT i Pr T L PrT T 4 C LScC ScKrC SoSc e L Pr (L Pr ) Sci C LScC ScKr C SoScT 4 () (3) (4) (5) (6) (7) (8) And the corresponding boundar conditions are t>, u=u=, T=T=, C=, C= at = u u, T T, C C as (9) Soling the aboe equations (3) to (8) and using boundar conditions (9), the solution of equations are expressed as u(, t) (A A A ) e ( A A ) e A e e ( (A A A ) e m8 L Pr m 4 i t m m m 6 ( A6 A8 ) e A7 e ) LPr it m T(,t) e e e C t A Ae e A e m4 L Pr it m m 6 (, ) ( )e (e ) () () ()

6 56 A. Mthree and K.S. Balamurugan Skin-friction: u m (A A A ) L Pr( A A ) m A (m (A A A ) m ( A A ) m A )e it (3) here i t Pr L Pr L i Pr LSc L Sc 4ScKr L Ae, m, m4, SoScL Pr N M i K, A, L Pr L Sc Pr ScKr 4 m 6 i LSc L Sc 4Sc Kr 4, SoScm A Sci m LScm ScKr Gc( A ) A4 m, m4 Lm4 M K GcA A m Lm N, A A Gr A, L Pr L Pr M K, 3 8 m Lm N, L L 4N Gr, A6 m 8 m m L N, L L 4M K 4. RESULT AND DISCUSSION: To assess the phsical depth of the problem, the effects of arious parameters like Schmidt number Sc, thermal Grashof number Gr, magnetic parameter M, chemical reaction parameter Kr, mass Grashof number Gc, soret effect number So and prandtl number Pr on elocit, temperature and concentration distribution are studied in figures -, the numerical alues of Prandtl number Pr are chosen as Pr =.7, and Pr =7, which corresponds to air and water at o C respectiel. The numerical alues of the remaining parameters are chosen arbitraril. Figure presents tpical elocit profiles in the boundar laer for arious alues of the thermal Grashof number. The thermal Grashof number defines the ratio of the species buoanc force to the iscous hdrodnamic force. As expected, the fluid

7 Chemical Reaction and Soret effect on MHD free Conectie Flow 57 elocit increases and the peak alue is more distinctie due to increase in the species buoanc force. The influence of the Solutal Grashof number Gc on the elocit is presented in Figure. The Solutal Grashof number signifies the relatie effect of the thermal buoanc force to the iscous hdrodnamic force in the boundar laer. As expected, it is obsered that there is a rise in the elocit due to the enhancement of thermal buoanc force. Here, the positie alues of Gc correspond to cooling of the plate. Also, as Gc increases, the peak alues of the elocit increases rapidl near the porous plate and then decas smoothl to the free stream elocit. Figure 3 displas the influence of thermodiffusion (soret) parameter oer the dimensionless elocit. It is eident that the thermodiffusion parameter accelerates the elocit of the flow field. Figures 4 and 5 displas the effects of the Schmidt number (Sc) on the elocit and concentration profiles, respectiel. As the Schmidt number increases, the concentration decreases. This causes the concentration buoanc effects to decrease ielding a reduction in the fluid elocit. The reduction in the elocit and concentration profiles are accompanied b simultaneous reductions in the momentum and concentration boundar laers thickens. These behaiors are clearl shown in Figures 4 and U 4 Gr 5 Gr Gr 5 Gr Fig.: Effects of thermal Grashof number Gr on elocit Gc U 4 Gc Gc 5 Gc Fig.: Effects of mass Grashof number Gc on elocit

8 58 A. Mthree and K.S. Balamurugan 4 3 U So So So 3 So Fig.3: Effects of Soret number So on elocit U.5 Sc. Sc.3 Sc.4 Sc Fig.4: Effects of Schimidt number Sc on elocit..8.6 C.4 Sc. Sc. Sc.3 Sc Fig.5: Effects of Schmidt number Sc on Concentration

9 Chemical Reaction and Soret effect on MHD free Conectie Flow U.5 M M M 3 M Fig.6: Effects of magnetic parameter M on elocit kr..5 kr.4. kr.6 U kr Fig.7: Effects of chemical reaction parameter Kr on elocit..8.6 C.4 Kr. Kr.4 Kr.6 Kr Fig.8: Effects of Chemical reaction parameter Kr on Concentration

10 6 A. Mthree and K.S. Balamurugan Pr.7.5 Pr. Pr 3 U Pr Fig.9: Effects of Prandtl number Pr on elocit..8.6 t.4 Pr.7 Pr Pr 4 Pr Fig.: Effects of Prandtl number Pr on temperature For different alues of the magnetic field parameter M, the elocit profiles are plotted in Figure. 6. It is obious that the effect of increasing alues of the magnetic field parameter results in a decreasing elocit distribution across the boundar laer. Figures 7 and 8, displa the effects of the chemical reaction parameter (Kr) on the elocit and concentration distributions respectiel. It is seen, that the elocit and concentration decreases with increasing the chemical reaction parameter. Figures 9 and illustrate the elocit and temperature profiles for different alues of the Prandtl number Pr. The Prandtl number defines the ratio of momentum diffusiit to thermal diffusiit. The numerical results show that the effect of increasing alues of Prandtl number results in a decreasing elocit (Figure 9). From Figure, it is obsered that an increase in the Prandtl number results a decrease of the thermal boundar laer thickness. The reason is that smaller alues of Pr are equialent to increasing the thermal conductiities, and therefore heat is able to diffuse awa from

11 Chemical Reaction and Soret effect on MHD free Conectie Flow 6 the heated plate more rapidl than for higher alues of Pr. Hence in the case of smaller Prandtl numbers as the boundar laer is thicker and the rate of heat transfer is reduced. 5. CONCLUSIONS: The goerning equations for unstead MHD conectie mass transfer past an infinite ertical porous plate with ariable suction were formulated. The plate temperature oscillates with the same frequenc as that of ariable suction elocit with the chemical reaction effects. The computed alues obtained from analtical solutions for the elocit, temperature, concentration fields and skin-friction coefficient are presented through graphs. In the absence of chemical reaction (Kr=), these results are in good agreement with the results of Seethamahalakshmi et al. [8]. e conclude the following after analzing the graphs:. The elocit decreases with increasing the Prandtl number, magnetic field parameter, chemical reaction parameter and Schmidt number whereas reerse trend is seen with increasing the Soret number, thermal and solutal Grashof numbers.. The temperature decreases as the alues of Prandtl number increase. 3. The concentration decreases as the alues of the chemical reaction parameter and Schmidt number increases. REFERENCES: [] Raptis, N.G. Kafoussias (98), Magneto hdrodnamic free conection flow and mass transfer through porous medium bounded b an infinite ertical porous plate with constant heat flux, Can. J. Phs., 6(), [] Sattar M.A. (993), Unstead hdromagnetic free conection flow with Hall current mass transfer and ariable suction through a porous medium near an infinite ertical porous plate with constant heat flux, Int. Energ Research, 7, -5. [3] Kim (4), Heat and mass transfer in MHD micropolar flow oer a ertical moing porous plate in a porous medium, Transport in Porous Media, 56(), 7 37, 4. [4] E.R.G. Eckert, R.M. Drake, Analsis of Heat and Mass Transfer, McGraw- Hill, New York, 97. [5] Dursunkaa, Z.M. (99) orek, Diffusion-thermo and thermal-diffusion effects in transient and stead natural conection from ertical surface, Int. J. Heat Mass Transfer, 35(8), [6] M. Anghel, H. S. Takhar, I. Pop,(). Dufour and Soret effects on freeconection boundar laer oer a ertical surface embedded in a porous medium, Studia Uniersities Babes-Bolai, Mathematica, XLV(4),.

12 6 A. Mthree and K.S. Balamurugan [7] Postelnicu, A (4), Influence of a magnetic field on heat and mass transfer b natural conection from ertical surfaces in porous media considering Soret and Dufour effects, Int. J. Heat Mass Transfer, 47(6 7), [8] Alam, M. S. and M.M. Rahman (5). Dufour and Soret effects on MHD free conectie heat and mass transfer flow past a ertical flat plate embedded in a porous medium, J. Naal Architecture and Marine Engng., (), [9] Kandasam, R., Periasam, K. and Siagnana Prabhu, K.K (5). Chemical reaction, heat and mass transfer on MHD flow oer a ertical stretching surface with heat source and thermal stratification effects, Int. J. Heat and Mass Transfer, 48 (-), [] Al-Odat, M. Q. and Al-Azab, (7). Influence of chemical reaction on transient MHD free conection oer a moing ertical plate, Emirates J. Engg. Res., (3), 5-. [] El-Aziz M.A. (9), Radiation effect on the flow and heat transfer oer an unstead stretching sheet, International Communications in Heat and Mass Transfer, 36, [] Prakash J, K.S. Balamurugan and S.V.K. Varma (5). Thermodiffusion and Chemical reaction effects on MHD three dimensional free conectie couette flow, alailak Journal of Science and Technolog, (9) [3] Singh, P, N.S. Tomer, S. Kumar and D. Sinha, (). MHD oblique stagnation point flow towards a stretching sheet with heat transfer, International Journal of Applied Mathematics and Mechanics, 6(3), 94-. [4] Elbashbesh, E. M. A. and D. M. Yassmin and A. A. Dalia, (). Heat transfer oer an unstead porous stretching surface embedded in a porous medium with ariable heat flux in the presence of heat source or sink, African Journal of Mathematics and Computer Science Research, 3(5), [5] Angirasa, D., Peterson, G. P. and Pop, (997). Combined heat and mass transfer b natural conection with opposing buoanc effects in a fluid saturated porous medium, Int. J. Heat Mass Trans., 4(), [6] Ahmed, S (7). Effects of unstead free conectie MHD flow through a porous medium bounded b an infinite ertical porous plate, Bull. Cal. Math. Soc., 99 (5), 5-5. [7] Mthree, A, J.P. Pramod and K.S. Balamurugan, (5), Chemical Reaction on unstead MHD conectie heat and mass transfer past a semi-infinite ertical permeable moing plate with heat absorption, Procedia Engineering 7, [8] Seethamahalakshmi, B.D.C.N. Prasad and G.V. Ramana Redd (). MHD free conectie mass transfer flow past an infinite ertical porous plate with ariable suction and soret effect, Asian J. of Current Engg. (),

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