Hall effects on unsteady MHD natural convective flow past an impulsively moving plate with ramped temperature and concentration

Transcription

1 Indin Jounl of Pue & Applied Phsics Vol. 54, August 16, pp Hll effects on unsted MHD ntul convective flow pst n impulsivel moving plte with mped tempetue nd concenttion S Ds *, R N Jn b & S K Ghosh c Deptment of Mthemtics, Univesit of Gou Bng, Mld 73 13, Indi b Deptment of Applied Mthemtics, Vidsg Univesit, Midnpoe 71 1, Indi c Deptment of Mthemtics, Njole Rj College, Njole, Midnpoe 71 11, Indi Received 8 Novembe 13; evised 31 Decembe 13; ccepted 5 Jnu 14 This ppe dels with the stud of n unsted mgnetohdodnmic ntul convective flow of viscous incompessible electicll conducting fluid pst n impulsivel moving infinite veticl plte with mped tempetue nd mss concenttion tken into ccount the Hll effects. A unifom mgnetic field is pplied tnsvesel to the diection of the flow. The flow considetion is subjected to smll mgnetic Renolds numbe. Induced mgnetic field is bsent. The Rosselnd ppoimtion is used to descibe the ditive het flu in the eneg eqution. Anlticl solution of the govening equtions hs been obtined b emploing the Lplce tnsfom technique. The influences of the petinent pmetes on the velocit field, tempetue distibution, mss concenttion in fluid, she stess nd te of het nd mss tnsfe e discussed with the help of gphs. Hll cuent is found to elevte the fluid velocit components. It is obseved tht significnt diffeence cn be obseved between the velocit pofiles due to mped nd isotheml bound conditions. Kewods: MHD ntul convection, Veticl plte, Theml dition, Rmped tempetue, Concenttion 1 Intoduction In mn engineeing pplictions, especill in het echnges, het tnsfe impovement is n impotnt issue. The plte het echnges e the most impotnt in most of the industies such s food pocessing, dug nd chemicl industies. Bound le theo is n impotnt spect in the stud of continuousl stetching sufce into quiescent fluid, flows cenio tht hs gneed much ttention ove sevel decdes. Some of pplictions tht involve this scenio include hot olling, ppe poduction, metl spinning, dwing plstic films, glss blowing, continuous csting of metls, nd spinning of fibes. Ntul o fee convection is phsicl pocess of het nd mss tnsfe involving fluids which oigintes when the tempetue s well s species concenttion chnge cuse densit vitions inducing buonc foces to ct on the fluid. Such flows eist bundntl in ntue nd due to its pplictions in engineeing nd geophsicl envionments, these hve been studied etensivel in pctice. Welt et l. 1 defines mss tnsfe s the tnspot of one constituent fom egion of highe concenttion to tht of lowe concenttion. Mss *Coesponding utho (E-mil: jn61171@hoo.co.in) tnsfe is the bsis fo mn biologicl nd chemicl pocesses. Biologicl pocesses include the ogention of blood nd the tnspot of ions coss membnes within the kidne. Mss tnsfe lso occus in mn othe pocesses such s bsoption, ding, pecipittion, membne filttion nd distilltion. Anothe pocess of het tnsfe is dition though electomgnetic wves. Mn engineeing pocesses such s fossil fuel combustion eneg pocesses, sol powe technolog, stophsicl flows nd spce vehicle e-ent occu t high tempetue, so ditive het tnsfe pls ve impotnt ole. Also theml dition on flow nd het tnsfe pocesses is of mjo inteest in the design of mn dvnced eneg convesion sstems opeting t high tempetue. Theml dition effects become impotnt when the diffeence between the sufce nd the mbient tempetue is lge. Rditive convective flows e encounteed in sevel industil nd envionmentl pocesses. The stud of dition intection with convection fo het nd mss tnsfe in fluids is quite significnt. The heting of ooms inside buildings using ditos is n emple of ppliction of het tnsfe b fee convection. Sometimes long with the fee convection cuents cused b diffeence in tempetue of the

2 518 INDIAN J PURE & APPL PHYS, VOL 54, AUGUST 16 flow is lso ffected b the diffeences in concenttion o mteil constitution. Thee e mn situtions whee convection het tnsfe phenomen e ccompnied b mss tnsfe lso. When mss tnsfe tkes plce in fluid t est, the mss is tnsfeed puel b molecul diffusion esulting fom concenttion gdients. Fo low concenttion of the mss in the fluid nd low mss tnsfe tes, the convective het nd mss tnsfe pocesses e simil in ntue. A numbe of investigtions hve led been cied out with combined het nd mss tnsfe unde the ssumption of diffeent phsicl situtions. The illusttive emples of mss tnsfe cn be found in the book of Cussle. Combined het nd mss tnsfe flow pst sufce nlzed b Chudh et l. 3, Muthucumswm et l. 4-6 nd Rjput et l. 7 with diffeent phsicl conditions. Juncu 8 pioneeed unsted het nd mss tnsfe flow pst sufce b numeicl method. Ds et l. 9 consideed the effects of fist ode chemicl ection on the flow pst n impulsivel stted infinite veticl plte with constnt het flu nd mss tnsfe. In the bove mentioned studies the effects of dition on the flow hs not been consideed. Actull, mn pocesses in new engineeing es occu t high tempetue nd knowledge of dition het tnsfe becomes impetive fo the design of the petinent equipment. Nucle powe plnts, gs tubines nd the vious populsion devices fo icft, missiles, stellites, nd spce vehicles e emples of such engineeing es. The unsted fee convection flow pst veticl plte with chemicl ection unde diffeent tempetue condition on the plte is elucidted b Neog et l. 1 nd Rjesh et l. 11 Theml dition effect on the flow pst veticl plte with mss tnsfe is emined b Mulidhn et l. 1 nd Rjput et l. 13. Ntul convective flow pst n oscillting plte with constnt mss flu in the pesence of dition hs been studied b Chudh et l. 14. The effects of dition on fee convection on the cceleted flow of viscous incompessible fluid pst n infinite veticl plte hs mn impotnt technologicl pplictions in the stophsicl, geophsicl nd engineeing poblem. Howeve, it seems tht less ttention ws pid on hdomgnetic fee convection flows ne veticl plte subjected to constnt het flu bound condition even though this sitution involves in mn engineeing pplictions. Ogulu et l. 15 nd Nhi et l. 16 hve emined the flow pst sufce with constnt het flu. The fee convection effects on flow pst n infinite veticl cceleted plte with constnt het flu is pioneeed b Chudh et l. 17. Ahmed nd Dutt 18 hve pesented the tnsient mss tnsfe flow pst n impulsivel stted infinite veticl plte with mped plte velocit nd mped tempetue. The flow of fluid in the pesence of ntul convection is lteed due to tempetue nd concenttion gdient sultimtel ffecting the momentum bound le thickness.wll suction nd wll motion is n efficient w to contol the thickness of momentum bound le. Besides this, electicll conducting fluid flow cn be contolled b electomgnetic foces. Whees, The motion of fluids with high electicl conductivit cn be contolled b clssicl MHD flow contol. In wekl conducting fluids howeve, the cuents induced b n etenl mgnetic field lone e too smll, nd n etenl electic field must be pplied to chieve n efficient flow contol. The stud of effects of mgnetic field on fee convection flow is impotnt in liquid metls, electoltes nd ionized gses. Geophsics encountes MHD phenomen in intection on conducting fluids nd mgnetic fields. MHD in the pesent fom is due to pionee contibution of sevel notble uthos like Alfven 19, Cowling. With the dvent of ve high-speed (hpesonic) flight, the subject of mgnetohdodnmics (MHD) hs ssumed get significnce. This is due to the fct tht hed of high-speed bod enteing the tmosphee shock wve, s mentioned elie is fomed between the wve nd bod sufce nd thee will be le of gs t etemel high tempetue due to shock compession s well s fictionl heting in the bound le. At such high tempetues, the gs becomes ionized nd hence becomes electicll conducting. Hence it cn be epected tht b the ppliction of suitbl oiented mgnetic field to the flow in the shock le, the flow ptten cn be modified nd this in tun cuses chnge in the te of het tnsfe to the sufce. When the ionized gs is sufficientl dense, the electon-tom collision fequenc is lge enough so tht the tendenc fo the electons to spil ound the mgnetic field lines is suppessed. It ws emphsized b Cowling tht when the stength of the pplied mgnetic field is sufficientl

3 DAS et l.: HALL EFFECTS ON UNSTEADY MHD NATURAL CONVECTIVE FLOW 519 lge, Ohm's lw needs to be modified to include Hll cuents. The Hll effect is due meel to the sidews mgnetic foce on the dfting fee chges. The electic field hs to hve component tnsvese to the diection of the cuent densit to blnce this foce. In mn woks of plsm phsics, it is not pid much ttention to the effect cused due to Hll cuent. Howeve, the Hll effects cn not be completel ignoed if the stength of the mgnetic field is stong nd numbe of densit of electons is smll s it is esponsible fo the chnge of the flow ptten of n ionized gs. Hll effect esults in development of n dditionl potentil diffeence between opposite sufces of conducto fo which cuent is induced pependicul to both the electic nd mgnetic field. This cuent is temed s Hll cuent. It ws discoveed in 1979 b Edwin Hebet Hll while woking on his doctol degee. Pop 1, Dutt et l. nd Mlique nd Stt 3 hve pesented some model studies on the effect of Hll cuent on MHD convection flow becuse of its possible ppliction in the poblem of MHD genetos nd Hll cuent. Hll cuents e of get impotnce in mn stophsicl poblems, Hll cceleto nd flight MHD s well s flows of plsm in MHD powe geneto (Alpein nd Sutton 4 ). Tkh et l. 5 hve studied the unsted fee convection flow ove n infinite veticl poous plte due to the combined effects of theml nd mss diffusion. The unsted MHD fee convective flow pst veticl poous plte immesed in poous medium with Hll cuent, theml diffusion nd het tnsfe hs been studied b Ahmed et l. 6. In ll the bove investigtions, the nlticl o numeicl solution is obtined ssuming tht the tempetue t the intefce ws continuous nd well defined. Howeve, thee eist sevel poblems of phsicl inteest which m equie non-unifom o bit wll conditions. Seth et l. 7 hve discussed the effect of ottion on unsted hdomgnetic ntul convection flow pst n impulsivel moving veticl plte with mped tempetue in poous medium with theml diffusion nd het bsoption. Jn et l. 8, 9 hve investigted the effects of ottion nd dition on the hdodnmic flow pst n impulsivel stted veticl plte with mped plte tempetue. The effects of dition on MHD ntul convection ne veticl plte with oscillto mped plte tempetue hve been pesented b Jn et l. 3. Sk et l. 31 hve nlzed the effects of Hll cuents nd dition on MHD fee convective flow pst n oscillting veticl plte with oscillto plte tempetue in poous medium. The effect of dition on MHD fee convection flow pst n impulsivel moving veticl plte with mped wll tempetue hve been studied b Gh et l. 3. Mnn et l. 33 hve emined the effects of dition on unsted MHD fee convective flow pst n oscillting veticl poous plte embedded in poous medium with oscillto het flu. Nhi 34 hs studied the tnsient fee convection flow between long veticl pllel pltes with mped wll tempetue t one bound in the pesence of theml dition nd constnt mss diffusion. Ahmed nd Ds 35 hve emined the Hll effects on tnsient MHD flow pst n impulsivel stted veticl plte in poous medium with mped tempetue, ottion nd het bsoption. Nndkeol et l. 36 hve emined the unsted hdomgnetic het nd mss tnsfe flow of het diting nd chemicll ective fluid pst flt poous plte with mped wll tempetue. The effects of theml dition nd ottion on unsted hdomgnetic fee convection flow pst n impulsivel moving veticl plte with mped tempetue in poous medium hve been studied b Seth et l. 37. Ahmed et l. 38 hve pesented the effects of Chemicl ection nd dition on n unsted MHD flow pst n cceleted infinite veticl plte with vible tempetue nd mss tnsfe. Seth et l. 39 hve emined the effects of Hll cuent, dition nd ottion on ntul convection het nd mss tnsfe flow pst moving veticl plte. An MHD ntul convection het nd mss tnsfe flow pst time dependent moving veticl plte with mped tempetue in otting medium with Hll cuents, dition nd chemicl ection hs been descibed b Seth nd Sk 4. Seth et l. 41 hve investigted the Soet nd Hll effects on n unsted MHD fee convection flow of diting nd chemicll ective fluid pst moving veticl plte with mped tempetue in otting sstem. Ht et l. 4 hve emined the influences of Hll cuent nd chemicl ection in mied convective peistltic flow of Pndtl fluid. Ht et l. 43 hve nlzed the Hll nd ion slip effects on peistltic flow of Jeffe nnofluid with Joule heting. The min pupose of the pesent investigtion is to stud the combined effect of Hll cuents, theml dition nd mped wll tempetue nd mss concenttion on n MHD fee convective flow pst moving plte when stong mgnetic field is

4 5 INDIAN J PURE & APPL PHYS, VOL 54, AUGUST 16 imposed. The mgnetic Renolds numbe hve ssumed to be so smll tht the induced mgnetic field cn be neglected. Rosselnd model of dition hs been chosen in the investigtion. The nondimensionl equtions govening the flow e solved b the Lplce tnsfom technique. Solutions e in tems of eponentil nd complement eo function. The esults obtined in this wok e consistent with the phsicl sitution of the poblem. Fomultion of the Poblem Conside the unsted hdomgnetic flow of viscous incompessible electicll conducting nd het diting fluid pst n impulsivel moving infinite veticl plte with mped plte tempetue nd mss tnsfe whee unifom mgnetic field B of stength B is pplied in the diection pependicul to the fluid flow. Choose Ctesin co-odinte sstem with the - is long the plte in the veticll upwd diection, the - is pependicul to the plte nd z - is is the noml of the -plne. The phsicl model of the poblem is pesented in Fig. 1. Initill, t time t, the plte nd the fluid e t est t unifom tempetue T nd species concenttion C. At time t >, the plte t = stts to move in its own plne with unifom velocit u nd the tempetue of the plte t = is ised o loweed to T + ( T T ) t, T T w w w t nd lso the mss concenttion t the plte = Fig. 1 Geomet of the poblem ised o loweed to C ( C C ) t, C C + when w w w t < t t nd the unifom tempetue w mss concenttion C w e mintined when t > t. It is ssumed tht the flow is lmin nd is such tht the effects of the convective nd pessue gdient tems in the momentum nd eneg equtions cn be neglected. It is lso ssumed tht the ditive het flu in the - diection is negligible s comped to tht in the - diection. As the plte is of infinite etent nd electicll nonconducting, ll phsicl quntities, ecept the pessue, e functions of nd t onl. Genelized Ohm's lw on tking Hll cuent into ccount is (Cowling ): ω τ B ( ) σ ( ) e e J + J B = E + q B, T nd unifom (1) whee q, B, E, J, σ, ω e nd τ e e espectivel the velocit vecto, the mgnetic field vecto, the electic field vecto, the cuent densit vecto, electic conductivit, ccloton fequenc nd electon collision time. In witing the Eq. (1), the ionslip nd the themoelectic effects s well s the electon pessue gdient e neglected. The ight hnd side is the electic field in the moving fme. The fist tem on the left hnd side comes fom the electon dg on the ions. The second tem is the Hll tem nd hs to do with the ide tht electons nd ions cn decouple nd move septel. The eqution of continuit q = with no-slip condition t the plte gives v = evewhee in the flow whee q ( u, v, w), u, v nd w e espectivel velocit components long the coodinte es. The fluid is metllic liquid whose mgnetic Renolds numbe is smll nd hence the induced mgnetic field poduced b the fluid motion is negligible in compison to the pplied one 44 so tht the mgnetic field B (, B,). The solenoidl eltion B = gives B = constnt = B evewhee in the flow. The consevtion of electic cuent J = ields j = J j, j, j. This constnt is zeo constnt whee ( z ) since j = t the plte which is electicll nonconducting. Hence, j = evewhee in the flow. In view of the bove ssumption, Eq. (1) ields: j mj = σ ( E wb ), () z

5 DAS et l.: HALL EFFECTS ON UNSTEADY MHD NATURAL CONVECTIVE FLOW 51 j + mj = σ ( E + u B ), (3) z z whee m = ωeτ e is the Hll pmete which epesets the tio of electon-ccloton fequenc nd the electon-tom collision fequenc. In genel, Hll cuents influence the mechnics of flow sstem when pplied mgnetic field is stong o when the collision fequenc is low.the effect of Hll cuents gives ise to foce in the -diection, which induces coss flow in tht diection. To simplif the poblem, we ssume tht thee is no vition of flow quntities in -diection. This ssumption is consideed to be vlid if the sufce be of infinite etent in the - diection. Hee m gives the esult of the hdodnmic fluid cse nd m = coesponds to MHD fluid in the bsence of Hll cuents. Since the induced mgnetic field is neglected, Mwell's eqution H becomes E = E = t E z = which gives E = nd. This implies tht E = constnt nd E z = constnt evewhee in the flow. We choose this constnts equl to zeo, i.e. E = E =. E z Solving fo j nd = E =, we hve: z σ B j = mu w 1+ m ( ), σ B jz = mw + u 1+ m ( ). j z fom () nd (3), on tking (4) (5) Tking into considetion the ssumptions mde bove, the govening equtions fo lmin ntul convection flow of viscous incompessible nd electicll conducting fluid with ditive het tnsfe, unde Boussinesq ppoimtion, i.e. the densit chnges with tempetue, which gives ise to the buonc foce, nd using (4) nd (5), e given b: u u σ B ( ) ( ) = ν u + mw + gβ T T gβ ( C C ), + t ρ(1 + m ) (6) w w σ B = ν + ( mu w ), t ρ(1 + m ) ρc p T T q = k t, (7) (8) C C = D t, (9) Whee u is the velocit in the -diection, T the tempetue of the fluid, g the cceletion due to gvit, ν the kinemtic viscosit, ρ the fluid densit, k the theml conductivit, C is concenttion in the fluid, D is mss diffusivit, β the theml epnsion coefficient, β the concenttion epnsion coefficient, c the specific het t constnt pessue nd q the ditive het flu. The heting due to viscous dissiption is neglected fo smll velocities in the eneg Eq. (9). Assuming tht thee is no-slip between the plte nd the fluid, the initil nd bound conditions fo the fluid flow poblem e: t : u =, w =, T = T, C = C fo, t > : u u, w, T = = = w t Tw + ( Tw T ) fo < t t t T fo t > t p (1) t Cw + ( Cw C ) fo < t t C = t t =, (11) Cw fo t > t t > : u, w, T T, C C t. In ode to simplif the phsicl poblem, the opticll thick dition limit is consideed in the pesent nlsis. Fo n opticll thick fluid, in ddition to emission thee is lso self-bsoption nd usull the bsoption coefficient is wvelength dependent nd lge so tht we cn dopt the Rosselnd ppoimtion fo ditive flu q. The ditive flu vecto q unde the Rosselnd ppoimtion is: * 4 4σ T q =, 3k (1) * whee σ is the Stefn-Boltzmn constnt nd k the spectl men bsoption coefficient of the medium. Assuming smll tempetue diffeence between the fluid tempetue T nd the fee stem tempetue

6 5 INDIAN J PURE & APPL PHYS, VOL 54, AUGUST 16 4 T, T is epnded in Tlo seies bout the fee stem tempetue T. Neglecting second nd highe ode tems in (T T ), we obtin: T = 4T T 3 T. (13) It is emphsized hee tht Eq. (13) is widel used in computtionl fluid dnmics involving dition 4 bsoption poblems in epessing the tem T s line function. In view of (1) nd (13), Eq. (8) educes: * 3 T T 16σ T T ρ cp = k +. t 3k (14) Intoducing non-dimensionless vibles: u t u ( u, w) T T C C =, τ =, ( u1, w1 ) =, θ =, φ =, ν ν u Tw T Cw C (15) Eqs. (6), (7), (9) nd (14) become: u 1 1 = u M ( u 1 + mw 1) + G θ + Gc φ, τ 1+ m w1 w1 M = + ( mu 1 w 1), τ 1+ m θ θ 3R P = ( 3R + 4 ), τ φ φ Sc =, τ whee M σ B ν ρ u (16) (17) (18) (19) = is the mgnetic pmete which is the tio of Loentz foce to viscous foce, g β ( Tw T ) ν = the theml Gshof numbe G u 3 chcteizes the eltive effect of the theml buonc foce to the viscous hdodnmic foce, g β ( Cw C ) ν Gc = the mss Gshof numbe 3 u detemines the eltive effect of the species buonc foce to the viscous hdodnmic foce, R k k 4σ T = is * 3 ρν cp the dition pmete, P = the Pndtl numbe tht mesuses tio of momentum diffusivit to the k ν D theml diffusivit, Sc = the Schmidt numbe which embodies the tio of theml diffusivit to mss diffusivit. Combining (16) nd (17), we hve: (1 ) τ 1+ m F = F M im F + G θ + Gc whee φ, () F = u1 + iw1 nd i = 1. (1) The coesponding initil nd bound conditions e: F =, φ =, θ = fo ll nd τ, τ fo < τ 1 τ fo < τ 1 F = 1, φ =, θ = 1 fo τ > 1 1 fo τ > 1 t = () F, φ, θ t fo τ >. On the use of the Lplce tnsfomtion, Eqs. (), (18) -(19) become: d F M (1 im) sf = F + G θ + Gc φ, d 1+ m d θ 3R P s θ 3R 4, d (3) = ( + ) (4) d φ s Sc φ =, (5) d whee sτ sτ F(, s) = F(, τ ) e dτ, θ (, s) = θ (, τ ) e dτ, sτ φ (, s) = φ(, τ ) e dτ (6) nd s > ( s being Lplce tnsfom pmete). The coesponding bound conditions fo u 1, θ nd ϕ e: 1 1 s 1 s F, θ (1 e = = ), φ = (1 e ) t =, s s s F, θ, φ t. (7) The solution of Eqs. (3)-(5) subject to the bound conditions (7) e esil obtined nd e given b:

7 DAS et l.: HALL EFFECTS ON UNSTEADY MHD NATURAL CONVECTIVE FLOW 53 φ 1 = (8) s s s Sc (, s) (1 e ) e, 1 s s α θ (, s) = (1 e ) e, (9) s s+ e s s 1 e G s+ α s + ( 1)( { e e s } α s β) Gc s+ Sc s + { e e } fo α 1, Sc 1, (Sc 1)( s γ ) F(, s) = s + e s s 1 e G s+ α s + ( 1)( { e e s α s β } ) Gc s+ s { e e } fo α 1, Sc = 1, s whee γ = Sc. 1 3RP α = 3R + 4, M (1 im) =, 1+ m β = (3) α 1 nd The invese tnsfoms of (8)-(3) give the solution fo the mss concenttion, tempetue distibution nd velocit field s: ϕ(, τ ) = f ( Sc, τ ) H ( τ 1) f ( Sc, τ 1), (31) 1 1 θ (, τ ) = f ( α, τ ) H ( τ 1) f ( α, τ 1), (3) 1 1 G f (, τ ) + f3(, β, τ ) f4 ( α, β, τ ) α 1 H ( τ 1){ f3(, β, τ 1) f4 ( α, β, τ 1)} Gc + f3(,, ) f4 ( Sc,, ) Sc 1 γ τ γ τ H ( τ 1){ f3(, γ, τ 1) f4 ( Sc, γ, τ 1)} fo α 1, Sc 1, F(, τ ) = G f (, τ ) + f3(, β, τ ) f4 ( α, β, τ ) α 1 H ( τ 1){ f3(, β, τ 1) f4 ( α, β, τ 1)} Gc [ f 5(,, τ ) f 1(, τ ) H ( τ 1){ f5(,, τ 1) f1(, τ 1)} ] fo α 1, Sc = 1, (33) whee f, 1 f, f 3, f 4 nd f 5 e dumm functions which e given in Appendi A, efc ( ξ ) being complement eo function nd nd H ( ξ ) is the unit step function..1 Solution when Pndtl numbe is unit In the bsence of theml dition (i.e. when R ), i.e. if pue convection pevils, it is obseved tht α = P nd the solution fo the tempetue given b Eq. (31) is vlid fo ll vlues of P, but the solution fo the velocit field given b Eq. (33) is not vlid fo P = 1. Since the Pndtl numbe is mesue of the eltive impotnce of the viscosit nd theml conductivit of the fluid, the cse P = 1 coesponds to those fluids whose momentum nd theml bound le thicknesses e of the sme ode of mgnitude. Theefoe, the solution fo the velocit field in the bsence of theml dition effects when P = 1 hs to be obtined subject to the initil nd bound conditions (). It cn be epessed in the following fom: G f (, τ ) + f3(, β, τ ) f4 ( P, β, τ ) P 1 H ( τ 1){ f3(, β, τ 1) f4 ( P, β, τ 1)} Gc [ f5(,, τ ) f1(, τ ) F(, τ ) = H ( τ 1){ f5(,, τ 1) f1(, τ 1)} ] fo P 1, Sc = 1 G + Gc f (, τ ) [ f5(,, τ ) f1(, τ ) H ( τ 1){ f5(,, τ 1) f1(, τ 1)} ] fo P = 1, Sc = 1 (34) whee f 1, f, f 3, f 4 nd f 5 e dumm functions which e given in Appendi A.. Solution fo isotheml plte o constnt plte tempetue In ode to highlight the effect of the mped bound conditions on the flow, it m be wothwhile to compe such flow pst moving plte with constnt tempetue. In this cse, the initil nd bound conditions () e the sme, 1 φ, τ = 1fo ecepting the condition θ ( τ ) = nd ( ) τ. Unde the ssumptions, it cn be esil shown tht the concenttion, tempetue nd velocit fields fo the flow pst moving plte with constnt tempetue cn be epessed s: α ϕ(, τ ) = efc, τ (35)

8 54 INDIAN J PURE & APPL PHYS, VOL 54, AUGUST 16 Sc θ (, τ ) = efc, τ (36) G F(, τ ) = f (, τ ) + f6 (, β, τ ) f7 ( α, τ ) α 1 Gc + f6 (, γ, τ ) f7 ( Sc, τ ) fo α 1, Sc 1, Sc 1 (37) whee f, f 6 nd f 7 e dumm functions which e given in Appendi A. 3 Results nd Discussion In this section, the obtined ect solutions e studied in ode to detemine the effects of embedded pmetes. Numeicl vlues of the non-dimensionl fluid velocit components u 1 nd w 1, fluid tempetue θ, concenttion φ fo sevel vlues of mgnetic pmete M, Hll pmete m, dition pmete R, theml Gshof numbe G, mss Gshof numbe Gc, Pndtl numbe P, Schmidt numbe Sc nd time τ e pesented in Figs The ghphicl esults e pesented using Mthemtic. 3.1 Velocit pofiles Figue shows the time evolution of the pim nd second velocit pofiles fo fied set of pmete vlues. As shown in Fig., to genete the plot, the vlues of nd τ e vied fom to 8 nd to 4, espectivel. The sufce plot helps to undestnd the vition in velocit components with nd τ. The vlue of the pim nd second velocities is zeo when is close to 8 in both mped tempetue nd isotheml cses. Also, the pim velocit enhnces unifoml with time to its sted stte vlue in both mped tempetue nd isotheml cses, with the eception of the oscillto behvio in the neighbohood of τ = 1. The second velocit enhnces gdull nd ssume pbolic shpe with time to its sted stte vlue in both mped tempetue nd isotheml cses, with the eception of the oscillto behvio in the neighbohood of τ = 1. It is eveled fom Fig. 3 the both pim nd second velocities subpess s mgnetic pmete M inceses in both mped nd isotheml cses. Tht is the pim o second fluid motion is etded due to ppliction of tnsvese mgnetic field. This phenomenon clel gees to the fct tht if n etenl mgnetic foce is pplied pependicul to the flow diection of n electicll conductive fluid, it epeiences n electic field nd poduces cuent pependicul to both mgnetic field nd flow diection. The poduct of electic cuent nd Fig. Sufce plot of pim nd second velocities when M = 1, m =.5, P =, R =.5, G = 5, Gc = 5 nd Sc =.3

9 DAS et l.: HALL EFFECTS ON UNSTEADY MHD NATURAL CONVECTIVE FLOW 55 mgnetic field cetes foce which is known s Loentz foce. The diection of the Loentz foce is lws opposite to the diection of fluid flow in the bsence of n pplied electic field. Tht is, the imposition of the tnsvese mgnetic field is helpful in stbilizing the flow. Effects of Hll cuents on velocit components e pesented in Fig. 4. Both the pim nd second velocities enhnce when Hll pmete m inceses in both mped tempetue nd isotheml cses. It is ttibuted tht lge vlues of Hll pmete m decese the effective conductivit nd thus decese the mgnetic dmping foce hence velocit components incese. In fct, the Hll effect blnces the esistive influence of pplied mgnetic field to some etent. It is eveled fom Fig. 5 tht the both pim nd second velocities decese fo incing vlues of dition pmete in both mped nd isotheml cses. This is consistent with the definition of R. An incese in the Fig. 3 Pim nd second velocities fo ving M when P =, m =.5, G = 5, G = 5, Sc =.3 nd τ =.5 Fig. 4 Pim nd second velocities fo ving m when M = 1, P =, R =.5, G = 5, Gc = 5, Sc =.3 nd τ =.5 Fig. 5 Pim nd second velocities fo ving R when M = 1, P =, m =.5, G = 5, Gc = 5, Sc =.3 nd τ =.5

10 56 INDIAN J PURE & APPL PHYS, VOL 54, AUGUST 16 vlue of R implies decesing dition effects. The cse R epesents the bsence of theml dition i.e. the pue convection. In Fig. 6, n incese in theml Gshof numbe G leds to ise in fluid velocit components in both mped nd isotheml cses. Incesing theml Gshoff numbe G deceses dg foces nd hence velocit pofiles incese. It is lso noticed tht fluid velocit components boost up ve ne the plte nd fte this fluid velocit components smptoticll decese to its zeo vlue s. This phenomenon is clel suppoted b the phsicl elit, since the buonc effectcs e significnt ne the plte, which esults in sudden ise of the fluid velocit components djcent to the plte. Fig. 7 elucidtes the effects of mss Gshof numbe Gc on velocit components. The velocit components incese when mss Gshof numbe Gc inceses in both mped nd isotheml cses. The mss Gshof numbe is defined s the tio of the species buonc foce to the viscous hdodnmic foce. As mss Gshof numbe inceses, the viscous hdodnmic foce deceses. As esult, the momentum of the fluid is highe. Figue 8 shows tht n incese in Schmidt numbe Sc leds to decese in the fluid velocit components in both mped nd isotheml cses. We hve chosen the Sc vlues s Sc =.,.64, 1. nd.3 which coespond to hdogen, wte vpo, sulfu dioide nd nphthlene espectivel. Schmidt numbe shows the eltive influence of momentum diffusion to species diffusion. Momentum diffusion is fste thn species when Sc > 1 nd opposite is tue when Sc < 1. When Sc = 1 both momentum nd species diffuse t the sme te in the bound le. In this cse both momentum nd species bound les e of the sme ode of mgnitude. As Sc inceses velocit components e s epected educed since incesingl momentum is diffused t lesse te thn species. As time pogesses, the both pim nd second velocities e getting cceleted shown in Fig. 9. This is due to incesing buonc effects s time pogesses. Fom the bove figues, it is noted tht thee is significnt diffeence cn be obseved between the velocit pofiles due to mped tempetue nd isotheml bound conditions. 3. Tempetue pofiles Figue 1 demonsttes the effects of of dition pmete R, Pndtl numbe P nd time on the Fig. 6 Pim nd second velocities fo ving G when M = 1, R =.5, m =.5, Gc = 5, Sc =.3 nd τ =.5 Fig. 7 Pim nd second velocities fo ving Gc when M = 1, P =, R =.5, G = 5, Sc =.3 nd τ =.5

11 DAS et l.: HALL EFFECTS ON UNSTEADY MHD NATURAL CONVECTIVE FLOW 57 Fig. 8 Pim nd second velocities fo ving Sc when M = 1, P =, R =.5, G = 5, Gc = 5, m =.5 nd τ =.5 Fig. 9 Pim nd second velocities fo ving time τ when fluid tempetue distibution. Figue 1() elucidtes tht the fluid tempetue θ deceses with n incese in dition pmete R in both the mped tempetue nd isotheml cses. In the pesence of dition, the theml bound le lws found to thicken which implies tht the dition povides n dditionl mens to diffuse eneg. This mens tht the theml bound le deceses nd moe unifom tempetue distibution coss the bound le. Figue 1(b) shows the impct of Pndtl numbe on the tempetue field θ. Pndtl numbe is the tio of momentum to theml diffusivities. Pndtl numbe hs invese eltionship with theml diffusivit. Incese in Pndtl numbe coesponds to stonge momentum diffusivit nd weke the- ml diffusivit. Hee weke theml diffusivit dominnt ove the stonge momentum diffusivit due to which lowe tempetue is noticed. Theefoe, the tempetue nd theml bound le thickness e decesed significntl when the vlues of Pndtl numbe e lged in both the mped tempetue nd M = 1, P =, R =.5, G = 5, Gc = 5, Sc =.3 nd m =.5 isotheml cses. The tempetue θ inceses in pogess of time τ shown in Fig. 1(c). It is cle tht the tempetue pofiles ech to unifom tempetue distibution due to incesing time in the cse of isotheml bound condition, but fo the tempetue pofiles due to mped bound condition equied moe time to ech the unifom tempetue distibution. 3.3 Concenttion pofiles This subsection dels with the vition in concenttion pofile φ fo diffeent vlues of embedded pmetes. To be moe elistic, the vlues of Schmidt numbe e chosen to epesent the diffusing chemicl species of most common inteest like hdogen ( Sc =.), wte vpo ( Sc =.64), sulfu dioide ( Sc = 1.) nd nphthlene ( Sc =.3). The vitions of Schmidt numbe Sc on species concenttion e dewn in Fig. 11(). It is obseved tht the species concenttion nd the ssocited bound le decese fo n incese in Schmidt numbe Sc R in both the mped concenttion nd isotheml

12 58 INDIAN J PURE & APPL PHYS, VOL 54, AUGUST 16 cses. As lge vlues of Sc minimize the theml diffusion nd less diffused species pevents the fluid to become dense enough to enhnce the concenttion distibution nd thus it dops with n incese in Schmidt numbe Sc. We ecll tht n incese in Schmidt numbe mens fll in mss diffusivit. This obsevtion is in geement with the outcome of Fig. 11(). It is eveled fom Fig. 11(b) tht the concenttion φ inceses with n incese in time τ in both mped concenttion nd isotheml cses. As time pogesses, pim nd second velocities e getting cceleted. 3.4 Rte of het nd mss tnsfe In non-dimensionl fom, the te of mss nd het tnsfes t the plte = e obtined s: φ Sc φ τ τ τ 1/ 1/ (, ) = = ( 1), π = (38) θ α θ τ τ τ 1/ 1/ (, ) = = ( 1). π = (39) Fo isotheml plte, the te of mss nd het tnsfes t the plte = e given b: φ Sc φ (, τ ) = =, π τ = (4) Fig. 1 Tempetue pofiles fo ving () R when P = nd τ =.5 (b) P when R =.5 nd τ =.5 (c) time τ when P = nd R =.5 θ θ (, τ ) = = = α (41) π τ Numeicl esults of the te of het tnsfe θ (, τ ) t the plte ( = ) e pesented in Fig. 1 fo sevel vlues of dition pmete R, Pndtl Fig. 11 Concenttion pofiles fo ving () Sc when τ =.5 (b) time τ when Sc =.3

13 DAS et l.: HALL EFFECTS ON UNSTEADY MHD NATURAL CONVECTIVE FLOW 59 Fig. 1 Rte of het tnsfe fo ving () P when τ =.5 (b) time τ when P = 7.1 numbe P nd time τ. It is seen fom Fig. 1() tht the te of het tnsfe θ (, τ ) enhnces when Pndtl numbe P inceses. in both the mped tempetue nd isotheml cses. Pndtl numbe is the tio of momentum diffusivit to theml diffusivit. It is themophsicl popet of fluid. Fo the cse P < 1, theml diffusivit eceeds momentum diffusivit. In othe wods, in this cse het will diffuse t quicke te thn momentum. Fo the cse P = 1, the viscous nd eneg diffusion tes e the sme i.e. the theml nd momentum bound les e of the sme ode of mgnitude. Fo the cse P > 1, momentum diffusivit is gete thn theml diffusivit. In othe wods, momentum will diffuse t quicke te thn het.this is consistent with the fct tht smlle vlues of P e equivlent to incesing theml conductivities nd theefoe het is ble to diffuse w fom the plte moe pidl thn highe vlues of P, hence the te of het tnsfe t the plte is educed. Fig. 1(b) evels tht the te of het tnsfe θ (, τ ) inceses with n incese in dition pmete R in both the mped tempetue nd isotheml cses. Thus, the theml dition tends to ccelete het tnsfe t the plte while theml diffusion hs the evese effect. The te of het tnsfe θ (, τ ) is decesing function of time fo isotheml bound condition, but it hs incesing behvio fo τ < 1, n opposite behvio is obseved fo τ > 1 in the cse of mped bound condition. Moeove, thee is shp ise in the te of het tnsfe θ (, τ ) in the neighbohood of τ = 1onl in the cse of mped wll tempetue. This is due to the tnsition of the tempetue fom mped to isotheml t τ = 1. This is n ecellent geement with Nhi 34. Negtive vlue of θ (, τ ) mens tht the het flows fom the moving plte to fluid. This is epected since the plte is hotte thn the fluid. The numeicl esults of the te of mss tnsfe φ (, τ ) t the plte ( = ) e pesented in Fig. 13 fo sevel vlues of Schmidt numbe Sc nd time τ. It cn be seen fom Fig. 13 tht lge vlues of Sc enhnc the te of mss tnsfe φ (, τ ) in both the mped concenttion nd isotheml cses. Infct Schmidt numbe is the tio of viscous diffusion te to molecul diffusion te. Theefoe highe vlues of Schmidt numbe enhnce the viscous diffusion te which in tun inceses the te of mss tnsfe φ (, τ ). Fom Figs. 1 nd 13, it is seen tht the te Fig. 13 Rte of mss tnsfe fo ving Sc of het nd mss tnsfe is not pesented t τ = in the isotheml cse becuse τ = is singul point, but the te of het nd mss tnsfe is pesented fo ll vlues of time in the cse of mped bound condition becuse it is well defined t ll τ. 3.5 She stesses In non-dimensionl fom, the she stesses t the plte = due to the fluid flows e given b: F τ + i τ = =

14 53 INDIAN J PURE & APPL PHYS, VOL 54, AUGUST 16 G f8(, τ ) + [ f9 ( β, τ ) + f1 ( α, β, τ ) α 1 H ( τ 1){ f9( β, τ 1) + f1 ( α, β, τ 1)} ] Gc + [ f9 ( γ, τ ) + f1 (Sc, γ, τ ) Sc 1 H ( τ 1){ f9( γ, τ 1) + f1 (Sc, γ, τ 1)} ] fo α 1, Sc 1, G = f8(, τ ) + [ f9 ( β, τ ) + f1 ( α, β, τ ) α 1 H ( τ 1){ f9( β, τ 1) + f1 ( α, β, τ 1)} ] Gc τ + f 11(, τ ) π τ 1 H ( τ 1) f11(, τ 1) fo α 1, Sc = 1, π (4) whee f 6, f 7, f 8 nd f 9 e dumm functions which e given in Appendi A. Fo isotheml plte, the she stesses t the plte = due to the fluid flows: F τ + i τ = = G 1 = f8(, τ ) + f1 ( β, τ ) f8(, τ ) + f13 ( α, β, τ ) α 1 β Gc 1 1 (, ) 8(, ) 13(Sc,, ) fo 1, Sc 1, Sc 1 f γ τ γ f τ f γ τ + + α (43) whee f 8, f 1 nd f 13 e dumm functions which e given in Appendi A. Numeicl vlues of the non-dimensionl she stesses τ nd τ due to the pim nd second flows t the plte = e pesented in Figs. 14- fo sevel vlues of mgnetic pmete M, Hll pmete m, Gshof numbe G, mss Gshof numbe Gc, Schmidt numbe Sc nd Pndtl numbe P. In Fig. 14, the mgnetic pmete M is found to elevte the she stesses τ nd τ in both mped nd isotheml cses, becuse the velocit gdient inceses ne the plte. Figue 15 shows tht the Fig. 14 She stesses τ nd τ fo ving M when R =.5, P = 7.1, m =.5, G = 5, Gc = 5 nd Sc =.3 Fig. 15 She stesses τ nd τ fo ving m when R =.5, P = 7.1, M = 1, = 5 G, Gc = 5 nd Sc =.3

15 DAS et l.: HALL EFFECTS ON UNSTEADY MHD NATURAL CONVECTIVE FLOW 531 she stesses τ nd τ enhnce when Hll pmete m is lgeed in both mped tempetue nd isotheml cses. As dition pmete R inceses, the she stesses τ nd τ e educed in both mped nd isotheml cses. This is due to the fct tht n incese in R cuses decese in the velocit components nd hence the she stesses t the plte educe in both mped nd isotheml cses. It is seen fom Fig. 16 tht the she stesses τ nd τ enhnce when theml Gshof numbe G enlges in both mped tempetue nd isotheml cses. Figue 17 evels tht the she stesses τ nd τ enhnce fo incesing vlues of mss Gshof numbe Gc in both mped nd isotheml cses. Figue 18 evel tht the she stesses τ nd τ decese with n incese in Pndtl numbe P in Fig. 16 She stesses τ nd τ fo ving R when M = 1,.5 m =, G = 5, Gc = 5, P = 7.1 nd Sc =.3 Fig. 17 She stesses τ nd τ fo ving G when R =.5, P = 7.1, M = 1,.5 m =, Gc = 5 nd Sc =.3 Fig. 18 She stesses τ nd τ fo ving Gc when R =.5, P = 7.1, M = 1,.5 m =, G = 5 nd Sc =.3

16 53 INDIAN J PURE & APPL PHYS, VOL 54, AUGUST 16 Fig. 19 She stesses τ nd τ fo ving P when R =.5, Sc =.3, M = 1,.5 m =, G = 5 nd Gc = 5 Fig. She stesses τ nd τ fo ving Sc when R =.5, Gc = 5, M = 1, m =.5, G = 5 nd P = 7.1 both mped nd isotheml cses. This is consistent with the fct tht n incese in the Pndtl numbe mens n incese of fluid viscosit, which cuses decese in she stesses t the plte. Fom Fig. 19, it cn be seen tht the she stesses τ nd τ decese when Schmidt numbe Sc inceses in both mped tempetue nd isotheml cses. The eson is due to the fct tht the incese in Schmidt numbe cuses decese in flow velocit nd hence she stesses educe t the plte. It is inteesting to note tht the she stesses τ nd τ e incesing functions of time fo isotheml bound condition, but it hs incesing behvio fo τ < 1, n opposite behvio is obseved fo τ > 1 in the cse of mped bound condition. Thee is shp ise in the she stesses τ nd τ in the neighbohood of τ = 1onl in the cse of mped wll tempetue. This is due to the tnsition of the tempetue fom mped to isotheml t τ = 1. Phsicll, negtive vlue of the she stess τ signifies the moving plte eets dg foce on the fluid long -is, nd positive vlue mens the opposite. This is epected since in the pesent poblem, we conside the cse of moving plte which induces the flow. 4 Conclusions In this ppe, the impcts of Hll effect, dition nd mped wll temetue nd mss concenttion on the unsted hdomgnetic fee convective flow of viscous incompessible electicll conducting nd het diting fluid pst n infinite veticl plte hsve been emined. The min findings obtined fom the pesent stud m be summized s follows: (i) The pim nd second velocities e etded unde the effects of tnsvese mgnetic field whees these e cceleted due to Hll effects in both mped tempetue nd isotheml cses. (ii) An incese in eithe dition pmete o theml Gshof numbe o mss Gshof numbe leds to ise in the velocit components. (iii) The fluid tempetue deceses with n incese in dition pmete. The fluid tempetue inceses when time pogesses.

17 DAS et l.: HALL EFFECTS ON UNSTEADY MHD NATURAL CONVECTIVE FLOW 533 (iv) The she stess t the plte enhnces fo incesing vlues of dition pmete. (v) The te of het tnsfe inceses when dition pmete enlges. (vi) Such fluid flow finds mn engineeing pplictions such s those in MHD devices nd in sevel ntul phenomen occuing subject to theml dition in the pesence of mss tnsfe. Refeences 1 Welt J R, Wicks C E, Wilson R E & Roe G L, Fundmentls of momentum, het mss Tnsfe, (John Wile nd Sons, USA), 7. Cussle E L, Diffusion mss tnsfe in fluid sstems, (Cmbidge Univesit Pess, Cmbidge), Chudh R C & Jin A, Rom J Phs, 5 (7) Muthucumswm R & Kume G S, Theo Appl Mech, 31(1) (4) Muthucumswm R & Vijlkshmi A, Int J Appl Mth Mech, 4 (8) Muthucumswm R, Sund R M & Submnin V S A, Int J Appl Mth Mech, 5 (9) Rjput U S & Kum S, Appl Mth Sci, 5(3) (11) Juncu Gh, Het Mss Tnsfe, 41 (5) Ds U N, Dek R K & Soundlgek V M, Fosch Ingenieuwes, 6 (1994) Bhben C N & Ds R K, Int J Eng Res Tech, 1 (1) Rjesh V, Int J Appl Mth Mech, 6(14) (1) 6. 1 Mulidhn M & Muthucumswm R, Indin J Sci Tech, 3 (1) Rjput U S & Kum S, Int J Appl Mth Mech, 8 (1) Chudh R C & Jin A, Act Tech CSAV, 5 (7) Ogulu A & Mkinde O D,Chem Eng Commun, 196 (9) Nhi M & Debnth L, Z Angew Mth Mech, 93, (13) Chudh R C & Jin A, Mtemtics, XVII (9) Ahmed N & Dutt M, Int J Phs Sci, 8(7) (13) Alfven H, Ntue, 15 (194) 45. Cowling T G, Mgnetohdodnmics, (Wile Inte Science, New Yok), Pop I, J Mth Phs Sci, 5 (1971) 375. Dtt N & Jn R N, J Phs Soc Jpn, 4 (1976) Mlique M A & Stt M A, Int J Het Mss Tn, 48 (5) Alpein M & Sutton G P, Advnced populsion sstems, (Pegmon Pess, New Yok), Tkh H S, Ro S & Nth G, Het Mss Tnsfe, 39(1) (3) Ahmed N, Klit H & Buh D P, Int J Eng Sci Technol, (6) (1) Seth G S, Nndkeol R & Ansi M S, Int J Appl Mth Mech, 7(1) (11) 5. 8 Jn M, Ds S & Jn R N, Int J Appl Info Sst, 3(4) (1) Jn M, Ds S & Jn R N, Int J Appl Eng, (1) Jn M, Ds S & Jn R N, Int J Eng Innov Res, 1(4) (1) Sk B C, Ds S & Jn R N, Bull Soc Mth Sev Stnd, 1(3) (1) 6. 3 Gh N, Ds S, Mji S L & Jn R N, Am J Sci Ind Res, 3(6) (1) Mnn S S, Ds S & Jn R N, Adv Appl Sci Res, 3(6) (1) Nhi, M, Meccnic, 47 (1) Ahmed N & Ds K K, Appl Mth Sci, 7(51) (13) Nndkeol R, Ds M & Sibnd P, Mth Pobl Eng, 13 (13) Seth G S, Nndkeol R & Ansi M S, J Appl Fluid Mech, 6(1) (13) Ahmed N, Goswmi J K & Bu D P, Indin J Pue Appl Mth, 44(4) (13) Seth G S, Sk S & Hussin S M, Ain Shms Eng J, 5 (14) Seth G S & Sk S, J Mech, 31(1) (15) Seth G S, Kumbhk B & Sk S, Int J Eng Sci Technol, 7() (15) Ht T, Zhi H, Tnvee A & Alsedi A, J Mgn Mgn Mte, 47 (16) Ht T, Shfique M, Tnvee A & Alsedi A, J Mgn Mgn Mte, 47 (16) Cme K & Pi S, Mgnetofluid dnmics fo enginees nd pplied phsicists, (McGw Hill, New Yok, USA), Appendi A The following constnt epessions e utilized in the esults: ξ ξ τ 4τ 1 f1( ξ, τ ) = τ + ξ efc ξ e, τ π 1 ξ ξ ξ ξ f ( ξ, τ ) = e efc + τ + e efc τ, τ τ λτ e ξ + λ ξ ξ + λ ξ f3( ξ, λ, τ ) = e efc + ( + λ) τ + e efc ( + λ) τ λ τ τ 1 1 ξ ξ ξ 1 ξ ξ ξ τ + + e efc + τ + τ + e efc τ, λ λ τ λ τ

18 534 INDIAN J PURE & APPL PHYS, VOL 54, AUGUST 16 λτ e ξ λ ξ ξ f4 ( ξ, λ, τ ) = e efc + λτ + e ξ λ efc λτ λ τ τ ξ ξ τ + λ π τ λ λ τ ξ τ 4τ ξ ξ e efc efc, 1 ξ ξ f5( ξ, λ, τ ) τ e λ ξ ξ ξ = + efc + λ τ + τ e λ ξ efc λ τ, λ τ λ τ λτ (,, ) e ξ + λ efc ξ ( f ) ξ + λ ξ 6 ξ λ τ = e + + λ τ + e efc ( + λ) τ λ τ τ 1 ξ ξ λ ξ e efc + τ + e efc τ, λ τ τ λτ e ξ λ ξ ξ λ ξ 1 ξ f7 ( ξ, λ, τ ) = e efc + λτ + e efc λτ efc, λ τ τ λ τ 1 f8( λ, τ ) = λ ef ( λ τ ) + e λ τ, πτ ( ) λτ e 1 f9 ( λ, τ ) = λ ef ( λ) τ e λ πτ f ( + λ) τ τ + τ + + ef ( τ ) + τ + e, λ λ πτ λ λτ e 1 λτ 1 ξ 1 ( ξ, λ, τ ) = λξ ef ( λτ ) + e 3 τ +, λ πτ λ πτ λ 1 ( ) 1 τ f11( λ, τ ) = λτ + ef λ τ + e λ τ, λ π ( ) λτ e 1 ( + λ) τ f1 ( λ, τ ) = + λ ef ( + λ) τ + e, λ πτ λτ e 1 λτ 1 ξ f13( ξ, λ, τ ) = λ ξ ef ( λτ ) + e λ πτ λ πτ