LAMINAR MHD MIXED CONVECTION FLOW OF A NANOFLUID ALONG A STRETCHING PERMEABLE SURFACE IN THE PRESENCE OF HEAT GENERATION

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1 In: International Journal o Microscale and Nanoscale hermal ISSN: Volume, Number Nova Science Publishers, Inc. LAMINAR MHD MIXED CONVECION FLOW OF A NANOFLUID ALONG A SRECHING PERMEABLE SURFACE IN HE PRESENCE OF HEA GENERAION OR ABSORPION EFFECS Ali J. Chamkha 1*, Abdelraheem M. Al and Humood F. Al-Mudha 3 1 Manuacturing Engineering Department, he Public Authorit or Applied Education and raining, Shueikh 70654, Kuait Civil Engineering Department, Facult o Engineering, Kushu Universit, Japan 3 Chemical Engineering Department, he Public Authorit or Applied Education and raining, Shueikh 70654, Kuait ABSRAC he boundar-laer lo o a nanoluid on a linearl moving permeable vertical surace in the presence o magnetic ield, heat generation or absorption, thermopherosis, Bronian motion and suction or injection eects is studied. Similarit solutions are obtained or the boundar-laer equations subject to poer-la all temperature, nanoparticles volume raction and velocit variations. he obtained equations are solved numericall b an eicient, iterative, tri-diagonal, implicit inite-dierence method. A detailed parametric stud is perormed to access the inluence o the various phsical parameters on the longitudinal velocit, temperature and nanoparticle volume raction proiles as ell as the local skin-riction coeicient, local Nusselt number and the local Sherood number and the results are presented in both graphical and tabular orms. Keords: MHD; Nanoluid; similarit solutions; ree convection; suction or injection; Bronian motion; thermophoresis; heat generation or absorption. Nomenclature B () magnetic ield strength C D B D local skin-riction coeicient Bronian diusion coeicient thermophoresis diusion coeicient * achamkha@ahoo.com

2 5 Ali J. Chamkha, Abdelraheem M. Al and Humood F. Al-Mudha suction or injection parameter g acceleration due to gravit M magnetic ield parameter N buoanc ratio parameter Nb Bronian motion parameter Ns local buoanc parameter Nt thermophoresis parameter Nu local Nusselt number k base luid thermal conductivit P base luid Pressure Pr Prandtl number Q () heat generation or absorption coeicient Sc Sh u Schmidt number local Sherood number temperature velocit in -direction u V v stretching speed o the plate velocit vector velocit in -direction Greek Smbols nanoparticles concentration p μ C C p densit o the base luid densit o the nanoparticles material heat capacit o the base luid heat capacit o the nanoparticles material base luid thermal diusivit ratio o heat capacities electrical conductivit o the base luid base luid dnamic viscosit volumetric volume epansion coeicient o the nanoluid heat generation or absorption parameter

3 Superscript Laminar MHD Mied Convection Flo o a Nanoluid 53 Dierentiation ith respect to η Subscripts p a?? base luid nanoparticles material all ree stream 1. Introduction Nanoluids are prepared b dispersing solid nanoparticles in luids such as ater, oil, or ethlene glcol. hese luids represent an innovative a to increase thermal conductivit and, thereore, heat transer. Unlike heat transer in conventional luids, the eceptionall high thermal conductivit o nanoluids provides or eceptional heat transer, a unique eature o nanoluids. Advances in device miniaturization have necessitated heat transer sstems that are small in size, light mass, and highperormance. Several authors have tried to establish convective transport models or nanoluids. Nanoluid is a to-phase miture in hich the solid phase consists o nanosized particles. In vie o the nanoscale size o the particles, it ma be questionable hether the theor o conventional to-phase lo can be applied in describing the lo characteristics o nanoluid [1]. Since the size o the particles is less than 100 nm, nanoluids behave like a luid than a miture [1 3]. Xuan and Roetzel [1] proposed homogeneous lo model here the convective transport equations o pure luids are directl etended to nanoluids. his means that all traditional heat transer correlations (e.g. Dittus Boelter) could be used or nanoluids provided the properties o pure luids are replaced b those o nanoluids involving the volume raction o the nanoparticles. he homogeneous lo models are hoever in conlict ith the eperimental observations o Maliga et al. [3], as the under predict the heat transer coeicient o nanoluids. Xuan et al. [4] have eamined the transport properties o nanoluid and have epressed that thermal dispersion, hich takes place due to the random movement o particles, takes a major role in increasing the heat transer rate beteen the luid and the all. his requires a thermal dispersion coeicient, hich is still unknon. Bronian motion o the particles, ballistic phonon transport through the particles and nanoparticle clustering can also be the possible reason or this enhancement [5]. Das et al. [6] has observed that the thermal conductivit or nanoluid increases ith increasing temperature. he have also observed the stabilit o Al O 3 ater and CuO ater nanoluid. Eperiments on heat transer due to natural convection ith nanoluid have been studied b Putra et al. [7] and Wen and Ding [8]. he have observed that heat

4 54 Ali J. Chamkha, Abdelraheem M. Al and Humood F. Al-Mudha transer decreases ith increase in concentration o nanoparticles. he viscosit o this nanoluid increases rapidl ith inclusion o nanoparticles as shear rate decreases. he problem o stead hdromagnetic lo and heat transer over a stretching surace could be ver practicable in man applications in the polmer technolog and metallurg. In particular, man metallurgical processes involve the cooling o continuous strips or ilaments b draing them though a quiescent luid and that in the process o draing, these strips are sometimes stretched. In the case o annealing and thinning o copper ires, the properties o the inal product depend to a great etent on the rate o cooling. B draing such strips in an electricall conducting luid subject to a magnetic ield, the rate o cooling can be controlled and inal products o desired characteristics might be achieved [9]. And also, in several engineering processes, materials manuactured b etrusion processes and heat treated materials traveling beteen a eed roll and a ind up roll on conve belts possess the characteristics o a moving continuous surace. he stead lo on a moving continuous lat surace as irst considered b Sakiadis [10] ho developed a numerical solution using a similarit transormation. Chiam [11] reported solutions or stead hdromagnetic lo over a surace stretching ith a poerla velocit ith the distance along the surace. sou et al. [1] studied a ide ranging analtical and eperimental investigation o the lo and heat transer characteristics o the boundar laer on a continuous moving surace. he to-dimensional lo caused solel b a linearl stretching sheet in an otherise quiescent incompressible luid hich has a ver simple closed rom eponential solution as established b Crane [13]. Gorla et al. [14] studied the MHD eect on a vertical stretching surace ith suction and bloing. Anjali Devi and Kandasam [15] studied the stead MHD laminar boundar laer lo over a all o the edge ith suction or injection in the presence o species concentration and mass diusion. Seddeek [16] studied the eects o heat generation or absorption on heat and mass transer o a viscoelastic luid ith a magnetic ield over a stretching sheet. he stud o heat generation has several phsical problems such as those concerned ith dissociating luids. Possible heat generation eects ma change the temperature distribution and, thereore, the particle deposition rate. his ma occur in such applications related to nuclear reactor cores, ire and combustion modeling, electronic chips and semi conductor aers. Representative studies dealing ith heat generation or absorption eects have been reported previousl b such authors as Achara and Goldstein [17], Vajravelu and Naeh [18] and Chamkha [19]. he objective o this paper is stud mied convection MHD lo o a nanoluid past a stretching permeable surace in the presence o magnetic ield, heat generation or absorption, thermopherosis, Bronian motion and suction or injection eects.. Mathematical Analsis Consider stead, to-dimensional lo o a nanoluid consisting o a base luid and small nanoparticles due to the stretching o a vertical permeable surace in the presence o magnetic ield, heat generation or absorption, thermopherosis, Bronian motion and

5 Laminar MHD Mied Convection Flo o a Nanoluid 55 suction or injection eects. he Oberbeck Boussinesq approimation is emploed. he governing equations are based on the balance las o total mass, momentum, thermal energ and nanoparticles modiied to include the various eects stated above. hese equations can be ritten respectivel as:. V 0, (1) B () V. V VP 1 1 g V, p C V. k C D.. Q(), p B D () (3) V. D B D. (4) here V u, is the velocit vector ith u and being the - and - components o velocit, is temperature and is the nanoparticles concentration. is the densit o the base luid and p is the densit o the nanoparticles material. D B and D are the Bronian diusion coeicient and the thermophoresis diusion coeicient respectivel. P is the pressure, is the electrical conductivit o the luid, B () is the strength o magnetic ield, Q () is the heat generation parameter such that Q>0 corresponds to heat generation hile Q<0 corresponds to heat absorption. g is the acceleration due to gravit., k and are the dnamic viscosit, thermal conductivit and volumetric volume epansion coeicient o the nanoluid, respectivel. C and C p are the heat capacit o the base luid and the eective heat capacit o the nanoparticles material, respectivel. is the ree stream temperature. It should be noted that details o the derivation o equations (3) and (4) are given in the papers b Buongiorno [0] and Kuznetsov and Nield [1] in the absence o magnetic ield and heat generation or absorption eects. he boundar conditions are as ollos: m u u U, 0, c n, n 1 D at 0 u 0,, as (5)

6 Ali J. Chamkha, Abdelraheem M. Al and Humood F. Al-Mudha 56 here m U u 0 is the stretching speed o the plate. and are the all temperature and nanoparticles concentration, respectivel. is the ree stream concentration o nanoparticles. c, D, n and n 1 are constants. For a similarit transormation o the governing equations, it ill be seen that 1 1 m n n. B assuming the nanoparticles concentration is dilute and that the magnetic ield is applied normal to the plate and using a suitable or the reerence pressure e can rite equation () as ollo: u B g P V V V p ) ( 1. (6) B using the standard boundar-laer approimation, based on a scale analsis, and rite the governing equations: 0, u (7), 1 u B g g u u u u P p (8) 0, p (9), C Q D D u B (10) B D D u (11) here C k and p C C are the base luid thermal diusivit and the ratio o heat capacities, respectivel. Substituting the olloing dimensionless variables:

7 Laminar MHD Mied Convection Flo o a Nanoluid 57 m1 m U 0 u U 0 m 1 m 1, m1, c, m 1 1 D m m 1 S, Re Re U 0 m1 m1 m1, B B, 0 Q Q (1) into equations (7)-(11), one obtains the olloing sel-similar ordinar dierential equations: m N NsS M 0, (13) 1 m (1 m) 4m Nb S Nt Pr Pr 0 (1 ) m 1 m (14) D B Nt 4m S Sc S S 0 (15) Nb 1 m here 1 g c p gd B 0 Q 0 N Ns M Pr U0, U0, U 0 k, U 0, and Sc are the buoanc ratio parameter, local buoanc parameter, magnetic ield parameter and heat generation or absorption parameter, Prandtl number and the Schmidt m1 D c m1 number, respectivel. Also, the parameters Nt and Nb D B D are the thermophoresis parameter and the Bronian motion parameter, respectivel such Nt D c that. Nb D D B he corresponding transormed dimensionless boundar conditions become:, ' 1, 1, S 1, at 0 ' 0, 0, S 0, as (16) 0 here 1m is the suction or in injection parameter. νu ( m 1) 0

8 58 Ali J. Chamkha, Abdelraheem M. Al and Humood F. Al-Mudha O special signiicance or this tpe o lo and heat and mass transer situation are the local skin-riction coeicient, local Nusselt number Nu and the local Sherood number Sh hich are deined b: C m 1 1 Re 0, (17a) m 1 Nu Re 0, m 1 Sh Re S 0 (17b) (17c) 3. Numerical Method he sel-similar equations (13) through (15) are nonlinear and possess no analtical solution and must be solved numericall. he eicient, iterative, tri-diagonal, implicit inite-dierence method discussed b Blottner [] has proven to be adequate or the solution o such equations. he equations are linearized and then descritized using three points central dierence quotients ith variable step sizes in the direction. he resulting equations orm a tri-diagonal sstem o algebraic equations that can be solved b the ell knon homas algorithm (see Blottner []). Due to the nonlinearities o the equations, an iterative solution ith successive over or under relaation techniques is required. he convergence criterion required that the maimum absolute error beteen to successive iterations be he computational domain as made o 196 grids in the direction. A starting step size o 01 in the direction ith an increase o 375 times the previous step size as ound to give ver accurate results. he maimum value o ( ) hich represented the ambient conditions as assumed to be 35. he starting step size and the groth actor emploed ere arrived at ater perorming numerical eperimentations to assess grid independence and ensure accurac o the results. he accurac o the aorementioned numerical method as validated b direct comparisons ith the numerical results reported earlier b Chiam [11] or a regular luid in the presence o a magnetic ield and the absence o all other eects. It should be noted that M in Chiam s [11] paper is deined dierentl and is equal to M/(1+m) in this paper. able 1 shos the comparison ith the results o Chiam [11] based on his equations. It is clear that ecellent agreement beteen the results eists. his avorable comparison lends conidence in the numerical results to be reported in the net section.

9 Laminar MHD Mied Convection Flo o a Nanoluid 59 m able 1. Comparison o values o "(0) or = 1.5, 5.0 and various values o M 1 m or =0, N=0 and Ns=0 m "(0) "(0) M 1 m Chiam [11] Present Results and Discussion In this section, representative numerical results are displaed ith the help o graphical illustrations. Computations ere carried out or various values o phsical parameters such as the magnetic ield parameter M, heat generation or absorption parameter, Schmidt number Sc, buoanc ratio N, local buoanc parameter Ns, thermophoresis parameter Nt, Bronian motion parameter Nb and the suction or injection parameter. ' M=0, 1,, 3 Pr=0.7 Pr=10 =0, m=0.5, N=1, Nb=0.3 Nt=0.3, Ns=0.5, Sc=10, = Figure 1. Eects o the magnetic ield parameter and the Prandtl number on the longitudinal velocit proiles.

10 60 Ali J. Chamkha, Abdelraheem M. Al and Humood F. Al-Mudha Pr=0.7 Pr=10 =0, m=0.5, N=1, Nb=0.3 Nt=0.3, Ns=0.5, Sc=10, =0 M=0, 1,, Figure. Eects o the magnetic ield parameter and the Prandtl number on the temperature proiles. Pr=0.7 Pr=10 =0, m=0.5, N=1, Nb=0.3 Nt=0.3, Ns=0.5, Sc=10, =0 S 0. M=0, 1,, Figure 3. Eects o the magnetic ield parameter and the Prandtl number on the nanoparticles volume raction. Figures 1-3 sho the eects o the magnetic ield parameter M ith to values o the Prandtl number Pr (Pr=0.7 and Pr=10) on the longitudinal velocit, temperature and the nanoparticles volume raction proiles. Application o a magnetic ield has the tendenc to slo don the movement o the luid causing its velocit to decrease as the magnetic ield parameter increases. his decrease in the lo movement as the magnetic ield parameter increases is accompanied b increases in both the temperature and nanoparticle volume raction proiles. In addition, as the Prandtl number increases, both o the longitudinal velocit and the temperature proiles decrease hereas the nanoparticles volume raction proiles increase.

11 Laminar MHD Mied Convection Flo o a Nanoluid 61 ' m=0.5, M=0, N=1, Nb=0.3 Nt=0.3, Ns=0.5, Pr=0.7, Sc=10 = -0.5, 0, 0.5 =-0.5 =0 = Figure 4. Eects o the heat generation or absorption parameter and the suction or injection parameter on the longitudinal velocit. =-0.5 =0 =0.5 m=0.5, M=0, N=1, Nb=0.3 Nt=0.3, Ns=0.5, Pr=0.7, Sc=10 = -0.5, 0, Figure 5. Eects o the heat generation or absorption parameter and the suction or injection parameter on the temperature proiles. S = -0.5, 0, 0.5 =-0.5 =0 =0.5 m=0.5, M=0, N=1, Nb=0.3 Nt=0.3, Ns=0.5, Pr=0.7, Sc= Figure 6. Eects o the heat generation or absorption parameter and the suction or injection parameter on the nanoparticles volume raction.

12 6 Ali J. Chamkha, Abdelraheem M. Al and Humood F. Al-Mudha Figures 4-6 present the eects o the heat generation or absorption parameter or various values o the suction or injection parameter on the longitudinal velocit, temperature and nanoparticles volume raction proiles, respectivel. Increasing the heat generation or absorption parameter has the tendenc to increase the thermal state o the luid. his increase in the luid temperature causes more induced lo toards the stretched plate through the thermal buoanc eect. Hoever, these increases in both the velocit and temperature proiles are accompanied b slight decreases in the nanoparticle volume raction proiles as the heat generation or absorption parameter increases. In addition the velocit, temperature and nanoparticles volume raction proiles are higher in the presence oall suction conditions ( 0 ) than or all injection conditions ( 0 ). It should be noted that distinctive peaks in the longitudinal velocit and temperature proiles or hich the values o the velocit and temperature are higher than their all values occur in the vicinit o the all or the case o moderate suction and internal heat generation conditions. his could also occur even in the absence o suction ( 0 ) i a higher value o than 0.5 is used. he presence o the peaks in the velocit temperature proiles ould mean that the values o 0 and 0 hich are related to the skin-riction coeicient and Nusselt number ould be negative as ill be seen later. ' =0, M=0, N=1, Nb=0.3 m=0.1, 0.5, 1, 1.5 Sc=1 Sc=10 Nt=0.3, Ns=0.5, Pr=0.7, = Figure 7. Eects o the stretching speed eponent and the Schmidt number on the longitudinal velocit.

13 Laminar MHD Mied Convection Flo o a Nanoluid =0, M=0, N=1, Nb=0.3 Sc=1 Sc=10 Nt=0.3, Ns=0.5, Pr=0.7, =0 m=0.1, 0.3, 0.5, 1, Figure 8. Eects o the stretching speed eponent and the Schmidt number on the temperature proiles. 1. =0, M=0, N=1, Nb=0.3 Sc=1 Sc=10 Nt=0.3, Ns=0.5, Pr=0.7, =0 S m=0.1, 0.5, 1, Figure 9. Eects o the stretching speed eponent and the Schmidt number on the nanoparticle volume raction. Figures 7-9 present the eects o the stretching speed eponent m and Schmidt number on the longitudinal velocit, temperature and nanoparticles volume raction proiles, respectivel. It is seen that as the stretching speed eponent m increases, all o the velocit, temperature and nanoparticles volume raction proiles decrease. In addition, increasing the Schmidt number Sc causes increases in the velocit proiles hile the temperature and nanoparticle volume raction proiles decrease. Furthermore, the nanoparticle volume raction boundar-laer thickness decreases as Sc increases. hese behaviors are clearl depicted in Figures 7-9.

14 64 Ali J. Chamkha, Abdelraheem M. Al and Humood F. Al-Mudha ' N=0.1, 0.5, 1 Nb=0.1 Nb=0.5 =0, m=0.5, M=0, Nt=0.3 Ns=0.5, Pr=0.7, Sc=10, = Figure 10. Eects o the buoanc parameter and the Bronian motion parameter on the longitudinal velocit. Nb=0.1 Nb=0.5 =0, m=0.5, M=0, Nt=0.3 Ns=0.5, Pr=0.7, Sc=10, =0 N=0.1, 0.5, Figure 11. Eects o the buoanc parameter and the Bronian motion parameter on the temperature proiles. Nb=0.1 Nb=0.5 =0, m=0.5, M=0, Nt=0.3 Ns=0.5, Pr=0.7, Sc=10, =0 S 0. N=0.1, 0.5, Figure 1. Eects o the buoanc parameter and the Bronian motion parameter on the nanoparticles volume raction.

15 Laminar MHD Mied Convection Flo o a Nanoluid 65 Figures 10-1 displa the eects o the buoanc ratio parameter N and the Bronian motion parameter Nb on the longitudinal velocit, temperature and nanoparticles volume raction proiles, respectivel. As the buoanc ratio parameter N increases, the velocit proiles increases, hile the temperature and nanoparticles volume raction proiles decrease. In addition, increasing the Bronian motion parameter Nb causes the velocit proiles to increase hile the temperature and nanoparticle volume raction proiles decrease. Furthermore, the nanoparticle volume raction boundar-laer thickness decreases as the Bronian motion parameter increases. ' Ns=0.1, 0.5, 1 =0, m=0.5, M=0, N=1 Nt=0.1 Nt=0.5 Nb=0.3, Pr=0.7, Sc=10, = Figure 13. Eects o the local buoanc parameter and the thermophoresis parameter on the longitudinal velocit. =0, m=0.5, M=0, N=1 Nt=0.1 Nt=0.5 Nb=0.3, Pr=0.7, Sc=10, =0 Ns=0.1, 0.5, Figure 14. Eects o the local buoanc parameter and the thermophoresis parameter on the temperature proiles.

16 66 Ali J. Chamkha, Abdelraheem M. Al and Humood F. Al-Mudha =0, m=0.5, M=0, N=1 Nt=0.1 Nt=0.5 Nb=0.3, Pr=0.7, Sc=10, =0 S 0. Ns=0.1, 0.5, Figure 15. Eects o the local buoanc parameter and the thermophoresis parameter on the nanoparticles volume raction. Figures sho the eects o local buoanc parameter Ns or to values o the thermophoresis parameter Nt on the longitudinal velocit, temperature and nanoparticles volume raction proiles, respectivel. As the local buoanc parameter Ns increases, the temperature proiles increase hile the velocit proiles decrease. he local buoanc parameter Ns has small eects on the nanoparticle volume raction proiles. In addition, as the thermophoresis parameter Nt increases, all o the velocit, temperature and nanoparticles volume raction proiles increase. able illustrates the eects o the suction or injection parameter, Schmidt number Sc, stretching speed eponent m, heat generation or absorption parameter and 0, 0 and the reduced local Sherood number S0. the magnetic ield parameter M on the reduced local skin-riction coeicient reduced local Nusselt number It is observed that the values o the reduced local skin-riction coeicient increase as either o the magnetic ield parameter, Suction or injection parameter or the stretching speed eponent m increases hile it decreases as the heat generation or absorption parameter increases. he reduced local Nusselt number increases as either o the stretching speed eponent m or the suction or injection parameter increases, hile it decreases as either o the heat generation or absorption parameter or the magnetic ield parameter increases. he reduced local Sherood number increases as either o the heat generation or absorption parameter, suction or injection parameter or the stretching speed eponent m increases hereas it decreases as the magnetic ield parameter increases. It should be mentioned that the negative values in 0 and 0 in the presence o heat generation ( 0. 5) are related to the distinctive peaks hich occur in the velocit and temperature proiles as discussed in Figures 4 and 5.

17 Laminar MHD Mied Convection Flo o a Nanoluid 67 0, reduced local 0 and the reduced local Sherood number S0 able. Values o the reduced local skin-riction coeicient Nusselt number values o or various, Sc, m, and M at Pr=0.7, N=1, Nb=0.3, Nt=0.3 and Ns=0.5. Sc M M 0 0 S , 0 and the reduced local Sherood number S0 able 3 presents the values o the reduced local skin-riction coeicient reduced local Nusselt number or dierent values o the buoanc ratio parameter N, local buoanc parameter Ns, Bronian motion parameter Nb and the thermophoresis parameter Nt. It is observed that the values o the reduced local skin-riction coeicient increase as the local buoanc parameter N increases hile it decreases as either o the buoanc ratio parameter N, S Bronian motion parameter Nb or the thermophoresis parameter Nt increases. he reduced local Nusselt number decreases as either o the Bronian motion parameter Nb, local buoanc parameter N or the thermophoresis parameter Nt increases hereas it S increases as the buoanc ratio parameter N increases. Finall, the reduced local Sherood number increases as either o the buoanc ratio parameter or the Bronian motion parameter increases. On the other hand, the reduced local Sherood number decreases as either o the local buoanc parameter or the thermophoresis parameter increases.

18 68 Ali J. Chamkha, Abdelraheem M. Al and Humood F. Al-Mudha 0, reduced local 0 and the reduced local Sherood number S0 able 3. Values o the reduced local skin-riction coeicient Nusselt number or various values o N, Nb, Ns and Nt at m 0.5, M 0, Pr 0.7, Sc 10, 0 and 0 N Nb Ns Nt 0 0 S Conclusion In this paper, the problem o mied convection MHD lo o a Nanoluid past a stretching permeable surace is considered. he eects o heat generation or absorption, buoanc, thermophoresis, Bronian motion and all suction or injection are investigated. he governing equations and boundar conditions are reduced to ordinar dierential equations (similarit equations) and conditions using appropriate scaling transormations. he obtained similarit equations are solved numericall subject to the transormed boundar conditions using an eicient, iterative, tri-diagonal implicit initedierence method. It as ound that the reduced local skin-riction coeicient increased as either o the magnetic ield parameter, suction or injection parameter, stretching speed eponent or the local buoanc parameter increased hile it decreased as either o the buoanc ratio parameter, Bronian motion parameter, thermophoresis parameter or the heat generation or absorption parameter increased. he reduced local Nusselt number increased as either o the stretching speed eponent, buoanc ratio parameter or the suction or injection parameter increased hile it decreased as either o the heat generation or absorption parameter, magnetic ield parameter, Bronian motion parameter, local buoanc parameter or the thermophoresis parameter increased. Finall, the reduced local Sherood number increased as either o the heat generation or absorption parameter, suction or injection parameter, buoanc ratio parameter, Bronian motion parameter or the stretching speed eponent increased hereas it decreased as either o the magnetic ield parameter, local buoanc parameter or the thermophoresis parameter increased.

19 Laminar MHD Mied Convection Flo o a Nanoluid 69 Reerences [1] Xuan, Y. and Roetzel, W. Conceptions or heat transer correlation o nanoluids, Int. J. Heat Mass ranser 43 (000) [] Lee, S., Choi, S. U. S., Li, S. and Eastman, J. A. Measuring thermal conductivit o luids containing oide nanoparticles, ASME rans., J. Heat ranser 11 (1999) [3] Maliga, S. E. B., Palm, S. M., Nguen, C.., Ro, G. and Galanis, Heat transer enhancement using nanoluid in orced convection lo, Int. J. Heat Fluid Flo 6 (005) [4] Xuan, Y., Yu, K. and Li, Q. Investigation on lo and heat transer o nanoluids b the thermal Lattice Boltzmann model, Progress in Computational Fluid Dnamics 5 (005) [5] Keblinski, P., Phillpot, S. R., Choi, S. U. S. and Eastman, J. A. Mechanisms o heat lo in suspensions o nano-sized particles (nanoluids), Int. J. Heat Mass ranser 45 (00) [6] Das, S. K., Putra, N., hiesen, P. and Roetzel, W. emperature dependence o thermal conductivit enhancement or nanoluids, J. Heat ranser 15 (003) [7] Putra, N., Roetzel, W. and Das, S. K. Natural convection o nano-luids, Heat Mass ranser 39 (003) [8] Wen, D. and Ding, Y. Natural convective heat transer o suspensions o titanium dioide nanoparticles (nanoluids), IEEE rans. Nanotechnol. 5 (006) 0 7. [9] Chakrabarti, A. and Gupta, A. S. Hdromagnetic lo and heat transer over a stretching sheet, Q Appl. Math. 37 (1979) [10] Sakiadis, B. C. Boundar laer behavior on continuous solid suraces I. Boundar laer equations or to-dimensional and aismmetric lo, AIChE J. 7 (1961) 6 8. [11] Chiam,. C. Hdrodnamic lo over a surace stretching ith a poer la velocit, Int. J. Eng. Sci. 33 (1995) [1] sou, F. K., Sparro, E. M. and Goldstein, R. J. Flo and heat transer in the boundar laer on a continuous moving surace, Int. J. Heat Mass ranser 10 (1967) [13] Crane, L. J. Flo past a stretching sheet, ZAMP 1 (1970) [14] Gorla, R. S. R., Abboud, D. E. and Sarmah, A. Magnetohdrodnamic lo over a vertical stretching surace ith suction and bloing, Heat Mass ranser, 34 (1998) [15] Anjali Devi, S. P. and Kandasam, R. Eects o chemical reaction, heat and mass transer on non-linear MHD laminar boundar laer lo over a edge ith suction or injection, Int. Commun. Heat Mass ranser 9 (00) [16] Seddeek, M. A. Heat and mass transer on a stretching sheet ith a magnetic ield in a viscoelastic luid lo through a porous medium ith heat source or sink, Comput. Mater. Sci. 38 (007)

20 70 Ali J. Chamkha, Abdelraheem M. Al and Humood F. Al-Mudha [17] Achara, S. and Goldstein, R. J. Natural convection in an eternall heated vertical or inclined square bo containing internal energ sources, ASME. J. Heat rans. 107 (1985) [18] Vajravelu, K. and Naeh, J. Hdromagnetic convection at a cone and a edge, Int. Commun. Heat Mass 19 (199) [19] Chamkha, A. J. Non-Darc ull developed mied convection in a porous medium channel ith heat generation/absorption and hdromagnetic eects, Numer. Heat ranser 3 (1997) [0] Buongiorno, J. Convective transport in nanoluids, ASME J. Heat ranser 18 (006) [1] Kuznetsov, A. V. and Nield, D. A. Natural convective boundar-laer lo o a nanoluid past a vertical plate, Int. J. hermal Sciences (009), in press. [] Blottner, F. G. Finite-dierence methods o solution o the boundar-laer equations, AIAA J. 8 (1970)