Mixed convection boundary layer flow past a horizontal circular cylinder in viscoelastic nanofluid with constant wall temperature

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1 RESEARH ARTILE Mixed convection boundary layer lo at a horizontal circular cylinder in vicoelatic nanoluid ith contant all temerature Rahimah Mahat ab Noraihan Aiqah Rai b Abdul Rahman Mohd Kaim c Sharidan Shaie b* a Univeriti Kuala Lumur Malayian Intitute Technology o Indutrial Technology Periaran Sinaran Ilmu Johor Bahru Johor Malayia b Deartment o Mathematic Faculty o Science Univeriti Tenologi Malayia 80 UTM Johor Bahru Johor Malayia c Univeriti Malayia Pahang Lebuhraya Tun Raza 600 Gambang Kuantan Pahang Malayia * orreonding haridan@utm.my Article hitory Received 4 October 07 Acceted 8 November 07 Abtract The teady o to-dimenional convection boundary layer lo o vicoelatic nanoluid over a circular cylinder ith contant all temerature i invetigated in thi aer. arboxymethyl celluloe olution-ater (M-ater) i choen a the bae luid and coer a a nanoarticle ith the Prandtl number Pr = 6.. The governing boundary layer artial dierential equation are tranormed into dimenionle orm. Then the obtained equation are olved numerically by uing the Keller-Box method. Thi aer ocu on the eect o elected arameter on the lo and heat traner characteritic and be reented in grah. The reult ho that the velocity roile are increaed hile the temerature roile are decreaed by increaing the value o nanoarticle volume raction and vicoelatic arameter reectively. Alo the value o reduced in riction are increaed by increaing mixed convection arameter but the value o heat traner coeicient roduce an ooite behavior ith an increaing in mixed convection arameter. Keyord: Vicoelatic nanoluid mixed convection horizontal circular cylinder contant all temerature 07 Penerbit UTM Pre. All right reerved INTRODUTION The concet o nanoluid a irt manieted by erie o reearch at Argonne National Laboratory and hoi [] a the irt to call the luid ith article o nanometer dimenion uended in them a Nano-luid hich ha gained oularity. Nanoarticle ued in nanoluid can be claiied by material. The nanoarticle are coniting o nano ized metal oxide and carbon nanotube. Recently tudy o nanoluid ha become mot oular among reearcher due to it variou alication in many indutrie engineering and medical cience a ell. Baically thi ind o luid have extremely high thermal conductivitie comared to the conventional liquid then nanoluid have been rooed a a route or uraing the erormance o heat traner liquid. Many reearcher had involved in the tudy o boundary layer lo roblem in nanoluid uch a Nield and Kuznetov [] Khan and Po [] Ahmad and Po[4] hamha et al. [5] Qaim et al. [6] Zaib et al. [7] and Noghrehabadi et al. [8]. Recently Sabir et al. [9] invetigated mixed convection lo o vicoelatic nanoluid by a cylinder ith variable thermal conductivity and heat ource/in. In reality mot luid are non-netonian. Non-Netonian luid mean that their vicoity i deendent on hear rate (thicening) or the deormation hitory. Noaday the non-netonian nanoluid have received much coniderable interet and concern by the reearcher due to the otential o nanoluid alication in many tye o indutry. Rai et al. [0] Mabood et al. [] Sarar et al. [] and Bouchoucha and Beaih [] have been conidered ome intereting tudie involving the non-netonian nanoluid ith variou condition and geometrie have been conidered. Motivated by the above-mentioned tudie thi aer conider the teady o to-dimenional mixed convection boundary layer lo at a horizontal circular cylinder in vicoelatic nanoluid ith contant all temerature.the Tiari and Da model [4] a ued in thi tudy. To the bet o our noledge thi roblem ha never been conidered beore and the reult reorted here are ne. The iluence o relevant arameter on the dimenionle luid velocity the temerature the nanoarticle volume raction the rate o heat traner and the rate o in riction coeicient are invetigated and hon grahically and dicued. MATHEMATIAL FORMULATION Fig. Phyical model and coordinate ytem. In thi aer the teady mixed convection boundary layer lo at an iothermal horizontal circular cylinder o radiu a laced in a vicoelatic nanoluid i tudied. Fig. illutrate the geometry o the roblem and the correonding coordinate ytem. It i aumed that 0

2 the contant temerature o the urace o the cylinder i o the ambient luid i T here cylinder (aiting lo) and T T T T T and that correond to a heated correond to a cooled cylinder (ooing lo) reectively. It i alo aumed that the vicou diiation i neglected. Further olloing Merin [5] e aume that the velocity o the reetream i U. Under thee aumtion along ith the Bouineq aroximation the boundary layer equation can be ritten a ollo Merin [5] Nazar [6]; u v 0 x y () Table Thermohyical roertie o nanoarticle and bae luid. Phyical Proertie M-ater u g m J g K Wm K K.67 By introducing the olloing non-dimenional variable; x y x y u u ue u u v ue u u u u x 0 u v g T T in x y y y xy a () x x a v y y a Re u u U Re / T T T T v U ue x ue x U / (6) here Re Ua v i the Reynold number. By ubtitution equation (6) into equation ()-() the dimenionle equation become u v x y y T T T () u v 0 x y (7) ubjected to the boundary condition: u 0 v 0 T T at y 0 x 0 u u ue x 0 T T a y x 0 (4) y here x and y are the arteian coordinate along the urace o the cylinder. The value i tarting rom the loer tagnation oint o the cylinder. While y i the coordinate meaured normal to the urace o the cylinder u and v are the velocity comonent ue x i the velocity outide the boundary layer T i the temerature o elected luid 0 0 i the contant o the vicoelatic material ith 0 0 and μ are the denity and dynamic vicoity o nanoluid β i the thermal exanion o nanoluid and ρ contant are deined by i the eective thermal conductivity o the nanoluid i the heat caacitance o nanoluid. Thee nanoluid.5 here i the nanoarticle volume raction o the nanoluid. The thermohyical roertie o nanoarticle and bae luid i given in Table [7].. (5) u u ue u u v ue.5 x y x y u u u u K u v in x x y y y xy (8) u v (9) x y Pr y ith the boundary condition u 0 v 0 at y 0 x 0 u u ue x 0 y 0 a y x 0 (0) here Pr i the Prandtl number K non a dimenionle vicoelatic arameter and i non a contant mixed convection arameter that are deined by U K a 0 here Gr g T T a / v Gr Re () i alo non a a Graho number. 0 i correlate to an auxiliary lo hile 0 i correlate to the ooing lo. But 0 i correlate to the orced convection lo reectively. In thi roblem i K 0 it i reerring to the cae o vicou (Netonian) luid. Further e introduce no the tream unction deined a uual a; xf x y xy here i the tream unction that deine a () u v () y x

3 Uing () and () into equation (8) and (9) to obtain F F F F F F x x F y y xy x y y in co x x F.5 x y in x x 4 F F F F F F F F F F F F F x x y x 4 y y y x y y x y y 4 y y y K 4 4 (4) F F x x F y x x y y Pr y (5) hich are ubject to the olloing boundary condition; F F 0 0 at y 0 x 0 y F in x y x F 0 y 0 a y x 0. (6) The hyical quantitie interet o thi roblem i the all temerature ditribution x and the in riction coeicient that are deined a Re U aq Re T T (7) here i the hear tre or in riction hile q i the heat lux rom the urace o cylinder [] that are given by u u u u u y xy y x y 0 u v y0 y0 q T y y0. (8) iv K 0 0 Pr and the boundary condition (9) a (0) () 0 0 () In thi cae obtained x () The reent tudy in equation (0)-() reduce to that o Nazar et al. [6] and Anar et al. [8] or a vicoelatic luid hen 0 (regular luid). RESULTS AND DISUSSION Both equation (4-5) ith the boundary condition (6) and the ordinary dierential equation (0-) ith the boundary condition () that ere olved numerically uing the Keller-box method. The eect o dierent value o the reviou arameter on the velocity roile temerature roile reduced in riction and heat traner coeicient are revieed in articularly. Figure and ho the comarion o the reent reult or the in riction coeicient and the heat traner rom the cylinder ith Nazar et al. [6] or a Netonian luid (K=0) ith no nanoluid eect and Pr=. It ho that the numerical olution obtained by the reent author are in good agreement ith Nazar et al. [6]. Thu e coident that the reent reult are very accurate. Figure 4 and 5 ho the local in riction and local heat traner or K= Pr=6. and variou value o ith the reence o nanoarticle volume raction hen From the grah it ho that by the reence o nanoarticle volume raction the value o local in riction i increaed by increaing mixed convection arameter. The reult ho the ame behavior or heat traner coeicient here the value o i increaed by increaing the value o. and it i non that i or dynamic vicoity hile i or thermal conductivity. By ubtitution o (6) (7) and (8) obtained F x y y0 x x 0. (9) When x 0 in the cae o loer tagnation oint introduced the imilarity variable x and a er belo. Thu it can be hon that equation (7)-(9) diminih to the ordinary dierential equation a:.5 Fig. omarion o the local in riction luid) Pr= 0 and variou o or K=0 (Netonian

4 Fig. omarion o the local heat traner luid) Pr= 0 and variou o or K=0 (Netonian Fig.6 Local in riction or K= Pr=6. 0. and variou o Fig.7 Local heat traner or K= Pr=6. 0. and variou o Fig.4 Local in riction or K= Pr= and variou o Fig.8 Velocity roile or variety value o nanoarticle volume raction or K= Pr=6..5 (ooing lo) Fig.5 Local heat traner or K= Pr= and variou o Further Figure 6 and 7 ho the local in riction local in riction and local heat traner ith the ame value o arameter but dierent value o nanoarticle volume raction hich i 0.. The ame trend behavior i obvere a in Figure 4 and 5. Finally Figure 8 to illutrate the velocity and temerature roile o the cylinder againt y or the value o mixed convection arameter.5 (ooing lo) and (aiting lo) hen K Pr 6. but ith dierent value o nanoarticle volume raction hich are varie rom Thee igure ho ho the nanoarticle volume raction aect the luid velocity and temerature roile. Thu rom the grah both velocity and temerature roile are increaed by increaing the value o nanoarticle volume raction and mixed convection arameter. Fig.9 Temerature roile or variety value o nanoarticle volume raction or K= Pr=6..5 (ooing lo)

5 the inancial uort through vote number STRG 5095 H74 4F7 and A. J or thi reearch. REFERENES Fig.0 Velocity roile or variety value o nanoarticle volume raction or K= Pr=6. (aiting lo) Fig. Temerature roile or variety value o nanoarticle volume raction or K= Pr=6. (aiting lo) ONLUSION Steady o mixed convection boundary layer lo o vicoelatic nanoluid at a horizontal circular cylinder ith a cae o contant all temerature roblem ha been olved numerically uing Kellerbox method. The numerical reult or the in riction heat traner coeicient velocity and temerature roile are reented or variety value o nanoarticle volume raction mixed convection arameter and vicoelatic arameter. The eect o velocity and temerature roile due to the iluence o nanoarticle volume raction and mixed convection arameter are alo analyed. The comarion o the reult ho that the reent reult agree very ell ith the reviou ublihed reult reorted by Nazar et al. [6] and Anar et al. [8]. AKNOWLEDGEMENT The author ould lie to acnoledge the enter or Reearch and Innovation UniKL Reearch Management entre UTM and Minitry o Higher Education (MOHE) STEM Grant Malayia or [] S. U. hoi Enhancing Thermal onductivity o Fluid ith Nanoarticle ASME Int. Mech. Eng. ongr. Exo. vol [] D. A. Nield and A. V. Kuznetov The heng-minoycz roblem or natural convective boundary-layer lo in a orou medium aturated by a nanoluid Int. J. Heat Ma Tran. vol. 5 no [] W. A. Khan and I. Po Boundary-layer lo o a nanoluid at a tretching heet Int. J. Heat Ma Tran. vol. 5 no [4] S. Ahmad and I. Po Mixed convection boundary layer lo rom a vertical lat late embedded in a orou medium illed ith nanoluid Int. ommun. Heat Ma Tran. vol. 7 no [5] A. hamha R. S. R. Gorla and K. Ghodear Non-imilar Solution or Natural onvective Boundary Layer Flo Over a Shere Embedded in a Porou Medium Saturated ith a Nanoluid Tran. Porou Media vol. 86 no.. 0. [6] M. Qaim I. Khan and S. Shaie Heat traner and ma diuion in nanoluid over a moving ermeable convective urace Math. Probl. Eng. vol [7] A. Zaib K. Bhattacharyya and S. Shaie Unteady boundary layer lo and heat traner over an exonentially hrining heet ith uction in a coer-ater nanoluid J. ent. South Univ. vol. no [8] A. Noghrehabadi R. Pourrajab and M. Ghalambaz Eect o artial li boundary condition on the lo and heat traner o nanoluid at tretching heet recribed contant all temerature Int. J. Therm. Sci. vol [9] Hayat T. Waqa M. Sabir Ali Shehzad Alaedi A. International Journal o Numerical Method or Heat & Fluid Flo Article iormation : Int. J. Numer. Method Heat Fluid FLo vol. 6 no [0] N. A. Rai M. R. Ilia Z. M. Ia and S. Shaie G-Jitter induced mixed convection lo and heat traner o microolar nanoluid lo over an inclined tretching heet AIP o. Proc. vol [] F. Mabood W. a. Khan and M. M. Yovanovich Forced onvection o Nanoluid Flo acro Horizontal ircular ylinder ith onvective Boundary ondition J. Mol. Liq. vol [] S. Sarar S. Ganguly G. Bia and P. Saha Eect o cylinder rotation during mixed convective lo o nanoluid at a circular cylinder omut. Fluid vol [] A. E. M. Bouchoucha and R. Beaïh Natural convection in a quare cavity illed ith nanoluid Fluid Dyn. Mater. Proce. vol. no [4] R. K. Tiari and M. K. Da Heat traner augmentation in a toided lid-driven dierentially heated quare cavity utilizing nanoluid Int. J. Heat Ma Tran. vol. 50 no [5] J. H. Merin Mixed convection rom a horizontal circular cylinder Int. J. Heat Ma Tran. vol. 0 no [6] R. Nazar N. Amin and I. Po Mixed convection boundary-layer lo rom a horizontal circular cylinder ith a contant urace heat lux Heat Ma Tran. vol. 40 no [7] Y. Lin L. Zheng and X. Zhang Radiation eect on Marangoni convection lo and heat traner in eudo-latic non-netonian nanoluid ith variable thermal conductivity Int. J. Heat Ma Tran. vol [8] I. Anar N. Amin and I. Po Mixed convection boundary layer lo o a vicoelatic luid over a horizontal circular cylinder Int. J. Non. Linear. Mech. vol. 4 no