The Combined Effect of Chemical reaction, Radiation, MHD on Mixed Convection Heat and Mass Transfer Along a Vertical Moving Surface

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1 Available at Appl. Appl. Math. ISSN: Vol. 5, Issue (Decembe ), pp (Peviously, Vol. 5, Issue, pp. 63 6) Applications and Applied Mathematics: An Intenational Jounal (AAM) The Combined Effect of Chemical eaction, Radiation, MHD on Mied Convection Heat and Mass Tansfe Along a Vetical Moving Suface Navneet Joshi and Manoj Kuma Depatment of Mathematics, Statistics and Compute Science G.B. Pant Univesity of Agicultue and Technology Pantnaga 63 5, India navneet.nimt@gmail.com;mnj_kuma@yahoo.com Received: August 5, ; Accepted: Novembe 3, Abstact This pape discusses the effect of Chemical eaction, Radiation and MHD on lamina mied convection bounday laye flow and heat and mass tansfe on continuously moving vetical suface. The fluid viscosity is assumed to vay as an invese linea function of tempeatue and local similaity solutions ae obtained fo the bounday laye equations subject to isothemally moving vetical suface with unifom speed. The system of non-linea patial diffeential equations developed in the pocess is finally tansfomed into a set of odinay diffeential equations with the help of similaity tansfomations involved in the poblem. This set of equations is fo diffeent values of the vaious paametes. The esults showing the effect of physical paametes on velocity, tempeatue and concentation have been computed and pesented gaphically to discuss thei in details. Keywods: MHD mied convection; vetical moving suface; chemical eaction; adiation; Runge-Kutta method shooting technique MSC No.: 76D5. Intoduction Continuously moving suface though an othewise quiescent medium has many applications in manufactuing pocesses, such as wie dawing, metal etusion, and pape poduction [Altan et al. (979) and Tadmo et al. (97)]. The pioneeing wok in this aea was caied out by 53

2 AAM: Inten. J., Vol. 5, Issue (Decembe ) [Peviously, Vol. 5, Issue, pp. 63 6] 535 Sakiadis (96) who developed a numeical solution fo the bounday laye flow field of a stetched suface. Many authos have eviewed this poblem to study the hydodynamic and themal bounday laye due to moving suface [Magyai (999), Kelle (999) and Tsou et al. (967)]. Suction o injection of a unifom suface was intoduced by Fo et al. (968) and Eickson et al. (966) fo stetched suface velocity and tempeatue and by Gupta and Gupta (977) fo linealy moving suface. Magyai et al. (999) have epoted analytical and computational solution when the suface moves with apidly deceasing velocities using the selfsimila method. Many eseaches consideed the effect of constant viscosity on bounday layes developed by continuously moving suface. The chemical equation is applicable to tansfomations of elementay paticles as well as nuclea eaction. Numeous applications of chemical eaction ae epeiment in chemical engineeing, in polyme poduction and manufactuing of ceamics etc. The impotance of themal adiation becomes intensified at high absolute-tempeatue levels due to basic diffeence between adiation and the convection and conduction enegy-echange mechanisms, some devices fo space applications ae designed to opeate at high tempeatue levels in ode to achieve high themal efficiency. Hence, adiation must often be consideed when calculating themal effects in devices such as a ocket nozzle, a nuclea powe plant, o a gaseous - coe nuclea ocket. It is known that the fluid viscosity changes with tempeatue [Hewig et al. (986)]. Recently, in Ali et al. (6), the effect of vaiable viscosity on a mied convection heat tansfe along a vetical moving suface was studied. Now in Pesent investigation we have seen the effects of MHD, chemical eaction and adiation and to get pecious infomation about the flow, tempeatue and concentation.. Mathematical Analysis Let us conside a steady two dimensional lamina flow due to vetically moving isothemal suface. Using boussinesq appoimation fo incompessible viscous fluid, the fluid viscosity is assumed to vay as an invese linea function of tempeatue., o a whee a and The equations govening fo convective vaiable viscosity fluid flow ae Equation of continuity: u u y () Figue. The suface moving upwads in the X-diection

3 536 Navneet Joshi and Manoj Kuma Equation of momentum: u u u v y Sg T T Sg C C * u B u y y () Equation of enegy: T u T v y T y C p q y (3) Equation of diffusion: C C C, u v D k C C y y () subject to the following bounday conditions: u v at y U w T T w Cw u T T C C at y C as y. (5) The -coodinate is measued along the moving suface fom the point whee the suface oiginates and y-coodinate is measued nomal to it (Figue), whee u and v ae the velocity components in -and y-diections espectively. S is a dummy paamete stands fo, +, -. The steam function and following tansfomation have been used. y Re f Re T T T T w C C C C w The adioactive heat flu q unde Rosseland appoimation by Bewste (99) has the fom: q T, 3 y whee σ is Stefan-Boltzmann constant and χ is the mean absoption coefficient. We assume that the tempeatue diffeences within the flow ae so small that T can be epessed as a linea

4 AAM: Inten. J., Vol. 5, Issue (Decembe ) [Peviously, Vol. 5, Issue, pp. 63 6] 537 function oft. This is obtained by epending highe ode tems. Thus we get: T in Taylo seies about T and neglecting the T 3 T T 3T. Hee, u and v is the velocity components along in the -and y-diection. Uw u Re f, Re, v f f Re, (6) whee f and ae the dimensionless velocity and tempeatue espectively, is similaity vaiable. Putting all the values in equations ()- (), we get f ff f Mf (7) P f R, (8) f L Sc. (9) The tansfomed bounday conditions ae given by: whee f, f,,, f,,, () is constant viscosity/tempeatue paamete defined by T T T T T T w w Intoducing the following non-dimensional paametes:. SG Re SG c, Re, G 3 g Tw T, G c 3 g C C w, M U w, 3 6T R, 3k P, Sc k, L, D U w

5 538 Navneet Joshi and Manoj Kuma whee λ is buoyancy paamete, M is the magnetic paamete, Sc is the Schmidt numbe, P is the pandtle numbe, L is the chemical eaction paamete, R is the adiation paamete, G and G c ae the Gashof numbe fo heat and mass tansfe. Keeping in view of engineeing aspects, the most impotant chaacteistics of the flow ae ate of skin-fiction and heat tansfe coefficient, which can be witten as C Re f,, f Nu Re,. 3. Results and Discussion The couple nonlinea odinay diffeential equation (7) to (9) ae solved numeically by using the fouth ode Runge-Kutta method. Local similaity solutions of the diffeential equation (7) to (9) subject to the bounday condition () wee obtained fo inceasing values of at each constant. At each new and we stat fom the known solution of the equations with ae known., and, whee f Fo a given value of the values of f wee estimated and the diffeential equations (7) to (9) wee integated using Runge-Kutta method until the bounday condition at infinity f, and decay eponentially to zeo ( whee the solution to be accepted and solution with f, and > will not be consideed). If the bounday condition ae not satisfied then the numeical outine uses a half inteval method to calculate coection to the estimated values of f, and. The pocess is epeated iteatively until eponentially decaying solution in f, and is obtained. The local solutions wee obtained fo diffeent values of by Ali et al. (6). The values of was chosen as lage as possible between 3.5 and 5 depending upon the Pandtl numbe and the viscosity / tempeatue paamete without causing numeical oscillations in the values f, and. The effects of these paametes on the velocity, tempeatue and concentation pofiles have been analyzed with the help of gaphical epesentation though Figues -6. Figues ()-() shows that the velocity inceases with an incease in the value of viscosity/tempeatue paamete, chemical eaction paamete, adiation paamete, and buoyancy paamete. While velocity deceases with an incease in the values of magnetic paamete, Pandtl numbe, Schmidt numbe in Figues (5)-(7). It is clea fom figues (8) and (9) tempeatue inceases with an incease in the value of adiation and magnetic paamete while figue () shows that tempeatue deceases with incease in the value of buoyancy paamete. And concentation pofile inceases with incease in the value of chemical eaction paamete and magnetic paamete while concentation pofile deceases with incease in the value of Schmidt numbe and buoyancy paamete. It is woth mentioning that small values of P (<<) physically coespond to liquid metals, which have high themal conductivity but low viscosity, while P ~,

6 AAM: Inten. J., Vol. 5, Issue (Decembe ) [Peviously, Vol. 5, Issue, pp. 63 6] 539 coesponds to di-atomic gases including ai. On the othe hand, lage values of P (>> ) coespond to high-viscosity oils. Table descibes the effects of numeical values of vaious paametes in skin fiction coefficient and local Nusselt numbe. Table : Skin fiction coefficient f ''() and local Nusselt numbe () Physical paametes Values f ''() '() L. P.7 7 δ.5 M ' Conclusion The authos have theoetically studied how the govening paametes viscosity/tempeatue paamete, chemical eaction paamete, adiation paamete, buoyancy paamete and Pandtl numbe influence the bounday laye flow and heat tansfe chaacteistics on the moving suface. The investigation of the effects of viscosity/tempeatue paamete and Pandtl numbe on the skin fiction coefficient and on local Nusselt numbe eveals that both the skin fiction coefficient and the Nusselt numbe incease as the viscosity/tempeatue paamete and the Pandtl numbe incease. The magnetic field also inceases the skin fiction, but educes the heat tansfe

7 5 Navneet Joshi and Manoj Kuma θ=,, 3 Figue. pofile fo vaious value of θ at M=, P=.7, R=,λ=,L=.,Sc=.5 δ= L=,.,. 3 Figue. pofile fo vaious value of L at M=, P=.7, R=,λ=,θ=,Sc=.5 δ= R=, 5, 3 Figue 3. pofile fo vaious value of R at M=, P=.7, L=.,λ=,θ=,Sc=.5 δ= δ=.5,, Figue.. pofile fo vaious value of R at M=, P=.7, L=.,λ=,θ=,Sc=.5 δ= M=,,3 3 Figue 5. pofile fo vaious value of R at M=, P=.7, L=.,λ=,θ=,Sc=.5 δ= P=.7, 7, 7 3 Figue 6. pofile fo vaious value of P at M=, R=, L=.,λ=,θ=,Sc=.5 δ=

8 AAM: Inten. J., Vol. 5, Issue (Decembe ) [Peviously, Vol. 5, Issue, pp. 63 6] Sc=.5,, Figue 7. pofile fo vaious value of Sc at M=, P=.7, L=., λ=, θ=,r= δ=, Sc=.5 Tempeatue R=, 5, Figue 8.Tempeatue pofile fo vaious value of R at M=, P=.7, L=., λ=, θ=, δ=, Sc=.5 Tempeatue M=,, 3 3 Figue 9. Tempeatue pofile fo vaious value of M at R=, P=.7, L=., λ=, θ=, δ=, Sc=.5 Tempeatue δ=.5,, Figue. Tempeatue pofile fo vaious value of δ at R=, P=.7, L=., λ=, θ=, M=, Sc=.5 Concentation δ=.5,, 3 Figue. Concentation pofile fo vaious value of δ at R=, P=.7, L=., λ=, θ=, δ=, Sc=.5 Concentation M=,, 3 Figue. Concentation pofile fo vaious value of M at R=, P=.7, L=., λ=, θ=, δ=, Sc=.5

9 5 Navneet Joshi and Manoj Kuma. P=.7, 7, 7. Sc=.5,, Concentation Concentation Figue 3. Concentation pofile fo vaious value of P at R=, M=, L=., λ=, θ=, δ=, Sc=.5 Figue. Concentation pofile fo vaious value of P at R=, M=, L=., λ=, θ=, δ= Tempeatue P=.7, 7, 7 3 Figue 5. Tempeatue pofile fo vaious value of P at R=, M=, L=., λ=, θ=, δ= Sc=.5 Concentation L=,.,. 3 Figue 6. Concentation pofile fo vaious value of L at R=, M=, Sc=.5, λ=, θ=, δ= REFERENCES Ali, M.E. (6). The Effect of vaiable viscosity on Mied Convection Heat Tansfe along a Vetical Moving Suface, Int. J of Themal Sciences, Vol. 5, pp Altan, T., Oh, S. and Gegel, H. (979). Metal Foming Fundamentals and Applications, Ameican Socicty of Metals, Metals Pak, OH. Bewste, M.Q. (99). Themal Radiative Tansfe and Popeties, John Wiley and Sons, New Yok.

10 AAM: Inten. J., Vol. 5, Issue (Decembe ) [Peviously, Vol. 5, Issue, pp. 63 6] 53 Eickson, L.E., Fan, L.T. and Fo, V.G. (966). Heat and Mass Tansfe on a Moving Continuous Flat plate with Suction o Injection, Indust. Engg. Chem. Fundamentals Vol.5, pp.9-5. Fo, V.G, Eickson, L.E. and Fan, L.T. (968). Method fo Solving the Bounday Laye Equations fo Moving Continuous Flat Sufaces with Suction and Injection, AIChE J. Vol., pp Gupta, P.S. and Gupta, A.S. (977). Heat and Mass Tansfe on a Stetching Sheet with Suction o Blowing, Cannad. J. Chem. Engg, Vol. 55 (6), pp Hewig, H. and Wicken, G. (986). The Effect of Vaiable Popeties on Lamina Bounday Laye Flow, Wame-and stoffubetagung, Vol., pp Magyai, E., Kelle, B. (999). Heat tansfe chaacteistics of the sepaation Appl. Phys. Vol.3, pp Magyai, E., Kelle, B. (999). Heat and mass tansfe in the bounday layes on an eponentially stetching continuous suface, J. Phys. D: Appl. Phys. Vol. 3, pp Magyai, E., Ali M.E., Kelle, B. (). Heat and mass tansfe chaacteistics of the selfsimila bounday-laye flows induced by continuous suface stetched with apidly deceasing velocities, Heat Mass Tansfe, Vol. 38, pp Sakiadis, B.C. (96). Bounday laye behavio on continuous solid sufaces: I. Bounday-laye equation fo two-dimensional and asisymmetic flow, AICh J. Vol., pp Tsou, F.K., Spaow, E.M, Goldstein, R.J. (967). Flow and heat tansfe in the bounday laye on continuous moving suface, Int. J. Heat Mass Tansfe Vol.,pp. 9-3 Tadmo, Z., Klein, I. (97). Engineeing Pinciples of plasticting Etusion, Polyme Science and Engineeing Seies, Van Nostand Reinhold, New Yok.