Dual Solutions for MHD Jeffery Hamel Nano-Fluid Flow in Non-parallel Walls Using Predictor Homotopy Analysis Method

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1 Journal o Applied Fluid Mechanic, Vol. 8, No. 4, pp. 9-99, 5. Available online at ISSN , EISSN DOI:.8869/acadpub.jam Dual Solution or MHD Jeery Hamel Nano-Fluid Flow in Non-parallel Wall Uing Predictor Homotopy Analyi Method N. Freidoonimehr and M. M. Rahidi, 3 Young Reearcher & Elite Club, Hamedan Branch, Ilamic Azad Univerity, Hamedan, Iran Mechanical Engineering Department, Engineering Faculty o Bu-Ali Sina Univerity, Hamedan, Iran 3 Mechanical Engineering Department, Univerity o Michigan-Shanghai Jiao Tong Univerity Joint Intitute, Shanghai Jiao Tong Univerity, Shanghai, People Republic o China Correponding Author nreidoonimehr@yahoo.com (Received July, 4; accepted Augut 6, 4) ABSTRACT The main purpoe o thi tudy i to preent dual olution or the problem o magneto-hydrodynamic Jeery Hamel nano-luid low in non-parallel wall. To do o, we employ a new analytical technique, Predictor Homotopy Analyi Method (PHAM). Thi eective method i capable to calculate all branche o the multiple olution imultaneouly. Moreover, comparion o the PHAM reult with numerical reult obtained by the hooting method coupled with a Runge-Kutta integration method illutrate the high accuracy or thi technique. For the current problem, it i ound that the multiple (dual) olution eit or ome value o governing parameter epecially or the convergent channel cae (α = -). The luid in the non-parallel wall, divergent and convergent channel, i the drinking water containing dierent nanoparticle; Copper oide (CuO), Copper (Cu) and Silver (Ag). The eect o nanoparticle volume raction parameter (φ), Reynold number (Re), magnetic parameter (Mn), and angle o the channel (α) a well a dierent type o nanoparticle on the low characteritic are dicued. Keyword: MHD; Nano-luid; Jeery Hamel low; Non-parallel wall; Predictor homotopy analyi method; Multiple olution. NOMENCLATURE B c i F ma Mn P r Re u α σ electromagnetic induction arbitrary contant el-imilar velocity dimenional contant magnetic parameter luid preure radial direction in cylindrical polar coordinate Reynold number velocity component in the radial direction dimenionle degree emi-angle between the two inclined wall electrical conductivity δ precribed parameter auiliary unction ћ auiliary nonzero parameter L auiliary linear operator N nonlinear operator φ nanoparticle volume raction θ tangential direction in cylindrical polar coordinate ρ denity ν kinematic vicoity μ vicoity Subcript luid n nano-luid olid. INTRODUCTION Working luid have great demand placed upon them in term o increaing or decreaing energy releae to ytem, and their inluence depend on thermal conductivity, heat capacity and other phyical propertie in modern thermal and manuacturing procee. A low thermal conductivity i one o the mot remarkable parameter that can limit the heat traner

2 N. Freidoonimehr et al. / JAFM, Vol. 8, No. 4, pp. 9-99, 5. perormance. Supending the ultraine olid metallic particle in technological luid caue an increae in the thermal conductivity. Thi i one o the mot modern and appropriate method or increaing the coeicient o heat traner. It i epected that the ultraine olid particle i able to increae the thermal conductivity and heat traner perormance, ince the thermal conductivity o olid metal i higher than that o bae luid. Choi and Eatman (995) were probably the irt to employ a miture o nanoparticle and bae luid that uch luid were deignated a Nano-luid. Eperimental tudie have diplayed that with -5% volume o olid metallic or metallic oide particle, the eective thermal conductivity o the reulting miture can be increaed by % compared to that o the bae luid, a tated by Eatman et al. (999). Xuan and Li (3) tated that the low and heat traner perormance o nano-luid under the turbulent low in tube. Their eperimental reult howed that the convective heat traner coeicient and Nuelt number o nano-luid are enhanced by increaing the Reynold number and volume raction o nanoparticle. A wide range o review paper on nano-luid and their dierent application can be ound in Bachok et al. (), Mahian et al. (), Rahidi et al. (3a), Rahidi et al. (4a), Sheikholelami and Ganji (4a). The tudy o low in converging/diverging channel i very important due to it vat engineering and indutrial application, uch a enhancing heat traner o heat echanger or milk lowing, cold drawing operation in polymer indutry, etruion o molten polymer through converging die, and many other, a tated by Kato and Shibanuma (98), Hooper et al. (98), Sadeghy et al. (7). In recent year, thi problem i etenively tudied by everal reearcher. Moghimi et al. () tudied the MHD Jeery Hamel low in nonparallel wall analytically uing Homotopy analyi method (HAM). Hatami et al. (4), Hatami and Ganji (4) invetigated the MHD Jeery Hamel nano-luid low in non-parallel wall uing Dierential Tranormation Method (DTM), Leat quare method (LSM) and Weighted Reidual Method (WRM). In another tudy, Moradi et al. (3) dicued the nonlinear Jeery Hamel low problem in a nano-luid. One o the mot important method or highlynonlinear problem i the homotopy analyi method (HAM) which wa irtly employed by Liao (4a), (4b) or the nonlinear problem, which i o undamental interet or practical uing in cience and engineering. Thi powerul method i being employed vatly by many reearcher in dierent practical apect o engineering and nonlinear problem. Rahidi et al. (4b) employed HAM to invetigate the ree convective heat and ma traner in a teady D magnetohydrodynamic luid low over a tretching vertical urace in porou medium. In another tudy, Rahidi et al. (4c) invetigated the MHD mied convective heat traner or an incompreible, laminar, and electrically conducting vicoelatic luid low pat a permeable wedge with thermal radiation via HAM. Hayat et al. (9) depicted the eect o MHD low o an upper-convected (UCM) luid over a tretching urace via HAM. Abba et al. () preented an analytical olution or the mied convective low in a Mawell luid over a tretching urace. Rahidi et al. (3b) tudied the irt and econd law analyze o an electrically conducting luid pat a rotating dik in the preence o a uniorm vertical magnetic ield analytically and then applied artiicial neural network and particle warm optimization algorithm to minimize the entropy generation. Recently, a new method related to the homotopy analyi method ha been preented by Abbabandy and Shivanian (), Shivanian and Abbabandy (4), Vooughi et al. (), Abbabandy and Shivanian (4) namely Predictor Homotopy Analyi Method (PHAM). The main idea o PHAM i to rebuild the homotopy analyi method by adding rule o multiplicity o olution and ocalled precribed parameter. Thi analytical technique can receive much more attention becaue o it accuracy and the ability to gain the olution or problem with multiplicity olution (dual, triple, etc. olution) (Bing-li and Yin-ping (3)). In thi article, we check and preent the dual olution or the MHD Jeery Hamel nano-luid low in non-parallel wall via a new analytical technique, PHAM. Although the problem o Jeery Hamel nano-luid in non-parallel wall are already conidered in everal article by Hatami et al. (4), Moradi et al. (3) and Sheikholelami et al. () uing dierent numerical/analytical technique a mentioned beore, dual olution or the problem o MHD Jeery Hamel nano-luid low in non-parallel wall are irtly preented in thi tudy. Moreover, the eect o the nanoparticle volume raction parameter, Reynold number, magnetic parameter, and angle o the channel a well a dierent type o nanoparticle; Cuo, Cu and Ag, on the low velocity are dicued. The content o thi article i divided up a ollow: in ection we derive the mathematical model that will be invetigated in thi tudy. In ection 3 we implement the PHAM to olve the reulting ytem o nonlinear dierential equation. Section 4 deal with the PHAM calculation o the multiplicity o olution. Reult are dicued in ection 5. Concluion ection i alo preented in ection 6.. PROBLEM FORMULATION Let u conider the ytem o cylindrical polar coordinate (r, θ, z) which teady D low o an incompreible conducting vicou luid rom a ource or ink at channel wall lie in plane, and interect in z-ai. It i aumed that there are no change with repect to z and the motion i purely in radial direction and jut depend on r and θ, which mean that there i no change in the low parameter along the z direction or v = (u(r, θ), ), and moreover there i no magnetic ield along 9

3 N. Freidoonimehr et al. / JAFM, Vol. 8, No. 4, pp. 9-99, 5. a b c) Fig.. Geometry o the MHD Jeery Hamel low with nano-luid in divergent channel; a) 3-D view, b) -D view and c) Schematic etup o problem. z-ai. We conider that the converging/diverging channel ha macro-cale and thereore no-lip condition hold at the wall. In addition, we aume that the bae luid and nanoparticle have the ame velocity in an incompreible laminar low. The coordinate ytem, D and 3D view o the problem geometry and chematic etup o problem are hown in igure. The equation o continuity and motion under the above conideration can be written a (Hatami and Ganji (4), Moradi et al. (3), Sheikholelami et al. (), Sheikholelami and Ganji (4b)): n ru( r, ) ru( r, ), () r r ur (, ) ur (, ) P r n r ur (, ) ur (, ) r r r n ur (, ) ur (, ) r r P n u( r, ), n r r n B n r ur (, ), Subjected to the bellow boundary condition: At the channel centerline: ur (, ) ur (, ) Uma () (3) At the plate, making the body o the channel: ur (, ) where ρ n i the luid denity, P i the luid preure, ν n i the coeicient o kinematic vicoity, σ n i the conductivity o the luid, and B i the electromagnetic induction. In addition, μ n and ρ n are the eective dynamic vicoity and eective denity o the nano-luid, where μ n ha been propoed by Brinkman (95). The mentioned phyical nano-luid parameter are introduced a (Hatami et al. (4)): ( ), n n n,,.5 n ( ) n 3 n, (4) where φ i the nanoparticle volume raction, the ubcript n, and are the thermo-phyical propertie o the nano-luid, bae luid and the olid nanoparticle, repectively. The phyical propertie o the bae luid (drinking water) and dierent nanoparticle are given in Table (Hatami et al. (4), Oztop and Abu-Nada (8)). Conidering u θ = or purely radial low, the velocity parameter can be deined a: ( ) ru( r) (5) Introducing the = θ / α a the dimenionle degree, the dimenionle orm o the velocity parameter can be obtained by dividing that to it maimum value ( ma ), a dimenional contant which can be related to the low rate per unit length, a: F( ) ( ) (6) ma Eliminating P between Eqn. () and (3), we obtain 93

4 N. Freidoonimehr et al. / JAFM, Vol. 8, No. 4, pp. 9-99, 5. the ollowing problem: Table Thermo-phyical propertie o the bae luid and dierent nanoparticle Fluid phae Phyical propertie CuO Cu Ag (Drinking water) ( kg m ) ( Sm ) Fig.. Precribed parameter δ via convergence controller parameter ћ in according to eq. () with M = 5 when α = -, Re =, φ =.5 and Mn =. F ( ).5 Re ( ) F ( ) F ( ).5 4 ( ) F( ), 3 Mn (7) where Mn B i the magnetic parameter and the Reynold number i: ma Umar Re divergent channel, ma (8) convergent channel, The reduced orm o boundary condition become: F(), F(), F(), (9) ma 3. PREDICTOR HOMOTOPY ANALYSIS METHOD (PHAM) In order to gain PHAM olution, the boundary condition in eq. (9) become a ollow: F(), F(), F(), () with the additional orcing condition which play an important role in PHAM: F (), () Now, we ue PHAM or the Eqn. (7) and () with precribed parameter δ. According to the initial condition: () F (, ) ( / ), In thi paper, the auiliary unction ha been choen to be ( ) and the linear operator i a ollow: 3 F(, ; q) F(, ; q), (3) F 3 with the propertie: F ( c c c 3 ), (4) Thereore, ater three ubequent integration, the M th order deormation equation o PHAM yield or M. Fm(, ) m Fm (, ) Rm Fm,, d d d (5) c c c, where 3 m j F (, ) Re ( ) m j m j.5 4 ( ) F F (, ) F (, ) m (, ),.5 3 Mn (6) and integration contant c, c and c 3 are obtained by the condition Fm(, ) F m(, ) F m(, ), (7) Uing eq.(5), we obtain the unction F (, ) m or m =,, 3, ucceully. Finally, we can obtain M th order approimate olution: M FM (,, ) Fm(, ), (8) m And the additional orcing condition () take the orm o: FM (,, ), (9) Now, or eample, we conider the convergent channel cae, α = -, in igure. Due to the eq 94

5 N. Freidoonimehr et al. / JAFM, Vol. 8, No. 4, pp. 9-99, 5. Fig. 3. Precribed parameter δ via convergence controller parameter ћ in according to eq. () with M = 5 when α = -, Re =, φ =.5 and Mn = (Magniication o Fig. ).. = F() - - =-6.66 Reidual error Fig. 4. Dual velocity proile when Re =, φ =.5 and Mn =. (9) in thi igure, δ (precribed parameter) a a unction o convergence parameter ћ, ha been igured. In order to obtain the value o δ and ћ with high accuracy, the magniied orm o precribed parameter a a unction o convergence parameter are plotted in igure 3. Two δ-land can be determined in thi igure, namely δ = -.57 in the range [-.6, -.4] o ћ and δ = -6.6 in the range [-.6, -.3] o ћ. The approimate PHAM olution o velocity proile correpond to δ = -.57 and δ = -6.6, given by (9), are diplayed in igure 4. In order to urvey the accuracy o thee dual approimate olution, the reidual error o the 5 th order o PHAM olution are illutrated in igure 5. In addition, we compare ome o our reult with the numerical reult obtained by the hooting method coupled with a Runge-Kutta integration method in table to highlight the validity o the applied method or the convergent and divergent channel cae. A very ecellent agreement can be oberved between them. It hould be mentioned that we preent two branche o olution via PHAM in table, the irt line reer to upper branch and the econd line reer to lower branch olution. 4. PHAM CALCULATION OF THE TWO BRANCHES OF SOLUTION In thi problem, the multiplicity o olution (dual Fig. 5. The reidual error or M = 5 when Re =, φ =.5 and Mn =. olution) o the eqn. (7) and (9) or equivalently (), or the convergent channel cae, α = -, in the parameter plane (ћ, δ), have been identiied, we may turn to calculate them eplicitly to any deired order M o PHAM-approimation according to eq. (8). We mention that both the upper and lower branche o olution are calculated at the ame time only by eq. (9) with dierent δ and ћ which are peciied rom Fig. 3, δ = -.57 and δ = -6.6, repectively or the irt and econd branche o olution. A one o the mot important advantage o PHAM, there i no need to employ more than one initial approimation gue, one auiliary linear operator, and one auiliary unction that i in a harp contrat to all approimation method which are applied to converge to one olution. 5. RESULTS AND DISCUSSION The nonlinear ordinary dierential equation (7) ubject to the boundary condition (9) i olved analytically via a deinitely new analytical technique, PHAM, or ome value o the nanoparticle volume raction parameter (φ), Reynold number (Re), magnetic parameter (Mn), and angle o the channel (α). We conidered three type o nanoparticle, namely, Copper oide (CuO), Copper (Cu) and Silver (Ag) with water a the bae luid. We remark that the copper 95

6 N. Freidoonimehr et al. / JAFM, Vol. 8, No. 4, pp. 9-99, 5. nanoparticle i ued in all o the cae in thi ection Table Comparion between the F() reult o PHAM method and the numerical method when Re = Mn = and φ =.5 α = - α = + PHAM Numerical PHAM Numerical F() F ().4 Divergent Channel.4 Convergent Ch annel. = = = = 3. = =- =- = a b Fig. 6. Eect o the angle between two plate on the velocity proile when Re = Mn = and φ =.5. Re = 5,, 5, - F() U pper branch a b Fig. 7. Eect o the Reynold number on the velocity proile when φ =.5 and Mn =. ecept or thoe igure which ocu on the inluence o the type o applied nanoparticle on the velocity component proile. For the preent invetigation, we conidered the value o the volume raction parameter φ vary rom (regular Newtonian luid) to.. Eect o the angle between two plate on the velocity proile i diplayed in igure 6. Note that the rigid wall are conidered divergent channel when α > and convergent when α <. A α increae, the divergent channel cae, the eect o wall on the luid low decreae, when we move away rom them which lead to an increae o velocity; however there i a revere behavior in velocity proile or the convergent channel cae. 96

7 N. Freidoonimehr et al. / JAFM, Vol. 8, No. 4, pp. 9-99, 5. Fig. 7 preent the eect o Reynold number or repectively. In the divergent channel cae, the the divergent and convergent channel with lope, a) b) Divergent Channel.8 - =,.5,.5,.75,.6 - F () F () =. =.5 =.5 =.75 = Fig. 8. Eect o the nanoparticle volume raction parameter on the velocity proile when Re = Mn =. a) b) Divergent Channel Mn =, 5,,, 3 F ().4 F () -5. Mn = Mn = 5 Mn = Mn = Mn = Fig. 9. Eect o the magnetic parameter on the velocity proile when Re = and φ = a) b) Divergent Channel.8 - CuO, Cu, Ag F ().6.4 F () CuO Cu Ag Fig.. Eect o dierent nanoparticle type on the velocity proile when Re = Mn = and φ =.5. reult how that increaing in Reynold number make a decreae in velocity proile. Furthermore, the low move reverely and a region o back low near the wall i oberved or the higher Reynold number (Re ). In addition, or convergent channel, reult are invered and by increaing the Reynold number, velocity proile are enhanced. A the reult illutrate, the dual olution eit in the convergent channel cae. Fig. 8 depict the eect o nanoparticle volume raction parameter on the velocity proile or a divergent and convergent channel with lope. It can be een that increaing nanoparticle volume raction parameter in divergent channel lead to decreae velocity proile. The reult are invered or the convergent channel cae. The eect o magnetic parameter or a divergent and convergent channel i illutrated in igure 9. The velocity proile how that the rate o tranport i coniderably reduced with increae o M. Thi clearly diplay that the tranvere magnetic ield oppoe the tranport phenomena. Becaue o thi act that the variation o M lead to the variation o the Lorentz orce due to magnetic ield and the Lorentz orce produce more reitance to tranport phenomena. Increaing in the magnetic parameter make an increae in velocity proile. In addition, the low reveral diappear by the increaing o M or the divergent channel cae. 97

8 N. Freidoonimehr et al. / JAFM, Vol. 8, No. 4, pp. 9-99, 5. Finally, the eect o dierent nanoparticle type on the velocity proile i diplayed in igure. It i obviou that electing copper oide (CuO) a a nanoparticle lead to maimum value in the velocity proile, but thi treatment o nano-luid tructure i completely vice vera or convergent channel. 6. CONCLUSION In thi tudy, Predictor Homotopy Analyi Method (PHAM) a a new analytical technique wa applied to olve the problem o MHD Jeery Hamel nanoluid low in non-parallel wall or the divergent and convergent channel. Thi applied method i very powerul epecially or thoe boundary value problem which admit multiple olution and alo i capable to calculate all branche o the olution imultaneouly. For the current problem, it wa ound that the dual olution eited or the convergent channel cae (α = -). The reult o current tudy were compared with the numerical olution obtained uing the hooting method, coupled with a Runge-Kutta cheme. We ound that the analytical olution matche the numerical olution quite well. In continued, the eect o the nanoparticle volume raction parameter, Reynold number, magnetic parameter, and angle o the channel a well a dierent type o nanoparticle on the low were dicued in detail. REFERENCES Abba, Z., Y. Wang, T. Hayat and M. Oberlack (). Mied convection in the tagnationpoint low o a Mawell luid toward a vertical tretching urace. Nonlinear Analyi: Real World Application (4), Abbabandy, S. and E. Shivanian (4). Chapter : Predictor Homotopy Analyi Method (PHAM). Advance in the Homotopy Analyi Method Abbabandy, S. and E. Shivanian (). Predictor homotopy analyi method and it application to ome nonlinear problem. Communication in Nonlinear Science and Numerical Simulation 6, Bachok, N., A. Ihak and I. Pop (). Boundarylayer low o nanoluid over a moving urace in a lowing luid. International Journal o Thermal Science 49(9), Brinkman, H. C. (95). The vicoity o concentrated upenion and olution. Journal o Chemical Phyic (4), 57. Choi, S. U. S. and J. A. Eatman (995). Enhancing thermal conductivity o luid with nanoparticle. Material Science 3, Eatman, J. A., U. S. Choi, S. Li, G. Soyez, L. J. Thompon and R. J. DiMeli (999). Novel Thermal Propertie o Nanotructured Material. Material Science Forum 3-34, Hatami, M. and D. D. Ganji (4). MHD nanoluid low analyi in divergent and convergent channel uing WRM and numerical method. International Journal o Numerical Method or Heat & Fluid Flow 4(5), 9-3. Hatami, M., M. Sheikholelami, M. Hoeini and D. D. Ganji (4). Analytical invetigation o MHD nanoluid low in non-parallel wall. Journal o Molecular Liquid 94(), Hayat, T., Z. Abba and M. Sajid (9). MHD tagnation-point low o an upper-convected Mawell luid over a tretching urace. Chao, Soliton & Fractal 39(), Hooper, A., B. R. Duy and H. K. Moatt (98). Flow o luid o non-uniorm vicoity in converging and diverging channel. Journal o Fluid Mechanic 7, JIANG Bing-li and L. Yin-ping (3). Predictor homotopy analyi method and it application to two nonlinear ytem. Journal o Eat China Normal Univerity (3), Kato, H. and H. Shibanuma (98). Diverging Converging Flow o Dilute Polymer Solution : t Report, Preure Ditribution and Velocity Proile. Japan Society o Mechanical Engineer 3(8), 47. Liao, S. J. (4a). Beyond perturbation: introduction to the homotopy analyi method. Chapman & Hall/CRC Liao, S. J. (4b). On the homotopy analyi method or nonlinear problem. Applied Mathematic and Computation 47(), Mahian, O., S. Mahmud and S. Z. Heri (). Analyi o entropy generation ween corotating cylinder uing nanoluid. Energy 44(), Moghimi, S. M., G. Domairry, S. Soleimani, E. Ghaemi and H. Bararnia (). Application o homotopy analyi method to olve MHD Jeery Hamel low in non-parallel wall. Advance in Engineering Sotware 4(3), 8-3. Moradi, A., A. Alaedi and T. Hayat (3). Invetigation o Nanoparticle Eect on the Jeery Hamel Flow. Arabian Journal or Science and Engineering 38(), Oztop, H. and E. Abu-Nada (8). Numerical tudy o natural convection in partially heated rectangular encloure illed with nanoluid. 98

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