CHEMICAL REACTION EFFECTS ON UNSTEADY MHD FREE CONVECTIVE FLOW PAST AN EXPONENTIALLY ACCELERATED VERTICAL POROUS PLATE

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1 Asian Jounal of Cuent Engineeing and Maths : Jan Feb (3) Contents lists available at ASIAN JOURNAL OF CURRENT ENGINEERING AND MATHS Jounal homepage: CHEMICAL REACTION EFFECTS ON UNSTEADY MHD FREE CONVECTIVE FLOW PAST AN EXPONENTIALLY ACCELERATED VERTICAL POROUS PLATE A.Venkata Sinivasa Rao, G. Venkata Ramana Redd, K. Jaaami Redd 3 Kishna Chaithana institute of Sciences and Technolog, Makapu, (India),3 Depatment of Mathematics, KL Univesit, Guntu (India) ARTICLE INFO Coesponding Autho A Venkata Sinivasa Rao Kishna Chaithana Institute of Sciences and Technolog, Makapu, AP,(India).5336 Ke Wods: MHD, themal adiation, chemical eaction, vaiable tempeatue, vaiable mass diffusion. ABSTRACT An analtical stud is pefomed to investigate the effects of chemical eaction and adiation on unstead MHD flow past an exponentiall acceleated vetical plate with vaiable tempeatue and vaiable mass diffusion in the pesence of applied tansvese magnetic field. It is assumed that the effect of viscous dissipation is negligible in the eneg equation and thee is a fist ode chemical eaction between the diffusing species and the fluid. The flow is assumed to be in x diection which is taken along the infinite vetical plate in the upwad diection and axis is taken nomal to the plate. At time t >, the both tempeatue and species concentation levels nea the plate ae aised lineal with time t. A geneal exact solution of the govening patial diffeential equations is obtained b usual closed analtical method. The velocit, tempeatue and concentation fields ae studied fo diffeent phsical paametes like themal Gashof numbe (G), mass Gashof numbe (Gm), Schmidt numbe (Sc), Pandtl numbe (P), adiation paamete (R), magnetic field paamete (M), acceleated paamete (a), heat absoption paamete, chemical eaction paamete (K) and time (t) gaphicall.. INTRODUCTION Fee convection flows ae of geat inteest in a numbe of industial applications such as fibe and ganula insulation, geothemal sstems etc. Buoanc is also of impotance in an envionment whee diffeences between land and ai tempeatues can give ise to complicated flow pattens. Magneto hdo-dnamic has attacted the attention of a lage numbe of scholas due to its divese applications. In astophsics and geophsics, it is applied to stud the stella and sola stuctues, intestella matte, adio popagation though the ionosphee etc. In engineeing it finds its applications in MHD pumps, MHD beaings etc. The phenomenon of mass tansfe is also ve common in theo of stella stuctue and obsevable effects ae detectable, at least on the sola suface. The stud of effects of magnetic field on fee convection flow is impotant in liquid metals, electoltes and ionized gases. The themal phsics of hdo magnetic poblems with mass tansfe is of inteest in powe engineeing and metallug. Diffusion ates can be alteed temendousl b chemical eactions. The effect of a chemical eaction depend whethe the eaction is homogeneous o heteogeneous. This depends on whethe the occu at an inteface o as a single phase volume eaction. In wellmixed sstems, the eaction is heteogeneous, if it takes place at an inteface and homogeneous, if it takes place in solution. In most cases of chemical eactions, the eaction 3, AJCEM, All Right Reseved. ate depends on the concentation of the species itself. A eaction is said to be of the ode n, if the eaction ate is popotional to the nth powe of the concentation. In paticula, a eaction is said to be fist ode, if the ate of eaction is diectl popotional to the concentation itself. A few epesentative fields of inteest in which combined heat and mass tansfe along with chemical eaction pla an impotant ole in chemical pocess industies such as food pocessing and polme poduction. Chambe and Young [] have analzed a fist ode chemical eaction in the neighbouhood of a hoizontal plate. Gupta [] studied fee convection on flow past a lineal acceleated vetical plate in the pesence of viscous dissipative heat using petubation method. Kafousias and Raptis [3] extended this poblem to include mass tansfe effects subjected to vaiable suction o injection. Fee convection effects on flow past an exponentiall acceleated vetical plate was studied b Singh and Naveen kuma [4]. The skin fiction fo acceleated vetical plate has been studied analticall b Hossain and Shao [5]. Basant kuma Jha [6] studied MHD fee convection and mass tansfom flow though a poous medium. Late Basant kuma Jha et al. [7] analzed mass tansfe effects on exponentiall acceleated infinite vetical plate with constant heat flux and unifom mass diffusion. Das et al. [8] have studied the effect of homogeneous fist ode chemical eaction on the flow past an impulsivel stated 3

2 Guampati et.al/chemical Reaction Effects On Unstead Mhd Fee Convective Flow Past An Exponentiall infinite vetical plate with unifom heat flux and mass tansfe. Again, mass tansfe effects on moving isothemal vetical plate in the pesence of chemical eaction studied b Das et al. [9]. The dimensionless govening equations wee solved b the usual Laplace Tansfom technique. Muthucumaaswam and Senthil Kuma [] studied heat and mass tansfe effects on moving vetical plate in the pesence of themal adiation. Recentl Muthucumaaswam et al. [] studied mass tansfe effects on exponentiall acceleated isothemal vetical plate. The object of the pesent pape is to stud the effects chemical eaction and a unifom tansvese magnetic field (fixed elative to the plate) on the fee convection and mass tansfom flow past an exponentiall acceleated vetical plate. The dimensionless govening equations ae solved using the closed analtical method and discussed qualitativel.. Fomulation of the Poblem We conside the unstead hdo magnetic adiative flow of viscous incompessible fluid past an exponentiall acceleated infinite vetical plate with vaiable tempeatue and also with vaiable mass diffusion and heat souce paamete in the pesence of chemical eaction of fist ode and applied tansvese magnetic field. Initiall, the plate and the fluid ae at the same tempeatue T in the stationa condition with concentation level C at all the points. At timet, the plate is exponentiall acceleated with a velocit u u exp( a t ) in its own plane and at the same time the tempeatue of the fluid nea the plate is aised lineal with time t and species concentation level nea the plate is also inceased lineal with time. All the phsical popeties of the fluid ae consideed to be constant except the influence of the bod foce tem. A tansvese magnetic field of unifom stength B is assumed to be applied nomal to the plate. Then unde usual Boussinesq s appoximation, the unstead flow is govened b the following set of equations. u u B g T T g C C u () t T T q CP k Q TT () t C C D K C C t (3) with the following initial and bounda conditions: t : u, T T, C C foall w w, u, T T, C C as t (4) u u exp( at), T T T T At, C ' C ( C C ) At at u whee A, u - velocit of the fluid in the x diection, - the dimensional distance along the coodinate axis nomal to the plate, g the gavitational acceleation, ρ - the fluid densit, and ae the themal and concentation expansion coefficients espectivel, B - the magnetic induction, T - tempeatue of the fluid in the x diection, C the 3 coesponding concentation, C concentation of the fluid fa awa fom the plate, σ is the electic conductivit, C - the specific heat at constant pessue, D -the diffusion p coefficient, q - the heat flux, Q -the dimensional heat absoption coefficient, and K is the chemical eaction paamete. The local adiant fo the case of an opticall thin ga gas is expessed b q 4 4 4a T T (5) It is assumed that the tempeatue diffeences within the 4 flow ae sufficientl small and that T ma be expessed as a linea function of the tempeatue. This is obtained b 4 expanding T in a Talo seies about T and neglecting the highe ode tems, thus we get T 4T T 3T (6) Fom equations (5) and (6), equation () educes to T T 3 Cp 6a T T T (7) On intoducing the following non-dimensional quantities u tu u T T C C a u, t,,,, a u T w T C w C u (8) g ( T w T) g ( C w C ) C p G, Gm,P, 3 3 u u k B 6a T K Q Sc, M, R, K, Q, D u u u C u 3 p we get the following govening equations which ae dimensionless u u G Gm Mu (9) t R Q t P P () K () t Sc The initial and bounda conditions in dimensionless fom ae as follows: t : u,, foall () u exp( at), t, t at t, u, as M G Gm R Q Sc K denote the whee,,,,p,, and magnetic field paamete, themal Gashof numbe, mass Gashof numbe, adiation paamete, Pandtl numbe, heat absoption paamete, Schmidt numbe and chemical eaction paamete espectivel. 3. Solution of the Poblem In ode to educe the above sstem of patial diffeential equations to a sstem of odina equations in dimensionless fom, we ma epesent the velocit, tempeatue and concentation as u(, t) u ( ) e it (3) e (4) e (5) (, t) ( ) i t (, t) ( ) i t

3 Guampati et.al/chemical Reaction Effects On Unstead Mhd Fee Convective Flow Past An Exponentiall Substituting Eqns (3), (4) and (5) in Eqns (9), () and (), we obtain: u N u G Gm (6) 3 (7) N C N C (8) Hee the pimes denote the diffeentiation with espect to. The coesponding bounda conditions can be witten as ai t it it u e, te, te at u,, as (9) The analtical solutions of equations (6) (8) with satisfing the bounda conditions (9) ae given b N 3 N N u ( ) A e e i t Ae A e te it 3 4 () N te e i t () ( ) e N i t t e () In view of the above solutions, the velocit, tempeatue and concentation distibutions in the bounda lae become N 3 N N u(, t) A e Ae A e t (3), t N te (, t) t e N 3 (4) (5) It is now impotant to calculate the phsical quantities of pima inteest, which ae the local wall shea stess, the local suface heat, and mass flux. Given the velocit field in the bounda lae, we can now calculate the local wall shea stess ( i.e., skin- fiction) is given b u w, and in dimensionless fom, we obtain u C f N3A3 NA NA t Fom tempeatue field, now we stud the ate of heat tansfe which is given in non -dimensional fom as: Nu tn Fom concentation field, now we stud the ate of mass tansfe which is given in non -dimensional fom as: Sh whee tn paamete at /, N Sc K i, N P i R, N M i, A G / N N, 3 3 A Gm N N A e t A A 3 3 RESULTS AND DISCUSSION 3 In ode to get phsical insight into the poblem, we have plotted velocit pofiles fo diffeent values of the phsical paametes a(acceleated paamete), M(magnetic field paamete) R(adiation paamete), G(themal Gashof numbe), Gm(mass Gashof numbe), P(Pandtl numbe), Sc(Schmidt numbe), t(time) and the chemical eaction paamete (K) is pesented gaphicall. The value of Sc (Schmidt numbe) is taken to be.6 which coesponds to the wate vapo. Also, the value of P (Pandtl numbe) ae chosen such that the epesent ai (P=.7). The effect of the magnetic paamete M is shown in Fig.. It is obseved that the tangential velocit of the fluid deceases with the incease of the magnetic field numbe values. The decease in the tangential velocit as the magnetic paamete M inceases is because the pesence of a magnetic field in an electicall conducting fluid intoduces a foce called the Loentz foce, which acts against the flow if the magnetic field is applied in the nomal diection, as in the pesent stud. This esistive foce slows down the fluid velocit component as shown in Fig.. Fig. shows the effect of the acceleated paamete a on the velocit distibution. It is found that the velocit inceases with an incease in an acceleated paamete a. Fo vaious values of the themal Gashof numbe G and mass Gashof numbegm, the velocit pofiles u ae plotted in Figs. (3) and (4). The themal Gashof numbe G signifies the elative effect of the themal buoanc foce to the viscous hdodnamic foce in the bounda lae. As expected, it is obseved that thee is a ise in the velocit due to the enhancement of themal buoanc foce. Also, as G inceases, the peak values of the velocit inceases apidl nea to the plate and then decas smoothl to the fee steam velocit. The mass Gashof numbe Gm defines the atio of the species buoanc foce to the viscous hdodnamic foce. As expected, the fluid velocit inceases and the peak value is moe distinctive due to incease in the species buoanc foce. The velocit distibution attains a distinctive maximum value in the vicinit of the plate and then deceases popel to appoach the fee steam value. It is noticed that the velocit inceases with inceasing values of the mass Gashof numbe. Fig.5 shows the effect of the time t on the velocit field. It is found that the velocit inceases with an incease in time t. Fig.6. illustates the velocit pofiles fo diffeent values of adiation paamete R, the numeical esults show that the effect of inceasing values of adiation paamete esult in a deceasing velocit. Fo diffeent values of the Pandtl numbe P the velocit pofiles ae plotted in Fig.7. It is obvious that an incease in the Pandtl numbe P esults in decease in the velocit within the bounda lae. Fig.8 illustates the behavio velocit fo diffeent values of chemical eaction K. It is obseved that an incease in leads to a decease in the values of velocit. Fo diffeent values of the Schmidt numbe Sc the velocit pofiles ae plotted in Fig.9. It is obvious that an incease in the Schmidt numbe Sc esults in decease in the velocit within the bounda lae. Fo diffeent values of fequenc of excitation on the tempeatue pofiles ae shown in Fig.. It is

4 Velocit Velocit Velocit Guampati et.al/chemical Reaction Effects On Unstead Mhd Fee Convective Flow Past An Exponentiall noticed that an incease in the fequenc of excitation esults a decease in the tempeatue field. Fig. shows the behavio of the tempeatue fo diffeent values Pandtl numbe P. It is obseved that an incease in the Pandtl numbe esults a decease of the themal bounda lae thickness and in geneal lowe aveage tempeatue within the bounda lae. The eason is that smalle values of P ae equivalent to incease in the themal conductivit of the fluid and theefoe, heat is able to diffuse awa fom the heated suface moe apidl fo highe values of P. Hence in the case of smalle Pandtl numbe as the themal bounda late is thicke and the ate of heat tansfe is educed. Fo diffeent values of themal adiation R the tempeatue pofiles ae shown in Fig.. It is obseved that an incease in the themal adiation paamete, the decease of the tempeatue The effect of the Schmidt numbe Sc on the concentation is shown in Fig.3. As the Schmidt numbe inceases, the concentation deceases. This causes the concentation buoanc effects to decease ielding a eduction in the fluid velocit. A eduction in the concentation distibution is accompanied b simultaneous eduction in the concentation bounda laes. The effect of chemical eaction paamete K on the concentation pofiles as shown in Fig.4. It is obseved that an incease in leads to a decease in the values of concentation. REFERENCES []. P. L. Chambe, J. D. Young, On the diffusion of a chemicall eactive species in a lamina bounda lae flow, The Phsics of Fluids, Vol., pp.48-54, 958. []. A. S. Gupta, I. Pop, V. M. Soundalgeka, Fee convection effects on the flow past an acceleated vetical plate in an incompessible dissipative fluid, Rev. Roum. Sci. Techn. -Mec. Apl., 4, pp , 979. [3]. N. G. Kafousias, A.A. Raptis, Mass tansfe and fee convection effects on the flow past an acceleated vetical infinite plate with vaiable suction o injection, Rev. Roum. Sci. Techn.- Mec. Apl., 6, pp.-, 98. [4]. A. K. Singh, N. Kuma, Fee convection flow past an exponentiall acceleated vetical plate, Astophsics and Space science, 98, pp , 984. [5]. M. A. Hossain, L. K. Shao, The Skin fiction in the unstead fee convection flow past an acceleated plate, Astophsics and Space Science, 5, pp , 986. [6]. B.K. Jha, MHD fee-convection and mass tansfom flow though a poous medium, Astophsics and Space science, 75, 83-89, 99. [7]. B.K. Jha, R. Pasad, S. RAI, Mass tansfe effects on the flow past an exponentiall acceleated vetical plate with constant heat flux, Astophsics and Space Science, 8, pp.5-34, 99. [8]. U. N. Das, R. K. Deka, V. M. Soundalgeka, Effects of mass tansfe on flow past an impulsivel stated infinite vetical plate with constant heat flux and chemical eaction, Foschung in Ingenieuwsen, Vol.6, pp.84-87, 994. [9]. U. N. Das, R. K. Deka, V. M. Soundalgeka, Effects of mass tansfe on flow past an impulsivel stated infinite vetical plate with chemical eaction, The Bulletin of GUMA, Vol.5, pp.3-, 999. []. R. Muthucumaaswam, G. Senthil kuma, Heat and Mass tansfe effects on moving vetical plate in the pesence of themal adiation, Theoet. Appl. Mech., Vol.3, No., pp.35-46, 4. []. R. Muthucumaaswam, K.E. Sathappan, R. Nataajan, Mass tansfe effects on exponentiall acceleated isothemal vetical plate, Int. J. of Appl. Math. and Mech., 4(6), 9-5, Sc=.6;P=.7;R=;K=5; t=.;a=.;g=5;gm=5;=; M=,, 3, Fig.. Effects of magnetic paamete on velocit a =.,.5,.9,.5 R=5;Sc=.6;M=4;P=.7; t=.4;k=5;g=;gm=5;=; Fig.. Effects of acceleated paamete on velocit R=5;Sc=.6;M=4;P=.7; t=.4;k=5;a=.;gm=5;=; G = 5,, 5, Fig. 3. Effects of Themal Gashof numbe on velocit 33

5 Velocit Velocit Velocit Velocit Velocit Velocit Guampati et.al/chemical Reaction Effects On Unstead Mhd Fee Convective Flow Past An Exponentiall.4.4. R=5;Sc=.6;M=4;P=.7; t=.4;k=5;a=.;g=;=;. R=5;Sc=.6;M=4;K=5; t=.4;a=.5;g=;gm=5;=; Gm = 5,, 5,.8.8 P =.7,. 3., Fig. 4. Effects of mass Gashof numbe on velocit Fig.7. Effects of Pandtl numbe on velocit.5 R=5;Sc=.6;M=4;P=.7; Gm=5;K=5;a=.;G=;=;.4. R=5;Sc=.6;M=4;P=.7; t=.4;a=.5;g=;gm=5;=; t = K =, 5,, Fig.5. Effects of time on velocit Fig.8. Effects of chemical eaction paamete on velocit.4. Sc=.6;P=.7;M=4;K=5; t=.;a=.;g=;gm=5;=;. R=5;P=.7;M=4;K=5; t=.4;a=.5;g=;gm=5;=; Sc =.,.3,.6,.78.4 R=,, 3, Fig.6. Effects of adiation paamete on velocit Fig.9. Effects of Schmidt numbe on velocit 34

6 Tempeatue Tempeatue Concentation Tempeatue Concentation Guampati et.al/chemical Reaction Effects On Unstead Mhd Fee Convective Flow Past An Exponentiall R=;Sc=.6;M=4;P=.7; Gm=5;K=5;a=.;G=;t=.;.8.6 P=.7;K=5;M=4;R=5; Gm=5;=;a=.;G=;t=.; =,, 3,4..8 Sc=.,.3,.6, Fig.. Effects of fequenc of excitation on tempeatue R=;Sc=.6;M=4;=; Gm=5;K=5;a=.;G=;t=.; P =.7,., 3., Fig.. Effects of Pandtl numbe on tempeatue Fig.3. Effects of Schmidt numbe on concentation P=.7;Sc=.6;M=4;R=5; Gm=5;=;a=.;G=;t=.; K =,, 3, Fig.4. Effects of chemical eaction paamete on concentation.8.6 P=.7;Sc=.6;M=4;=; Gm=5;K=5;a=.;G=;t=.; R=,, 3, Fig.. Effects of adiation paamete on tempeatue 35