The effect of variable viscosity on the flow and heat transfer of a viscous Agwater and Cu-water nanofluids *

Transcription

1 1 2013,25(1):1-9 DOI: /S (13) The eect o vaiable viscosity on the low and heat tanse o a viscous Agwate and Cu-wate nanoluids * VAJRAVELU Kuppalapalle Depatment o Mathematics, Univesity o Cental Floida, Olando, Floida 32816, USA, kuppalapalle.vajavelu@uc.edu PRASAD Keehalli Vinayaka Depatment o Mathematics, VSK Univesity, Vinavaka Naga, Bellay, Kanataka, India NG Chiu-On Depatment o Mechanical Engineeing, The Univesity o Hong Kong, Hong Kong, China (Received May 15, 2012, Revised August 30, 2012) Abstact: A numeical study is caied out to study the eects o the tempeatue dependent viscosity on the low and heat tanse o a nanoluid ove a lat suace in the pesence o viscous dissipation. The govening nonlinea patial dieential equations ae tansomed into nonlinea odinay dieential equations, and ae solved numeically by the Kelle-box method. The numeical esults indicate that the eect o nanopaticle volume action is to incease the heat tanse and hence enhance the themal bounday laye thickness. This is tue even in the pesence o vaiable viscosity and the viscous dissipation. Futhemoe, the esults obtained o heat tanse chaacteistics with nanopaticles eveal many inteesting behavios that waant uthe study on the eects o the nano-solid-paticles. Key wods: nanoluid paticles, vaiable viscosity, bounday layes, viscous dissipation, Kelle-box method Intoduction In ecent yeas a geat deal o inteest has been geneated to study the convective heat tanse o nanoluids due to its inceasing impotance in nanotechnological industies. Examples o these convectional heat tanse nanoluids include oil, wate, and ethylene glycol mixtue. These luids exhibit poo heat tanse ates because the themal conductivity o these luids play a pominent ole in calculating the coeicient o heat tanse between the heat tanse medium and the heat tanse suace. Seveal methods have been poposed to impove the themal conductivity o these luids by suspending nano/mico lage sized paticle mixtues in liquids. An innovative * Poject suppoted by the Reseach Gants Council o the Hong Kong Special Administative Region, China (Gant No. HKU E). Biogaphy: VAJRAVELU Kuppalapalle (1949-), Male, Ph. D., Poesso Coesponding autho: NG Chiu-On, cong@hku.hk technique to impove the heat tanse ate is by using nano-size paticles in the base luids. Nano technology has been widely used since the mateial with size o nanometes possesses unique physical and chemical popeties. Nano sized paticle luids ae called nanoluids, and is ist initiated by Choi [1]. In his pape, Choi [1] showed that the addition o small amount o nanopaticles to convectional heat tanse luids would enhance the themal conductivity o the luids by appoximately two times. Ate the wok o Choi [1], nano technology wok is consideed by many eseaches to be one o the signiicant oces that would dive the next majo industial evolution o this centuy. It epesents the most elevant technological cutting edge cuently being exploed. It aims at manipulating the stuctue o matte at molecula level with goal o innovation in vitually evey industy and public endeavo including biological sciences, physical sciences, electonics cooling, tanspotation, the envionment and national secuity. Some numeical and expeimental studies on nanoluids include themal conductivity behavio [2] and convective heat tanse [3-13]. A compehensive suvey o convective

2 2 tanspot in nanoluids was made by Kakaç and Pamuanjaoenkij [13]. The numeical study o nanopaticle deposition in a ully developed lamina low unde dieent conditions was pesented by Sun et al. [14] and via moment method by Liu et al. [15]. Recently, Wang and Lin [16] showed that the cuves o paticle size distibution move towad the aeas with lage paticle diametes, the cuve peak becomes lowe and the ange that paticle diametes cove becomes wide as time elapses. Nield and Kuznetsov [17] have studied the poblem o natual convection past a vetical plate, in a poous medium satuated by a nanoluid. In all the above studies, the physical popeties o the nanoluids wee assumed to be constant. Howeve, it is well known that these physical popeties o the nanoluids may change with tempeatue. Fo heat tanse in luids, heat geneated by intenal iction and the coesponding ise in the tempeatue aects the viscosity o the luid, so that the luid viscosity no longe be assumed constant. Mukhopadhyay and Layek [18] analyzed the eect o themal adiation and vaiable luid viscosity on ee convective and heat tanse past a poous stetching suace. An incease in tempeatue leads to an incease in the tanspot phenomena. Theeoe, to pedict the low and heat tanse ates in the nanoluids, it is necessay to conside the eects o the tempeatue-dependent viscosity o the base luid. Available liteatue on the vaiable luid popeties and nanoluids [19-21] shows that the wok is not caied out o nanoluid low ove a lat suace. Hence, we conside the eect o vaiable luid viscosity on the viscous Ag-wate and Cu-wate nanoluids ove a semi-ininite impemeable lat suace in the pesence o viscous dissipation: Tempeatue-dependent viscosity, nanopaticle-size volume action and viscous dissipation yield a non-linea coupled system o patial dieential equations. The coupled non-linea patial dieential equations govening the poblem ae educed to a system o coupled non-linea odinay dieential equations. The esulting bounday value poblem is then conveted into a system o ive simultaneous equations o ist ode with ive unknowns. The system o equations is then solved by a second ode implicit inite dieence scheme known as the Kelle-box method. Numeical computation has been caied out o tempeatue and the velocity poiles, Nusselt numbe and skin iction coeicient o the case when the walls ae maintained with Constant Suace Tempeatue (CST). Attempts have been made to investigate the eects o the vaiable viscosity, the nanopaticle-size volume action, the Ecket numbe with ixed Pandtl numbe (o wate) on the convectional low behavio and the heat tanse pocess. The outline o this wok is highlighted on studying the nanopaticle-size volume action behavio which deceases the bounday laye thickness in the pesence o vaiable viscosity paamete and the viscous dissipation. Fig.1 Physical model and coodinate system 1. Flow analysis Conside a steady, lamina, two-dimensional bounday-laye low with heat tanse o a nanoluid ove a semi-ininite, impemeable lat suace as shown in Fig.1. The low takes place in the positive xy-plane with the suace being at the plane y = 0 and is maintained at a constant tempeatue T w. The woking luid is wate-based nanoluid containing dieent type o nano-solid-paticles (namely coppe Cu, o silve Ag). The nanoluid is a two component mixtue with the ollowing assumptions: (1) incompessible low, (2) no-chemical eaction, (3) negligible intenal heat geneation/absoption, and (4) negligible adiative heat tanse. The themo-physical popeties o the luid with nanopaticles ae given in Table 1. The themo-physical popeties o the ambient luids ae assumed to be constant except o the base luid viscosity which is assumed to be as an invese unction o the tempeatue. Unde these assumptions with the usual bounday laye appoximation, the govening equations o the consevation o mass, momentum and the enegy o the poblem unde consideation (in the pesence o vaiable luid viscosity and viscous dissipation) can be witten as u v + =0 x y u u 1 u u + v = n x y ρn y y 2 2 T T T n u u + v = α n + 2 x y y ( ρcp ) n y The appopiate bounday conditions ae (1) (2) (3)

3 3 Table 1 Numeical esults o wall-tempeatue gadient o dieent values o the physical paametes Themo physical popeties [7] Base luid+ nanosolid paticles Wate with coppe solid paticles Base luid+ nanosolid paticles Wate with silve solid paticles C p (J/kg K) C p (J/kg K) ρ (kg/m 3 ) ρ (kg/m 3 ) k (w/m K) k (w/m K) α (10 7 m 2 /s) α (10 7 m 2 /s) θ Ec =0 Ec = 0.25 Ec = 0.5 Ec =0 Ec = 0.25 Ec = u( x, 0) = 0, v( x, 0) = 0, T( x, 0) = T w u( x, y), T( x, y) as y (4) U whee u and v ae the luid velocity components in the steamwise diection o x and the cosssteamwise diection, espectively. Hee T is the tempeatue o the luid, U and T ae the ee steam velocity and the tempeatue o the luid a T away om the suace. The eective density o the nanoluid is given as ( ρ) = (1 ) ρ + ρ (5) n s whee is the solid volume action o nanopaticles. The eective dynamic viscosity o the nanoluid given by Binkman [22] is

4 4 n = (1 ) 2.5 (6) nanoluid, which o spheical nano-paticles, accoding to Maxwell [23], is Hee is the coeicient o viscosity which is consideed to vay as an invese unction o tempeatue [19] as ollows 1 1 = [1 + δ ( T T )] i.e., whee δ a =, T = T 1 δ 1 = at ( T ) (7) The viscosity o a luid usually deceases with an incease in the tempeatue. To uthe demonstate the appopiateness o Eq.(7) coelations between the viscosity and tempeatue o ai and wate ae given below because these two ae the most commonly used woking luids in engineeing: Fo ai K n = K Ks +2K 2 ( K Ks) Ks +2 K + ( K Ks) (10) Hee suixes n, and s epesent espectively the themo-physical popeties o the nano-luids, base luid and the nano-solid-paticles. 2. Similaity tansomation and the two-point bounday value poblem The speciic oms o the velocity and the constant suace tempeatue ae chosen to allow the coupled non-linea patial dieential Eqs.(1)-(3) to be conveted into a set coupled, non-linea odinay dieential equations by the similaity tansomation (see Chamkha and Issa [24] ) U η = 2 γ x 1/2, 1/2, ψ = (2 γ U x) ( η) 1 = 123.2( T 742.6) based on T o = 293 K(20 C) ψ u = y, v = ψ, θη ( )= x T T T T w (11) and o wate 1 = 29.83( T 258.6) based on T o = 288 K(15 C) The eeence tempeatues thus selected o the coelations ae vey pactical in most applications. Hee both a and T ae constants, and thei values depend on the eeence state and the themal popety o the luid, i.e., δ (a constant) in geneal, a > 0 coesponds to liquid and a < 0 o gases. T is the tempeatue, T and ae the constant values o the tempeatue and the coeicient o viscosity espectively, a away om the sheet. Themal diusivity o the nanoluid is whee η is the similaity vaiable, and θ ae the dimensionless velocity and tempeatue, espectively. The velocity components u and v in Eq.(11) automatically satisy the continuity Eq.(1). In tems o ( η ) and θη ( ), the momentum Eq.(2) and the enegy Eq.(3) can be witten as θ + = θ 2 1 θ θ θ 1 θ 2.5 ρ s (1 ) 1 + ρ (12) α n Kn = ( ρc ) p n (8) K ( C ) p s θ + P 1 + θ + Kn ( Cp ) whee, the heat capacitance o the nanoluid is given as ( ρc ) =(1 )( ρc ) + ( ρc ) (9) p n p p s K Ec 2 P ( ) =0 Kn 2.5 θ (1 ) 1 θ (13) In Eq.(8), K n is the themal conductivity o the The coesponding bounday conditions ae

5 5 ( η )=0 at η =0 (14a) ( η)=0 at η =0 (14b) θη ( )=1 at η =0 (14c) ( η) 1, θη ( ) 0 as η (14d) The dimensionless paametes in Eqs.(12)-(13) ae θ = 1/[ δ ( Tw T )] the luid viscosity paamete, P = ν / α the Pandtl numbe and 2 = / Ec U [( T T ) c ] the Ecket numbe. The value o θ w p is detemined by the viscosity o the luid unde consideation and the opeating tempeatue dieence. I θ is lage, in othe wods, i ( Tw T ) is small, the eects o vaiable viscosity on the low can be neglected. On the othe hand, o smalle values o θ, eithe the luid viscosity changes makedly with tempeatue o the opeating tempeatue dieence is high. In eithe case, the vaiable luid viscosity eect is expected to become vey impotant. Also beaing in mind that the liquid viscosity vaies dieently with tempeatue than that o gas, it is impotant to note that θ is negative o liquids and positive o gases. It is woth mentioning hee that in the absence o nano-paticles =0 and θ as δ 0. In this case, Eqs.(12), (13) educe to those o Chamkha and Issa [24]. Fo pactical puposes, the unctions ( η ) and θη ( ) allow us to detemine the skin iction coeicient C u = = ρ n 2 uw y y= /2 1 ( ) (0) 2.5 Rex (1 ) θ and the Nusselt numbe Nu x xkn T = = k ( T T ) y w y=0 (15) k 1/2 n ( Rex ) θ (0) (16) k Hee Rex = uwx / ν is the local Reynolds numbe. 3. Numeical solution method Bounday laye nanoluid low and heat tanse o a lat suace with Cu-wate and Ag-wate nanopaticles unde the inluence o vaiable luid viscosity and the viscous dissipation is consideed o investigation. The solutions o the bounday value poblem ae obtained numeically by a second ode implicit inite dieence scheme known as the Kelle-box method [25-28]. The numeical solutions ae obtained in ou steps ae as ollows: (1) educing Eqs.(12), (13) to a system o ist-ode equations, (2) witing the dieence equations using cental dieences, (3) lineaizing the algebaic equations by Newton s method, and wite them in matix-vecto om, and (4) solving the linea system by the block ti-diagonal elimination technique. Fo numeical calculations, a uniom step size o η = 0.01 is ound to be satisactoy and the solutions ae obtained with an eo toleance o 10 6 in all the cases. We need to be vey caeul in choosing the values o these paametes. It is also impeative that the computational time o each set o input paametic values should be shot. One souce o the eo is inevitable due to the unbounded physical domain, wheeas the computational domain has to be inite. The step size η and the position o the edge o the bounday laye η ae to be adjusted o dieent values o the paametes to maintain the accuacy. We an ou bulk computations with the value o η = 10 which is suicient to achieve the a ield bounday conditions asymptotically o all values o the paametes consideed hee. Fo bevity, the details o the solution pocedue ae not pesented hee. Since the bounday laye low and heat tanse o the nanoluid ae aected by the vaiable viscosity paamete and the nanoluid paticle volume action paamete, the main ocus o the pesent study is to bing out the eects o these paametes though the numeical solutions. It is woth mentioning that we have used the data pesented in Table 1 o the themophysical popeties o the luid and nanopaticles. Fo all calculations, pue wate is consideed as the base luid (with P = ) 4. Results and discussion Numeical computations ae being caied out to obtain the low and heat tanse chaacteistics with nano-paticles. The pesent study involves thee moe paametes in addition to the usual Pandtl numbe: namely, the luid viscosity paamete, the Ecket numbe and the nanopaticle volume action paamete. In ode to discuss the eects o dieent physical paametes on the nanoluid low and heat tanse chaacteistics with Ag-wate and Cu-wate nanopaticles, the numeical esults ae epesented in Figs.2-9 and in Table 1. Hee in the pesent pape we have consideed two dieent types o nanopaticle volume a-

6 6 ction namely, Cu-wate and Ag-wate nano-paticles. o the cuves with the nanopaticle volume action paamete eveals that the eect o non-zeo values o the nanopaticle volume action paamete is to decease the bounday laye thickness and to incease the skin iction as shown in Fig.3. The numeical esults o the tempeatue poile θη ( ) ae depicted in Figs.4-7 o dieent values o the govening paametes. The geneal tend obseved om these igues is that the tempeatue distibution is unity at the wall, and with the change in physical paametes it tends asymptotically to zeo in the ee steam egion. Fig.2 Velocity poiles o dieent values o o wate and coppe Fig.4 Tempeatue poiles o dieent values o o wate and coppe Fig.3 Velocity poiles o dieent values o wate and coppe θ and o Figue 2 shows the eect o nanopaticle volume action paamete on the velocity poile in the bounday laye. It is obseved that the velocity poile is zeo at the wall, inceases and tends to unity as the distance inceases om the bounday. The eect o inceasing values o the nanopaticle volume action paamete is to decease the velocity poile. This is due to the act that the pesence o nanopaticle volume action leads to uthe thinning o the bounday laye. This is even tue o the negative values o the vaiable viscosity paamete (see Fig.3). The eect o inceasing values o luid viscosity paamete θ is to decease the momentum bounday laye thickness. It is clealy seen that as θ 0 the bounday laye thickness deceases and the velocity distibution becomes linea o lage values o the Pandtl numbe. This is due to the act that o a given luid, when δ is ixed, smalle θ implies highe tempeatue dieence between the wall and the ambient luid. The esults pesented in this pape demonstate quite clealy that θ, which is an indicato o the vaiation o luid viscosity with tempeatue, has a stong eect on the velocity poile η ( η ) within the bounday laye and hence on the skin iction chaacteistics. Compaison Fig.5 Tempeatue poiles o dieent values o and o wate and coppe Figue 4 depicts the tempeatue poiles o dieent values o the nanopaticle volume action paamete. Analysis o the gaph shows that the eect o the inceasing values o the nanopaticle volume action paamete is to enhance the tempeatue and tends asymptotically to zeo as the distance inceases om the bounday. Also the themal bounday laye o nano-solid-paticles, namely Cuwate, is geate than that o pue wate. This is because nano-solid-paticles volume action paamete (o Cu) has high themal conductivity, so the thickness o the themal bounday laye inceases. This obsevation can also be noticed o inceasing values o θ

7 7 the luid viscosity paamete θ (see Fig. 5). shows that o dieent values o, (0) is positive which means that the suace has inluence on the luid. Also, as and θ incease, the local skin iction inceases. Futhemoe, it is obseved om Fig.9 that o a paticula value o θ the local Nusselt numbe inceases as incease om 0% to 20%. This obsevation is even tue with anothe nanoluid, namely wate with silve nanopaticles. Fig.6 Tempeatue poiles o dieent values o and Ec o wate and coppe Fig.8 Vaiation o Skin iction o dieent values o θ with the themo-physical popeties deined in Table 1 when Ec =0 Fig.7 Tempeatue poiles o dieent values o and Ec o wate and coppe Fom the gaphical epesentation we see that the eect o inceasing values o the luid viscosity paamete θ is to enhance the tempeatue. This is due to the act that an incease in the luid viscosity paamete θ inceases the themal bounday laye thick ness, which esults in incease in the tempeatue. The non-dimensional tempeatue poiles with the space vaiable η o dieent values o the Ecket numbe Ec and nanopaticle volume action paamete ae pesented in Fig.6. Fom this igue we notice that the eect o an incease in the Ecket numbe Ec is to incease the tempeatue distibution θη ( ). This is in conomity with the act that the enegy is stoed in the luid egion as a consequence o dissipation due to viscosity and elastic deomation. Futhe it is obseved om Fig.5 that the values o the tempeatue ae always less in the absence o nano paticle volume action than in its pesence. This tend continues even in the pesence o luid viscosity paamete (see Fig.7). Numeical esults o the local skin iction coeicient in tems o (0) and the local Nusselt numbe in tems o θ (0) with changes in ϕ and θ ae shown in Figs.8 and 9 espectively. Figue 8 Fig.9 Vaiation o wall tempeatue gadient o dieent values o θ with the themo-physical popeties dei ned in Table 1 when Ec =0 The impact o all the physical paametes on θ (0), which signiies the wall tempeatue gadient, is ecoded in Table 1. Fom Table 1 it is inteesting to note that the eect o inceasing values o the nanopaticle volume action paamete is to incease the wall tempeatue gadient. Quite opposite is the phenomenon with the luid viscosity paamete. The eect o Ecket numbe is to incease the wall tempeatue gadient. Also, the simila eect is obseved with anothe viscous nanoluid, namely the Ag-wate paticles. 5. Concluding emak In this pape, the eect o vaiable viscosity paamete on the low and heat tanse o a nanoluid ove

8 8 a lat suace has been analyzed numeically. In paticula, we ocus on Ag-wate and Cu-wate nanoluids and investigate the eects o the nanopaticle volume action on the low and heat tanse chaacteistics unde the inluence o vaiable luid popety in the pesence o viscous dissipation. The luid viscosity is assumed to vay as an invese unction o tempeatue. The govening equations ae tansomed into a system o coupled non-linea odinay dieential equations, and the esulting system is solved numeically by a inite dieence scheme (known as the Kelle-box method). A paametic study is peomed to exploe the eects o the govening paametes (the vaiable viscosity paamete, the nanoluid paticle volume action paamete, the Ecket numbe and the Pandtl numbe) on the low and heat tanse chaacteistics. Numeical esults ae pesented though a table and gaphs. The esults obtained o the low and heat tanse chaacteistics eveal many inteesting behavios that waant uthe study o nanoluids. Acknowledgment We thank the anonymous eeee o the helpul comments, which have geatly impoved the quality o the manuscipt. Reeences [1] CHOI S. U. S., ZHANG Z. G. and YU W. et al. Anomalously themal conductivity enhancement in nanotube suspensions[j]. Applied Physics Lettes, 2001, 79(14): [2] KANG H. U., KIM S. H. and OH J. M. Estimation o themal conductivity o nanoluid using expeimental eective paticle volume[j]. Expeimental Heat Tanse, 2006, 19(3): [3] MAIGA S. E. B., PALM S. J. and NGUYEN C. T. et al. Heat tanse enhancement by using nanoluids in oced convection low[j]. Intenational Jounal o Heat Fluid Flow, 2005, 26(4): [4] TIWARI R. K., DAS M. K. Heat tanse augmentation in a two-sided lid-diven dieentially heated squae cavity utilizing nanoluids[j]. Intenational Jounal o Heat Mass Tanse, 2007, 50(9-10): [5] TZOU D. Y. Themal instability o nanoluids in natual convection[j]. Intenational Jounal o Heat Mass Tanse, 2008, 51(11-12): [6] ABU-NADA E. Application o nanoluids o heat tanse enhancement o sepaated lows encounteed in a backwad acing step[j]. Intenational Jounal o Heat and Fluid Flow, 2008, 29(1): [7] OZTOP H. F., ABU-NADA E. Numeical study o natual convection in patially heated ectangula enclosues illed with nanoluids[j]. Intenational Jounal o Heat and Fluid Flow, 2008, 29(5): [8] KUZNETSOV A. V., NIELD D. A. Natual convective bounday-laye low o a nanoluid past a vetical plate[j]. Intenational Jounal o Themal Sciences, 2010, 49(2): [9] AMINOSSADATI S. M., GHASEMI B. Natual conve- ction cooling o a localised heat souce at the bottom o a nanoluid-illed enclosue[j]. Euopean Jounal o Mechanics B/Fluids, 2009, 28(5): [10] KHANAFER K., VAFAI K. and LIGHTSTONE M. Buoyancy-diven heat tanse enhancement in a twodimensional enclosue utilizing nanoluids[j]. Intenational Jounal o Heat Mass Tanse, 2003, 46(19): [11] BUONGIORNO J., HU W. Nanoluid coolants o advanced nuclea powe plants[c]. Intenational Congess on Advances in Nuclea Powe Plants. Seoul, Koea, Pape 5705, [12] KHAN W. A., Pop I. Bounday-laye low o a nanoluid past a stetching sheet[j]. Intenational Jounal o Heat Mass Tanse, 2010, 53(11-12): [13] KAKAÇ S., PRAMUANJAROENKIJ A. Review o convective heat tanse enhancement with nanoluids[j]. Intenational Jounal o Heat Mass Tanse, 2009, 52(13-14): [14] SUN Lei, LIN Jian-zhong and BAO Fu-bing. Numeical simulation on the deposition o nanopaticles unde lamina conditions[j]. Jounal o Hydodynamics, Se. B, 2006, 18(6): [15] LIU Song, Lin JIAN-zhong. Numeical simulation o nanopaticle coagulation in a Poiseuille low via a moment method[j]. Jounal o Hydodynamics, 2008, 20(1): 1-9. [16] WANG Yu-ming, LIN Jian-zhong. Evolution o numbe concentation o nano-paticles undegoing Bownian coagulation in the tansition egime[j]. Jounal o Hydodynamics, 2011, 23(4): [17] NIELD D. A., KUZNETSOV A. V. The Cheng- Minkowycz poblem o natual convective bounday laye low in a poous medium satuated by a nanoluid[j]. Intenational Jounal o Heat Mass Tanse, 2009, 52(25-26): [18] MUKHOPADHYAY S., LAYEK G. C. Eect o themal adiation and vaiable luid viscosity on ee convective and heat tanse past a poous stetching suace[j]. Intenational Jounal o Heat Mass Tanse, 2008, 51(9-10): [19] PRASAD K. V., VAJRAVELU K. and DATTI P. S. The eects o vaiable luid popeties on the hydomagnetic low and heat tanse ove a non-linealy stetching sheet[j]. Intenational Jounal o Themal Sciences, 2010, 49(3): [20] SEEDBEEK M. A. Finite element method o the eects o chemical eaction, vaiable viscosity, themophoesis and heat geneation / absoption on a bounday laye hydo magnetic low with heat and mass tanse ove a heat suace[j]. Acta Mechanic, 2005, 177(1-4): [21] ALI M. E. The eect o vaiable viscosity on mixed convection heat tanse along a vetical moving suace[j]. Intenational Jounal o Themal Sciences, 2006, 45(1): [22] BRINKMAN H. C. The viscosity o concentated suspensions and solution[j]. Jounal o Chemical Physics, 1952, 20(4): 571. [23] MAXWELL J. A teatise on electicity and magnetism[m]. Second Edition, Cambidge, UK: Oxod Univesity Pess, [24] CHAMKHA A. J., ISSA C. Eects o heat geneation/ absoption and themophoesis on hydomagnetic low with heat and mass tanse ove a lat suace[j]. Intenational Jounal o Numeical Methods Fo Heat

9 9 and Fluid Flow, 2000, 10(4): [25] CEBECI T., BRADSHAW P. Physical and computational aspects o convective heat tanse[m]. New Yok, USA: Spinge-Velag, [26] KELLER H. B. Numeical methods o two-point bounday value poblems[m]. New Yok, USA: Dove, [27] PRASAD K. V., PAL D. and DATTI P. S. MHD low and heat tanse in the low o a powe law luid ove a non-isothemal stetching sheet[j]. Communications in Nonlinea Science Numeical Simulation, 2009, 14(5): [28] PRASAD K. V., DATTI P. S. and VAJRAVELU K. Hydomagnetic low and heat tanse o a non- Newtonian powe law luid ove a vetical stetching sheet[j]. Intenational Jounal o Heat Mass Tanse, 2010, 53(5-6):