E. GEETHA Department of Mathematics Sri Chandrasekharendra Saraswathi Viswa Mahavidyalaya University Enathur, Kanchipuram , INDIA

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1 In. J. of Applied Mechanics and Engineering, 13, vol.18, No.3, pp DOI: 1.478/ijame CHEMICAL REACTION EFFECTS ON MHD FLOW PAST A LINEARLY ACCELERATED VERTICAL PLATE WITH VARIABLE TEMPERATURE AND MASS DIFFUSION IN THE PRESENCE OF THERMAL RADIATION R. MUTHUCUMARASWAMY * Deparmen of Applied Mahemaics, Sri Venkaeswara College of Engineering Sriperumbudur 6 15, INDIA msamy@svce.ac.in E. GEETHA Deparmen of Mahemaics Sri Chandrasekharendra Saraswahi Viswa Mahavidyalaya Universiy Enahur, Kanchipuram , INDIA An exac soluion of firs order chemical reacion effecs on a radiaive flow pas a linearly acceleraed infinie isohermal verical plae wih variable mass diffusion, under he acion of a ransversely applied magneic field has been presened. The plae emperaure is raised linearly wih ime and he concenraion level near he plae is also raised o C w linearly wih ime. The dimensionless governing equaions are ackled using he Laplace-ransform echnique. The velociy, emperaure and concenraion fields are sudied for differen physical parameers such as he magneic field parameer, radiaion parameer, chemical reacion parameer, hermal Grashof number, mass Grashof number, Schmid number, Prandl number and ime. I is observed ha velociy increases wih decreasing magneic field parameer or radiaion parameer. Bu he rend is jus reversed wih respec o he chemical reacion parameer. Key words: acceleraed, isohermal, radiaion, verical plae, hea and mass ransfer, magneic field, chemical reacion. 1. Inroducion Thermal radiaion is an imporan facor in he hermodynamic analysis of many high emperaure sysems like solar collecors, boilers and furnaces. The simulaneous effec of hea and mass ransfer in he presence of hermal radiaion plays an imporan role in manufacuring indusries for he design of fins, seel rolling, nuclear power plans, cooling of owers, gas urbines and various propulsion device for aircraf, maerials processing, energy uilizaion, emperaure measuremens, remoe sensing for asronomy and space exploraion, food processing and cryogenic engineering, as well as numerous agriculural, healh and miliary applicaions. England and Emery (1969) sudied he hermal radiaion effecs of an opically hin gray gas bounded by a saionary verical plae. Radiaion effecs on mixed convecion along an isohermal verical plae were sudied by Hossain and Takhar (1996). The governing equaions were solved analyically. Das e al. (1996) analyzed radiaion effecs on he flow pas an impulsively sared infinie isohermal verical plae. Chemical reacions can be codified as eiher heerogeneous or homogeneous processes. This depends on wheher hey occur a an inerface or as a single phase volume reacion. In well-mixed sysems, he reacion is heerogeneous, if i akes place a an inerface and homogeneous, if i akes place in a soluion. * To whom correspondence should be addressed

2 78 R.Muhucumaraswamy and E.Geeha Chambre and Young (1958) analyzed a firs order chemical reacion in he neighborhood of a horizonal plae. Das e al. (1994) sudied he effec of a homogeneous firs order chemical reacion on he flow pas an impulsively sared infinie verical plae wih uniform hea flux and mass ransfer. Again, mass ransfer effecs on a moving isohermal verical plae in he presence of chemical reacion were sudied by Das e al. (1999). The dimensionless governing equaions were solved by he usual Laplace-ransform echnique. MHD plays an imporan role in agriculure, peroleum indusries, geophysics and in asrophysics. Imporan applicaions in he sudy of geological formaions, in exploraion and hermal recovery of oil,and in he assessmen of aquifers,geohermal reservoirs and underground nuclear wase sorage sies. MHD flow has applicaions in merology, solar physics and in moion of earh core. Also i has applicaions in he field of sellar and planeary magneospheres, aeronauics, chemical engineering and elecronics. In he field of power generaion, MHD is receiving considerable aenion due o he possibiliies i offers for much higher hermal efficiencies in power plans. Gupa e al. (1971) sudied free convecion on he flow pas a linearly acceleraed verical plae in he presence of viscous dissipaive hea using he perurbaion mehod. Kafousias and Rapis (1981) exended he above problem o include mass ransfer effecs subjeced o variable sucion or injecion. Free convecion effecs on he flow pas an acceleraed verical plae wih variable sucion and uniform hea flux in he presence of a magneic field were sudied by Rapis e al. (1981). MHD effecs on he flow pas an infinie verical plae for boh he classes of impulse as well as acceleraed moion of he plae were sudied by Rapis and Singh (1981). Mass ransfer effecs on he flow pas an uniformly acceleraed verical plae were sudied by Soundalgekar (198). Again, mass ransfer effecs on he flow pas an acceleraed verical plae wih uniform hea flux were analyzed by Singh and Singh (1983). Basanh and Prasad (199) analyzed mass ransfer effecs on he flow pas an acceleraed infinie verical plae wih hea sources. Recenly, Muhucumaraswamy e al. (11) sudied an exac soluion of a hydromagneic flow pas an acceleraed isohermal verical plae in he presence of variable mass diffusion. Hence, i is proposed o sudy hermal radiaion and MHD effecs on an unseady flow pas a linearly acceleraed infinie verical plae wih variable emperaure and mass difusion in he presence of a chemical reacion of firs order. The dimensionless governing equaions are solved using he Laplace-ransform echnique. The soluions are in erms of exponenial and complemenary error funcions.. Mahemaical formulaion Here he unseady flow of a viscous incompressible fluid which is iniially a res and surrounds an infinie verical plae wih emperaure T and concenraion C is sudied. The x-axis is aken along he plae in he verically upward direcion and he y-axis is aken normal o he plae. A ime, he plae is 3 u acceleraed wih a velociy u=, in is own plane agains he graviaional field and he emperaure from he plae is raised linearly wih ime and he concenraion level near he plae is also raised o C w. The plae is also subjeced o a uniform ransverse magneic field of srengh B. The fluid considered here is a gray, absorbing-emiing radiaion bu a non-scaering medium. I is assumed ha he effec of viscous dissipaion is negligible in he energy equaion and here is a firs order chemical reacion beween he diffusing species and he fluid. Then by usual Boussinesq's approximaion, he unseady flow is governed by he following equaions u u B =g TT g C C u y *, (.1) T T C p =k y qr, y (.)

3 Chemical reacion effecs on MHD flow pas a linearly acceleraed 79 C C D Kl CC. y (.3) In mos cases of chemical reacions, he rae of reacion depends on he concenraion of he species iself. A reacion is said o be of he order n, if he reacion rae is proporional o he n h power of he concenraion. In paricular, a reacion is said o be firs order, if he rae of reacion is direcly proporional o concenraion iself. The prescribed iniial and boundary condiions are as follows u=, T=T, C =C for all y,, 3 u >: u=, TT TwTA, CCw a y=, (.4) u, T T, CC as y where, u A= v. On inroducing he following non-dimensional quaniies u u yu T T U=, =, Y=, =, u T T w * C C g Tw T g Cw C Gr =, C=, Gc =, 3 C C 3 u u w (.5) * 3 16a v T C v B v vk R=, Pr =, Sc=, M =, K= k D ku u u p l The local radian for he case of an opically hin gray gas is expressed by y q r = 4a * T 4 T 4. (.6) 4 I is assumed ha he emperaure differences wihin he flow are sufficienly small such ha T 4 may be expressed as a linear funcion of he emperaure. This is accomplished by expanding T in a Taylor series abou T and neglecing higher-order erms, hus T 4T T 3T. (.7) By using Eqs (.6) and (.7), Eq.(.) reduces o

4 73 R.Muhucumaraswamy and E.Geeha T T * 3 C p =k 16a T T T, y (.8) in Eqs (.1) o (.4), leads o U U = Gr Gc C MU, Y 1 R =, Pr Y Pr C 1 C = KC. Sc Y (.9) (.1) (.11) The iniial and boundary condiions in non-dimensional quaniies are U=, =, C= for all Y,, >: U=, =, C=1 a Y=, U,, C as Y. (.1) 3. Mehod of soluion The resuling soluions are in erms of exponenial and complemenary error funcion. The relaion beween he error funcion and is complemenary error funcion is as follows erfc(x) =1-erfc(x). The dimensionless governing Eqs (.9) o (.11), subjec o he iniial and boundary condiions (.1), are solved by he usual Laplace-ransform echnique and he soluions are derived as follows = exp R erfc Pr a exp R erfc Pr a (3.1) R a R a Pr exp erfc Pr exp erfc Pr, R 1 C = exp KSc erfc Sc K exp KSc erfc Sc K, (3.)

5 Chemical reacion effecs on MHD flow pas a linearly acceleraed 731 Merfc M erfc exp U= 1 bd e exp M M where and bd exp M erfc M exp M erfc M M exp M cerfc M c eexpc exp M cerfc M c exp ( M b) erfc M b dexpb exp M b erfc Mb exp Pr aberfc Pr ab dexpb exp Pr aberfc Pr ab exp ScKcerfc Sc Kc eexpc exp Sc( Kc) erfc Sc Kc eexp KSc erfc Sc K exp KSc erfc Sc K exp.erfc Pr exp.erfc Pr Pr exp R erfc Pr a exp R erfc Pr a d 1b R a R a bd R R M R M KSc Gr Gc a, b, c, d, e, Pr Pr1 Sc 1 b ( 1 Pr) c( 1Sc) =Y /. (3.3) 4. Discussion of resuls To inerpre he resuls for a beer undersanding of he problem, numerical compuaions are carried ou for differen physical parameers Gr, Gc, Sc, K, R, M and upon he naure of he flow and ranspor. The value of he Schmid number Sc is aken o be.6 which corresponds o waer-vapor. The value of he Prandl number Pr is chosen such ha i represens air (Pr =.71).The numerical values of he velociy, emperaure and concenraion are compued for he above menioned parameers. The concenraion profiles for differen values of he chemical reacion parmeer (K=.,,5), Sc=.6 and =. are shown in Fig.1. The effec of he chemical reacion parameer plays an imporan role in he concenraion field. I is observed ha he plae concenraion decreases wih increasing values of he chemical reacion parameer. Figure illusraes he effec of he concenraion profiles for differen values of he Schmid number (Sc=.16,.6,.1), K= and =.. The profiles have he common feaure ha he

6 73 R.Muhucumaraswamy and E.Geeha concenraion decreases in a monoone fashion from he surface o a zero value far away in he free sream. I is observed ha he concenraion increases wih he decreasing Schmid number C K Fig.1. Concenraion profiles for differen values of K. C Sc Fig.. Concenraion profiles for differen values of Sc. The emperaure profiles are calculaed for differen values of he hermal radiaion parameer (R=.,, 5) a ime =. and hese are shown in Fig.3. I is observed ha he emperaure increases wih he decreasing radiaion parameer. The rend shows ha here is a fall in plae emperaure due o higher hermal radiaion.

7 Chemical reacion effecs on MHD flow pas a linearly acceleraed R Fig.3. Temperaure profile for differen values of R. The velociy profiles for differen values of ime (., 4., 6. ), K, Gr, Gc 5, R 5 and M are sudied and presened in Fig.4. I is observed ha velociy increases wih increasing values of ime U Fig.4. Velociy profiles for differen values of. Figure 5 demonsraes he effec of velociy profiles for differen values of he chemical reacion parameer ( K 51,, ), Gr,Gc 5, R5, M and =.. I is observed ha velociy increases wih decreasing values of he chemical reacion parameer. The rend shows ha here is a fall in velociy due o increasing values of he chemical reacion parameer.

8 734 R.Muhucumaraswamy and E.Geeha.35.3 U K Fig.5. Velociy profiles for differen values of K. Figure 6 illusraes he effecs of he magneic field parameer on velociy when ( M=47),,, R=K=1, Gr=, Gc=5 and =.. I is observed ha velociy increases wih decreasing values of he magneic field parameer. This shows ha he increase in he magneic field parameer leads o a fall in velociy. This agrees wih he expecaions, since he magneic field exers a rearding force on he free convecive flow. Figure 7 demonsraes he effecs of he radiaion parameer on velociy when ( R =, 15, 45), M=1, K=1, Gr=Gc=1 and =.. I is observed ha velociy increases wih he decreasing hermal radiaion parameer. The rend shows ha velociy is supressed due o higher hermal radiaion..5. U.15.1 M Fig.6. Velociy profiles for differen values of M.

9 Chemical reacion effecs on MHD flow pas a linearly acceleraed R U Concluding remarks Fig.7. Velociy profiles for differen values of R. An exac soluion of a hermal radiaion and hydromagneic flow pas a linearly acceleraed infinie verical plae wih variable emperaure and uniform mass diffusion, in he presence of a chemical reacion of firs order is given. The dimensionless governing equaions are solved by he usual Laplace-ransform echnique. The effec of differen parameers such as hermal Grashof number, mass Grashof number, chemical reacion parameer, radiaion parameer, magneic field parameer and are sudied graphically. The conclusions of he sudy are as follows: (I) The emperaure of he plae decreases wih increasing values of he hermal radiaion parameer. (II) The concenraion near he plae increases wih decreasing values of he chemical reacion parameer or Schmid number. (III) Velociy increases wih decreasing values of he magneic field parameer or chemical reacion parameer or hermal radiaion parameer. Bu he rend is jus reversed wih respec o ime. Nomenclaure a A consan absorpion coefficien C dimensionless concenraion C species concenraion in he fluid specific hea a consan pressure C p C w concenraion of he plae C concenraion of he fluid far away form he plae D mass diffusion coefficien erfc complemenary error funcion Gr mass Grashof number Gc hermal Grashof number

10 736 R.Muhucumaraswamy and E.Geeha g accelerraed due o graviy K chemical reacion parameer k hermal conduciviy M magneic field parameer Pr Prandl number q r radiaive hea flux in he y-direcion Sc Schmid number T emperaure of he fluid near he plae T w concenraion of he plae T concenraion of he fluid far away form he plae dimensionless ime ' ime U dimensionless velociy u velociy of he fluid in he x-direcion u velociy of he plae x spaial coordinae along he plae y dimensionless coordinae axis normal o he plae y coordinae axis normal o he plae m volumeric coefficien of hermal expansion volumeric coefficien of expansion wih concenraion similariy parameer dimensionless emperaure cofficien of viscosiy densiy of he fluid dimensionless skin-fricion kinemaic viscosiy References Basanh K.J. and Prasad R. (199): Free convecion and mass ransfer effecs on he flow pas an acceleraed verical plae wih hea sources. Mechanics Research Communicaions, vol.17. pp Chambre P.L. and Young J.D. (1958): On he diffusion of a chemically reacive species in a laminar boundary layer flow. The Physics of Fluids, vol.1, pp Das U.N., Deka R.K. and Soundalgekar V.M. (1999): Effecs of mass ransfer on flow pas an impulsively sared infinie verical plae wih chemical reacion. The Bullein of Guma, vol.5, pp.13-. Das U.N., Deka R.K. and Soundalgekar V.M.(1996): Radiaion effecs on flow pas an impulsively sared verical infinie plae. J. Theo. Mech., vol.1, pp Das U.N., Deka R.K. and Soundalgekar V.M. (1994): Effecs of mass ransfer on flow pas an impulsively sared infinie verical plae wih consan hea flux and chemical reacion. Forschung im Ingenieurwesen, vol.6, pp England W.G. and Emery A.F. (1969): Thermal radiaion effecs on he laminar free. Convecion boundary layer of an absorbing gas. J. Hea Transfer, vol.91,pp Gupa A.S, Pop I. and Soundalgekar V.M. (1979): Free convecion effecs on he flow pas an acceleraed verical plae in an incompressible dissipaive fluid. Rev. Roum. Sci. Techn.-Mec. Apl., vol.4, pp Hossain M.A. and Takhar H.S. (1996): Radiaion effec on mixed convecion along a verical plae wih uniform surface emperaure. Hea and Mass Transfer, vol.31, pp Kafousias N.G. and Rapis A. (1981): Mass ransfer and free convecion effecs on he flow pas an acceleraed verical infinie plae wih variable sucion or injecion. Rev. Roum. Sci. Techn.-Mec. Apl., vol.6, pp.11-.

11 Chemical reacion effecs on MHD flow pas a linearly acceleraed 737 Muhucumaraswamy R., Sundar R. M. and Subramanian V.S.A. (11): Magneohydrodynamic convecive flow pas an acceleraed isohermal verical plae wih variable mass diffusion. Inernaional Journal of Applied Mechanics and Engineering, vol.16, pp Rapis A. and Singh A.K. (1981). MHD free convecion flow pas an acceleraed verical plae. Leers in Hea and Mass Transfer, vol.8. pp Rapis A., Tzivanidis G.J. and Peridikis C.P. (1981): Hydromagneic free convecion flow pas an acceleraed verical infinie plae wih variable sucion and hea flux. Leers in Hea and Mass Transfer, vol.8, pp Singh AK. and Singh J. (1983): Mass ransfer effecs on he flow pas an acceleraed verical plae wih consan hea flux. Asrophysics and Space Science, vol.97. pp Soundalgekar V.M. (198): Effecs of mass ransfer on flow pas a uniformly acceleraed verical plae. Leers in Hea and Mass Transfer, vol.9. pp Received: December 31, 11 Revised: May 15, 13

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