Entropy Generation due to MHD Mixed Convection of Nanofluid in a Vertical Channel with Joule Heating and Radiation effects

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1 IN (Print) : IN (Online) : Entropy Generation due to MHD Mixed Convection o Nanoluid in a Vertical Channel with Joule Heating and Radiation eects D. rinivasacharya #1, Md. haeeurrahan # # Departent o Matheatics, National Institute o Technology, Warangal , India 1 dsc@nitw.ac.in; dsrinivasacharya@yahoo.co second.author@second.co Abstract In this article, the eects o Joule heating and radiation on entropy generation due to lainar ixed convective low o an incopressible electrically conducting nanoluid in vertical channel is studied. The governing equations are non-diensionalized and then solved by using hootopy analysis ethod. The eects o agnetic, Joule heating, Brinkan nuber and radiation on the velocity in low direction, teperature, nanoparticle concentration, Bejan nuber and entropy generation are analyzed and presented geoetrically. Keyword - Entropy generation, Mixed convection, Radiation, Joule heating, MHD, Nanoluid, Vertical channel I. INTRODUCTION The study o convective heat transer in nanoluids has received considerable theoretical and practical interest in view o its use in a wide variety o engineering applications. These applications include cooling o nuclear reactor, power generating systes, autoobile engines, welding equipent, heat exchanging in electronics devices and coputers. Nanoluids are prepared by the unior dispersion and suspension o nanoeter sized etallic particle into convective heat transer luids such as water, oil, or ethylene glycol [1]. Choi [1] veriied that the nanoluids have ore theral-conductivity proportionate to the base luids. Mixed convection heat transer and nanoluid low in a vertical channel, a requently encountered geoetry in theral engineering equipent, has been concentrating on vast investigation or any years. Nuber o studies were conducted in recent years, on vertical channel illed with nanoluid by considering distinct types o convectional base luids with particular nanoparticles. Hang and Pop [] veriied the ixed convective nanoluids low in a vertical channel. Dehkordi and Hajipour [3] analyzed the ixed convective heat transer o nanoluid low between parallel plates. Grosan and Pop [4] analyzed the ully developed and ixed convective nanoluid low in a vertical channel. Hang et al [5] studied the ixed convective nanoluid low in a vertical channel using the Buongiorno ethod. Nield and Kuznetsov [6] discussed the orced convectional nanoluid low in a channel. Fakour and Vahabzadeh [7] studied the review o ixed convective low o a nanoluid in a vertical channel. Makhataret et al. [8] analyzes low reversal o ixed convective low o nanoluids with porous ediu in a vertical channel. Das et al. [9] analyzed the nanoparticle volue raction on ixed convective nanoluid low in a vertical plates. everal authors studied the eect o radiation paraeter on convective low o nanoluid together with heat and ass transer ro bodies o dierent geoetries under various physical conditions due to its vast range o applications involving high teperatures such as gas turbines issiles, satellites, nuclear power plant, aircrat and space vehicles etc. Chakha et al. [10] nuerically studied the eects o theral radiation on ixed convective low around a cone inscribed in a porous ediu illed with a nanoluid. heikholeslai and Davood [11] investigated the eect o theral radiation and agnetic ield on the unsteady low o a nanoluid low and heat transer. On the other hand nuerous studies have been carried out on the eect o Joule heating on luid low and heat transer at various conditions and ound that it plays notable eect on MHD low and heat transer. Anand et al. [1] analyzed the theral radiation and Joule heating eects on MHD low o a nanoluid in boundary layer. rinivasacharya and Vijay [13] described the ixed convective low o a nanoluid in a non-darcy porous ediu considering radiation eect. Hayat et al. [14] presented a odel or Joule heating and solar radiation in MHD convection low o nanoluid. DOI: /ijet/017/v9i4/ Vol 9 No 4 Aug-ep

2 IN (Print) : IN (Online) : The study o entropy generation, the aount o irreversibility associated with the real processes, has been growing interest as it is a powerul and useul optiization tool or a high range o theral applications. The entropy generation destroys the syste energy. Hence, the perorance o the syste can be iproved by decreasing the entropy generation. Bejan [15, 16] introduced the entropy generation iniization ethod and developed its applications in engineering sciences. ince then several researchers have been studying the entropy generation analysis or dierent types o geoetries with diverse luids. Oid et al. [17] analyzed the signiicance o radiation paraeter eect on entropy generation within nanoluids. Dehsara et al. [18] nuerically analyzed the entropy generation in nanoluid low in presence o variable agnetic ield, viscous dissipation and solar radiation. Govindaraju et al. [19] investigated the entropy generation in MHD low o a nanoluid. Rashidi et al. [0] presented the inluence o theral radiation on entropy generation in MHD blood low o a nanoluid. The survey o the literature reveals that the interaction o Joule heating and radiation eects with nanoparticles in a ixed convectional heat transer low in a vertical channel occupied by nanoluid has not been considered. Hence, the ain ai o this article is to investigate the ixed convective low o a nanoluid passing through a vertical channel with Joule heating and theral radiation eect. The hootopy analysis ethod is used to solve the nonlinear dierential equations. HAM [1] was irst introduced by Liao, which is one o the ost powerul technique to ind the solution o strongly nonlinear equations. The eect o radiation, Joule heating and agnetic paraeter on the velocity along the luid direction, teperature, nanoparticle concentration, Bejan nuber and entropy generation is investigated.. II. PROBLEM FORMULATION Consider a steady, lainar incopressible electrically conducting nanoluid low through a vertical channel o width d. The x-axis is taken to be in vertically upward direction o the low through the central line o the channel and y- axis in the orthogonal direction to the low as shown in igure 1. The plate y=-d is aintained at teperature T 1 and nanoparticle volue raction 1, whereas the plate y=d is aintained at T and respectively. Rate o low through the plates is assued to be constant v 0 and uniorly equal or both the plates. A unior agnetic ield B 0 is taken in y-direction. A unior pressure gradient in x-direction and buoyancy orces caused the ixed convective low. The induced agnetic ield is ignored as the agnetic Reynolds nuber is very low and all the luid properties are considered as constant apart ro the density in the buoyancy ter. The luid is considered to be a gray, absorbing/eitting radiation, but non-scattering ediu. The Rosseland approxiation [] is used to describe the radiative heat lux in the energy equation. u is the velocity along x- direction, T denotes the teperature and denotes the nanoparticle concentration. Figure 1: Geoetry o the proble Under these assuptions, the governing equations o the low by considering the nanoluid odel proposed by Buongiorno [3] are as ollows: v 0 (1) y u p u (1 ) ( 0 T 1) ( p ) ( 0 1) 0 v g T T g B u y x y () DOI: /ijet/017/v9i4/ Vol 9 No 4 Aug-ep

3 IN (Print) : IN (Online) : T T u 16T T T DT T 1 v D B B u y y cp y 3K cp y y y T y c p DT T v D B y T y y (4) where the density is ρ, the pressure is p, the electrical conductivity is σ, Brownian diusion coeicient is D B, the speciic heat capacity is Cp, the viscosity coeicient is μ, the coeicients o theral expansion is β T, the eective theral diusivity is α, the acceleration due to gravity is g, the therophoretic diusion coeicient is D T, tean-boltzan constant is σ 1 and coeicient o ean absorption is χ. The conditions on the boundary are: u 0, v v0, T T1, 1 on y d, u 0, v v0, T T, on y d, (5) Introducing the ollowing non-diensional variables y, u, u0, T T1, 1 p P, d u0 d T T1 1 (6) in equations (1) - (5), we get the nonlinear dierential equations as Gr R Nr M A 0 Re (7) 4 (1 Rd) RPr PrNb Pr Nt Br( ) J 0 3 (8) Nt RLe 0 Nb (9) where the kineatic viscosity coeicient is, Pr CP / k represents the Prandtl nuber and Le / DB represents the Lewis nuber, R v0 d / is the suction/injection paraeter, 3 Gr (1 ) g T ( T T1) d / is the Grasho nuber, Re u0 d / is the Reynold's nuber, M B0 d / is the agnetic paraeter, A ( d / u0) p/ x is a constant pressure gradient, Br u0 /( k ( T T1)) represents the Brinkan nuber, the Brownian otion paraeter is Nb D B ( 1)/, Nt DT( T T1)/( T ) is the theroporesis paraeter and buoyancy ratio is ( p )( 0 1) 3 u0 B0 d 4 Nr 1 T, Joule heating paraeter is J and Rd ( 0 T T T1)(1 ) ( T T1) k 3 K is the Radiation paraeter. The corresponding conditions (5) on boundary are 0, 0, 0 at 1, 1, 1, 0 at 1. (10) III. HOMOTOPY OLUTION To obtain the HAM solution (For ore details on hootopy analysis ethod see the works o Liao [1, 4-6] ), irst we guess the initial values as 0( ) 0, 1 1 0( ), and 0( ). (11) and the auxiliary linear operators asl i = /y, or i = 1,,3 such that L 1 (c 1 + c y) = 0, L (c 3 + c 4 y) = 0 and L 3 (c 5 + c 6 y) = 0, where c i,( i = 1,,...6) are constants. The zero th order deoration, which is given by (1 p) L1[ ( ; p) 0( )] ph1n1[ ( ; p)], (1 pl ) [ ( ; p) 0( )] phn [ ( ; p)], (1) (1 p) L [ ( ; p) ( )] ph N [ ( ; p)] (3) DOI: /ijet/017/v9i4/ Vol 9 No 4 Aug-ep

4 IN (Print) : IN (Online) : where Gr N1[ ( ; p), ( ; p), ( ; p)] R NrM A, Re 4 N[ ( ; p), ( ; p), ( ; p) ] (1 Rd) RPr PrNb Pr Nt Br ( ) J, 3 Nt N3[ ( ; p), ( ; p), ( ; p)] RLe. Nb (13) Here p [0, 1] is the ebedded paraeter and hi,( i1,, 3.) are auxiliary paraeters. Fro p = 0 to p = 1, we can have ( ;0) 0, ( ;0) 0, ( ;0) 0, (14) ( ;1) ( ), ( ;1) ( ), ( ;1) ( ). Thus, as p varying ro initial value 0 to inal value 1,, θ and changes ro 0, θ 0 and 0 to the inal solution (η), θ(η) and (η). Using Taylor's series one can write 1 ( ; p) ( ; p) 0 ( ) p, ( ),! p 1 p0 1 ( ; p) ( ; p) 0 ( ) p, ( ), 1! p 1 ( ; p) ( ; p) 0 ( ) p, ( ) 1! p and choose the values o the auxiliary paraeters or which the series (15) are convergent at p=1 i.e., p0 p ( ) ( ), ( ) ( ), ( ) ( ). (16) is convergent. The equivalent boundary conditions are ( 1, p) 0, (1, p) 1, ( 1, p) 0, (1, p) 1, ( 1, p) 0, (1, p) 0. (17) The th -order deoration is given by L1[ ( ) 1( )] h1r( ), L[ ( ) 1( )] hr( ), L3[ ( ) 1( )] h3r( ). where Gr R( ) R NrM (1 ) A, Re 4 R Rd RPr PrNb PrNt or an integer 1 1 ( ) 1 1 n n 1 nn 3 n0 n0 1 1 Br 1 n nj 1 n n, n0 n0 Nt R ( ) RLe. Nb 1 or 1 0 or 1 (19). (15) (18) DOI: /ijet/017/v9i4/ Vol 9 No 4 Aug-ep

5 IN (Print) : IN (Online) : IV. CONVERGENCE In HAM, it is essential to see that the series solution converges. Also, the rate o convergence o approxiation or the HAM solution ainly depend on the values o h. To ind the adissible space o the auxiliary paraeters, h-curves are drown or 0 th -level o approxiation and shown in igure. It is oserved ro this igure that the perissible range or h 1, h and h 3 is - < h 1 < -0, -0.7< h < -0 and - < h 3 < -0., respectively. In order to obtain the optial values o the auxiliary paraeters, the ollowing average residual errors [5] are coputed and ound that these errors are least at h 1 = , h = -4 and h 3 = k 1 k, 1 j, j k ik j0 k ik j0 E N ( i t), E N ( i t), (0) 1 k E, N3 j( i t). k ik j0 where t 1/ k and k = 5. Also, or dierent values o the series solutions are calculated and noticed that the series (16) converges in the total area o η. Further, the graphs o the ratio ( h) ( h) ( h),,. ( h) ( h) ( h) (1) in contrast to the nuber o ters in the hootopy series are calculated and observed that the series (16) converges to the exact solution. V. ENTROPY GENERATION The voluetric rate o local entropy generation o a nanoluid in vertical channel can be expressed as 3 K T 16 T T B0 u u RD RDT G. () T y 3 K 1 y T1 T1 y 1 y T1 y y where R is the universal gas constant and D is the species diusivity through the luid. The entropy generation nuber Ns is the ratio o the voluetric entropy generation rate to the characteristic entropy generation rate according to Bejan[16]. Thereore Ns is given by 4 1 Ns (1 Rd) B r J 3 (3) 3 1 The diensionless coeicients are and 3, called irreversibility distribution ratios which are related to diusive irreversibility, RD RD given by 3. (4) K. 1 1 K. 1 T where, 1 are the concentration and teperature ratios, respectively and 1 T1 characteristic entropy generation rate. The Eq.(3)can be expressed as K ( T) d T 1 is the Ns Nh Nv (5) The entropy generation due to heat transer irreversibility is denoted by the irst ter on the right hand side o the eq.(5) and the entropy generation due to viscous dissipation is represented by second ter o eq.(5). The ratio o the entropy generation (Nh) and the total entropy generation (Nh+Nv) is called Bejan nuber (Be). To understand the entropy generation echaniss the Bejan nuber Be is speciied. The Bejan nuber or this proble can be expressed as Nh Be (6) Nh Nv In general the liits o Bejan nuber is 0 to 1. Finally, the irreversibility due to viscous dissipation doinant represent by Be = 0, whereas Be = 1 represents the doination o heat transerirreversibility on Ns. It is clear that the heat transer irreversibility is equal to viscous dissipation at Be=0.5. DOI: /ijet/017/v9i4/ Vol 9 No 4 Aug-ep

6 IN (Print) : IN (Online) : VI. REULT AND DICUION The eects o radiation, Joule heating, agnetic and Brinkan nuber on non-diensional velocity, teperature and nanoparticle volue raction, Bejan nuber Be and entropy generation Ns are presented graphically in igures 3-7. To study these eect o the paraeters taking coputations asnr = 1, Nb =0.5, Gr = 10, Re =, R = 1, Pr = 1, A = 1, Le = 1, Tp = 0.1. Figure (3) displays the eect o the radiation paraeter Rd on velocity in low direction, teperature, nanoparticle concentration, entropy generation and Bejan nuber. Figure (3a) reveals that the velocity increased as enhanceent in the radiation Rd. This indicates that Rd have a retarding ipact on the ixed convective low. Fro (3b), it is noticed that θ(η) increased with an increent in the radiation Rd. A raise in the radiation Rd paraeter leads to release o heat energy in the low direction, thereore the luid teperature increased. Figure (3c) depicts that the nanoparticle concentration (η) decays with an enhanceent in the radiation Rd. Figure (3d) shows that entropy generation reduces with an enhanceent in the radiation paraeter Rd. It is noticed ro igure (3e) that Be (Bejan nuber) is increasing near the lower plate o the channel, eanwhile ar away ro the plate the trend is reversed due to ore contribution o the heat transer irreversibility on Ns and Be decreasing near the upper plate o the channel with enhance in Rd. The variation o velocity in low direction, teperature, nanoparticle concentration, Ns and Be with agnetic paraeter M is presented in igure (4). It is noticed ro igure (4a) that, the diensionless velocity decreases with an increase in the agnetic paraeter M. Figure (4b) reveals that θ(η) decreases with a rise in agnetic paraeter M. There is a raise in the nanoparticle concentration (η) with a rise in the agnetic paraeter M as depicted in igure (4c). Figure (4d) shows that entropy generation decreased with enhanceent in the agnetic paraeter M. It is clear ro igure (4e) that Be is decreasing near the lower plate o the channel, eanwhile ar away ro the plate the trend is reversed due to ore contribution o the heat transer irreversibility on Ns and the Be is decreasing near the upper plate o the channel with enhance in M. The inluence o the Brinkan nuber Br on (η), θ(η), (η), Ns and Be is depicted in Figure (5). The diensionless velocity (η) increased with the rise in Brinkan nuber Br as shown in igure (5a). Figure (5b) reveals that the θ(η) increased with a raise in the paraeter Br. Fro igure (5c) it is noticed that nanoparticle concentration (η) decay with an increent in the Brinkan nuber Br. Figure (5d) shows that the increase in the Brinkan nuber Br causes a increent in entropy generation. As increase in Br, the Bejan nuber is decreasing near the lower plate o the channel, eanwhile ar away ro the plate the trend is reversed due to ore contribution o heat transer irreversibility on Ns and Be increasing near the upper plate o the channel as represented in igure (5e). The inluence o the Joule heating paraeter on (η), θ(η), ( η), Ns and Be is shown in Figure (6). The velocity in low direction increased with the rise in Joule heating paraeter J as shown in igure (6a). Figure (6b) explains that the θ(η) raises with enhanceent in the Joule heating paraeter J. Fro igure (6c) it is noticed that nanoparticle concentration (η) decreased with an enhanceent in the Joule heating paraeter J. Figure (6d) shows that the increent in Joule heating paraeter J raises the entropy generation. As Joule heating paraeter J increases, the Bejan nuber is decreasing near the lower plate o the channel, eanwhile ar away ro the plate the trend is reversed due to ore contribution o heat transer irreversibility on Ns and then Be increasing near the upper plate o the channel as represented in igure (6e). Figure (7) represents the eect o theroporesis paraeter Nt on velocity in low direction (η), nanoparticle concentration (η), teperature θ(η), Bejan nuber Be and entropy generation Ns. The velocity (η) decreased with enhanceent in the theroporesis paraeter Nt as shown in igure (7a). Figure (7b) reveals that, the teperature θ(η) increased with enhanceent in theroporesis paraeter Nt. Increase in the theroporesis paraeter Nt leads to increase in the eective-conductivity, hence the nanoparticle concentration (η) is increases as depicted in igure (7c). It is noticed ro igure (7d) that as increase in the theroporesis paraeter Nt the entropy generation is also increased. It is seen ro igure (7e) that the Be is decreasing near the lower plate o the channel, eanwhile ar away ro the plate the trend is reversed due to ore contribution o heat transer irreversibility on Ns and then Be is increasing near the upper plate o the channel as Nt increases. VII. CONCLUION In this article the lainar entropy generation in ixed convective nanoluid low in a vertical channel has been investigated by including agnetic, Joule heating, Brinkan nuber and cheical reaction paraeter eects. The non-diensional non-linear equations are solved by the HAM procedure. The ain observations are suarized below: The diensionless velocity, teperature increased whereas the nanoparticle concentration and entropy generation decreased with raise in the theral-radiation Rd. An increase in the Brinkan nuber leads to increase the velocity, teperature, entropy generation and leads to decrease the nanoparticle concentration. DOI: /ijet/017/v9i4/ Vol 9 No 4 Aug-ep

7 IN (Print) : IN (Online) : As the Joule heating paraeter increases, the diensionless teperature, velocity and entropy generation increases but the nanoparticle concentration decreases. The axiu values o Bejan nuber are observed at upper plate o the channel due to ore heat transer irreversibility on Ns and iniu value near the lower plate o channel due to ore contribution o the luid riction irreversibility on Ns with the increase in Rd, M, Br, J and Nt. REFERENCE [1] Choi,.U..: Enhancing theral conductivity o luids with nanoparticles. In inger, D and Wang, H (Eds.), Developent and Applications o Non-Newtonian Flows, Aerican ociety o Mechanical Engineers, New York, , [] Hang, Xu. and Pop, I. : Fully developed ixed convection low in a vertical channel illed with nanoluids, Int. Coun. Heat Mass Transer, vol.39, , 01. [3] Hajipour, M. and Dehkordi, A.M. : Analysis o nanoluid heat transer in parallel plate vertical channels partially illed with porous ediu, Int. J. Theral ciences, vol.55, , 01. [4] Grosan, T. and Pop, I. : Fully developed ixed convection in a vertical channel illed by a nanoluid, J. Heat Transer, vol.134(8), , 01. [5] Hang, X., Tao, F. and Pop, I. : Analysis o ixed convection low o a nanoluid in a vertical channel with the Buongiorno atheatical odel, International Counications in Heat Mass Transer, vol.44, pp. 15-, 013. [6] Nield, D.A. and Kuznetsov, A.V. : Forced convection in a parallel -plate channel occupied by a nanoluid or a porous ediu saturated by a nanoluid, Int. J. Heat Mass Transer, vol.70, , 014. [7] Fakour, M., Vahabzadeh, A. and Ganji, D.D. : crutiny o ixed convection low o a nanoluid in a vertical channel, Case tudies in Theral Engineering vol.4, 15 3, 014. [8] Makhatar, N.A.M., Habibis,. and Ishak, H. : Flow reversal o ully developed ixed convection o nanoluids in a vertical channel illed with porous ediu, International Journal o Pure and Applied Matheatics, vol.103(1), 81-97, 015. [9] Das,., Jana, R.N. and Makinde, O.D. : Mixed convective agneto-hydrodynaic low in a vertical channel illed with nanoluids, Engineering cience and Technology, an International Journal, vol.18, 44-55, 015. [10] Chakha, A.J., Abbasbandy,., Rashad, A.M. and Vajravelu, K. : Radiation eects on ixed convection about a cone ebedded in a porous ediu illed with a nanoluid, Meccanica vol.48(), 75 85, 013. [11] heikholeslai, M. and Ganji, D.D. : Unsteady nanoluid low and heat transer in presence o agnetic ield considering theral radiation, J Braz. oc. Mech. ci. Eng., vol.37, , 015. [1] Anand, R. J., Vasuathi, G. and Mounica, J. : Joule Heating and Theral Radiation Eects on MHD Boundary Layer Flow o a Nanoluid over an Exponentially tretching heet in a Porous Mediu, World Journal o Mechanics, vol.5, , 015. [13] rinivasacharya, D. and Vijaykuar, P. : Mixed Convection over an Inclined Wavy urace in a Nanoluid aturated Non-Darcy Porous Mediu with Radiation Eect, International Journal o Cheical Engineering, 015. [14] Hayat T., Waqas M., hehzad,.a. and Alsaedi A. : A odel o solar radiation and Joule heating in agnetohydrodynaic(mhd) convective low o thixotropic nanoluid, Journal o Molecular Liquids, vol.15, , 016. [15] Bejan, A., ``econd-law analysis in heat transer and theral design'', \eph{adv. Heat Transer}, Vol. 15, pp.1-58, 198. [16] Bejan, A., ``Entropy Generation Miniization'', New York: CRC Press. Boca Raton [17] Oid, M., Ali, K., Cleent, K., Al-Nir, M.A., Ioan, P., Ahet, Z.. and ochai, W. : A review o entropy generation in nanoluid low, International Journal o Heat and Mass Transer, vol.65, , 013. [18] Dehsara, M. Neat, D.and Mohaad, R.H.N. : Nuerical analysis o entropy generation in nanoluid low over a transparent plate in porous ediu in presence o solar radiation, viscous dissipation and variable agnetic ield, Journal o Mechanical cience and Technology, vol.8(5), , 014. [19] Govindaraju M., Vishnuganesh N., Ganga B. and Hakee A.K. : Entropy generation analysis o agnetohydrodynaic low o a nanoluid over a stretching sheet, Journal o the Egyptian Matheatical ociety, vol.3, , 015. [0] Rashidi, M.M., Bhatti,M.M., Munawwar, A.A. and Mohaed, E..A. : Entropy Generation on MHD Blood Flow o Nanoluid Due to Peristaltic waves, Entropy, vol.18, 016. [1] Liao.J. : Beyond perturbation. Introduction to hootopy analysis ethod. Boca Raton: Chapan and Hall/CRC Press, 003. [] parrow E M and Cess, RD., 1978, Radiation Heat Transer, Washington, DC: Heisphere. [3] Buongiorno, J. : Convective transport in nanoluids, AME J Heat Transer, vol.18, 40-50, 006. [4] Liao.J. : On the hootopy analysis ethod or nonlinear probles. Appl Math Coput., vol.147(), , 004. [5] Liao.J. : An optial hootopy-analysis approach or strongly nonlinear dierential equations. Coun Nonlinear ci Nuer iul 15:003-16, 010. [6] Liao,.J. : Advances in the Hootopy Analysis Method. World cientiic Publishing Copany, ingapore, 013. DOI: /ijet/017/v9i4/ Vol 9 No 4 Aug-ep

8 IN (Print) : IN (Online) : (0) h 1 (a) (η) h (b) θ(η) 8 6 (0) h 3 (c) (η) Figure : The h-curves or (η), θ(η), (η) when Nr=1, Nt=1, Gr=10, R=1, Nb=0.5, Pr=1, A=1, Re=, Le=1, Rd=1.5, M=, Br=0.5, J=, Tp=0.1. DOI: /ijet/017/v9i4/ Vol 9 No 4 Aug-ep

9 IN (Print) : IN (Online) : Rd=0.1 Rd=0.5 Rd= Rd=1.5 Rd=0.1 Rd=0.5 Rd= Rd= (a) (η) (b) θ(η) Rd=0.1 Rd=0.5 Rd= Rd= Rd=0.1 Rd=0.5 Rd= Rd= Ns (c) (η) 0.9 Rd=0.1 Rd=0.5 Rd= Rd=1.5 (d) Ns Be (e) Be Figure 3: Eect o Radiation paraeter on velocity (η), teperature θ(η), nanoparticle concentration (η), Entropy generation Ns and Bejan nuber Be. DOI: /ijet/017/v9i4/ Vol 9 No 4 Aug-ep

10 IN (Print) : IN (Online) : M=0.1 M=0.5 M= M=.0 M=0.1 M=0.5 M= M= (a) (η) (b) θ(η) M=0.1 M=0.5 M= M= M=0.1 M=0.5 M= M=.0 30 Ns (c) (η) 0 (d) Ns 0.9 M=0.1 M=0.5 M= M=.0 Be (e) Be Figure 4: Eect o Magnetic paraeter on velocity (η), teperature θ(η), nanoparticle concentration (η), Entropy generation Ns and Bejan nuber Be. DOI: /ijet/017/v9i4/ Vol 9 No 4 Aug-ep 017 7

11 IN (Print) : IN (Online) : Br=0.3 Br=0.5 Br=0.7 Br= Br=0.3 Br=0.5 Br=0.7 Br= (a) (η) (b) θ(η) Br=0.3 Br=0.5 Br=0.7 Br= Br=0.3 Br=0.5 Br=0.7 Br= 30 Ns (c) (η) 0.9 Br=0.3 Br=0.5 Br=0.7 Br= (d) Ns Be (e) Be Figure 5: Eect o Brinkan nuber on velocity (η), teperature θ(η), nanoparticle concentration (η), Entropy generation Ns and Bejan nuber Be. DOI: /ijet/017/v9i4/ Vol 9 No 4 Aug-ep

12 IN (Print) : IN (Online) : J=1 J= J=3 J=4 J=1 J= J=3 J= (a) (η) (b) θ(η) J=1 J= J=3 J= J=1 J= J=3 J=4 1 Ns (c) (η) 0.9 J=1 J= J=3 J=4 (d) Ns Be (e) Be Figure 6: Eect o Joule heating paraeter on velocity (η), teperature θ(η), nanoparticle concentration (η), Entropy generation Ns and Bejan nuber Be. DOI: /ijet/017/v9i4/ Vol 9 No 4 Aug-ep

13 IN (Print) : IN (Online) : Nt=0.1 Nt=0.3 Nt=0.5 Nt=0.7 Nt=0.1 Nt=0.3 Nt=0.5 Nt= Nt=0.1 Nt=0.3 Nt=0.5 Nt=0.7 (a) (η) Nt=0.1 Nt=0.3 Nt=0.5 Nt=0.7 (b) θ(η) Ns (c) (η) 0.9 Nt=0.1 Nt=0.3 Nt=0.5 Nt=0.7 (d) Ns Be (e) Be Figure 7: Eect o Theroporesis paraeter on velocity (η), teperature θ(η), nanoparticle concentration (η), Entropy generation Ns and Bejan nuber Be. DOI: /ijet/017/v9i4/ Vol 9 No 4 Aug-ep