TIME-FRACTIONAL FREE CONVECTION FLOW NEAR A VERTICAL PLATE WITH NEWTONIAN HEATING AND MASS DIFFUSION

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1 Vieru, D., e al.: Time-Fracional Free Convecion Flow near a Verical Plae wih THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S85-S98 S85 TIME-FRACTIONAL FREE CONVECTION FLOW NEAR A VERTICAL PLATE WITH NEWTONIAN HEATING AND MASS DIFFUSION by Dumiru VIERU a, Consanin FETECAU b,c, and Corina FETECAU a a Deparmen of Theoreical Mechanics, Technical Universiy of Iasi, Iasi, Romania b Deparmen of Mahemaics, Technical Universiy of Iasi, Iasi, Romania c Academy of Romanian Scieniss, Bucuresi, Romania Original scienific paper DOI: 1.98/TSCI15S1S85V The ime-fracional free convecion flow of an incompressible viscous fluid near a verical plae wih Newonian heaing and mass diffusion is invesigaed in presence of firs order chemical reacion. The dimensionless emperaure, concenraion, and velociy fields, as well as he skin fricion and he raes of hea and mass ransfer from he plae o he fluid, are deermined using he Laplace ransform echnique. Closed form expressions are esablished in erms of Robonov-Harley and Wrigh funcions. The similar soluions for ordinary fluids are also deermined. Finally, he influence of fracional parameer on he emperaure, concenraion and velociy fields is graphically underlined and discussed. Key words: ime-fracional free convecion flow, Newonian heaing, mass diffusion, chemical reacion Inroducion Free or naural convecion flow of an incompressible viscous fluid near a verical plae was exensively sudied due o is vas indusrial applicaions [1, ]. Since 1971, Gebhar and Pera [3] have sudied he verical naural convecion flow resuling from he combined buoyancy ecs of hermal and mass diffusion. Laer, Chamkha e al. [4] and Ganesan and Loganahan [5] sudied he radiaion ecs on he free convecion flow wih mass ransfer near a semi infinie verical plae, respecively, pas a moving cylinder. Over ime, differen publicaions of his ype appeared bu an increasing ineres has been evinced in he radiaion ineracion wih convecion and chemical reacion. In many engineering processes he chemical reacions play an imporan role in hea and mass ransfer. A chemical reacion is said o be of he firs order if is rae of reacion is direcly proporional o he concenraion [6]. Many researchers have sudied he ecs of chemical reacion under differen condiions on he convecive flow wih hea and mass ransfer. Some of he mos recen and ineresing sudies of his kind are hose of Mahapara e al. [7], Sharma e al. [8], Muhucumaraswamy and Shankar [9], Reddy e al. [1, 11], Ahmed and Dua [1], Reddy e al. [13], and Srihari e al. [14]. However, in all hese sudies he flow is driven by a prescribed surface emperaure or by a surface hea flux. In his work we assume ha he flow is se up by Newonian heaing, i. e. he hea ransfer from he surface is proporional o he local surface emperaure. The Newonian heaing, wih imporan applicaions in engineering, was iniiaed by Merkin [15] and is ecs on he free convecion flow over an infinie plae have been invesigaed by Corresponding auhor; dumiru_vieru@yahoo.com

2 S86 Vieru, D., e al.: Time-Fracional Free Convecion Flow near a Verical Plae wih THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S85-S98 many auhors. A few of he mos recen and imporan resuls in his field seem o be hose of Narahari e al. [16], Narahari and Dua [17], Ramzan e al. [18], Hussanan e al. [19, ], and Vieru e al. [1]. In he las ime, he fracional calculus has been exensively used o describe he viscoelasic behavior of maerials. I is increasingly seen as an icien ool hrough which useful generalizaions of physical conceps can be obained []. Usually, he governing equaions for fracional fluids are obained from hose of ordinary fluids by replacing ime derivaives of an ineger order wih fracional derivaives of order. In he case of diffusion phenomena [3], for insance, = 1 corresponds o he classical diffusion while for < < 1 or > 1 he ranspor phenomenon exhibis sub-diffusion, respecively, super-diffusion. To he bes of our knowledge, he fracional calculus has no been used in convecion problems wih Newonian heaing and mass ransfer. The purpose of his work is o sudy he ime-fracional free convecion flow of an incompressible viscous fluid near a verical plae wih Newonian heaing and chemical reacion. The radiaive ecs are no aken ino consideraion bu, according o Magyari and Panokraoras [4], hey can be easy included by a simple re-scaling of he Prandl number. The Laplace ransform echnique is used o deermine closed-form expressions for velociy, emperaure, concenraion, skin fricion and he raes of hea and mass ransfer from he plae o he fluid. The soluions corresponding o ordinary fluids are also deermined. In he absence of chemical reacion, as expeced, he expressions of emperaure and concenraion reduce o he corresponding soluions of Narahari and Dua [17]. Finally he influence of fracional parameer on he emperaure, concenraion and velociy is graphically underlined and discussed. Mahemaical formulaion The governing equaions corresponding o he unseady free convecion flow of an incompressible viscous fluid over an infinie verical plae wih mass diffusion and chemical reacion, as i resuls from [1] and [17], are given by: uy (, ) uy (, ) = ν + g β [ Ty (, ) T] + g g[ Cy (, ) C] (1) Ty (, ) Ty (, ) ρcp = k () Cy (, ) Cy (, ) = D K [ C( y, ) C ] (3) Equaions (1)-(3) are obained under he usual Boussinesq approximaion [17, 4] when he viscous dissipaion of energy is negligible. Furhermore, as i was shown in [4], he ecs of hermal radiaion in he linearized Rosseland approximaion are quie rivial boh physically and compuaionally and hey will be finally included by a simple re-scaling of he Prandl number wih a facor involving he radiaion parameer. The appropriae iniial and boundary condiions are: uy (,) =, Ty (,) = T, Cy (,) = C ; y (4) y= Ty (, ) h u(, ) =, = T(, ), C(, ) = Cw; > k (5)

3 Vieru, D., e al.: Time-Fracional Free Convecion Flow near a Verical Plae wih THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S85-S98 S87 uy (, ), Ty (, ) T, Cy (, ) C as y (6) In order o develop a model wih fracional derivaives, we firsly muliply eqs. (1)- (3) by λ = νh/ gk and hen replace λ and he parial derivaives wih respec o from he lef pars of he obained equaions by λ, respecively, D where: 1 f () s d f() D f( ) = ds if < < 1; D f( ) = if = 1 Γ(1 ) (7) ( s) d is he Capuo fracional differenial operaor of order. A simple analysis clearly shows ha λ has he dimension of he ime. In order o deermine soluions ha are independen of he geomery of flow regime we also inroduce he following dimensionless variables and parameers: h k T T C C µ C p y = y, =, u = u, T =, C =, Pr =, k λ νh T C C k w are: 3 Re ν h h, Gr T, Gm ( Cw C ), Sc ν = = β g, K ν = = = K g k D gk The dimensionless forms of he governing equaions, dropping ou he * noaion, (8) uy (, ) Duy (, ) = Re + Gr Ty (, ) + Gm Cy (, ); y, > Ty (, ) Pr DTy (, ) = ; y, > Cy (, ) Sc DCy (, ) = KSc Cy (, ); y, > (9) (1) (11) The noion of ecive Prandl number Pr, bu wih a lile differen significaion, has been firsly inroduced by Magyari and Panokraoras [4] showing ha a wo parameer approach is superfluous. The corresponding iniial and boundary condiions are: uy (,) =, Ty (,) =, Cy (,) = ; y (1) Ty (, ) u(, ) =, = [ T(, ) + 1 ], C(, ) = 1; > y y = (13) uy (, ), Ty (, ), Cy (, ) as y, (14) Soluion of he problem The parial differenial eqs. (1) and (11) are no coupled o he momenum eq. (9). Consequenly, we shall firsly deermine he emperaure and concenraion fields by means of he Laplace ransform echnique and hen he fluid velociy.

4 S88 Vieru, D., e al.: Time-Fracional Free Convecion Flow near a Verical Plae wih THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S85-S98 Calculaion of he emperaure field Applying he Laplace ransform wih respec o he emporal variable o eq. (1) and using he corresponding iniial and boundary condiions, we find ha: T( yq, ) Pr qt( yq, ) = ; y> (15) where q is he ransform parameer and he Laplace ransform T( yq, ) of T( y,) has o saisfy he condiions: T( yq, ) 1 = T(, q) + and T( yq, ) as y q y y= (16) The soluion of he ordinary differenial eq. (15) wih he boundary condiions (16) can be wrien under he suiable form: 1 e T( yq, ) = (17) Pr q 1 q Now, applying he inverse Laplace ransform o eq. (17) and using eqs. (A1) and (A) from Appendix, as well as he convoluion heorem, we find ha: 1 1 / Ty (, ) = F /, s 1, ; y Pr s F ds Pr Pr (18) where F(,) µ a is he F-funcion of Robonov and Harley [5] and Φ( abc, ; ) is he Wrigh funcion [6]. In he special case when = 1, we recover he soluion: y Pr q y Pr y Pr Ty (, ) = exp y+ erfc erfc Pr Pr (19) obained by Narahari and Dua [17, eq. (14)]. The local coicien of he rae of hea ransfer from he plae o he fluid, in erms of Nussel number, namely: 1 / 1 / Nu = 1+ E or Nu = exp erfc Pr, + 1 Pr Pr Pr () for (,1), respecively, = 1 is obained by inroducing T( yq, ) ino relaion (see [1, eq. (45)]): T( y, ) 1 1 T( yq, ) Nu L = = [ T( yq, )] = L (1) y y= y= y= and using eq. (A3) from he Appendix. Here, Eab, ( z ) is he well-known Miag-Ller funcion.

5 Vieru, D., e al.: Time-Fracional Free Convecion Flow near a Verical Plae wih THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S85-S98 S89 Calculaion of he concenraion field Applying he Laplace ransform o eq. (11) and using he corresponding iniial and boundary condiions, i resuls ha: 1 C( yq, ) ( q + KC ) ( yq, ) = ; y> Sc where he Laplace ransform C( yq, ) of C( y,) has o saisfy he condiions: 1 C(, q) = and C( yq, ) as y (3) q Now, in order o deermine he expression of C( y,), we wrie he soluion of eq. () wih he boundary condiions (3) under he suiable form: () q C( yq, ) = + K exp y Sc ( q + K) q q + K (4) Applying he inverse Laplace ransform o eq. (4) and using again he convoluion heorem as well as eqs. (A), (A4), and he propery (A5) from he Appendix, we find ha: Ku y Sc 1 1 C( y, ) = e erfc K + (, ; us )dsd u, Φ u s Γ(1 )( s) if (,1) (5) In he case of = 1, he corresponding soluion, see [7]: 1 y KSc y Sc y KSc y Sc = + + C( y, ) e erfc K e erfc K as expeced, reduces as form o eq. (13a) from [17], when K =. The local coicien of he rae of mass ransfer from he plae o he fluid, in erms of he Sherwood number, namely: Sh = Sc G, -1,1/ (- K, ) + KG,-1,1/ (-K, ) or K e Sh = Sc + K erf ( K ) (7) π for (,1), respecively, = 1 is obained inroducing C( yq, ) ino relaion [1, eq. (47)]: C( y, ) 1 1 C( yq, ) Sh L = = C( yq, ) = L y y= y= y= and using eq. (A6) from he Appendix where Gabc,,( d,) is he G-funcion of Lorenzo- Harley [8]. (6) (8)

6 S9 Vieru, D., e al.: Time-Fracional Free Convecion Flow near a Verical Plae wih THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S85-S98 Calculaion of velociy Applying he Laplace ransform o eq. (9) and bearing in mind he associaed iniial and boundary condiions as well as he previous expressions of T( yq, ) and C( yq, ), i resuls ha: ( Pr 1) ( ) u( yq, ) Gr q u( yq, ) = Re y + exp y Pr q + q q Gm exp y Sc ( q + + K) q (9) where he Laplace ransform u( yq, ) of u( y,) has o saisfy he condiions: u(, q) = and u( yq, ) as y (3) The soluion of eq. (9) subjec o he condiions (3) is: Gr Re 1 Pr q u( yq, ) = exp y q exp y + 1 Pr + q Re Re 1 ( Pr q Re ) Gm Sc q + exp y ( q K) exp y + q[(1 Sc) q KSc Re Re Finally, in order o obain he ( y,)-domain soluion for velociy, namely: (31) Gr Re Re Pr uy (, ) = F /, s F + 1, ; y (1 Pr) Pr Pr Re s y F + 1, ; s ds+ Re s Gm Ku y Sc 1 + e erfc F(, ; us )dsdu 1 Sc Re u s Gm KSc y F, s F 1, ; ds + 1 Sc 1 Sc Re s Ku e erfc E, 1 ( s) (, ; us )dsdu + u s 1 Sc (3) Gm K y Sc ( s) KSc + Φ (1 Sc) Re we apply he inverse Laplace ransform o eq. (31) and use eqs. (A1)-(A3) and (A5) from he Appendix. The soluion corresponding o = 1, namely:

7 Vieru, D., e al.: Time-Fracional Free Convecion Flow near a Verical Plae wih THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S85-S98 S91 Gr Re Pr Re y Re u( y, ) = ϕ y,, ϕ,, (1 Pr) Pr Re Pr Re Pr Gm Sc Sc K Sc Ψ y, K,, y, K,, K Sc Ψ + Re Re 1 Sc y KSc y +Ψ,,, Ψ,,, Re 1 Sc Re (33) is obained by means of he eqs. (A7) and (A8) from he Appendix. The skin fricion in non-dimensional form is: ha: u( y, ) 1 1 u( yq, ) = = L [ u( yq, )] = L y y= y= y= Inroducing u( yq, ) from eq. (31) ino (34) and using eqs. (A1) and (A6), we find (34) Gr Re Gm K Sc = G, G, + Pr( Pr + 1) Pr Re(1 Sc) 1 Sc, 1,1, 1,1 Gm Sc K Sc + G 1( K, s) + KG 1( K, s) F, s ds (35) Re(1 Sc), 1,, 1, 1 Sc In he case = 1, lenghy bu sraighforward compuaions lead o he simpler expression, see (A9) from he Appendix: Gr Pr Re Re = + 1 exp erfc + Re( Pr + 1) p Re Pr Pr Gm 1 K Sc K K Sc + exp erfc erf erf ( K ) K ReSc 1 Sc 1 Sc 1 Sc 1 Sc (36) Numerical resuls and discussions In order o ge some physical insigh of presen resuls, some numerical calculaions have been carried ou for differen values of he fracional parameer, he ime and physical parameers. However, in order o avoid repeiion, only he mos significan graphical represenaions regarding he ecs of fracional parameer will be here included. The numerical values compued from analyical soluions of he problem have been visualized in figs. 1-6 boh for (,1) and = 1. As a confirmaion of he validiy of resuls ha have been obained, in all cases, he diagrams corresponding o fracional soluions end o superpose over hose of ordinary soluions when 1.

8 S9 Vieru, D., e al.: Time-Fracional Free Convecion Flow near a Verical Plae wih THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S85-S98 Figures 1 and presen he dimensionless emperaure and concenraion profiles a wo imes for differen values of he fracional parameer. As expeced, he fluid emperaure and concenraion are increasing funcions wih respec o ime. Their values, ha are maxima near he plae, smoohly decrease o zero for increasing y. The influence of fracional parameer is significan and boh he emperaure and he concenraion increase for increasing. Furhermore, heir values a any disance y from he plae are always higher for 1 han hose for if 1 > and his resul clearly confirms ha for < < 1 he ranspor phenomenon exhibis sub-diffusion in comparison o he classical diffusion corresponding o = 1. Figure 1. Temperaure profiles vs. y for Pr = 15; (a) = 1, (b) =, and differen values of Figure. Concenraion profiles vs. y for Sc = 1 and K =.5; (a) = 5, (b) = 4, and differen values of The influence of ecive Schmid number Sc and chemical reacion parameer K on he fluid concenraion is presened in figs. 3 and 4. I is clearly seen from hese figures ha he concenraion level of he fluid decreases whenever Sc or K is increased. In he case of he ecive Schmid number, his is possible because an increase of Sc means an increase of he Schmid number ha implies a fall in he mass diffusiviy [1]. The dimensionless velociy profiles a wo imes for differen values of are depiced in figs. 5(a) and 5(b). I clearly resuls from hese graphs ha he fluid velociy agains y is an increasing funcion wih respec o and bu a sronger increase appears wih regard o he fracional parameer. Near he surface of he plae he fluid velociy increases, becomes maximum and hen decreases o an asympoic value for large values of y. In all cases he val

9 Vieru, D., e al.: Time-Fracional Free Convecion Flow near a Verical Plae wih THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S85-S98 S93 Figure 3. Concenraion profiles vs. y for = 1, K =.5, and Sc = 1; (a) Sc = 1, (b) Sc = 4, and differen values of Figure 4. Concenraion profiles vs. y for = 1 and Sc = 1; (a) K =.5, (b) K =., and differen values of Figure 5. Velociy profiles vs. y for Re = 5, Pr = 5, Gr = 4, Gm =.3, Sc =.1, and K = 1.5; (a) = 5, (b) = 1, and differen values of ues of velociy a any disance y from he plae are always higher or lower for disinc values of or. Figures 6(a) and 6(b) reveal he ecs of he ecive Schmid number and of fracional parameer on he Sherwood number vs.. The Sherwood number, as i resuls from hese figures, is an increasing funcion wih respec o Sc and decreases in ime from a

10 S94 Vieru, D., e al.: Time-Fracional Free Convecion Flow near a Verical Plae wih THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S85-S98 maximum value near he plae o an asympoic value for large values of he ime. I also increases wih respec o up o a criical value of he ime (abou.) and decreases laer. Figure 6. Sherwood number vs. for K =.5; (a) Sc = 4, (b) Sc = 5, and differen values of Finally, in order o have a clear idea abou he accuracy of analyical soluions ha have been here esablished in he fracional case, a comparison beween he numerical and exac resuls was prepared for he dimensionless emperaure and concenraion. The corresponding resuls for he fluid emperaure have been included in ab. 1. The emperaure values resuling from eq. (18), where 35 erms of he sums have been aken ino consideraion, are compared wih hose obained using he Sehfes s numerical algorihm for calculaing he inverse Laplace ransform [9]. This algorihm is based on he relaion: n 1 ln ln T( y, ) = L T ( yq, ) dt j y, j (37) j= 1 Table 1. Values of he dimensionless emperaure T an (y) resuling from he analyic soluion (18), and he numerical values T n (y) a = 3, =.65, and Pr = 16 y T an (y) T n (y) y T an (y) T n (y) y T an (y) T n (y)

11 Vieru, D., e al.: Time-Fracional Free Convecion Flow near a Verical Plae wih THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S85-S98 S95 where n is a posiive ineger, d j = ( 1) min( jn, ) j+ n k ( k)! (38) ( n k)! k!( k 1)!( j k)!( k j)! j+ 1 k = and [r] denoes he ineger par of he real number r. According o his able, he absolue error being of order 1 5, here exiss a very good agreemen of analyical and numerical resuls. Conclusions The presen work represens a heoreical sudy of he ime-fracional free convecion flow near a verical plae wih Newonian heaing, mass diffusion and chemical reacion. The radiaive ecs are no aken ino consideraion because, in he case of he Rosseland diffusion approximaion [4], he hea ransfer characerisics can be brough o ligh by means of a parameer only. More exacly, hey can be included by re-scaling he ecive Prandl number o be Pr /[Re(1 + R)] where R is he radiaion parameer. Consequenly, a wo parameer approach is superfluous. The fracional model was firsly normalized and closed-form soluions for velociy, emperaure, concenraion, skin fricion and he raes of hea and mass ransfer from he plae o he fluid have been deermined using he Laplace ransform echnique. I is worh poining ou ha, in he absence of chemical reacion, he emperaure and concenraion depend on ecive Prandl number and he ecive Schmid number, respecively, which are ranspor parameers represening he hermal diffusiviy and mass diffusiviy. However, our ineres here is on he special characerisics of he fracional model and he influence of fracional parameer on he hea and mass ransfer as well as on he fluid moion ha are graphically underlined. The main findings are as follows. Exac soluions corresponding o he ime-fracional free convecion flow wih Newonian heaing, mass diffusion and chemical reacion are esablished in erms of some known funcions. The dimensionless emperaure of he fluid and is concenraion in he absence of chemical reacion depend of only one essenial parameer Pr and Sc, respecively. The fracional parameer has a significan influence on he dimensionless emperaure, concenraion and velociy fields. They are increasing funcions wih respec o his parameer. The rae of mass ransfer from he plae o fluid in erms of Sherwood number is an increasing funcion wih regard o he ecive Schmid number and monoonically decreases in ime. Appendix n n ( n+ 1) µ a L Fµ ( a, ) ; µ µ = = > (A1) q a n= Γ [( n + 1) µ ] b aq n 1 e c 1 b z L ( c, b, a ); ( x, y, z) c = Φ Φ = (A) q n= Γ ( n + 1) Γ ( x + ny)

12 S96 Vieru, D., e al.: Time-Fracional Free Convecion Flow near a Verical Plae wih THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S85-S98 a b 1 q b 1 a zk L E ab, ( c ); E ab, ( z) ; a, b a = = > > (A3) q + c k = Γ ( ak + b) L a q a 1 a + a, < a < 1 = L + 1 a = Γ(1 a) q q q δ() + a, a = 1 a [ uw q ] 1 1 ( ) 1 (A4) If F( ) = L [F( q)] and g( u, ) = L e hen L {F[w( q)]} = f ( u)g( u, )du (A5) G + ( c) ( n 1) [( n ca ) b] b n ( n+ ca ) b 1 1 q d ( n c ) abc,,(,) = a c = ( q d) n= G G + G + G d L if Re( ac b) >, Re( q) >, and d < q (A6) a L a q+ b c 1 e e a b+ c a L = e erfc ( b + c ) + q c a b c a + e erfc + ( b+ c ) =Ψ( abc,,, ) + a q 1 e 1 a a 1 a ( + ab) a erfc exp = q ( q + b) b b b b π 4 1 a 3 exp( ab b )erfc b ϕ ( a, b, ) + + = b a q 1 e 1 a 1 ab+ b a L = erfc e erfc + b q( q b) b b + (A7) (A8) (A9) Nomenclaure C concenraion of he fluid, [kgm 3 ] C p specific hea a a consan pressure, [Jkg 1 K 1 ] C w concenraion level a he plae, [kgm 3 ] C concenraion of he fluid far away from he plae, [kgm 3 ] D mass diffusiviy, [m s 1 ] g acceleraion due o graviy, [ms ] Gm mass Grashof number, [= γ(c w C )], [ ] Gr hermal Grashof number, [= βt ] [ ] h hea ransfer coicien, [Wm K 1 ] K chemical reacion parameer, [s 1 ] k hermal conduciviy of he fluid, [Wm K 1 ] Nu Nussel number, [ ] Pr Prandl number (= μc p /k), [ ] Sh Sherwood number, [ ] T emperaure of he fluid, [K] T fluid emperaure far away from he plae, [K] u velociy of he fluid along he x-axis, [ms 1 ] Greek symbols β he volumeric coicien of hermal expansion, [K 1 ] γ he volumeric coicien of mass expansion, [m 3 kg 1 ] μ dynamic viscosiy, [kgm 1 s 1 ] ν kinemaic viscosiy, [m s 1 ] ρ fluid densiy, [kgm 3 ] τ skin fricion, [Nm ]

13 Vieru, D., e al.: Time-Fracional Free Convecion Flow near a Verical Plae wih THERMAL SCIENCE, Year 15, Vol. 19, Suppl. 1, pp. S85-S98 S97 Pr ecive Prandl number (= Pr/Re), [ ] Re Reynolds number (= ν h 3 /gk 3 ), [ ] Sc Schmid number (= ν/d), [ ] Sc ecive Schmid number (= Sc/Re), [ ] Subscrips ecive w condiion a he wall free sream condiions Acknowledgmens The auhors would like o express heir graiude o reviewers for heir careful assessmen and fruiful suggesions regarding he firs form of he manuscrip. References [1] Ghoshdasidar, P. S., Hea Transfer, Oxford Universiy Press, Oxford, UK, 4 [] Nield, D. A., Bejan, A., Convecion in Porous Media, Springer, New York, USA, 6 [3] Gebhar, B., Pera, L., The Naure of Verical Naural Convecion Flows Resuling from he Combined Buoyancy Effecs of Thermal and Mass Diffusion, Inernaional Journal of Hea and Mass Transfer, 14 (1971),, pp. 5-5 [4] Chamkha, A. J., e al., Radiaion Effecs on Free-Convecion Flow pas a Semi Infinie Verical Plae wih Mass Transfer, Chemical Engineering Journal, 84 (1), 3, pp [5] Ganesan, P., Loganahan, P., Radiaion and Mass Transfer Effecs on Flow of an Incompressible Viscous Fluid pas a Moving Cylinder, Inernaional Journal of Hea and Mass Transfer, 45 (), 1, pp [6] Cussler, E. L., Diffusion Mass Transfer in Fluid Sysems, Cambridge Universiy Press, London, 1988 [7] Mahapara, M., e al., Effecs of Chemical Reacion on Free Convecion Flow hrough a Porous Medium Bounded by a Verical Surface, Journal of Engineering Physics and Thermophysics, 83 (1), 1, pp [8] Sharma, P. R., e al., Influence of Chemical Reacion and Radiaion on Unseady MHD Free Convecion Flow and Mass Transfer hrough Viscous Incompressible Fluid pas a Heaed Verical Plae Immersed in Porous Medium in he Presence of Hea Source, Applied Mahemaical Sciences, 5 (11), 46, pp [9] Muhucumaraswamy, R., Shankar, M. R., Firs Order Chemical Reacion and Thermal Radiaion Effecs on Unseady Flow pas an Acceleraed Isohermal Infinie Verical Plae, Indian Journal of Science and Technology, 4 (11), 5, pp [1] Reddy, T. S., e al., The Effecs of Slip Condiion, Radiaion and Chemical Reacion on Unseady MHD Periodic Flow of a Viscous Fluid hrough Sauraed Porous Medium in a Planar Channel, Journal on Mahemaics, 1 (1), 1, pp [11] Reddy, T. S., e al., Unseady MHD Radiaive and Chemically Reacive Free Convecion Flow near a Moving Verical Plae in Porous Medium, Journal of Applied Fluid Mechanics, 6 (13), 3, pp [1] Ahmed, V., Dua, M., Transien Mass Transfer Flow pas an Impulsively Sared Infinie Verical Plae wih Ramped Plae Velociy and Ramped Temperaure, Inernaional Journal of Physical Sciences, 8 (13), 7, pp [13] Reddy, T. S., e al., Chemical Reacion and Radiaion Effecs on MHD Free Convecion Flow hrough a Porous Medium Bounded by a Verical Surface wih Consan Hea and Mass Flux, Journal of Compuaional and Applied Research, 3 (13), 1, pp [14] Srihari, K., Reddy, C. K., Effecs of Sore and Magneic Field on Unseady Flow of a Radiaing and Chemical Reacing Fluid: A Finie Difference Approach, Inernaional Journal of Mechanical Engineering, 3 (14), 3, pp. 1-1 [15] Merkin, J. H., Naural Convecion Boundary-Layer Flow on a Verical Surface wih Newonian Heaing, Inernaional Journal of Hea and Fluid Flow, 15 (1994), 5, pp [16] Narahari, M., e al., Newonian Heaing and Mass Transfer on Free Convecion Flow pas an Acceleraed Plae in he Presence of Thermal Radiaion, AIP Conference Proceedings, 148 (1), 1, pp [17] Narahari, M., Dua, B. K., Effecs of Thermal Radiaion and Mass Diffusion on Free Convecion Flow near a Verical Plae wih Newonian Heaing, Chemical Engineering Communicaions, 199 (1), 5, pp

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