Appled and Computatonal Mathematcs 014; 3(): 48-56 Publshed onlne Aprl 30 014 (http://www.scencepublshnggroup.com/j/acm) do: 10.11648/j.acm.014030.1 Effect of varable thermal conductvty on heat and mass transfer flow over a vertcal channel wth magnetc feld ntensty Ime Jmmy Uwanta Halma Usman Department of Mathematcs Usmanu Danfodyo Unversty Sokoto Ngera Emal address: meuwanta@yahoo.com(i. J. Uwanta) hhalmausman@yahoo.com (H. Usman) To cte ths artcle: Ime Jmmy Uwanta Halma Usman. Effect of Varable Thermal Conductvty on Heat and Mass Transfer Flow over a Vertcal Channel wth Magnetc Feld Intensty. Appled and Computatonal Mathematcs. Vol. 3 No. 014 pp. 48-56. do: 10.11648/j.acm.014030.1 Abstract: The objectve of ths paper s to study thermal conductvty and magnetc feld ntensty effects on heat and mass transfer flow over a vertcal channel both numercally and analytcally. The non-lnear partal dfferental equatons governng the flow are non-dmensonalsed smplfed and solved usng Crank Ncolson type of mplct fnte dfference method. To check the accuracy of the numercal soluton steady state solutons for velocty temperature and concentraton felds are obtaned by usng perturbaton method. Graphcal results for velocty temperature concentraton skn frcton Nusselt number and Sherwood number have been obtaned to show the effects of dfferent parameters enterng n the problem. Results from these study shows that velocty temperature and concentraton ncreases wth the ncrease n the dmensonless tme untl they reach steady state value. Also t was observed that the analytcal and numercal solutons agree very well at large values of tme. Keywords: Thermal Conductvty Heat and Mass Transfer Magnetc Feld Thermal Radaton 1. Introducton The nfluence of magnetc feld ntensty and thermal conductvty on the study of heat and mass transfer flow over a vertcal channel s currently attractng the attenton of a growng number of researchers because of ther mmense potental use n scence technology and engneerng applcatons. The effect of magnetc feld on vscous ncompressble flow of electrcally conductng flud has ts mportance n many applcatons such as metallurgcal processes whch nvolves the coolng of flaments n the felds of stellar and planetary magneto spheres aeronautcs plasma physcs nuclear scence etc. Magnetc feld ntensty also plays mportant role n agrculture petroleum ndustres geologcal formatons thermal recovery of ol and many other places that are mportant for our ndustral and fnancal developments. In most of the studes on hydromagnetc heat and mass transfer thermal conductvty has been taken as constant. In metallurgcal engneerng processng the thermal conductvty s n fact temperature dependent. That s the numercal value of thermal conductvty changes wth temperature. Therefore to predct the flow of heat and mass transfer accurately mathematcal models must consder the varaton of thermal conductvty wth temperature. Comprehensve revews on the subject to the above problems have been made by Evans [6] and Sparrow et al. [17]. Many researchers have studed the effects of thermal conductvty and magnetc feld ntensty on free convecton flow of heat and mass transfer. Cheng [4] studed the effect of magnetc feld on heat and mass transfer by natural convecton from a vertcal surface n porous medum. Venkateswalu et al. [19] nvestgated fnte dfference analyss on convectve heat transfer flow through a porous medum n a vertcal channel wth magnetc feld. Sharma and Sngh [16] have dscussed n detal the effect of varable thermal conductvty n MHD flud flow over a stretchng sheet consderng heat source and snk parameter. In another artcle Mahmoud [8] studed the effect of varable thermal conductvty and radaton on the mcropolar flud flow n the presence of a transverse magnetc feld. Noreen et al. [9] have presented a mathematcal model nvestgatng the mxed convectve heat and mass transfer effects on perstaltc flow of magneto hydrodynamc pseudo plastc
Appled and Computatonal Mathematcs 014; 3(): 48-56 49 flud n a symmetrc channel. Addtonally Zhang and Huang [0] descrbed the effect of local magnetc feld on electrcally conductng flud flow and heat transfer. Also work on the effect of varable vscosty and thermal conductvty of mcropolar flud n a porous channel n presence of magnetc feld has been developed by Patowary [1]. Smlarly Seddek et al. [15] studed the effects of temperature dependent vscosty and thermal conductvty on unsteady MHD convectve heat transfer past a semnfnte vertcal porous plate takng nto account the effect of a magnetc feld n the presence of varable sucton. Elbashbeshy et al. [5] have presented a theoretcal study on the effect of magnetc feld on flow and heat transfer over a stretchng horzontal cylnder n the presence of a heat source/snk wth sucton/njecton. Ahmed [1] analyzed a mathematcal model of nduced magnetc feld wth vscous/magnetc dsspaton bounded by a porous vertcal plate n the presence of radaton. Salem [14] nvestgated varable vscosty and thermal conductvty effects on MHD flow and heat transfer n vsco elastc flud over a stretchng sheet. In another artcle a steady twodmensonal magneto hydrodynamc heat and mass transfer free convecton flow along a vertcal stretchng sheet n the presence of magnetc feld has been examned by Hosan and Samand [7]. A numercal nvestgaton on the study of the effect of thermal conductvty on MHD flow past an nfnte vertcal plate wth Soret and Dufour effects has been carred out by Usman and Uwanta [18]. Parvn [11] presented a revew of magneto hydrodynamc flow heat and mass transfer characterstcs n a flud. In a related development Oahmre and Olajuwon [10] descrbed the study of hydro magnetc flow of a vscous flud near a stagnaton pont on a lnearly stretchng sheet wth varable thermal conductvty and heat source. Recently Qasm et al. [13] have consdered MHD boundary layer slp flow and heat transfer of Ferro flud along a stretchng cylnder wth prescrbed heat flux. Most recently Azzan et al. [] nvestgated the effect of magnetc feld on lamnar convectve heat transfer of magnetc nano fluds. In spte of all these studes the present nvestgaton focuses on the effect of thermal conductvty on heat and mass transfer flow over a vertcal channel takng nto account the nduced magnetc feld ntensty. The present study may have useful applcatons to several transport processes as well as magnetc materal processng.. Mathematcal Formulaton Consder an unsteady flow of a vscous ncompressble flud past a vertcal channel wth varable thermal conductvty and magnetc feld ntensty. A magnetc feld B 0 of unform strength s appled transversely to the drecton of the flow. The x ' axs s taken along the plate n the vertcally upward drecton and the y ' axs s normal to the plate n the drecton of the appled unform magnetc feld. The flud beng electrcally conductng the magnetc Reynolds number s assumed to be very small and hence the nduced magnetc feld can be neglected n comparson wth the appled magnetc feld n the absence of any nput electrc feld. It s also assumed that the effect of vscous dsspaton s neglgble n the energy equaton. Under the above assumptons as well as the Boussnesq s approxmaton the equatons of conservaton of mass momentum energy and speces governng the free convecton boundary layer flow past a vertcal channel can be expressed as: ν' 0 y' u ' t ' ν u ' gβ T ' T ' 0 y σb 0 ' ρ u' ν k * u' ( ) gβ * ( C ' C ' 0 ) T ' k 0 1α T ' T ' t ' ρc p y ' 0 1 q r ρc p y ' ( ) T' y ' (1) () (3) C ' D C ' ( t ' y ' R* C ' C ' 0 ) (4) wth the followng ntal and boundary condtons as follows: t 0 u ' 0T ' T ' w C' C ' w for all y ' t > 0 u ' 0T ' T ' w C' C ' w at y ' 0 u ' 0T ' T 0 C ' C 0 at y ' H (5) The thermal radaton s assumed to be present n the form of a undrectonal flux n the y drecton that s transverse to the vertcal surface. Usng the Rosseland approxmaton by Brewster [3] the radatve heat flux q r s gven by: q r 4σ ' 0 T '4 (6) 3k ' y ' where u ' and v ' are the Darcan velocty components n the x and y drectons respectvely t s the tme ν s the knematc vscosty g s the acceleraton due to gravty βs the coeffcent of volume expanson wth temperature ρ s the densty of the flud σ s the scalar electrcal conductvty β * s the volumetrc coeffcent of expanson wth concentraton C p s the specfc heat capacty at constant pressure k * s the dmensonless permeablty of the porous medum k 0 s the dmensonless thermal conductvty of the ambent flud α s a constant
50 Ime Jmmy Uwanta and Halma Usman: Effect of Varable Thermal Conductvty on Heat and Mass Transfer Flow over a Vertcal Channel wth Magnetc Feld Intensty dependng on the nature of the flud R * s the dmensonless chemcal reacton Ds the coeffcent of molecular dffusvty B 0 s the magnetc nducton of constant strength q r s the radatve heat flux n the y drecton σ0 ' s the Stefan-Boltzmann constant k ' s the mean absorpton coeffcent T ' and T ' 0 are the temperatures of the flud nsde the thermal boundary layer and the flud temperature n the free stream respectvely whle C ' and C ' are the correspondng concentratons. 0 To obtan the solutons of equatons () (3) and (4) subject to the condtons (5) n non-dmensonal forms we ntroduce the followng non-dmensonal quanttes: u u' t t' y' H y H θ T' T ' 0 T ' w T' 0 C C' C ' 0Pr ρc p M σb 0H C ' w C' 0 k 0 ρ Sc D k k* νh K r R* H λ α ( T ' T ' 0)R 16aσ' 0 HT'3 0 k ' Gr H gβ ( T ' w T' 0 ) ( ) Gc H gβ * C ' w C ' 0. Applyng these non-dmensonless quanttes (7) the set of equatons () (3) (4) and (5) reduces to the followng: u t u y Mu 1 k u Grθ GcC Pr θ 1 λθ t ( ) θ y λ θ Rθ y (9) (7) (8) C t 1 C Sc y K rc (10) The ntal and boundary condtons n non-dmensonal quanttes are: t 0 u 0θ C 0 for all y t > 0 u 0θ 1C 1 at y0 (11) u 0θ 0C 0 at y 1 where M s the magnetc feld parameter k s the porous parameter Gr s the thermal Grashof number Gcs the solutal Grashof number Prs the Prandtl number λs the varable thermal conductvty parameter Rs the radaton parameter Krs the chemcal reacton parameter Sc s the Schmdt number t s the dmensonless tme whle uand vare the dmensonless velocty components n x and y drectons respectvely. The skn frcton Nusselt number and Sherwood number are the mportant physcal parameters for ths type of boundary layer flow whch n non- dmensonal form respectvely are gven by: The skn-frcton coeffcent Nusselt number and Sherwood number are the mportant physcal parameters for ths type of boundary layer flow whch n nondmensonal form respectvely are gven by: C f u y y0 Nu θ y 3. Analytcal Solutons y0 Sh C y y0 (1) The governng equatons presented n ths problem are hghly coupled and non lnear and exhbt no closed-form solutons. In order to check the accuracy of the present numercal scheme of ths model there s need to compare numercal solutons wth the analytcal solutons. Snce the governng equatons are non-lnear t s therefore of nterest to reduce the governng equatons of the present problem to a form that can be solved n closed form. At steady state the physcal parameters do not have any effect hence the steady state equatons and boundary condtons for the problem can be wrtten as u y M 1 u Grθ Gc 0 (13) k The boundary condtons are: θ y Rθ 0 (14) C y ScK r 0 (15) u 0 θ 1 C 1 at y 0 u 0 θ 0 C 0 at y 1. (16) To fnd the approxmate soluton to equatons (13)-(15) subject to equaton (16) we use perturbaton method whch s a method that s used to approxmate the soluton to a dfferental equaton analytcally. Therefore the physcal varables u θ and C can be expanded n the power of ( R << 1). Ths can be possble physcally as R for the flow s always less than unty. Hence we can assume soluton of the form
Appled and Computatonal Mathematcs 014; 3(): 48-56 51 ( ) ( ) ( ) ( ) ( ) ( ) u u y Ru y R 0 1 0 1 0 1 0( ) θ θ y Rθ y R 0( ) C C y RC y R 0( ) (17) Usng equaton (17) n equatons (13)-(16) and equatng the coeffcent of lke powers of R we have: u '' 0 M 1 Grθ 0 GcC 0 (18) k θ '' 0 0 (19) C '' 0 ScK r C 0 0 (0) M 1 1 k u 1 Grθ 1 GcC 1 (1) u '' θ '' 1 θ 0 () C '' ScK C 1 r 1 0 (3) The correspondng boundary condtons are u 0 θ 1 C 1 0 0 0 u 0 θ 0 C 0 at y 0 1 1 1 u 0 θ 0 C 0 0 0 0 u 0 θ 0 C 0 at y 1 1 1 1 (4) Solvng equatons (18)-(3) wth the help of equaton (4) we get: H 1 e y p H e y p E 1 E y E 3 e y ScK r E 4 e y ScK r (5) θ 0 1 y (6) C 0 Ae y ScK r Be y ScK r (7) u 1 H 3 e y p H 4 e y p E 5 y 3 E 6 y E 7 y E 8 (8) θ 1 1 3 y 1 y 1 6 y3 (9) C 1 0 (30) In vew of the above equatons the solutons are: u H 1 e y p H e y p E 1 E y E 3 e y ScK r E 4 e y ScK r ( ) R H 3 e y p H 4 e y p E 5 y 3 E 6 y E 7 y E 8 (31) θ 1 y R 1 3 y 1 y 1 6 y3 (3) C Ae y ScK r Be y ScK r (33) 4. Numercal Soluton Procedure In order to solve the unsteady non-lnear coupled partal dfferental equatons (8)-(10) wth the assocated ntal and boundary condtons (11) an mplct fnte dfference technque of Crank-Ncolson type whch s known to converge rapdly and uncondtonally stable has been employed. The fnte dfference equatons correspondng to equatons (8)-(10) usng the method are as follows: u j1 j u 1 t y ( ) u j1 u j1 u j1 u j u j u j ( 1 ) 1 1 1 Mu j 1 k u j Grθ j j GcC Pr θ t j1 θ j H y ( ) θ j1 θ j1 θ j1 θ j θ j θ j ( 1 ) 1 1 1 λ ( y) θ 1 j j ( θ ) j Rθ C j1 j C 1 t Sc y ( ) C j1 C j1 C j1 C j C j C j ( 1 ) 1 1 1 K r C j (34) (35) (36) The ntal and boundary condtons may be expressed as u j 0 θ j 0 C j 0 j 0 θ 0j 1 C 0j 1 u Hj 0 θ Hj 0 C Hj 0 (37) where H corresponds to 1. Equatons (34) (35) and (36) may be wrtten respectvely as follows: d 1 u j1 1 d u j1 d 1 u j1 1 d 1 u j 1 ( d 3 d 4 )u j d 1 u j 1 d 5 θ j d 6 C (38) j d 7 θ j1 1 d 8 θ j1 d 7 θ j1 1 d 7 θ j 1 d 9 θ j d 7 θ j 1 d 10 θ j j 1 θ ( ) d 11 θ j (39) d 1 C j1 1 d 13 C j1 d 1 C j1 1 d 1 C j 1 (40) ( d 14 d 15 )C j j d 1 C 1 The ndex corresponds to space y and j corresponds to
5 Ime Jmmy Uwanta and Halma Usman: Effect of Varable Thermal Conductvty on Heat and Mass Transfer Flow over a Vertcal Channel wth Magnetc Feld Intensty tme t. yand t are the mesh szes along y-drecton and tme t-drecton respectvely. The fnte dfference equatons (38)-(40) at every nternal nodal pont on a partcular n- level consttute a tr-dagonal system of equatons whch are solved by usng the Thomas algorthm. In each tme step the temperature and concentraton profles have been computed frst from equatons (39) and (40) and then the computed values are used to obtan the u velocty profle at the end of tme steps that s j 1 computed from equaton (38). Ths process s carred out untl the steady state s reached. The steady-state soluton of the convergence crtera for stablty of the scheme s assumed to have been reached when the absolute dfferences between the values of velocty temperature and concentraton at two consecutve tme steps are less than 5 10 at all grd ponts. Computatons are carred out for dfferent values of physcal parameters nvolved n the problem. 5. Results and Dscusson In order to report on the analyss of the flud flow numercal computatons are carred out for varous values of magnetc parameter ( M ) thermal Grashof number ( Gr ) solutal Grashof number ( Gc ) permeablty parameter ( k ) thermal conductvty parameter ( λ ) radaton parameter (R) Prandtl number (Pr) Schmdt number ( Sc) chemcal reacton parameter ( K r ) and dmensonless tme (t). Therefore ths study s focused on the effects of these governng parameters on the transent velocty temperature as well as concentraton profles. Here the man dscusson s restrcted to the adng of favourable case only for fluds wth Prandtl number ( Pr 0.711.03.07.0 ) that represent ar atmospherc pressure Freon and water respectvely. The dffusng chemcal speces of most common nterest n ar has Schmdt number and s taken for water (Sc 0.60) Carbon doxde ( Sc 0.94) Methanol ( Sc 1.0 ) and Propyl benzene (Sc.6). The value of thermal Grashof number s taken to be postve whch correspond to the coolng of the plate. The default values of the thermo physcal parameters are specfed as follows: Gr 5Gc 5M Pr 0.71k 0.5 R 5 λ 0.5K r 1Sc 0.60 All graphs therefore correspond to these values unless otherwse ndcated. Fgures (1) and () llustrate the nfluence of thermal Grashof number Grn case of coolng of the plate and the solutal Grashof number Gc. It s notced that an ncrease n Grand Gcresults n an ncrease n the velocty. It s due to the fact that ncrease n the values of Grand Gchas the tendency to ncrease the thermal and mass buoyancy effect. Ths gves rse to an ncrease n the nduced flow. In fgure (3) t s observed that ncreasng permeablty parameter k enhances the velocty of the flud. The effect of magnetc feld parameter Mn case of coolng the plate on the velocty profle s depcted n fgure (4). It s found that the velocty decreases wth ncreasng magnetc parameter. The effect of radaton parameter Ron the temperature varaton for two workng fluds ar (Pr 0.71) and water (Pr 7.0) s graphcally llustrated n fgure (5). It s evdent from ths fgure that as Rncreases consderable reducton s observed n temperature profles. Fgure (6) reveals the transent temperature profles aganst y (dstance from the plate) for dfferent values of thermal conductvty parameter (λ V) n case of ar (Pr 0.71) and water (Pr 7.0). It s seen that as λ ncreases for both ar and water the temperature ncreases whch s physcally true because thermal conductvty of flud ncreases wth ncreasng Prandtl number resultng n thermal boundary layer thckness thereby ncreasng the temperature profles. Fgures (7) and (8) descrbe the behavor of concentraton profles for dfferent values of Schmdt number Sc and chemcal reacton parameter K r. A decrease n concentraton wth ncreasng Scas well as K r s observed from these fgures. Also t s noted that the concentraton boundary layer becomes thn as the Schmdt number as well as the chemcal reacton parameter ncreases. Fgures (9-11) show the velocty dstrbuton temperature and concentraton profles wth varaton of dmensonless tme trespectvely. It can be seen that the velocty temperature and concentraton of the flud ncreases wth tme and ultmately reaches ther steady state values for ar (Pr 0.71). The valdty of the present model has been verfed by comparng the numercal solutons and the analytcal solutons through fgures (1)-(14) for velocty temperature and concentraton profles respectvely. These results are presented to llustrate the accuracy of the numercal soluton. It s observed that the agreement between the results s good because the curves correspondng to analytcal and numercal solutons are lyng close to the other. Ths has establshed confdence n the numercal results reported n ths paper. Fgure (15) shows the varaton of skn frcton coeffcent aganst magnetc parameter M for dfferent values of thermal Grashof number Gr solutal Grashof number Gcand permeablty parameter k. It depcts that the skn frcton ncreases wth an ncrease n any of these parameters Gr Gcand k. The effects of Prandtl number Pr and thermal conductvty λ on the Nusselt number aganst radaton parameter Rare presented n fgure (16). It s observed that the rate of heat transfer decreases wth an ncreasng Pr. Also t s found that the rate of heat transfer falls wth ncreasng thermal conductvty. Fnally fgure (17) demonstrates the effect of Schmdt number Sc and chemcal reacton parameter K r on Sherwood number. It dsplays that the rate of concentraton transfer ncreases wth ncreasng values of Scand K r.
Appled and Computatonal Mathematcs 014; 3(): 48-56 53 Fg (1). Velocty profle for dfferent values of thermal Grashof number Fg (6). Temperature profle for dfferent values of varable thermal conductvty Fg (). Velocty profle for dfferent values of solutal Grashof number Fg (7). Concentraton profle for dfferent values of Schmdt number Fg (3). Velocty profle for dfferent values of porous parameter Fg (8). Concentraton profle for dfferent values of chemcal reacton parameter Fg (4). Velocty profle for dfferent values of magnetc parameter Fg (9). Velocty profle for dfferent values of dmensonless tme Fg (5). Temperature profle for dfferent values of radaton parameter Fg (10). Temperature profle for dfferent values of dmensonless tme
54 Ime Jmmy Uwanta and Halma Usman: Effect of Varable Thermal Conductvty on Heat and Mass Transfer Flow over a Vertcal Channel wth Magnetc Feld Intensty Fg (11). Concentraton profle for dfferent values of dmensonless tme Fg (16). Nusselt number profle for dfferent values of Prandtl number and varable thermal conductvty Fg (1). Comparson of numercal and analytcal solutons for velocty profle Fg (17). Sherwood number profle for dfferent values of Schmdt number and chemcal reacton parameter 6. Conclusons Fg (13). Comparson of numercal and analytcal solutons for Temperature profle Fg. (14). Comparson of numercal and analytcal solutons for Concentraton profle Fg (15). Skn frcton profle for dfferent values of porous parameter thermal and solutal Grashof numbers In ths paper the study on the effect of varable thermal conductvty on heat and mass transfer flow over a vertcal channel wth magnetc feld ntensty usng Crank-Ncolson type of mplct fnte dfference method has been carred out. The expressons for the velocty temperature and concentraton felds have been constructed and the effects of varous parameters on heat and mass transfer characterstcs of the flud flow are dscussed graphcally. From the present numercal nvestgaton the followng conclusons have been drawn: 1. The velocty of the flud ncreases wth an ncrease n thermal Grashof number solutal Grashof number permeablty parameter and dmensonless tme whle t decreases wth an ncrease n magnetc feld parameter as shown n Fgs. (1-4) and Fg. (9.). Increasng thermal conductvty parameter and dmensonless tme leads to ncrease the flud temperature. Ths s clearly ndcated n Fgs. (6) and (10). 3. A decrease n concentraton profle wth ncreasng Schmdt number as well as chemcal reacton parameter s observed n Fgs. (7) and (8). 4. The skn frcton coeffcent ncreases wth ncreasng thermal Grashof number solutal Grashof number and permeablty parameter as llustrated n fgure (15). 5. The rate of heat transfer n terms of Nusselt number falls wth ncrease n Prandtl number and thermal conductvty parameter s notced n fgure (16). 6. It s marked n fgure (17) that the rate of concentraton transfer ncreases wth ncreasng values of Schmdt number and chemcal reacton parameter.
Appled and Computatonal Mathematcs 014; 3(): 48-56 55 7. The accuracy of the present model has been verfed by comparng numercal and analytcal solutons and the agreement between the results s excellent. Ths s clearly shown n Fgs. (1-14). The solutons presented n ths paper for varous thermo physcal effects would be useful for subsequent analyss n heat and mass transfer n polymer processng metallurgcal transport modelng and many geophyscal processes lke crude ol recovery. Appendx r 1 t ( y) r tmr 3 tgrr 4 tgc H 1 λθr 5 Hr 1 r 6 λr 1 r 7 tr r 8 tsck r r 9 t 1 k d rd 1r 1 1 ( 1) d 3 r 1 r 9 d 4 r d 5 r 3 d 6 r 4 d 7 r 5 d 8 ( Prr 5 )d 9 ( Pr r 5 )d 10 r 6 d 11 r 7 d 1 r 1 d 13 ( r 1 Sc)d 14 ( Sc r 1 )d 15 r 8 A 1 BB e ScK r e ScK r e ScK r ( ) p M 1 k E 1 Gr p E Gr p E 3 GcA ScK r p E 4 GcB ScK r p E Gr 5 6p E Gr 6 p E 7 6E 5 p Gr 3p E E 6 8 p H 1 ( H E 1 E 3 E 4 ) H 3 ( H 4 E 8 ) 1 E 1 1 e p H ( e p e ) p E 4 e ScK r e p H 4 ( ) E E e ScK r 3 ( e ) p ( ) 1 ( e p e ) ( E p 8 ( 1 e ) p ( E 5 E 6 E 7 E 8 )). Nomenclature C concentraton C p specfc heat at constant pressure D - mass dffusvty g acceleraton due to gravty Gr Grashof number Gc solutal Grashof number k porous parameter Nu Nusselt number Pr Prandtl number Sc Schmdt number R- radaton parameter K - chemcal reacton parameter r T temperature C -Skn frcton f Sh- Sherwood number u v veloctes n the x and y-drecton respectvely x y Cartesan coordnates along the plate and normal to t respectvely B 0 - magnetc feld of constant strength M magnetc feld parameter Greek Letters β* - coeffcent of expanson wth concentraton β - coeffcent of thermal expanson ρ- densty of flud θ- dmensonless temperature υ- knematc vscosty - Stefan Boltzmann constant σ ' 0 λ - varable thermal conductvty σ- electrcal conductvty of the flud Subscrpts w- condton at wall - condton at nfnty References [1] S. Ahmed Mathematcal model of nduced magnetc feld wth vscous/magnetc dsspaton bounded by a porous vertcal plate n presence of radaton Internatonal Journal of Appled Mathematcs and Mechancs vol. 8 no.1 pp. 86-104 01. [] R. Azzan E. Doroodch T. Mckrell J. Buongomo L. W. Hu and B. Moghtaden Effect of magnetc feld on lamnar convectve heat transfer of magnetc nano fluds Internatonal Journal of Heat and Mass Transfer vol. 68 pp. 94-109 014. [3] M. Q. Brewster. Thermal radatve transfer and propertes John Wley and Sons. Inc New York 199. [4] C. Y. Cheng Effect of magnetc feld on heat and mass transfer by natural convecton from vertcal surfaces n porous meda- an ntegral approach Internatonal Communcatons n Heat and Mass Transfer vol. 6 no. 7 pp. 935-943 1999. [5] E. M. Elbashbeshy T. G. Emam M. S. El-Azaba and K. M. Abdelgaber Effect of magnetc feld on flow and heat transfer over a stretchng horzontal cylnder n the presence of a heat source/snk wth sucton/ njecton Appled Mechancal Engneerng vol. 1 no. 1 pp. 1-5 01. [6] H. L. Evans Mass transfer through lamnar boundary layers Internatonal Journal of Heat and Mass Transfer vol. 5 pp. 35-57 196. [7] M. S. Hosan and M. A. Samad Heat and mass transfer of an MHD free convecton flow along a stretchng sheet wth chemcal reacton radaton and heat generaton n presence of magnetc feld Research Journal of Mathematcs and Statstcs vol. 5 no. 1- pp. 5-17 013.
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