Department of Mathematics, K L University, Vaddeswaram, India

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1 RADIATION AND CHEMICAL REACTION EFFECTS ON MHD FREE CONVECTIVE FLOW PAST A SEMI-INFINITE VERTICAL PLATE V.Si Hai Babu, G.V.Ramana Redd and 3 K.Jaaami Redd Reseach Schola, Depatment of Mathematics, Raalaseema Univesit, Kunool, India-58,3 Depatment of Mathematics, K L Univesit, Vaddeswaam, India-55. sihai.veeavalli@gmail.com Abstact - In this aticle, we studied the effects of vaiable viscosit and themal conductivit on an unstead twodimensional lamina flow of a viscous incompessible electicall conducting fluid past a semi-infinite vetical plate taking into account the mass tansfe. The fluid viscosit is assumed to va as a linea function of tempeatue. The govening equations fo the flow ae tansfomed into a sstem of non-linea odina diffeential equations ae solved b a petubation technique. The effects of the vaious paametes on the velocit, tempeatue, concentation and skin-fiction pofiles ae pesented gaphicall and discussed qualitativel. Kewods Radiation paamete, Themal conductivit, MHD, poous medium and viscosit. I. INTRODUCTION Natual convection flow ove vetical sufaces immesed in poous media has paamount impotance because of its potential applications in soil phsics, geohdolog, and filtation of solids fom liquids, chemical engineeing and biological sstems. Stud of fluid flow in poous medium is based upon the empiicall detemined Dac s law. Such flows ae consideed to be useful in diminishing the fee convection, which would othewise occu intensel on a vetical heated suface. In addition, ecent developments in moden technolog have intensified moe inteest of man eseaches in studies of heat and mass tansfe in fluids due to its wide applications in geothemal and oil esevoi engineeing as well as othe geo-phsical and astophsical studies. Came, K. R. and Pai, S. I. [] taken tansvese applied magnetic field and magnetic Renolds numbe ae assumed to be ve small, so that the induced magnetic field is negligible Muthucumaaswam et al. [] have studied the effect of homogenous chemical eaction of fist ode and fee convection on the oscillating infinite vetical plate with vaiable tempeatue and mass diffusion. Shama [3] investigate the effect of peiodic heat and mass tansfe on the unstead fee convection flow past a vetical flat plate in slipflow egime when suction velocit oscillates in time. Chaudha and Jha [4] studied the effects of chemical eactions on MHD micopola fluid flow past a vetical plate in slip-flow egime. Anjalidevi et al. [5] have examined the effect of chemical eaction on the flow in the pesence of heat tansfe and magnetic field. Muthucumaaswam et al. [6] have investigated the effect of themal adiation effects on flow past an impulsivel stated infinite isothemal vetical plate in the pesence of fist ode chemical eaction. Moeove, Al-Odat and Al-Azab [7] studied the influence of magnetic field on unstead fee convective heat and mass tansfe flow along an impulsivel stated semi-infinite vetical plate taking into account a homogeneous chemical eaction of fist ode.. The effect of adiation on the heat and fluid flow ove an unstead stetching suface has been analzed b El-Aziz [8]. Singh et. al. [9] studied the heat tansfe ove stetching suface in poous media with tansvese magnetic field. Singh et. al. [] and [] also investigated MHD oblique stagnation-point flow towads a stetching sheet with heat tansfe fo stead and unstead cases. Elbashbesh et. al. [] investigated the effects of themal adiation and magnetic field on unstead bounda lae mixed convection flow and heat tansfe poblem fom a vetical poous stetching suface. Ahmed Sahin studied influence of chemical eaction on tansient MHD fee Convective flow ove a vetical plate. Recentl, the chemical eaction, heat and mass tansfe on MHD flow ove a vetical stetching suface with heat souce and themal statification have been pesented b Kandasam et al.[3]. The opposing buoanc effects on simultaneous heat and mass tansfe b natual convection in a fluid satuated poous medium investigated b Angiasa et al.[4]. Ahmed [5] investigates the effects of unstead fee convective MHD flow though a poous medium bounded b an infinite vetical poous plate. Ahmed Sahin [6] studied the Magneto hdodnamic and chemical eaction effects on unstead flow, heat and mass tansfe chaacteistics in a viscous, incompessible and electicall conduction fluid ove a semi-infinite vetical poous plate in a slip-flow egime. The objective of the pesent stud is to investigate the effect of vaious paametes like chemical Reaction paamete, themal Gashof numbe, mass Gashof numbe, magnetic field paamete, adiation paamete on convective heat tansfe along an semi-infinite vetical plate in poous medium. The govening non-linea patial diffeential equations ae fist tansfomed into a dimensionless fom and thus esulting non-simila set of equations has been solved using the petubation technique. Results ae pesented gaphicall and discussed quantitativel fo paamete values of pactical inteest fom phsical point of view. II. MATHEMATICAL FORMULATION In a situation of two dimensional unstead lamina natual convection flows of a viscous, incompessible, electicall conducting, adiating fluid past an impulsivel stated semiinfinite vetical plate in the pesence of tansvese magnetic field with viscous dissipation is consideed. The fluid is assumed to be 5 Page

2 ga, absobing-emitting but non-scatteing. The - axis is taken along the plate in the upwad diection and the - axis is taken nomal to it. The fluid is assumed to be slightl conducting and hence the magnetic Renolds numbe is much less than unit and the induced magnetic field is negligible in compaison with the tansvese applied magnetic field. Initiall, it is assumed that the plate and the fluid ae at the same tempeatue and concentation level evewhee in the fluid. At time, the plate stats moving impulsivel in the vetical diection with constant velocit against the gavitational field. Also, the tempeatue of the plate and the concentation level nea the plate ae aised to and, espectivel and ae maintained constantl theeafte. It is assumed that the concentation of the diffusing species in the bina mixtue is ve less in the compaison to the othe chemical species, which ae pesent and hence the Soet and Dufou effects ae negligible. It is also assumed that thee is no chemical eaction between the diffusing species and the fluid. Then, unde the above assumptions, in the absence of an input electic field, the govening bounda lae equations with Boussinesq s appoximation ae Continuous equation u u x Momentum consevation () u u u * u B u v g T T g C C u u t x K Eneg consevation T T T T q u u v t x cp cp Species consevation C C C C u v D K C C t x () (3) The initial and bounda conditions ae as follows: (4) t, u, v, T T, C C t u u v T T C C,, w, w at u, T T, C C as (5) Themal adiation is assumed to be pesent in the fom of a unidiectional flux in the - diection i.e., q (tansvese to the vetical suface). B using the Rosseland appoximation, the adiative heat flux is given b q 4 4 s T q 3kc (6) whee s is the Stefan-Boltzmann constant and K - c the mean absoption coefficient. In the Rosseland appoximation, the pesent analsis is limited to opticall thick fluids. If tempeatue diffeences within the flow ae sufficientl T 4 small, then equation (6) can be lineaized b expanding into the Talo seies aboutt, which afte neglecting highe ode tems takes the fom: T 4T T 3T (7) In view of equations (6) and (7), equation (3) educes to T T T T T u u v t x k c c (8) The nondimensionless quantities intoduced in these equations ae defined as 3 6 st 3 c p p xu u tu K u v X Y t K U V,,,, v v v u u u * g T w T g C w C kck v G, Gm, N, M u u 4 st u T T C C Ku v T, C, K P, Sc, T w T C w C D (9) In a situation whee onl one dimensional flow is consideed, the above set of equations (), (), (8) and (4) ae educed to the following non-dimensional fom: V V V Y (whee V ) U U U GT GmC M U t Y Y K t Y P 3N Y T T 4 T t Y Sc Y C C C KC () () () (3) The coesponding initial and bounda conditions ae as follows t : U, T C t : U, T, C at Y U, T, C as Y III. SOLUTION OF THE PROBLEM (4) In ode to solve equations () - (3) with espect to the bounda conditions (4) fo the flow, let us take, ( ) U t U U e t (5) 6 P a g e

3 , ( ) T t T T e t, ( ) C t C C e t (7) Substituting the Equations (5) - (7) in Equations () (3), we obtain: / u u M K u GT GmC / u u M K u GT GmC T K T T K TK T C ScC ScK C () C ScC Sc K C (3 Skin-fiction: The dimensionless sheaing stess on the suface of a bod, due to the fluid motion, is known as skin-fiction and is defined b the Newton's law of viscosit. Them skin-fiction is U m3 A3 A m Am IV. RESULTS AND DISCUSSION In ode to get a phsical insight in to the poblem the effects of vaious govening paametes on the phsical quantities ae computed and epesented in Figues -3 and discussed in detail. The fomulation of the effects of chemical eaction and heat absoption on MHD convective flow and mass tansfe of an incompessible, viscous fluid along a semi-infinite vetical (6) poous moving plate in a poous medium has been pefomed in the peceding sections. This enables us to ca out the numeical calculations fo the distibution of the velocit, tempeatue and concentation acoss the bounda lae fo vaious values of the paametes. The influence of Magnetic field on the velocit pofiles has been studied in Fig.. It is seen that the incease in the applied magnetic intensit contibutes to the decease in the velocit. (8) Futhe, it is seen that the magnetic influence does not contibute significantl as we move awa fom the bounding (9) suface. The influence of the poosit of the bounda on the velocit of the fluid medium has been shown in Fig. It is seen that as the poosit of the fluid bed inceases, the velocit also () inceases which is in tune with the ealistic situation. Futhe, the poosit of the bounda does not influence of the fluid motion as we move fa awa fom the bounding suface. The () contibution of adiation paamete on the velocit pofiles is noticed in Fig.3. It is obseved that the adiation paamete inceases, the velocit field is an inceasing. Futhe, it is noticed that the velocit deceases as we move awa fom the plate whee pime denotes odina diffeentiation with espect to. which is found to be independent of adiation paamete.the effect of Pandtl numbe on the velocit pofiles has been The coesponding bounda conditions can be witten as illustated in Fig.4. It is obseved that as the Pandtl numbe u, T, T, C, C at inceases, the velocit deceases in geneal. The dispesion in the velocit pofiles is found to be moe significant fo smalle u, T T, C C as values of P and not that significant at highe values of Pandtl (4) numbe. Solving equations (9) (3) unde the bounda conditions The influence of Schmidt numbe Sc on velocit pofiles has (4), we obtain the velocit, tempeatue and concentation been illustated in Fig.5. It is obseved that, while all othe distibution in the bounda lae as: paticipating paametes ae held constant and Sc is inceased, it m 3 m m U, t A3 e A e Ae is seen that the velocit deceases in geneal. Futhe, it is T, t e m noticed that as we move fa awa fom the plate, the fluid velocit goes down. The effect of Gashof numbe on the C, t e m velocit pofiles as shown in Fig. 6. Incease in G contibutes to an incease in velocit when all othe paametes that appea in the velocit field ae held constant. Also it is noticed that as we whee move awa fom the plate the influence of G is not that m.5sc Sc 4 KSc significant. The effect of modified Gashof numbe Gc on the, n 4 / 3 N, m P/ n ; velocit pofiles is obseved in Fig.7. Incease in Gc is found to influence the velocit to incease. Also, it is seen that as we A G / m m M / K, A Gm / m m M / K move fa awa fom the plate it is seen that the effect of Gc is P A3 A A ; K 4 / 3N found to be not that significant. Fig.8 shows that the effect of inceasing the chemical eaction paamete on velocit pofiles. It is noticed that velocit of flow field ae deceasing, as the values of chemical eaction ae inceasing. The Effect of Pandtl numbe on the tempeatue field has been illustated in Fig.9. It is obseved that as the Pandtl numbe inceases, the tempeatue in the fluid medium deceases. Also, as we move awa fom the bounda, the Pandtl numbe has not much of significant influence on the tempeatue. The dispesion is not found to be significant. Fig. illustates the influence of the adiation paamete on the tempeatue pofiles in the bounda lae. As adiation paamete inceases, tempeatue distibutions incease when the othe phsical paametes ae fixed. The influence of Schmidt numbe on the concentation is 7 P a g e

4 illustated in Fig.. It is obseved that incease in Sc contibutes to decease of concentation of the fluid medium. Futhe, it is seen that Sc does not contibutes much to the concentation field as we move fa awa fom the bounding suface. Fig. shows that the effect of inceasing the chemical eaction paamete on concentation pofiles. It is noticed that species concentation ae deceasing, as the values of chemical eaction ae inceasing. The effect of chemical eaction on velocit and tempeatue is less dominant in compaison to concentation. Skin fiction fo vaious values of magnetic field stength is potaed though Fig 3. It is seen that skin fiction deceases, as magnetic paamete inceases, wheeas it is inceasing when adiation paamete is inceasing. Fig.4. Effects of Pandtl numbe on velocit pofiles. Fig.. Effects of magnetic paamete on velocit pofiles. Fig.5. Effects of Schmidt numbe on velocit pofiles. Fig.. Effects of pemeabilit paamete on velocit pofiles. Fig.6. Effects of Gashof numbe on velocit pofiles Fig.3. Effects of adiation paamete on velocit pofiles Fig.7. Effects of modified Gashof numbe on velocit pofiles. 8 P a g e

5 .4. N=;P=.7;Gm=5; Sc=.6;G=5;K=;M=; K =,, 3, 4 Velocit Fig.8. Effects of magnetic paamete on velocit pofiles. Fig.. Effects of Schmidt numbe on concentation pofiles. Fig.9. Effects of pandtl numbe on tempeatue pofiles. Fig.. Effects of adiation paamete on tempeatue pofiles. Concentation K =,, 3, 4 N=;P=.7;Gm=5; Sc=.6;G=5;K=;M=; Fig.. Effects of chemical eaction paamete on concentation pofiles Fig.3. Effects of magnetic paamete on skin-fiction. REFERENCES [] Came, K. R. and Pai, S. I. Magneto fluid Dnamics fo Engineeing and Applied Phsicists (Mc Gaw Hill, New Yok), (973). [] Muthucumaaswam.R and Meenakshisundaam.S. Theoetical stud of chemical eaction effects on vetical oscillating plate with vaiable tempeatue. Theoet. Appl. Mech., vol.33, No.3 (6), pp [3] Shama, P. K. Fluctuating themal and mass diffusion on unstead fee convective flow past a vetical plate in slip-flow egime, Latin Ameican Applied Reseach, 35, 33-39,(5). [4] Anjalidevi, S.P. and R. Kandasam. Effects of a chemical eaction heat and mass tansfe on MHD flow past a semi infinite plate. Z.Angew. Math. Mech.,() 8: [5] Chaudha, R. C. and Jha, A. K.,Effects of chemical eactions on MHD micopola fluid flow past a vetical plate in slipflow egime, Applied Mathematics and Mechanics, 9 (9), 79-94, (8). [6] Muthucumaaswam, R. and Chandakala, P. Radiation heat and mass tansfe effects on moving isothemal vetical plate in the pesence of chemical eaction. J. of Applied Mechanics and Engineeing, vol. NO.3, (6), pp [7] Al-Odat, M. Q. and Al-Azab. Influence of chemical eaction on tansient MHD fee convection ove a moving vetical plate, Emiates J. Engg. Res., (3), 5-. (7). [8] M.A. El-Aziz, Radiation effect on the flow and heat tansfe ove an unstead stetching sheet, Intenational Communications in Heat and Mass Tansfe, Vol. 36, pp 5-54, (9). [9] P. Singh, N.S. Tome and D. Sinha, Numeical stud of heat tansfe ove stetching suface in poous media with tansvese magnetic field, Poceeding of Intenational Confeence on Challenges and application of Mathematics in Sciences and Technolog (), ISBN X, pp 9 P a g e

6 4-43. [] P. Singh, N.S. Tome, S. Kuma and D. Sinha, MHD oblique stagnation-point flow towads a stetching sheet with heat tansfe, Intenational Jounal of Applied Mathematics and Mechanics, Vol. 6, no.3, pp 94-, (). [] P. Singh, A. Jangid, N.S. Tome, S. Kuma and D. Sinha, Effects of Themal Radiation and Magnetic Field on Unstead Stetching Pemeable Sheet in Pesence of Fee Steam Velocit, Intenational Jounal of Infomation and Mathematical Sciences, Vol.6, no.3, pp-63-69, (). [] E. M. A. Elbashbesh, D. M. Yassmin and A. A. Dalia, Heat Tansfe Ove an Unstead Poousv Stetching Suface Embedded in a Poous Medium with Vaiable Heat Flux in the Pesence of Heat Souce o Sink, Afican Jounal of Mathematics and Compute Science Reseach Vol. 3, no.5, pp 68-73, (). [3] Kandasam, R., Peiasam, K. and Sivagnana Pabhu, K.K. Chemical eaction, heat and mass tansfe on MHD flow ove a vetical stetching suface with heat souce and themal statification effects, Int. J. Heat and Mass Tansfe, 48 (- ), ,(5). [4] Angiasa, D., Peteson, G. P. and Pop, I. Combined heat and mass tansfe b natual convection with opposing buoanc effects in a fluid satuated poous medium, Int. J. Heat Mass Tans., 4(), ,(997). [5] Ahmed, S. Effects of unstead fee convective MHD flow though a poous medium bounded b an infinite vetical poous plate, Bull. Cal. Math. Soc., 99 (5), 5-5,(7). [6] Ahmed Sahin. Influence of chemical eaction on tansient mhd fee Convective flow ove a vetical plate in slip-flow Regime. Emiates Jounal fo Engineeing Reseach, 5 (), 5-34 (). 3 P a g e