International Journal of Technical Research and Applications e-issn: , Special Issue 19 (June, 2015), PP.

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1 Intentionl Jounl of Technicl Resech nd Applictions e-issn: 3-863,.ijt.com Specil Issue 9 (June, 5), PP HEAT AND MASS TRANSFER FOR SORET, DUFOUR S AND MAGNETCI EFFECTS IN TRANSIENT FLOW OF CONDUCTING FLUID OVER A STRETCHING SHEET EMBEDDED IN A POROUS MEDIUM WITH CHEMICALLY REACTIVE SPICES D. K. Phukn Pincipl cum Associte pofesso, Demo College, Sivsg, Assm, Indi ddevknt@yhoo.com. Abstct- An investigtion is mde to cy out to study the theml-diffusion nd diffusion themo-effects in hydo-mgnetic tnsient flo by mixed convection boundy lye pst n impemeble veticl stetching sheet embedded in conducting fluid-stuted poous medium in the pesence of chemicl ection effect. The velocity of stetching sufce, the sufce tempetue nd the concenttion e diectly popotionl to the distnce long the sufce. The flo is impulsively set into motion est, nd both the tempetue nd concenttion t the sufce e lso suddenly chnged fom tht of the mbient fluid. An extenl mgnetic field of stength is pplied pependicul to the stetching sheet. Intoducing non- dimensionl pmetes, the govening set of ptil diffeentil eqution e tnsfomed into the self-simil unstedy boundy lye equtions. These equtions e solved by Runge-kutt integtion scheme ith shooting method fo the hole tnsient flo fom initil stte( ) to finl stedy stte flo ( fo the velocity, tempetue, nd Concenttion pofiles e pesented gphiclly fo diffeent existing flo pmete. A specil cse of ou esults is in good geement ith n elie published ok. Key ods: Het nd mss tnsfe, boundy lye flo, poous medi, mgnetic field Soet numbe nd Dufou s numbe. ). Numeicl esults I. INTRODUCTION Duing ecent yes of studies, the effect of mgnetic field on the flo of viscous fluid ith het nd mss tnsfe though unifom poous medi hs become the subject of get inteest due to ide ppliction in the fst going field of science nd technology. Numeous publictions hs been ppeed in the leding jounl in developed county. It ppes tht knoledge of the effect of n pplied mgnetic field on flo, mss nd het tnsfe is useful fo cooling pocesses in the pesence of n electolytic bth. In some metllugicl pocesses, such s ding, nneling nd those tht involve the cooling of continuous stips of filment by ding tinning of coppe ies etc, the popeties in quiescent fluid of the finl poduct depend to get extent on the te of cooling. The te of cooling cn be contolled by ding such stips in n electiclly conducting fluid subject to mgnetic field, nd the finl poduct of desied chcteistic cn be chieved. The ecent studies of physics of fluid flo though poous medi hs become bsic fo science nd technology nd of get inteest in pesent dys due to thei engineeing ppliction. One my efe the bnches of ppliction s quife systems in studies of gound te hydology, chemicl engineeing soil mechnics, te puifiction, industil filttion etc. The coupled het nd mss tnsfe phenomenon in poous medi hs dn the ttention of glxy schols/uthos due to it inteesting nd temendous ppliction. The pocesses involving het nd mss tnsfe in poous medi e often encounteed in the chemicl industy, in esevoi engineeing in connection ith theml ecovey pocesses, nd in the study of dynmics of hot nd slty sping of se, undegound speding of chemicl ste nd othe pollutnts, gin stoge, evpotion cooling, nd solidifiction e fe othe ppliction e hee combined themosolutl convection in poous medi is obseved. The exhustive volume of ok devoted to this e by the most ecent books by Nield nd Bejn(999), Vfi(). Futhemoe, the pesence of foeign mss in i o te cuses some kind of chemicl ection. Duing chemicl ection beteen to spices, het is lso geneted. Duffusion nd chemicl in n isotheml lmin flo long soluble flt plte s discussed by Fibnks nd ike(95). Ds et.l. (994) discussed the effects of mss tnsfe on the flo pst n impulsively stted infinite veticl plte ith constnt het flux nd chemicl ection. The flo of mss diffusion of chemicl spices ith fist ode nd {(Pop nd Inghm () nd Inghm nd Pop (998-)} highe ode ections ove linely stetching sufce s investigted by Andesson et.l.(994). The mixed convective het nd mss tnsfe ove hoizontl moving plte ith chemicl ection effect s studied by Fn et.l.(998). Anjlidevi nd Kndsmy (999) studied the stedy lmin flo long semi-infinite hoizontl plte in the pesence of spices concenttion nd chemicl ection. The flo nd mss diffusion of chemicl spices ith fist ode nd highe ode 39 P g e

2 Intentionl Jounl of Technicl Resech nd Applictions e-issn: 3-863,.ijt.com Specil Issue 9 (June, 5), PP ections ove continuously stetching sheet ith n pplied mgnetic field s studied by Thk et.l.(). Muthucumsmy () investigted the effects of chemicl ection on moving isotheml veticl infinitely long sufce ith suction. Chmkh et.l.(4) studied the double-diffusive convective flo of mico-pol fluid ove veticl plte embedded in poous medium ith chemicl ection. The combined effects of the fee convective het nd mss tnsfe on the unstedy boundy lye flo ove stetching sufce in the pesence of species concenttion nd chemicl ection s investigted by Aboeldhb nd Azm(6). Postelnicu (7) nlysed numeiclly the het nd mss tnsfe chcteistics of ntul convection bout veticl sufce embedded in stuted poous medium subjected to chemicl ection. Rshd nd El-kbei() ecently investigted the het nd mss tnsfe in tnsient flo by mixed convection boundy lye ove stetching sheet embedded in poous medium ith chemiclly ective spices. Sllm.N() nlysed the theml-diffusion nd diffusion-themo effects on mixed convection het nd tnsfe in poous medium. The pupose of the pesent ppe is to study the simultneous het nd mss tnsfe by n unstedy mixed convection boundy lye pst n impemeble veticl stetching sheet embedded in conducting fluid stuted poous medium ith chemiclly ective species in the pesence of mgnetic field. II. FORMULATION OF THE PROBLEM We conside unstedy to-dimensionl lmin het nd mss tnsfe by mixed convection boundy lye flo of viscous, incompessible, Netonin conducting fluid pst n impemeble veticl plte stetching in the diection ith positive velocity e U x x of stuted poous medium in the pesence of mgnetic field of unifom stength B hich is pependicul to the diection of flo. The chemicl ection is tking plce in the flo ove the poous medium ith effective mss diffusivity D nd the te of chemicl ection K thoughout the fluid. The mgnetic Reynolds numbe of the flo is tken to be smll enough so tht the induced mgnetic field cn be neglected. In ddition, Joule heting is neglected but Soet nd Dufou s effects e exmined. Fo the mthemticl modeling, e tke Ctesin coodintes (x, y) s shon in fig., hee the positive x-xis is extended long the sheet in the upd diection hile the y-xis is noml to the sufce of the sheet nd is positive in the diection fom the sheet to the fluid. The sttiony coodinte system hs its oigin locted in the cente of the sheet. The sheet is mintined tempetue ( ) concenttion C ( x) C bx, e T x T bx nd nd the mbient medium tempetue nd concenttion f y fom the sufce of the sheett nd C e ssumed to be unifom. Fo T T nd C ( x) C x, n upd (ssisting)flo is induced s esult of the theml nd concenttion buoyncy effects. Initilly (t<), the mbient fluid stuted poous medium is quiescent nd hs tempetue T nd concenttion C espectively. At t=>, the fluid is impulsively stted in motion ith the velocity U( x ), nd both the tempetue nd the concenttion t the sheet suddenly chnged to constnt vlues nd C ( x) C x, espectively. The fluid is ssumed to hve constnt popeties, except fo the influence of the density nd chemicl ection vitions ith tempetue nd concenttion hich e consideed in the body foce tem. Unde the peceding ssumption, the physicl vibles e functions of y nd t only nd the govening boundy lye equtions of mss, momentum, enegy nd diffusion unde Boussinesq ppoximtion could be itten s follos T T u v, x y u u u u B u v g ( ) ( ), T T T gc C C u u t x y y K T T T T DK C u v t x y y C C y e T, s p C C C C DK T u v D e K ( C C ), t x y y T y e T m () () (3) (4) The initil nd boundy conditions of equtions ()-(4) e t : u( x, y, t), v( x, y, t), T( x, y, t) T C( x, y, t) C (5),, t : u( x,, t) Ue( x) x, v( x,, t), T( x,, t) T T bx C( x,, t) C C bx, u( x,, t), T( x,, t) T C( x,, t) C,, (5b) 4 P g e

3 Whee u nd v e the velocity components long the -xis nd -xis espectively;t nd C e the tempetue nd concenttion of the conducting fluid, is the electicl y B conductivity, is the intensity of the unifom mgnetic field, is the density, is the kinemtic viscosity, is the theml diffusivity, T Intentionl Jounl of Technicl Resech nd Applictions e-issn: 3-863,.ijt.com Specil Issue 9 (June, 5), PP x is the theml expnsion coefficient, is the concenttion expnsion co-efficient, K is C the pemebility of the poous medium, K b, ( ) is the dimensionl chemicl ection pmete nd e constnt. Intoducing the floing chnge of vibles s Rsd nd El Kbei() nd Ishk et.l.(6) T T C C t, y, e, x f (, ),,, TT CC Whee is the stem function hich is defined s u y nd v. x The eqution of continuity () is identiclly stisfied hich cn be esily veified. It is convenient fo one to select the time scle so tht the egion of time integtion occu. Using (6) nd (7), the eqution (), (3) nd (4) convets to f f ( ) f ff ( f ) f ( N) Mf ( ), D (8) ( ) f f Df ( ), P (9) ( ) f ( f ) S ( ), Sc () Whee pime denotes diffeentitions ith espect to only, to the viscous foce, is the Pndtl numbe, is the nd the suffix denotes the ptil deivtives ith espect to Dufou s numbe, is the Schmidt numbe fo poous, is the mixed convection pmete, N is the tio of medium, is the dimensionless pmete of chemicl nd buoyncy foce due to mss diffusion to the buoyncy foce due to theml diffusion, is the Dcy numbe, M is the is the Soet numbe hich e defined, espectively s D hydo-mgnetic pmete hich is the tio of Loentz foce gt( T T ) x K ( ) C C C Gx D, N,, P,, Sc T ( T T ) U ( ) Re e x x D () ( C C) De KT DeKT C K DeKT ( T T) DeKT T Df,, S, CsCp ( T T) CsCp T Tm ( C C) TmC Whee De is the effective mss diffusivity, T m is the men f,, f,,,,,,. () fluid density, f C is the concenttion expnsion co-efficient,,,,,, It is noted tht the eqution (8)-() fo T is the theml expnsion co-efficient, KT is the theml M, Df, S educe to those of Rshd nd El-Kbei diffusion tio, D is the fluid mss diffusivity, Gx is the (). Futhemoe, fo M, Df, S, D, N, locl gshof numbe, Re x is the locl Reynolds numbe. educe to those of Ishk et.l, (6) It is obseved tht hen is positive, i.e., it coesponds to the (iding flo) ssisting flo cse. When is negtive i.e., then it coesponds to the opposing flo cse. The boundy condition 5() nd 5(b) in vie of (6) e educed to S S c P III. NUMERICAL SOLUTION The eqution (8)-() togethe ith boundy condition() e the pbolic ptil diffeentil equtions. Insted of solving these ptil diffeentil eqution diectly, e look fo the pticul cse of the poblem hich e the system of (6) (7) 4 P g e

4 Intentionl Jounl of Technicl Resech nd Applictions e-issn: 3-863,.ijt.com Specil Issue 9 (June, 5), PP odiny diffeentil equtions ith set of constints t the A. Unstedy solution t initil stge (stte) hen boundy nd cn be esily solved by the shooting method. The solution pocedue fo the entie time domin When time scle, i.e. fo initil unstedy flo, it is explined in the folloing pt. coesponds to, the eqution (7)-(9) convets to f f, () Df, P (3) S, S c (4) Subject to the boundy conditions f, f,,,. f,, (5) 3.. Solution fo stedy stte hen, coesponding to, i.e. finl stedy flo., When, equtions(8)-() becomes f ff ( D f ) f ( N) Mf, (6) f f Df, P (7) f ( f ) S, S c (8) Subject to the boundy conditions f,, f,,,,,,. f,,,,, (9) 3.3. Solution fo smll ( o ) The ppoximte solutions of eqution (8)-() subject to the boundy condition (), hich e vlid fo the egion, equivlent to the smll time solution, cn be expessed s f (, ) f ( ) f ( ) f ( ) () (, ) ( ) ( ) ( ) () (, ) ( ) ( ) ( ) () Whee f f, (3) Df, P (4) S, S c (5) Subject to the boundy conditions f,,, f,,,,, f (6) 4 P g e

5 D Intentionl Jounl of Technicl Resech nd Applictions e-issn: 3-863,.ijt.com Specil Issue 9 (June, 5), PP f ( f f ) f f f ( N ) f Mf f, (7) ( ) f f D, (8) f P ( ) f f S, S c (9) Subject to the boundy conditions f, f,,,, f,, (3) And f ( f f ) ff ff f f ( N ) f Mf f f, D (3) ( ) f f f f Df, P (3) ( ) f f f f S, S c (33) Subject to the boundy conditions f, f,,,, f,, (34) IV. NUMERICAL SOLUTION The numeicl solution to the boundy vlue poblem of odiny diffeentil equtions is obtined by the Runge kutt method in ssocition ith the shooting technique. It my be noted tht the souce of eo in the simultion my come fom the pescibed boundy conditions t the infinity. The eson is tht the physicl domin unde considetion is unbounded hees the computtionl domin is finite. In fct, the f field boundy condition usully depends on the physicl pmetes of the poblem, nd its vlue needs to be djusted s the vlue of the pmetes chnge. In pctice, the computtionl domin is chosen to be sufficiently lge, so tht the numeicl solution closely ppoximtes the teminl boundy conditions t infinity. Hee boundy condition t the f end hs been fixsed to nd is suitbly less thn depending on the choice of the pmetes. V. RESULTS AND DISCUSSION A epesenttive gphicl esults of velocity pofiles, tempetue pofiles nd concenttion pofiles e pesented in Figs. 3 fo vious existing flo pmetes coss the boundy lye of the conducting fluid. These Figs-3. Demonstte the dvncement of Velocity, tempetue nd concenttion pofiles fom initil to finl stedy stte hen D, S, f P.7, S., D, N,,, M c. Fom figs & 3, it is cle tht the vitions of velocities nd tempetue e obseved to be ise moe fom initil stte )to finl stedy stte ( ) espectively hee s ( the esults pesented in the fig.4 fo the concenttion of the fluid, e obseved to be decline fom initil stte ( ) to finl stedy stte ( f vs ). The vition of velocity pofiles fo vious mgnetic pmete D.5, S.5, P.7, S., D 5, f c M fo fixed N, 8, e shon in Fig.5.The effects of mgnetic field e seen to decese the velocity f coss the boundy lye of the conducting fluid s the Loentz foce etds the motion of the conducting fluid thoughout the boundy lye. The vition of tempetue pofiles vs fo vious mgnetic pmete M fo fixed D.5, S.5, P.7, S., D 5, f c N, 8, e shon in Fig.6. The effects of mgnetic field e seen to incese the tempetue coss the boundy lye of the conducting fluid. The concenttion pofiles vs fo vious mgnetic pmete M fo fixed D.5, S.5, P.7, S., D 5, f c N, 8,, e plotted in Fig.7. The effects of mgnetic field e seen to decese the concenttion coss the boundy lye of the conducting fluid s the Loentz foce etds the motion of the conducting fluid coss the boundy 43 P g e

6 Intentionl Jounl of Technicl Resech nd Applictions e-issn: 3-863,.ijt.com Specil Issue 9 (June, 5), PP lye. Vitions of velocity pofiles f vs nd tempetue pofile vs fo vious Dfou numbe fixed pmete S.5, P.7, D 5, N,,, M fo S.,, e plotted in Figs.8 nd Fig.9 espectively. The effects of Dfou numbe e seen to incese the velocity nd tempetue of the conducting fluid coss the boundy lye. The vitions of Concenttion pofiles vs fo vious Dfou numbe fo fixed pmete S.5, P.7, c S., D 5, N,,, M, e plotted in Figs.. The concenttion e seen to incese ith the incese of Dfou numbe. Vitions of velocity pofiles, vs f, tempetue pofile nd Concenttion pofiles fo vious Soet numbe D.5, P.7, S., f c S c fo fixed pmete D 5, N, 8,, M e plotted in Figs., & 3 espectively. It is obseved fom Fig. tht the velocity distibutions inceses ith the inceses of Soet numbe S f, coss the boundy lye of the conducting fluid. Fig. shos tht the tempetue distibutions inceses ith the inceses of Soet numbe S coss the boundy lye of the conducting fluid. The effects of Soet numbe S e seen to incese the Concenttion pofile coss the boundy lye hich e shon in Fig.3. Fig. Vitions of velocity pofiles fom initil ( to finl stedy stte ( ) When D, S, P.7, S., D, N,,, M. f c f ) Fig.3 Vitions of tempetue pofiles fom initil ( ) to finl stedy stte ( D, S, P.7, S., D, N,,, M f c ) hen Fig. Sketch of flo geomety Fig.4 Vitions of Concenttion pofiles fom initil ( ) to finl stedy stte ( ) hen D, S, P.7, S., D, N,,, M f c 44 P g e

7 Intentionl Jounl of Technicl Resech nd Applictions e-issn: 3-863,.ijt.com Specil Issue 9 (June, 5), PP Fig.5 Vitions of velocities pofiles fom initil ( ) to finl stedy stte ( pmete M hen D.5, S.5, P.7, S., f ) fo vious mgnetic f c D 5, N, 8,, ( Fig.8 Vitions of velocities pofiles )to finl stedy stte ( numbe f fom initil ) fo vious Dfou fo fixed pmete S.5, P.7, S., c D 5, N,,, M. Fig.6 Vitions of tempetue pofiles fom initil ( )to finl stedy stte ( pmete M hen D.5, S.5, P.7, S., ) fo vious mgnetic f c D 5, N, 8,, Fig.9 Vitions of tempetue pofiles fom initil ( ) to finl stedy stte ( numbe ) fo vious Dfou fo fixed pmete S.5, P.7, S., D 5, N,,, M. c Fig.7 Vitions of Concenttion pofiles fom initil ( )to finl stedy stte ( ) fo vious mgnetic pmete M hen D.5, S.5, P.7, S., D 5, N, 8,, f c Fig. Vitions of Concenttion pofiles fom initil ( )to finl stedy stte ( ) fo vious Dfou numbe D f fo fixed pmete S.5, P.7, S., D 5, N,,, M. c 45 P g e

8 Intentionl Jounl of Technicl Resech nd Applictions e-issn: 3-863,.ijt.com Specil Issue 9 (June, 5), PP Fig. Vitions of velocities pofiles ( )to finl stedy stte ( numbe S f fom initil ) fo vious Soet fo fixed pmete D.5, P.7, S., D 5, N, 8,, M. f c Fig. Vitions of tempetue pofiles fom initil ( )to finl stedy stte ( numbe S ) fo vious Soet fo fixed pmete D.5, P.7, S., D 5, N, 8,, M. f c Fig.3 Vitions of Concenttion pofiles fom initil ( )to finl stedy stte ( ) fo vious Soet numbe S fo fixed pmete D.5, P.7, S., D 5, N, 8,, M. f c REFERENCES []. Andesson, K. I., Hnsen, O. R., nd Holmedl, B., Diffusionof chemiclly ective species fom stetching sheet, Int.J. Het Mss Tnsfe, vol. 37, pp , 994. []. Anjlidevi, S. P. nd Kndsmy, R., Effect of chemicl ection, het nd mss tnsfe on lmin flo long semi infinite hoizontl plte convection in poous medium, Het Mss Tnsfe, vol. 8, pp , 999. [3]. Aboeldhb, E. M. nd Azzm, G. E. A., Unstedy theedimensionl combined het nd mss fee convective floove stetching sufce ith time-dependent chemicl ection, Act Mech., vol. 84, pp. 36, 6. [4]. Chmkh, A. J., Al-Mudhf, A., nd Al-Ytm, J., Double diffusive convective flo of mico-pol fluid ove veticl plte embedded in poous medium ith chemicl ection, Int. J. Fluid Mech. Res., vol. 6, pp , 4. [5]. Ds, U. N., Dek, R., nd Soundlgek, V. M., Effects of mss tnsfe on flo pst n impulsive stted infinite veticl plte ith constnt het flux nd chemicl ection, Fosch Ingenieuesen Eng. Res. Bd., vol. 6, pp , 994. [6]. Fibnks, D. F. ndwike, C. R., Diffusion nd chemicl ection in n isotheml lmin flo long soluble flt plte, Ind. Eng. Chem. Res., vol. 4, pp , 95. [7]. Fn, J. R., Shi, J. M., nd Xu, X. Z., Simility solution of mixed convection ith diffusion nd chemicl ection ove hoizontl moving plte, Act Mech., vol. 6, pp , 998. [8]. Inghm, D. nd Pop, I. (eds), Tnspot Phenomen in Poous Medi, vols., Pegmon, Oxfod, 998. [9]. Muthucumsmy, R., Effects of chemicl ection on moving isotheml veticl sufce ith suction, Act Mech., vol. 55, pp. 65 7,. []. Nield, D. A. nd Bejn, A., Convection in Poous Medi, nd ed., Spinge, Belin, 999. []. Pop, I. nd Inghm, D., Convective Het Tnsfe: Mthemticl nd Computtionl Modelling of Viscous Fluids nd Poous Medi, Pegmon, Oxfod,. []. Postelnicu, A., Influence of chemicl ection on het nd mss tnsfe by ntul convection fom veticl sufces in poous medi consideing soet nd dufou effects, Het Mss Tnsfe, vol. 43, pp , 7. [3]. Rshd, A.M., & El- Kbei, S.M.M. Het nd mss tnsfe in tnsient flo by mixed convection boundy lye ove stetching sheet embedded in poous medium ith chemiclly ective species. Jounl of poous medi, Vol.3 No.,pp ,. [4]. Sllm, N. Slm, Theml-diffusion nd diffusion- Themo effects on mixed convection het nd mss tnsfe in poous medium, Jounl of poous medi, Vol.3 No.4, pp ,. [5]. Tkh, H. S., Chmkh, A. J., nd Nth, G., Flo nd mss tnsfe on stetching sheet ith mgnetic field nd chemiclly ective species, Int. J. Eng. Sci., vol. 38,pp ,. [6]. Vfi, K. (ed), Hndbook of Poous Medi, Mcel Dekke, Ne Yok,. 46 P g e