Effect of thermal conductivity on Mhd heat and mass transfer: flow past an infinite vertical plate with Soret and Dufour effects

Transcription

1 Amercan Jornal of Appled Mathematcs 3; (3): 8-38 Pblshed onlne Agst 3 3 ( do:.648/j.ajam.33. Effect of thermal condctvty on Mhd heat and mass transfer: flo past an nfnte vertcal plate th Soret and Dfor effects Halma Usman Ime Jmmy Uanta Department of Mathematcs Usman Danfodyo Unversty Sokoto Ngera Emal address: hhalmasman@yahoo.com(h. Usman) meanta@yahoo.com(i. J. Uanta) To cte ths artcle: Halma Usman Ime Jmmy Uanta. Effect of Thermal Condctvty on Mhd Heat and Mass Transfer: Flo Past an Infnte Vertcal Plate th Soret and Dfor Effects. Amercan Jornal of Appled Mathematcs. Vol. No. 3 3 pp do:.648/j.ajam.33. Abstract: Ths paper nvestgates the stdy of the effect of thermal condctvty on MHD flo past an nfnte vertcal plate th Soret and Dfor effects. The governng non-lnear Partal Dfferental Eqatons are solved nmercally by mplct fnte dfference scheme of Crank-Ncolson. Comptatons are performed for a de range of the governng flo parameters sch as thermal Grashof nmber soltal Grashof nmber magnetc feld parameter Prandtl nmber Dfor nmber Soret nmber thermal condctvty and Schmdt nmber. The effects of these flo parameters on the velocty temperatre and concentraton are shon graphcally. It s observed that the velocty ncreases th the ncrease n Gr Gc D Ec k and λ and t decreases th the ncrease n M Sr and b. Temperatre decreases th an ncrease n γ and Pr and concentraton decreases th the ncrease n Sr and γ. Fnally the effects of varos parameters on the skn frcton coeffcent Nsselt nmber and Sherood nmber are shon on Tables. Keyords: Thermal Condctvty MHD Soret and Dfor Effects. Introdcton Combned heat and mass transfer by free convecton n a poros medm has attracted consderable attenton n the last decades de to t s many mportant engneerng and geophyscal applcatons. Sch flos have attracted the attenton of a nmber of scholars de to ther applcatons n many branches of scence and technology sch as n the feld of agrcltre engneerng to stdy the ndergrond ater resorces seepage of ater n rver-beds n petrolem technology to stdy the movement of natral gas ol and ater throgh ol reservors n chemcal engneerng for fltraton and prfcaton processes thermal and nslatng engneerng modelng of packed sphere beds solar poer collector polltant dsperson n aqfers coolng of electronc systems ventlaton of rooms crystal groth n lqds chemcal catalytc reactors gran storage devces petrolem reservors grond hydrology fber and granlar nslaton nclear aste repostores hgh-performance bldng nslaton postaccdent heat removal from pebble-bed nclear reactors concepts of aerodynamcs heat sheldng th transpraton coolng etc. Bejan et al. [] Ingham et al. [6] and Vafa [4]. The convecton problem n poros medm has other applcatons n geothermal reservors and geothermal energy extractons. A comprehensve reve of the stdes of convectve heat transfer mechansm throgh poros meda has been made by Neld and Bejan [9]. In ve of the above mentoned effects Alam and Rahman [6] have stded Soret and Dfor effects on mxed convecton flo past a poros flat plate th varable scton. Kafosass and Wllams [7] examned Soret and Dfor effects on mxed free-forced condctve and mass transfer bondary layer flo th temperatre dependent vscosty. Anghel et al [9] nvestgated the Dfor and Soret effects on free convecton bondary layer over a vertcal srface embedded n a poros medm. Postelnca [3] stded nmercally the nflence of a magnetc feld on heat and mass transfer by natral convecton from vertcal srfaces n poros meda consderng Soret and Dfor effects. Ajal and Uma [5] nvestgated thermal radaton effect over an electrcally condctng Netonan fld n a steady lamnar magneto hydrodynamc convectve flo over a poros rotatng nfnte dsk th the consderaton of heat and mass transfer n the presence of Soret and Dfor

2 Amercan Jornal of Appled Mathematcs 3; (3): nmbers. Olarenaj and Gbadeyan [] examned the effects of Soret Dfor chemcal reacton thermal radaton and volmetrc heat absorpton/generaton on mxed convecton stagnaton pont flo on an sothermal vertcal plate n a poros meda. Gbadeyan et al. [4] nvestgated thermal dffson and dffson-thermo effects on combned heat and mass transfer on mxed convecton bondary layer flo over a stretchng vertcal srface n a poros medm flled th a vsco elastc fld n the presence of magnetc feld. Emmanel et al. [3] stded nmercally the effects of thermal dffson and dffsonthermo on combned heat and mass transfer of a steady hydromagnetc convectve and slp flo de to a rotatng dsk th vscos dsspaton and ohmc heatng. Anar et al. [] examned the combned effects of Soret and Dfor dffson and poros mpedance on lamnar MHD mxed convecton heat and mass transfer of an electrcallycondctng Netonan Bossnesq fld from a vertcal stretchng srface n a Darcan poros medm nder nform transverse magnetc feld. Abdlraheem et al. [] have nvestgated a nmercal solton for doble dffsve free convectve flo over a vertcal stretchng srface embedded n a poros medm n the presence of a homogeneos frst order chemcal reacton radaton Soret and Dfor effects. Bhpendra et al. [] stded Soret and Dfor effects on nsteady MHD mxed convecton flo past an nfnte radatve vertcal poros plate embedded n a poros medm n the presence of chemcal reacton. Alam et al. [7] examned a steady to-dmensonal free convecton and mass transfer flo past a contnosly movng sem-nfnte vertcal poros plate n a poros medm by takng nto accont the Dfor and Soret effects. Adran [] nvestgated heat and mass transfer characterstcs of natral convecton abot a vertcal srface embedded n a satrated poros medm sbjected to a chemcal reacton by takng nto accont the dffsonthermo (Dfor) and thermal-dffson (Soret) effects. Very recently Aff [3] presented an analyss of the non-darcy MHD free convectve heat and mass transfer past a vertcal sothermal srface by ncldng the analyss varos effects sch as Soret and Dfor effects thermal dsspaton and temperatre-dependent vscosty. Olanreaj [] stded Dfor and Soret effects of a transent free convectve flo th radatve heat transfer past a flat plate movng throgh a bnary mxtre. Affy [4] stded smlarty solton n MHD: effects of thermal dffson and dffson-thermo on free convectve heat and mass transfer over a stretchng srface consderng scton or njecton. More recently Hayat et al. [5] nvestgated heat and mass transfer for Soret and Dfor s effect on mxed convecton bondary layer flo over a stretchng vertcal srface n a poros medm flled th a vsco elastc fld. Combned heat and mass transfer problems th Soret and Dfor effects are of mportance n many processes and have receved consderable focs n recent years. In processes sch as dryng evaporaton at the srface of ater body energy transfer n et coolng toer and the flo n a desert cooler heat and mass transfer occr smltaneosly Vafa [5]. In ve of the above stdes the present ork tends to stdy the effect of thermal condctvty on MHD heat and mass transfer flo past an nfnte vertcal plate takng nto accont the Dfor and Soret effects.. Mathematcal Formlaton An nsteady to-dmensonal flo of an ncompressble and electrcally condctng vscos fld along an nfnte vertcal poros flat plate embedded n a poros medm s consdered. The x-axs s taken on the nfnte plate and parallel to the free-stream velocty hch s vertcal and the y-axs s taken normal to the plate. A magnetc feld B of nform strength s appled transversely to the drecton of the flo. Intally the plate and the fld are at same temperatre T n a statonary condton th concentraton level C at all ponts. For t > the plate starts movng mplsvely n ts on plane th a velocty U ts temperatre s rased to T and the concentraton level at the plate s rased toc.the fld s assmed to have constant propertes except that the nflence of the densty varatons th temperatre and concentraton hch are consdered only n the body force term. Under the above assmptons the physcal varables are fnctons of y and t only. Assmng that the Bossnesq and bondary-layer approxmaton hold and sng the Darcy-Forchhemer model the eqatons governng the problem are (see Alam and Rahman [8]): v = σb ν v = ν t ρ k ( ) gβ( T T ) gβ C C t c c c T v = D D T T T T v = K( T) t ρcp DM C b ρcp ρcp [ γ T T ] () () (3) (4) K( T) = k ( ) (5) The correspondng ntal and bondary condtons are prescrbed as follos:

3 3 Halma Usman et al.: Effect of Thermal Condctvty on Mhd Heat and Mass Transfer: Flo past an Infnte Vertcal Plate th Soret and Dfor Effects t = T T C C for all y. t > = T T C C at y =. T T C C as y. Here and v are the velocty components along and normal to the plateν s the knematc vscosty g s the acceleraton de to gravty β s the coeffcent of volme expanson ρ the densty β s the volmetrc coeffcent of expanson th concentraton K(T) s the thermal condctvty C s the specfc heat capacty at constant pressre p (6) k s the permeablty of the poros medm D M s the coeffcent of mass dffsvty k s the thermal condctvty of the ambent fld D s the coeffcent of dffsvty D s the coeffcent of temperatre dffsvty T γ s the scton parameter b s the jole heatng parameter B s the magnetc feld of constant strength. T andt are the temperatre of the fld nsde the thermal bondary layer C and C are the correspondng concentratons. On ntrodcng the follong non-dmensonal qanttes: t y T T = t = y = = v v T T θ C C νρcp ν C = Pr = Sc = C C D b ν σb ν b = M = Ec = T T ( ) ρ ρcp ( T T ) β ( ) ν gβ ( C C ) ν g T T Gr = Gc = 3 3 ( ) ( C C ) = γ ( T T ) α = ν ( ) ( ) DT T T Dm C C Sr = ν D = ν νρ Cp T T λ Applyng these non-dmensonless qanttes (7) the set of eqatons () (3) (4) (5) and (6) redces to the follong: α = M t Grθ GcC k C C C θ α = Sr t Sc (7) (8) (9) θ θ λ θ θ α = ( λθ ) t Pr Pr C D b Ec Wth the follong ntal and bondary condtons = θ = C = for all y = θ = C = at y = = θ = C = as y. () () Where Ec s the Eckert nmber Pr s the Prandtl nmber Sc s the Schmdt nmber Sr s the Soret nmber D s the Dfor nmber M s the Magnetc feld parameter Gr s the thermal Grashof nmber Gc s the Soltal Grashof nmber b s the jole heatng parameter λ s the varable thermal condctvty α s the varable scton parameter hle and v are dmensonless velocty components n x- and y-dmensonless drectons respectvely and t s the dmensonless tme. 3. Nmercal Procedre To solve the nsteady copled non-lnear partal dfferental eqatons a fnte dfference technqe of mplct type namely Crank Ncolson mplct fnte dfference method hch s alays convergent and stable s employed. Ths method has been sed to solve eqatons (8) - () sbject to the bondary condtons (). The partal dfferental eqatons are converted to dfference eqaton. The Crank-Ncolson method has been sed n several heat transfer radaton and convecton problems. The fnte dfference approxmatons to these eqatons are as follos: j j j j α = t y j j j j j j ( y) j j j j M K j j j j θ θ C C Gr Gc j j j j C C C C α t y C C C C C C = Sc ( y) j j j j j j θ θ θ θ θ θ Sr ( y) j j j j j j () (3)

4 Amercan Jornal of Appled Mathematcs 3; (3): j j j j θ θ θ θ α = t y ( ) j j j j j j λθ θ θ θ θ θ θ Pr ( y) j j λ θ θ Pr ( y) C C C C C C D ( y) j j j j j j j j j j b Ec y (4) The ntal and bondary condtons take the follong forms: = θ = C = for all except = j j j = θ = C = j j j = θ = C = M j M j M j (5) here M corresponds to. Eqatons () (3) and (4) are rtten n the follong form: DU DU DU = DU j j j DU DU j j j 3 4 j j j j ( θ θ ) ( ) D D C C 5 6 EC EC EC = EC j j j EC EC j j j 3 4 j j j j j j ( ) E θ θ θ θ θ θ 5 Fθ F θ Fθ = j j j j j j j j ( ) j j j j ( ) ( ) j j j j j j ( ) Fθ F θ F θ F θ θ F F 6 7 F C C C C C C 8 Where: t t h = h = h 3 = t h4 = M y ( y) K ( ) D = h D = h hh D = αh h hh D = αh h D = hgr D = hgc E = h E = ( Sc h ) E = ( h Sc hαsc) 3 E = hαsc h E = hscsr 4 5 F = h F F = (Pr h F) F = ( h F Prhα Pr) 3 F = h F rα Pr F = h λ F = h Pr Ec F = hb Pr F = h DPr (6) (7) (8) Here the sffx refers to the space and j refers to the tme. y and t are the mesh szes along y drecton and tme t drecton respectvely. Conseqently t = tj tj and y = y y. The skn frcton Nsselt nmber and Sherood nmber are mportant physcal parameters for ths type of bondary layer flo. From the velocty the skn frcton s gven by τ = µ y= and n ve of eqaton (7) eqaton (6) redces to = τ y= (9) () τ τ = here ρ. The rate of heat transfer coeffcent n terms of the Nsselt nmber s gven by T q = k y = y In ve of eqaton (7) eqaton (8) redces to ql θ N = = k t y= o (). () The rate of mass transfer coeffcent hch n the dmensonal form n terms of the Sherood nmber s gven by C y= Sh = x C C ShRe x C = y= here Vx Re x = s the Local Reynolds nmber. ν 4. Nmercal Solton (3) Knong the vales of T and C at a tme t e can calclate the vales at a tme t t as follos. We sbsttte = M n eqaton () hch reslts n a tr-dagonal system of eqatons n nknon vales of. Usng ntal and bondary condtons the system can be solved by Gassan elmnaton method (Carnahan et al (8)). Ths s knon at all vales of y at tme t t y s chosen as y mn= and y max = 3.5. Then knong the vales of and applyng the same procedre and sng bondary condtons e can calclate C and T from eqatons (3) and (4) respectvely. Ths procedre s contned to obtan the solton tll desred

5 3 Halma Usman et al.: Effect of Thermal Condctvty on Mhd Heat and Mass Transfer: Flo past an Infnte Vertcal Plate th Soret and Dfor Effects tme t. Comptatons are carred ot for ater vapor (Pr =.6) for ar (Pr =.7) for salt ater (Pr =.) for hydrogen (Sc =.) and methanol (Sc =.). 5. Reslts and Dscsson 5.. Velocty Profles The effects of varos thermo physcal parameters on the fld velocty are llstrated n fgres to. Fgres and depct the effect of the soltal and thermal Grashof s nmbers Gc and Gr respectvely. The velocty profle ncreases th ncrease n the vales of Gr and Gc respectvely. It s clearly seen n fgre 6 that the effect of ncreasng the magnetc feld strength on the momentm bondary layer thckness s llstrated. Increasng ths parameter leads to a decrease n the velocty hch confrmed the fact that the magnetc feld reslts n a dampng effect on the velocty by creatng a drag force that opposes the feld moton. Fgre 5 llstrate the effect of on the velocty profle. varable thermal condctvty λ( A) λ > corresponds to assstng flo It s clearly seen that (heated plate). Increase n the mxed convecton parameter leads to an ncrease n the fld velocty. Fgre 4 dsplays the velocty profles across the bondary layer from dfferent vales of dmensonless vscoelastc parameter K and t s observed that ncreasng K leads to an ncrease n the fld velocty and thckenng the velocty bondary layer. In fgre 8 t s clearly seen that an ncrease n the vale of the Schmdt nmber leads to a decrease n the velocty profle. Ths means that the larger the vale of Sc the thnner the momentm bondary layer sze hence decrease n the velocty. The Prandtl nmber Pr Soret nmber Sr and Scton parameter γ ( V) has lttle or no nflence on the fld velocty (see fgres 9 and 7). It s also nterestng to note that ncrease n Dfor nmber D and Eckert nmber Ec ncreases the fld velocty (see fgres and 3). Fg. : Velocty profles for V =. Pr=.7 Gr = 5A = 3 Sc =. Ec =. b =. Sr =. K = M = D =.. Fg. 3: Velocty profles for V =. Pr=.7 Gr = 5A = 3 Sc =. Gc = 5 b =. Sr =. K = M = D =.. Fg. 4: Velocty profles for V =. Pr=.7 Gr = 5 Gc = 5A = 3 Sc =. Ec =. Sr =. b =. M = D =.. Fg.: Velocty profles for V =. Pr=.7 Gc = 5A = 3 Sc =. Ec =. b =. Sr =. K = M = D =..

6 Amercan Jornal of Appled Mathematcs 3; (3): Fg. 5: Velocty profles for V =. Pr=.7 Gr = 5 K = Sc =. Ec =. b =. Sr =. M = D =. Gc = 5. Fg. 8: Velocty profles for V =. Pr=.7 Gr = 5 A = 3 Ec =. b =. Sr =. K = M = D =.Gc = 5. Fg. 6: Velocty profles for V =. Pr=.7 Gr = 5A = 3 Sc =. Ec =. b =. Sr =. K = Gc = 5 D =.. Fg. 9: Velocty profles for V =. Pr=.7 Gr = 5A = 3 Sc =. Ec =. b =. K = M = D =. Gc = 5. Fg. 7: Velocty profles for Pr=.7 Gr = 5A = 3 Sc =. Ec =. b =. Sr =. K = M = D =. Gc = 5. Fg. : Velocty profles for V =. Gr = 5A = 3 Sc =. Ec =. b =. Sr =. K = M = D =. Gc = 5.

7 34 Halma Usman et al.: Effect of Thermal Condctvty on Mhd Heat and Mass Transfer: Flo past an Infnte Vertcal Plate th Soret and Dfor Effects Fg. 3: Temperatre profles for Gr = 5 Gc = 5 D = Sc =. Ec = K = 3 M = b = Pr =.7 Sr =. A =. Fg. : Velocty profles for V =. Pr=.7 Gr = 5A = 3 Sc =. Ec =. b =. Sr =. K = M = Gc = Temperatre Profles The nflences of varos embedded parameters on the fld temperatre are llstrated n fgres to 7. Fgre gves the varatons of varable thermal condctvty λ on the temperatre. Increasng λ leads to an ncrease n the temperatre. Fgre 3 shos the solton of temperatre profles across the bondary layer thckeness for dfferent vales of scton parameter γ. Increase n ths parameter leads to a decrease n the temperatre profle. It s clearly seen n fgre 4 the effect of Prandtl nmber Pr on the temperatre. The thermal bondary layer thkceness s fond to ncrease pon ncreasng the prandtl nmber. Hoever fgre 5 and 7 gves the effects of Dfor nmber D and Eckert nmber Ec on temperatre profles. Increasng these parameters leads to an ncrease n the thermal bondary layer thckeness. Smlarly n fgre 6 ncrease n Jole heatng parameter b decreases the thermal bondary layer. Fg. 4: Temperatre profles for V =. Gr = 5 Gc = 5 D = Sc =. Ec = K = 3 M = b = Sr =. A =. Fg. 5: Temperatre profles forv =. Gr = 5 Gc = 5 Sc =. Ec = K = 3 M = b = Pr =.7 Sr =. A =. Fg. : Temperatre profles for V =. Gr = 5 Gc = 5 D = Sc =. Ec = K = 3 M = b = Pr =.7 Sr =.. Fg. 6: Temperatre profles for V. = Gr = 5 Gc = 5 Sc =. Ec = K = 3 M = Pr =.7 Sr =. A = D =.

8 Amercan Jornal of Appled Mathematcs 3; (3): Fg. 7: Temperatre profles for V =. Gr = 5 Gc = 5 D = Sc =. K = 3 M = b = Pr =.7 Sr =. A =.5.3. Concentraton Profles Fg. 9: Concentraton profles for V =. Gr = 5 Gc = 5 Ec = b =. K = 3 M = D = 3 Sc =. A = Pr =.7. The nflences of varos embedded thermophyscal parameters on the fld concentraton had been llstrated n fgres 8 to. Fgre 8 reflects that th ncrease n Sc the fld concentratom ncreases. Fgre 9 s the graph of concentraton profle aganst spanse co-ordnate y for dfferent vales of Soret nmbers. It s note orthy that ncrease n ths parameter decreases the concentraton bonary layer thckenes. It s clearly seen that for a destrctve scton parameter γ the concentraton bondary layer thckeness decreases ( see fgre ). The order of the reacton Gr has lttle effects on the concentraton bondary layer thckeness. Increase n Gr leads to small decrease n the cocentraton profle across the bondary layer ( see fgre ). Fg. : Concentraton profles for V =. Sc =. Gc = 5 Ec = b =. K = 3 M = D = 3 Sr =. A = Pr =.7. Fg. 8: Concentraton profles for V =. Gr = 5 Gc = 5 Ec = b =. K = 3 M = D = 3 Sr =. A = Pr =.7. Fg. : Concentraton profles for Gr = 5 Gc = 5 Ec = b =. K = 3 M = D = 3 Sr =. A = Pr =.7 Sc =.. The effects of varos governng parameters on the skn frcton Nsselt nmber and Sherood nmber are shon n tables to 3 t s observed that as Gr Gc Pr K Ec and D ncreases the skn frcton coeffcent ncreases here as the skn frcton coeffcent decreases as M decreases.

9 36 Halma Usman et al.: Effect of Thermal Condctvty on Mhd Heat and Mass Transfer: Flo past an Infnte Vertcal Plate th Soret and Dfor Effects From table t s notced that the Nsselt nmber ncreases as Pr and Ec are ncreased hle as the jole heatng parameter b ncreases the Nsselt nmber decreases. From table 3 t s fond that as Sc Sr and scton parameter ncreases the Sherood nmber decreases. Table : Effect of Gr Gc Pr M K Ec and D on Skn frcton( τ ). Gr Gc Pr M K Ec D τ (A = V =. Sc =. Sr =. b =.) Table : Effect of Gr Pr D Ec and b on Nsselt nmber (N). Pr D b Ec Gr N (Gc = 5 A = M =.5 k = Sr =. Sc =. V =.) Table 3: Effect of Gr Sc Sr Ec and V on Sherood nmber (Sh). Gr Sc Sr Ec V Sh (Pr =.7 Ec =. M =.5 A = K = D = b =. Gc = 5) 6. Conclsons In ths paper e have stded nmercally the effect of thermal condctvty on MHD heat and mass transfer flo past an nfnte vertcal plate th Soret and Dfor effects. The non-dmensonal governng eqatons are solved th the help of mplct fnte dfference method of Crank- Ncolson type. The conclsons of the stdy are as follos:. The velocty ncreases th the ncrease n thermal Grashof nmber and soltal Grashof nmber.. The velocty decreases th an ncrease n the magnetc parameter. 3. The velocty as ell as the temperatre ncreases th an ncrease n Dfor nmber Eckert nmber and thermal condctvty. 4. Increasng the scton parameter redces velocty temperatre and concentraton. 5. The velocty as ell as the concentraton decreases th an ncrease n Soret nmber. 6. Increasng the Prandtl nmber sbstantally decreases the translatonal velocty and ncreases the temperatre profle. 7. An ncrease n the Schmdt nmber leads to ncrease n the concentraton. 8. Increasng the jole heatng parameter decreases the translatonal velocty. 9. The velocty ncreases th an ncrease n the permeablty of the poros medm parameter. Nomenclatre C Concentraton C Specfc heat at constant pressre p D - Mass dffsvty D Coeffcent of Mass dffsvty M D - Coeffcent of temperatre dffsvty T D Dfor nmber g Acceleraton de to gravty Gr Grashof nmber Gc Soltal Grashof nmber K Permeablty of the poros medm N Nsselt nmber Pr Prandtl nmber Sc Schmdt nmber Sr Soret nmber T Temperatre v veloctes n the x and y-drecton respectvely x y Cartesan coordnates along the plate and normal to t respectvely ρ Densty of the fld Bo - Magnetc feld of constant strength Ec Eckert nmber M Magnetc feld parameter b Jole heatng parameter

10 Amercan Jornal of Appled Mathematcs 3; (3): Greek Letters β - coeffcent of expanson th concentraton β - Coeffcent of thermal expanson µ -coeffcent of vscosty ρ- Densty of fld θ- dmensonless temperatre υ- Knematc vscosty α - Chemcal reacton parameter γ - Scton parameter λ - Varable thermal condctvty Sbscrpts - condton at all - condton at nfnty References [] Abdlraheem M. A. Mansor M. A. and Chamkha A. J. (). Effects of Soret and Dfor nmbers on free convecton over Isothermal and Adabatc stretchng srfaces embedded n poros meda. Jornal of poros meda DOI: 65/ Jornal Poros Meda 4() pp [] Adran P. (7). Inflence of chemcal reacton on heat and mass transfer by natral convecton from vertcal srfaces n poros meda consderng Soret and Dfor effects. Heat Mass Transfer 43: pp [3] Aff A. (6). Effects of Thermal dffson and dffson Thermo on non-darcy MHD free convectve flo and mass transfer past a vertcal sothermal sothermal srface embedded n a poros medm th thermal dsperson and temperatre dependent vscosty. Appled Math Model. [4] Affy A.A. (9). Smlarty solton n MHD: Effects of thermal-dffson and dffson-thermo on free convectve heat and mass transfer over a stretchng srface consderng scton or njecton. Commncaton n Non-Lnear Scence Nmercal Smlaton 4: pp. -4. [5] Ajal D. S. P. and Uma D. R. (). Soret and Dfor effects on MHD slp flo th thermal radaton over a poros rotatng nfnte dsk. Commncatons n Non Lnear Scence and Nmercal Smlaton 6(4) pp [6] Alam M. S. and Rahman M. M. (6). Dfor and Soret effects on mxed convecton flo past a vertcal poros plate th varable poros th varable scton. Non-lnear Analyss: Modelng and control () pp. 3-. [7] Alam M.S. Ferdos M. Ota M. and Maleqe M.A. (6). Dfor and Soret effects on Steady Free Convecton and Mass transfer Flo past a sem-nfnte Vertcal Poros Plate n a Poros Medm. Internatonal Jornal of Appled Mechancs and Engneerng (3) pp [8] Alam M. S. and Rahman M. M. (99). Dfor and Soret effects on MHD free convectve and mass transfer flo past a vertcal flat plate embedded n a poros medm. Jornal of Naval Archtectre and Marne Engneerng. (): pp [9] Anghel M. Takher H. S. and Pop I. (). Dfor and Soret effects on free convecton bondary layer over a vertcal srface embedded n a poros medm. Stda Unverstats Basas-BolyalMathematca XLV (4) pp. -. [] Anar B.O. Baker A.Y.and Prasad V.R. (9). Nmercal stdy of free convecton magneto hydrodynamc heat and mass transfer from a stretchng srface to a satrated poros medm th Soret and Dfor effects. Comptatonal Materal Scence 46 pp [] Bejan A. Dncer I. Lorente S. Mgel A. F. and Res A. H. (4). Poros and Complex Flo Strctres n Modern Technologes.Sprnger. Ne York NY. [] Bhpendra K. S. Kalash Y.Ndhsh K. Mshra R. C. and Chadhary R. C. (). Soret and Dfor Effects on Unsteady MHD Mxed Convecton Flo past a Radatve Vertcal Poros Plate Embedded n a Poros Medm th Chemcal Reacton. Appled Mathematcs 3 pp do:.436/am. [3] Emmanel O Jonathan S.and Robert H. (). Thermal-dffson and dffson-thermo effects on combned Heat and Mass transfer of a steady MHD hydro convectve and slp flo de to a rotatng dsk th vscos dsspaton and ohmc heatng. Global Jornal of Scence Fronter Research Internatonal Commncatons n Heat and Mass Transfer 35: pp [4] Gbadeyan J. A. Ido A. S. Ognsola A. W. Agbola O. O. and Olanreaj P. O. (). Heat and Mass transfer for Soret and Dfor s effect on mxed convecton bondary layer flo over a stretchng vertcal srface n a poros medm flled th a vsco elastc fld n the presence of a magnetc feld. Global Jornal of Scence Fronter Research. (8) pp [5] Hayat T. Mstafa M. and Pop I. (). Heat and Mass transfer for Soret and Dfor s effect on mxed convecton bondary layer flo over a stretchng vertcal srface n a poros medm flled th a vsco elastc fld. Commncatons n Non Lnear Scence and Nmercal Smlaton. 5 pp [6] Ingham D. B. and Pop I. (998). Transport phenomena n poros meda. Vol. II. Pergamon: Oxford UK. [7] Kafosass N. G. and Wllams E. M. (995). Thermaldffson and dffson-thermo effects on mxed free-forced convectve and Mass transfer bondary layer flo th temperatre dependent Vscosty. Internatonal Jornal of Engneerng Scence 33 pp [8] Carnahan H. Lther H. A. and Wlkes J.O. (969). Appled Nmercal Methoda John Wlley neyork. [9] Neld D. A. and Bejan A. (998). Convecton n Poros meda Second edton Sprnger Ne York. [] Olarenaj P. O. and Gbadeyan J.A. (). Effects of Soret Dfor and Chemcal reacton Thermal Radaton and Volmetrc Heat Generaton / Absorpton on mxed convecton stagnaton pont flo on an Isothermal Vertcal plate n poros meda. Pacfc Jornal of Scence and Technology. (): pp [] Olanreaj P. O. (). Dfor and Soret Effects of a Transent free convectve flo th Radatve heat transfer past a flat plate movng throgh a bnary mxtre. Pacfc Jornal of Scence and Technology (): pp

11 38 Halma Usman et al.: Effect of Thermal Condctvty on Mhd Heat and Mass Transfer: Flo past an Infnte Vertcal Plate th Soret and Dfor Effects [] Postelnca A. (7). Inflence of a magnetc feld on Heat and Mass transfer by natral convecton from vertcal srfaces n poros meda consderng Soret and Dfor effects. Internatonal Jornal of Heat Mass transfer 43 pp [3] Vafa K. (). Hand book of poros meda second edton. Taylor and Francs: Ne York. [4] Vafa K. (5). Hand book of poros meda second edton. Taylor and Francs: Ne York.