1. Introduction. Nabil T. M. Eldabe 1, Ahmed M. Sedki 2,3,*, I. K. Youssef 3

Transcription

1 Amercan Journal of Computatonal and Appled Mathematcs 4, 4(4): 4-53 DOI:.593/.acam Numercal Solutons for Boundary Layer Flud Flow wth Mass Transfer over a Movng Permeable Flat Plate Embedded n Porous Medum wth Varable Wall Concentraton n Presence of Chemcal Reacton Nabl T. M. Eldabe, Ahmed M. Sedk,3,*, I. K. Youssef 3 Mathematcs Department, Faculty of Educaton, An Shams Unversty, Helopols, Caro, Egypt Mathematcs Department, Faculty of Scence, Jazan Unversty, Jazan, Saud Araba 3 Mathematcs Department, Faculty of Scence, An Shams Unversty, Abbassa, Caro, Egypt Abstract An analyss s made to study the mass transfer n boundary layer flow past a movng permeable flat plate embedded n porous medum wth varable wall concentraton n presence of chemcal reacton. The governng nonlnear partal dfferental equatons are transformed nto a set of ordnary dfferental equatons by usng smlarty transformatons. The numercal computatons are carred out for several values of physcal parameters nvolved n the transformed equatons. The resultng nonlnear system of partal dfferental equatons are solved numercally by both Keller-Box method whch s an mplct fnte dfference method and by the numercal method based on fourth order Runge-Kutta teraton scheme wth shootng method. The features of the flow and mass transfer characterstcs for dfferent values of the governng parameters are analyzed and dscussed. To support the accuracy of the numercal results, a comparson s made wth known results from the open lterature for some partcular cases of the present study and the results are found to be n excellent agreement for the used numercal methods. It s found that the exstence of dual solutons exsts when the surface and the flud move n opposte drectons. The results ndcate that the ncrease of porous parameter decreases the varaton of a velocty profles and the varaton of a skn frcton coeffcent whle t ncreases both concentraton profles and concentraton gradent at the surface. It s due to the presence of a porous medum whch ncreases the resstance to flow resultng n decrease n the flow velocty and ncrease n the solute concentraton. Keywords Mass transfer, Chemcal reacton, Movng permeable plate, Varable wall concentraton, Porous medum, Smlarty transformatons, Keller box method. Introducton The mass transfer n lamnar boundary layer flow over a movng surface n porous medum has many mportant applcatons n modern ndustry. Thus the boundary layer flow problems have been wdely studed over the past few decades and the earler nvestgators were nterested n fndng the smlarty solutons for the boundary layer flow problems, vz. Blasus [], Howarth [], Sakads [3], Sekman [4], Klemp [5] and Abussta [6]. For the boundary layer flow on a movng flat plate n a quescent flud, Sakads [3] obtaned the same equatons as obtaned by Blasus [] wth dfferent boundary condtons. Abdulhafez [7] and Hussan et al. [8] reported the flud flow * Correspondng author: a.m.sedk@hotmal.com (Ahmed M. Sedk) Publshed onlne at Copyrght 4 Scentfc & Academc Publshng. All Rghts Reserved characterstcs for movng wall lamnar boundary layer problems. Smlar problems wth varous boundary condtons and n dfferent stuatons have been consdered by Ln [9], Wedman et al. [] and Cortell []. Ishak [] extended the classcal problems of Blasus [] by consderng a flat plate movng n the same or opposte drectons to a parallel free stream, all wth constant veloctes. Heat transfer n a movng flud over a movng non-sothermal flat surface s nvestgated numercally by Mukhopadhyay [3] and dual solutons for boundary flow of movng flud over a movng surface wth power law surface temperature s studed by Mukhopadhyay and Gorla [4]. By ntroducng composte velocty, Afzal et al. [5] combned Blasus and Sakads problems successfully and obtaned a sngle set of equatons. Andersson [6] nvestgated the transport of mass and momentum of chemcally reactve speces n the lamnar flow over a lnearly stretchng surface and solved the nonlnear ordnary dfferental equatons governng the

2 4 Nabl T. M. Eldabe et al.: Numercal Solutons for Boundary Layer Flud Flow wth Mass Transfer over a Movng Permeable Flat Plate Embedded n Porous Medum wth Varable Wall Concentraton n Presence of Chemcal Reacton self-smlar flow. Takhar et al. [7] studed the flow and mass dffuson of chemcal speces wth frst and hgher order reactons over a contnuously stretchng sheet wth an appled magnetc feld. Uddn M.S. [8] studed the boundary layer flow and reactve solute transfer wth frst order reacton past a stretchng surface, the varable ntal solute dstrbuton along the surface s taken nto account. Bhattacharyya [9] nvestgated the mass transfer wth frst order chemcal reacton on a contnuous flat plate movng wth constant velocty n parallel or reversely to a unform free stream; the varable reacton rate s consdered. Merkn [] studed dual soluton through porous medum, whle mass transfer over permeable surface s nvestgated by Magyar []. Elbashbeshy and Bazd [3] studed the mxed convecton along a vertcal plate wth varable surface heat flux embedded n porous medum. Nabl T. M. Eldabe et al [4] studed the effects of chemcal reacton and flud flow through a porous medum over a horzontal stretchng flat plate. The boundary layer flow and mass transfer over a stretchng sheet embedded n porous medum s nvestgated by Hossen [5]. Elbashbeshy and Sedk [6] studed the effect of chemcal reacton on mass transfer over a stretchng surface embedded n a porous medum. The am of the present study s to nvestgate the effects of chemcal reacton and dffuson n boundary layer flow wth mass transfer over permeable flat plate movng reversely or parallel to a free stream of a movng flud embedded n porous medum. In ths analyss, the wall concentraton s varable. The present study may be regarded as an extenson to Ishak []. The numercal computatons are carred out for several values of parameters nvolved n the transformed equatons vz. the sucton or necton parameter (s), the Schmdt number (Sc), the chemcal reacton rate parameter (B), the power law exponent (n), Porous parameter (N) and the velocty rato parameter (q). The features of the flow and mass transfer characterstcs vz. varaton of a velocty profles velocty f ( η), Varaton of a skn frcton coeffcent f (), Concentraton profles φ (η) and concentraton gradent at the surface () for dfferent values of the governng parameters are dscussed and analyzed.. The Governng Equatons We consder the two-dmensonal steady lamnar boundary-layer flow of an ncompressble, vscous flud and mass dffuson wth chemcal reacton over a flat surface subect to sucton or necton wth varable wall concentraton. The surface moves wth constant velocty U w n the same or opposte drecton to the free stream embedded n porous medum. The x -axs extends parallel to the surface, whle the y -axs extends upwards, normal to the surface. Usng boundary layer approxmaton, the governng equatons for the flow and concentraton dstrbuton may be wrtten as u v + = x u u v u u υ + = υ u x k c c c u + v = D Rc ( c ) x y Where (x, y) are the dmensonal coordnates along and normal to the tangent of the surface and (u, v) are the velocty components parallel to (x, y). υ = µ ρ s the knematc vscosty where ρ s the densty and µ s the dynamc vscosty of the flud. C s the concentraton, D s the dffuson coeffcent and C s the concentraton n the free stream. R(x) s the varable reacton rate of the solute and s gven as R( x) = LR x, L s the reference length and R s constant. The boundary condtons for the velocty components and the concentraton are u = u, v= v, c= c = c + c x n at y = (4) w w w () () (3) u u and c c as y (5) Where u the free stream velocty s c w s the varable plate concentraton and C s a postve solute constant. n s a power-law exponent whch sgnfes the change of amount of solute n the x-drecton. v w s the varable sucton or necton through the permeable plate and s gven by vw = v ( x ), v s a constant wth v < for sucton and v > for necton. The stream functon ψ (x, y) that satsfes the contnuty equaton and s related to the velocty components n the usual way as ψ ψ u =, and v = (6) y x Usng boundary layer approxmaton, the followng dmensonless varables for ψ and C can be ntroduced x ψ = υ ( Re ) f ( η ) and c = c + ( c c ) ϕ, (7) then we have ψ u = = Uf / ( η), ψ v= = f f x x Uυ ( ) / [ ( ) / η η ( η )] w (8) (9)

3 Amercan Journal of Computatonal and Appled Mathematcs 4, 4(4): u U = f // ( ) x x η η 3 / // u U = ( ) f ( η) υx u U = f υx /// ( η). () C Cw C = / ( )[ nϕ ( η) ηϕ ( η)] x x C U = / / ( ) ( Cw C ) ϕ ( η) y υx C U // = ( ) ( ) C w C ϕ η υx () Where Re = ( Ux / υ) s the local Reynolds number x and ƞ s the smlarty varable defned as η = yu ( / xυ) where U s the composte velocty defned as U=U +U w (Afzal et al. [5]). f ( η ) s the dmensonless stream functon and φ s the dmensonless concentraton functon. Substtutng n equatons (6-9) to obtan the set of ordnary dfferental equatons f ''' + f. f '' N f ' = () ϕ '' + ' sc f. ' sc(. n f B) ϕ + ϕ = (3) Where N = ( υ / ku ) Rex = υx / ku s the porous parameter, sc = υ D s the Schmdt number and B = LRo U s the chemcal reacton rate number. The boundary condtons fnally become f ( η) = s, f '( η) = q and ϕη ( ) = at η= (4) f '( η) q and ϕη ( ) as η (5) Where the velocty rato parameter q = u w /U and s s the sucton or necton parameter where S = f() = (- v / U)(Re ) = v / ( U), w / / x υ S > (.e. ν < ) corresponds to sucton and S < (.e. ν > ) corresponds to necton. The physcal quanttes of nterest n ths problem are the local skn-frcton coeffcent f ''() and rate of mass transfer - ϕ '() whch are defned as υ / u f ''() = (Re ) ( ) x y= (6) U υ C / ϕ '() = (Re x) ( ) y= UC ( w C ) 3. Numercal Method of Soluton (7) The system of the nonlnear ordnary dfferental equatons (-3) along wth the boundary condtons (4-5) s solved by usng the followng methods () Fourth order Rung Kutta Method (RKM) () Keller Box Method (KBM) 3.. Fourth Order Rung Kutta Method (RKM) The numercal method (RKM) s based on fourth order Runge-Kutta teraton scheme wth shootng method [9]. The system (-5) s solved by RKM, by convertng t nto an ntal value problem. In ths method we have to choose a sutable fnte value of η, sayη. We set the followng frst order system: f = p, p = g, g' = fg. + N p (8) ϕ ' = z, z ' = sc f. z + sc(. n p + B) ϕ, (9) Wth the boundary condtons f() = s, p() = q, ϕ() =, p( η) q and ϕη ( ) () To solve the system of the ntal value problem (8-9) wth () we need values for g() = f () and z() = ϕ () but no such values are gven n the boundary condtons. The ntal guess values for f () and ϕ () are chosen and applyng fourth order Runge Kutta method then soluton s obtaned. To get accurate soluton, t s mportant for shootng method to choose the approprate fnte value ofη. In order to determne η for the ntal value problem (8-), we start wth some ntal guess values for some partcular set of the physcal parameters to obtan f () and ϕ (). The soluton procedure s repeated wth another value of η untl two successve values of f () and ϕ () dffer only by the specfed sgnfcant dgt. The last value of η s fnally chosen to be the most approprate value of the lmt η for that partcular set of parameters. The value of η may change for another set of physcal parameters. After determnng the value η, we compare the calculated values of f ( η) and ϕη ( ) at η wth the gven boundary condtons

4 44 Nabl T. M. Eldabe et al.: Numercal Solutons for Boundary Layer Flud Flow wth Mass Transfer over a Movng Permeable Flat Plate Embedded n Porous Medum wth Varable Wall Concentraton n Presence of Chemcal Reacton f ( η ) = q and ϕη ( ) = and adust the values of f () and ϕ () usng Secant method to gve better approxmaton for the soluton. The step sze s taken as η =.. The process s repeated untl we get the results correct up to the desred accuracy -6 level. 3.. Keller Box Method (KBM) The system of the nonlnear ordnary dfferental equatons (-5) s solved numercally by Keller-Box method [7, 8] that s an mplct fnte dfference method. One of the basc deas of the Keller-box method s to wrte the governng system of equatons n the form of a frst order system (8-). We use centered dfference dervatves and averages at the mdpont of net rectangles to get fnte dfference equatons wth a second order truncaton error. The method allows for non-unform grd dscreton and converts the dfferental equatons nto algebrac ones that are then solved usng Thomas algorthm. Thomas algorthm s essentally the result of applyng Gauss elmnaton to the tr-dagonal system of equatons. The number of grd ponts n both drectons affects the numercal results. To obtan accurate results, a mesh senstvty study was performed The Fnte-Dfference Scheme We now consder the net rectangle n the plane and the net ponts defned as follows: x =, x = x + k, =,,..., I η =, η = η + h, =,,... J, η = η, where J k s the x - spacng and h s the η -spacng. Here and are ust sequence of numbers that ndcate the coordnate locaton, not tensor ndces or exponents. The dervatves n the η -drecton are replaced by fnte dfference, for example the fnte- dfference form for any ponts are ( ) = ( ) + ( ) ( ) = ( ) + ( ) f η,, / f / f / / = h We start by wrtng the fnte dfference of equatons for the mdpont ( x, η /) usng centered dfference dervatves, we get. f h f f = p = p + p p h p p = g = g + g ϕ h ϕ ϕ = z = z + z, If we assume f, p, g, ϕ, z to be known for J, then we have to obtan the soluton of the unknown ( f, p, g, ϕ, z ) for J. The system can be wrtten as f f h p + p = p p h g + g = ϕ ϕ h z z + = (a) g g + h( f + f )( g + g) 8 h. N.( p + p) = M /, where M / = ( g g ) h( fg) / + h. N. p /. z z + h. Sc.( f + f )( z + z ) 8 h. Sc. n.( p + p )( ϕ + ϕ ) 4 + h. Sc. B.( ϕ + ϕ ) = N /, where N z z h Sc f z + h. Sc. n( p ) / = ( )..(. ) / ϕ / h Sc Bϕ / We note that M / and N / nvolves only known quanttes f we assume that the soluton s known on

5 Amercan Journal of Computatonal and Appled Mathematcs 4, 4(4): x= x /. The transformed boundary layer thckness η J s to suffcently large so that t s beyond the edge of the boundary layer [9, 3]. The boundary condtons at x= x yelds f = s, p = q, ϕ =, pj = q and ϕj = (b) 3... Newton s Method To lnearze the nonlnear system of equatons usng Newton s method, we ntroduce the followng terates k k k f + k k k = f +δ f, p + = p + δ p, k k k + = + δ g g g k k k + = + δ z z z., k k k + = + ϕ ϕ δϕ, After droppng the quadratc and hgher order terms n k δ f, δ p k, g k k δ, δϕ, and δ z k. We have also dropped the superscrpt (k) for smplcty. Ths procedure yelds the followng lnear tr-dagonal system. ( δ f δ f ) h( δ f + δ f ) = ( R) ( δ p δ p ) h( δg + δg ) = ( R) ( δϕ δϕ ) h( δ z + δ z ) = ( R3) (a) aδg + aδg + a3δ f + a4δ f + a5δz + a6δz + a7δ p + a8δ p + a9δϕ + aδϕ = ( R4 ) bδg + bδg + b3δ f + b4δ f + b5δz + b6δz + b7δ p + b8δ p + b9δϕ + bδϕ = ( R5 ), where a = [ + h. f ], a = a, / 4 a = h. g, a = a, a = a =, 3 / a7 = h. N, a8 = a7, a9 = a =, b = b =, b = h. z, b = b 3 / b = [ + h. f ], b = b 5 / = ϕ / 8 = 7 b h. Sc.. n, b b, b9 = hsc.. n p / h. Sc. B, b9 = b, ( R ) = ( f f ) + h. p / / ( R ) = ( p p ) + h. g / / ( R ) = ( ϕ ϕ ) + h. z 3 / / ( R ) = M ( g g ) 4 / / h( f. g) / + h.. np / ( R) = N ( z z ) h( fz.) + h. Sc. n.( pϕ) + h. Sc. B.( ϕ). 5 / / / / / To complete the system (a), we recall the boundary condton (b), whch can be satsfed exactly wth no teraton. So, to mantan these correct values n all the terates, we take δ f =, δ p =, δϕ =, δ p = and δϕ =, J J (b) The Block Tr-dagonal Matrx The lnearzed dfference equatons () has a block tr-dagonal structure conssts of varables or constants, but here t conssts of block matrces. The elements of the matrces are defned as follows, [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ δ ] [ δ ] [ δ ] [ δ ] [ ] [ ] A C R B A C R = B A C R [ ] [ ] J J J J J BJ A J J R J That s: [ A ] [ A][ ] [ R] δ = (3) d d = d d, ( a ) ( a8 ) ( a3 ) ( a ) ( a5 ) ( b ) ( b8 ) ( b3) ( b ) ( b5 )

6 46 Nabl T. M. Eldabe et al.: Numercal Solutons for Boundary Layer Flud Flow wth Mass Transfer over a Movng Permeable Flat Plate Embedded n Porous Medum wth Varable Wall Concentraton n Presence of Chemcal Reacton d d A d = ( a8 ) ( a) ( a3) ( a ) ( a5) ( b8 ) ( b) ( b3 ) ( b ) ( b5 ) h where d =, and J, d B = d, J ( a4) ( a) ( a6) ( b4) ( b) ( b6) d d C = d, ( a8 ) ( a) ( a3) ( a ) ( a5) ( b8 ) ( b) ( b3 ) ( b ) ( b5 ) J g δ p δ δ z δϕ δ f, δ δ f = =, for J δ g δ g δ z δ z ( R ) / ( R ) / and R ( R3 ) = / for J ( R4 ) / ( R5 ) / [ δ ] To solve the system (3), we assume that A s nonsngular A= LU, where and can factorzed nto [ ] [ ] [ L] [ α] [ B ] [ α ] =, [ α J ] [ BJ] [ α J] [ I ] [ Γ] [ I ] [ Γ ] [ U ] [ I ] [ ΓJ ] [ α J ] Now we have [ LU δ ] = [ R], f we defne = [ U ] [ W] δ =, (4) then we have [ LW ] = [ R], (5) the elements W can be solved from equaton (5) [ α ][ W] = [ R], [ α ][ W] = [ R] [ B][ W ], < J. Where [ α ] = [ A], step n whch Γ, α and [ α ] = [ A ] [ B ][ Γ ], the W are calculated, s usually referred to as the forward sweep. Once the elements of W are found, equaton (4) then gves the soluton δ n the so called backward sweep, n whch the elements are obtaned by the followng relatons: [ δ J] = [ WJ], [ δ ] = [ W ] Γ [ ][ δ ], + < J, these calculatons are repeated untl some convergence crteron s satsfed and calculatons are stopped when () δg < ε, where ε s small prescrbed value 4. Results and Dscusson The nonlnear system of dfferental equatons (8-) s solved numercally by both Keller-Box method whch s an mplct fnte dfference method and also by the numercal method based on fourth order Runge-Kutta teraton scheme wth shootng method and the computer programmng methods are done n MATLAB. For the confrmaton of the accuracy of appled numercal methods we compare our results correspondng to the values of numercally obtaned skn-frcton coeffcent f () for varous of q wth prevously reported by Ishak [] and excellent agreement are found for the used numercal methods as shown n Table. The numercal computatons are carred out for several values of parameters nvolved n the equatons vz. the sucton or necton parameter (s), the Schmdt number (sc), the chemcal reacton rate parameter (B), the power law exponent (n), porous parameter (N) and the velocty rato parameter (q).

7 Amercan Journal of Computatonal and Appled Mathematcs 4, 4(4): f () Table. Skn-frcton coeffcent for varous of q wth N= and S= f () Ishak [] q Upper Branch Lower Branch The Present Work Fourth order Rung Kutta Method Keller Box Method Upper Lower Upper Lower Branch Branch Branch Branch q = q = q = q = q = q = q =.5 q = The computed results are explaned by plottng some fgures for varaton of a velocty profles f ( η), Varaton a skn frcton coeffcent f (), Concentraton profles φ(η) and concentraton gradent at the surface () for dfferent values of the each parameter and physcal meanng are also gven. The external sucton or necton parameter (S) effects are demonstrated n Fgures (-6), t s found that at fxed, varaton of a velocty profles f ( η) and varaton of a skn frcton coeffcent f () ncrease wth the ncrease of the sucton (S) whle both Concentraton profles φ(η) and concentraton gradent at the surface () reduce. It s due to the fact that the momentum as well as concentraton boundary layer thcknesses decrease wth sucton. f sc =.5, B =.3, n =, N=., q =. s = -.4, -.,.,.,.4 φ f sc =.5,.3, n =, N =., q =. s = -.4, -.,.,.,.4 Fgure. ø( ) for dfferent values of s N =, Sc =.5, B =, n =, q= S =.4,.,.,-., Fgure. f ( η) η for dfferent values of s Fgure 3. f ( η) aganst for dfferent values of s

8 48 Nabl T. M. Eldabe et al.: Numercal Solutons for Boundary Layer Flud Flow wth Mass Transfer over a Movng Permeable Flat Plate Embedded n Porous Medum wth Varable Wall Concentraton n Presence of Chemcal Reacton f ().35 N =, Sc =.5,B =, n=, q = S= -.4, -.,.,., Fgure 4. () f for dfferent values of s N =, Sc =.5, B =, n =, q= momentum and boundary layer thcknesses ncrease. The Schmdt number (Sc) effects are dsplayed n Fgures (7-8), t s observed that The Schmdt number has maor effects on the dstrbuton of solute. At fxed the ncrease of Schmdt Sc reduces quckly both concentraton profles φ(η) and concentraton gradent at the surface (). Ths s due to the fact that the rate of solute transfer from the surface ncreases when the Schmdt number ncreases. The negatve value of the concentraton profle for large Sc s because of substantal ncrease n the rate of solute transfer from the plate to the flud n the chemcal reacton. It s observed that the magntude of the concentraton gradent ntally ncreases wth Sc, but for greater values of η t decreases wth Sc N=, S=., B=.3, n=, q=.8 φ(η).6.5 Sc=.,.,.3,.5 φ.6 S = -.4,-.,.,., Fgure 5. φ(η) for varous values of s N =,Sc =.5,B =,n =, q= S =.4,.,.,-., Fgure 6. () for dfferent values of s On the other hand for the external necton case, wth the ncreasng necton, varaton of a velocty profles f ( η) and varaton of a skn frcton coeffcent f () decrease and Concentraton profles φ (η) and concentraton gradent at the surface () ncrease. It s due to the necton, both Fgure 7. φ (η) for varous values of Sc N=, S=., B=.3, n=, q= Sc=.,.,.3, Fgure 8. () for dfferent values of sc The effects of the reacton rate parameter (B) are llustrated n Fgures (9-), t s found that the reacton rate parameter affects the solute profles n smlar way as that of the Schmdt number.e., the ncrease of the reacton rate parameter B at fxed reduces both the concentraton profles φ (η) and concentraton gradent at the surface () and thus the chemcal reacton enhances the mass transfer.

9 Amercan Journal of Computatonal and Appled Mathematcs 4, 4(4): N=, Sc=.3, S=., n=,q= B =,,, 3 whle the ncrease of N ncrease both concentraton profles φ(η) and concentraton gradent at the surface (). Also t s shown n Fgures (-3, 5, 3) that the presence of a porous medum ncreases the resstance to flow resultng n decrease n the flow velocty and ncrease n the solute concentraton whch ncreases the solute boundary layer thckness. φ(η).4 N=,Sc=.5,B=.3,S=., n=,q= Fgure 9. φ(η) for varous values of B N=, Sc=.3, S=., n=,q= B =,,, Fgure. () for dfferent values of B The effects of the power law exponent (n) are shown n Fgures (-), t s observed that the ncrease of the power law exponent n wth n > reduces Concentraton profles φ (η) at fxed. Whle the concentraton profle ncreases wth the ncrease n the magntude of n wth n < and for large negatve values of n, the overshoot of solute s observed near the surface. The magntude of the concentraton gradent at the surface () ncreases wth the ncrease n postve n but decreases wth the ncrease n the magntude of n. wth n <. Thus, the effect of ncrease of n when the surface concentraton s Cw = C + Cx n whch s completely opposte to the effect of the ncrease of n when the surface concentraton s / n Cw = C + C x. Note that, the wall concentraton s constant when n=. Porous parameter (N) effects are demonstrated n Fgures (3-6), t s found that at fxed, varaton of a velocty profles f ( η) and varaton of a skn frcton coeffcent f () decrease wth the ncrease n porous parameter φ(η) f Fgure. φ(η) for varous values of n Fgure. () for dfferent values of n Fgure 3. n = 3,,, -, -3 N=,Sc=.5,B=.3,S=., n=,q= n = 3,,, -, S=, Sc=, B=, n=, q= N =.5,,.5, 3 4 f ( η) for dfferent values of N

10 5 Nabl T. M. Eldabe et al.: Numercal Solutons for Boundary Layer Flud Flow wth Mass Transfer over a Movng Permeable Flat Plate Embedded n Porous Medum wth Varable Wall Concentraton n Presence of Chemcal Reacton. S=, Sc=, B=, n=, q=.7.65 N=., Sc=, B=, n=, s=..5.6 N =.5,,.5,.55 f (). f ( η) Fgure 4. ().8 f for dfferent values of N S=, Sc=, B=, n=,q=.4 q=.3.35 q=.3.3 q=.35 q= Fgure 7. f ( η) for dfferent values of q N=., Sc=, B=, n=, s=. φ(η).6.4. N =,.5,, Fgure 5. φ(η) for varous values of N f () q=.3 q=.3 -. q=.35 q= Fgure 8. f () for dfferent values of q -.4 S=, Sc=, B=, n=,q= N=., Sc=, B=, n=, S= N =,.5,,.5 φ(η).6.4. q=.3 q=.3 q=.35 q= Fgure 6. () for dfferent values of N The velocty rato parameter (q) effects are llustrated n Fgures (7-), t s observed that at fxed, varaton of a skn frcton coeffcent f () decreases wth the ncreasng n q > whle varaton of a velocty profles f ( η) ncreases but far away from the plate f ( η) decreases. The ncrease of q > reduce both concentraton profles φ(η) and concentraton gradent at the surface () Fgure 9. φ(η) for varous values of q It s seen n Fgures (-) that the soluton are unque when q >, whle dual solutons are found to exst when q <,.e. when the plate and the free stream move n the opposte drectons. t s notced n Fgure () that the momentum boundary layer thckness ncreases n the upper branch soluton wth the ncreasng of the magntude of the velocty rato parameter and for the lower branch soluton t decreases wth magntude of q. The concentraton φ(η) at

11 Amercan Journal of Computatonal and Appled Mathematcs 4, 4(4): Fgure ncreases wth magntude of q for the upper branch n any pont and for the lower branch soluton ntally ncreases wth magntude of q and for large changng the nature, t decreases. It s shown n table () that the values of f () are postve when q <.5, and they become negatve when the value of q exceeds.5. The concentraton profles φ(η) and concentraton gradent () for some values of the chemcal reacton rate parameter (B) are presented n Fgures 3 and 4 respectvely. These profles satsfy the boundary condtons (3), whch support the numercal results besdes supportng the dual nature of the solutons presented n Fgures (-3). They show that φ(η) ncreases for both solutons wth the decreasng of B n any pont whle Fgure 4 shows that the concentraton gradent (η) for both solutons ntally decreases wth the ncreasng of B but far away from the plate t ncreases N=., Sc=, B=, n=, s=. -. q=.3 q=.3 q=.35 q= φ(η) φ Sc=,B=, n=,s=,n= Upper branch Lower branch q= -.5 q= -.5 q= -.4 q= -.4 q= -.3 q= Fgure. φ() for dfferent negatve values of q Sc=.5,q= -.5,n=,S=,N= Upper branch Lower branch B=. B=. B=-. B=-. B= B= f.5 Fgure. () for dfferent values of q q= -.5 q= -.5 q= -.4 q= -.4 q= -.3 q= -.3 Upper branch Fgure 3. φ( ) for dfferent negatve values of B Sc=.5,q= -.5,n=,S=,N= Upper branch B=. B=, B=-. B=-. B= B=.5 -. Lower branch Sc=,B=, n=,s=,n= Fgure. f ( η) for dfferent negatve values of q Lower branch Fgure 4. φ ( ) for dfferent negatve values of B

12 5 Nabl T. M. Eldabe et al.: Numercal Solutons for Boundary Layer Flud Flow wth Mass Transfer over a Movng Permeable Flat Plate Embedded n Porous Medum wth Varable Wall Concentraton n Presence of Chemcal Reacton 5. Conclusons The development of boundary layer flow wth mass transfer through porous medum over a movng permeable flat plate wth varable wall concentraton n the presence of frst order chemcal reacton s nvestgated. The classcal Blasus (98) and Sakads (96) problems are partcular cases of the present problem. Also Ishak (7) s extended by the present work. The nonlnear system of nonlnear dfferental governng equatons s solved numercally by both Keller-Box method and the method based on fourth order Runge-Kutta teraton scheme wth shootng method. A comparson s made wth partcular case of the present study prevously reported by Ishak [] and the results are found to be n excellent agreement. We found that, the Varaton of a velocty profles f ( η) and varaton of a skn frcton coeffcent f () decrease wth the ncreasng n porous parameter N whle both concentraton profles φ(η) and concentraton gradent at the surface () ncrease. The ncreasng of the velocty rato parameter (q), sucton or necton parameter S, the reacton rate parameter B, the Schmdt number Sc and the power law exponent (n) reduce both Concentraton profles φ(η) and concentraton gradent at the surface (). The ncreasng of velocty rato parameter (q) ncreases the varaton of a velocty profles f ( η) but far away from the plate t decreases. Dual solutons are found to exst when q <.e. the plate and the free stream move n the opposte drectons. Consequently, the solute boundary layer thckness s found to ncrease wth the ncrease of magntude of q for the upper branch and n the lower branch t decreases. It s found that both φ(η) and the concentraton gradent (η) ncrease for dual solutons wth the decreasng of B n any pont but far away from the plate the concentraton gradent (η) decreases for dual solutons. REFERENCES [] Blasus H., Grenzschchten n Flüssgketen mt klener Rebung, Zetschrft für Mathematk und Physk, 98, vol.56, -37. [] Howarth L., on the soluton of the lamnar boundary layer equatons, Proc. Roy. Soc. London A, 938, vol.64, [3] Sakads B.C., Boundary-layer behavor on contnuous sold surfaces: Boundary-layer equatons for two dmensonal and axsymmetrc flows, AIChE J., 96, vol.7, 6-8. [4] Sekman J., The lamnar boundary layer along a flat plate, Z. Flugwss.,96, vol., [5] Klemp J.B. and Acrvos A., The movng-wall boundary layer wth reverse flow. J. Flud Mech., 976, vol.76, [6] Abussta A.M. M., A note on a certan boundary-layer equaton, Appl. Math. Comp., 994, vol.64, [7] Abdulhafez T. A., Skn frcton and heat transfer on a contnuous flat surface movng n a parallel free stream, Int. J. Heat Mass Transf., 985, vol.8, [8] Hussan M.Y., Lakn W. D. and Nachman A., on smlarty solutons of a boundary-layer problem wth an upstream movng wall, SIAM J. Appl. Math., 987, vol.47, [9] Ln H.T. and Hang S. F., Flow and heat transfer of plane surface movng n parallel and reversely to the free stream, Int. J. Heat Mass Transf.,994, vol.37, [] Wedman PD, Kubtschek DG, Davs AMJ, The effect of transpraton on self-smlar boundary layer flow over movng surfaces, Int. J Eng Sc., 6, Vol. 44, [] Cortell R., A Numercal Tacklng on Sakads Flow wth Thermal Radaton, CHIN. PHYS. LETT., 8, Vol. 5, No. 4, 34. [] Ishak A, Nazar R, and Pop I, Boundary layer on a movng wall wth sucton and necton, Chn Phys Let., 7, Vol. 4, No.8, [3] Mukhopadhyay S., Heat transfer n a movng flud over a movng non-sothermal flat surface, Chn. Phys. Lett., vol.8, No., 476. [4] Mukhopadhyay and Gorla R.S.R., dual solutons for boundary flow of movng flud over a movng surface wth power law surface temperature, Int. J. of Appled Mechancs and Engneerng, 3, vol.8, No., 3-4. [5] Afzal N., Badaruddn A., Elgarv A.A., Momentum and transport on a contnuous flat surface movng n a parallel stream, Int. J Heat Mass Transfer,993, Vol. 36,, [6] Andersson H. I., Hansen O.R., Holmedal B.,994, Dffuson of a chemcally reactve speces from a stretchng sheet, Int. J. Heat Mass Transfer 37, [7] Takhar H.S., Chamkha A.J. and Nath G.,, Flow and mass transfer on a stretchng sheet wth a magnetc feld and chemcally reactve speces, Int. J. of Engneerng Scence 38, [8] Uddn M. S., Bhattacharyya K., Layek G.C. and W.A. Pk., Chemcally reactve solute dstrbuton n a steady MHD boundary layer flow over a stretchng surface, J. of Appled Flud Mechancs,,Vol. 4, No. 4, [9] Bhattacharyya K., Mass transfer on a contnuous flat plate movng n parallel or reversely to a free stream n the presence of a chemcal reacton,, Internatonal Journal of Heat and Mass Transfer 55, [] Glat A. and Subramanan V., 7, Numercal Methods for Engneers and Scentsts, Wley. [] Merkn J.H., on dual solutons occurrng n mxed convecton n a porous medum, J Eng Math, 985, Vol [] Magyar E. and Keller B., Exact solutons for self-smlar boundary-layer flows nduced by permeable stretchng walls, Eur. J. Mech. B Fluds,, vol.9, 9-. [3] Elbashbeshy EMA, Bazd MA, The mxed convecton along a vertcal plate wth varable surface heat flux embedded n porous medum, Appl Math Comp,, Vol. 5, [4] Eldabe N.T., Elsaka A. G., Radwan A. E. and Eltaweel M.A.,

13 Amercan Journal of Computatonal and Appled Mathematcs 4, 4(4): , Effects of chemcal reacton and heat radaton on the MHD flow of vsco-elastc flud through a porous medum over a horzontal stretchng flat plate, J. of Amercan Scence, 6(9). [5] Hossen M.A., Applcatons of Scalng Group of Transformaton on Boundary Layer Flow and Mass Transfer Over a Stretchng Sheet Embedded n a Porous Medum, Journal of Physcal Scences,, Vol. 5, [6] E. M. A. Elbashbeshy, A. M. Sedk, effect of chemcal reacton on mass transfer over a stretchng surface embedded n a porous medum, Internatonal Journal of Computatonal Engneerng Research, 4, Vol.4, Issue, -8. [7] Keller H. B., A New Dfference Scheme for Parabolc Problems, In Bramble, J. Numercal Solutons of Partal Dfferental Equatons, New York, Academc Press,97. [8] Keller H. B., A New Dfference Scheme for Parabolc Problems, In Hubbard, B. Numercal Solutons of Partal Dfferental Equatons, New York, Academc Press, 97. [9] Keller H. B. and Cebec T., Accurate Numercal Methods for Boundary Layer flows. II, Two-dmensonal Turbulent Flows, AIAA Journal ,97. [3] Cebec T., and Bradshaw P., Momentum Transfer n boundary Layers, Hemsphere Publshng Corporaton, New York, 977.