Temperature dependent viscosity effect on MHD mixed convective dissipating flow of cylinder shaped Cu-water nanofluid past a vertical moving surface

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1 Proceeding o ICFM 15 International Conerence on Frontier in Mathematic 15 Temperature dependent vicoity eect on MHD mied convective diipating low o cylinder haped Cu-water nanoluid pat a vertical moving urace T. POORNIMA, N. BHASKAR REDDY and P. SREENIVASULU Department o Mathematic, Sri Venkatewara Univerity-5175,Tirupati, A.P. (India poonima.anand@gmail.com, nbrvu@gmail.com, preddyvu11@gmail.com Abtract: An analyi i carried out to inpect the uphot o variable vicoity and on hydromagnetic two dimenional boundary layer low o nanoluid with cylinder haped Cu nanoparticle over a vertical moving urace with vicou diipation. Uing the imilarity tranormation, the governing tranport equation along with the appropriate boundary condition are tranormed to a et o ordinary dierential equation. The conequential ytem o equation i then olved by applying Runge-Kutta ourth order cheme along with hooting technique. The inluence o variou pertinent parameter uch a the magnetic parameter, volume raction parameter, mied convection parameter,, the vicoity/temperature parameter θ r and the Eckert number Ec on the velocity o the low ield and heat traner characteritic are calculated numerically and portrayed graphically. The preent reult are compared with the available literature and ound to be in good agreement. The preent problem ha a great cope in the ield o engineering and indutry, etc. The numerical reult indicate that the eect o nanotube volume raction i to increae the heat traner rather than pherical nanoparticle. Thi i true even in the preence o variable vicoity and the vicou diipation. Moreover, the reult obtained or heat traner characteritic with nanotube reveal many intereting behavior that warrant uture tudy on the eect o the nano-olid-tube. Keyword: Boundary layer, Magneto-hydrodynamic, Mied convection, Nanoluid, Shear tre, Variable vicoity Vicou diipation. I. INTRODUCTION Mot reearcher aume the phyical propertie o the nanoluid a contant. But, the phyical propertie o the nanoluid may change with temperature. Mainly, vicoity, one o the important phyical propertie o the luid low, depend on the temperature. The luid vicoity i no longer being aumed contant becaue heat generated by internal riction rie the temperature which in turn ha an eect on the vicoity o the luid. Thereore, to predict the low and heat traner rate in the nanoluid, it i neceary to conider the eect o the temperature-dependent vicoity o the bae luid. Available literature on the variable luid propertie o nanoluid (Praad et al. 1; Sedeek 5; Ali 6 how that the work i not carried out or nanoluid low over a lat urace. The eect o thermal radiation and variable luid vicoity on ree convective and heat traner pat a porou tretching urace wa analyzed by Mukhopadhyay and Layek (8. Conumption o energy production ha undoubtedly become one o the mot important global problem that we are going to ace in the orthcoming day, in particular, to control global warming which occur due to the emiion o greenhoue gae, or epected decreae in global oil production. Conidering the rapid increae in energy demand worldwide, inteniying heat traner procee and reducing energy loe due to ineective ue have become increaingly important tak. Nano-cience and technology i epected to play a igniicant role in revitalizing the traditional energy indutrie and timulating the emerging renewable energy indutrie (Wen et al. 9; Wen et al. 1. During lat two decade, the term nanoluid coined by Choi (1995 i more amiliar becaue o our increaed tate in miniaturization o object and uage o compact ubtance. Nanoluid have attracted the attention o the heat traner community ince traner o heat rom the compact ubtance i inevitable. Nanoluid are engineered colloid coniting nanometer-ized particle upended in traditional heat traner luid have been tudied etenively to enhance heat traner. According to Prodanovi et al.(1 nanoluid containing an ultraine nanoparticle ha the capability o lowing in porou media, and thee low can improve oil recovery; hence, nanoparticle are able to control the procee o oil recovery. To improve oil recovery o vicou oil, a luid, or eample water, i injected into the porou medium to diplace the oil, ince water vicoity i inerior to that o oil. However, increaing the injected luid vicoity uing nanoluid would dratically increae the recovery eiciency. A benchmark tudy on the thermal conductivity o nanoluid wa made by Buongiorno et al. (9. Oztop and Abu-Nada (8 howed an enhancement in heat traner wa regitered by the addition o nanoparticle. Uing AlO3 and CuO water nanoluid, Putra et al. (3 gave contrary report through eperimental inding. They reported that the natural convection heat traner coeicient wa lower than that o a clear low. Another eperimental work on natural convection wa done by Wen and Ding (6 who eplained the deterioration in heat traner i caued by the addition o nanoparticle. When a conductive luid move through a magnetic ield, an ionized ga i electrically conductive, the luid may be inluenced by the magnetic ield. There are everal application uch a the aerodynamic etruion o platic heet, geothermal application, high-temperature plama applicable to nuclear uion energy converion, liquid metal luid, and (MHD power generation ytem. Dameh (6 tudied the magnetohydrodynamic-mied convection heat traner problem rom a vertical urace embedded in a porou media. Sparrow and Ce (1961 tudied the eect o magnetic ield on the natural convection heat traner. Al-Odat et al. (6 analyzed the thermal boundary layer on an eponentially tretching continuou urace in the preence o magnetic ield eect. Takhar and Ram ( 1994 tudied the magnetohydrodynamic ree convection low o water through a porou medium. Shakhaoath Khan et al.(13 tudied the eect o magnetic ield on radiative low o a nanoluid pat a tretching heet. EL-Kabeir et al.( 7 tudied the unteady MHD combined convection over a moving vertical heet in a luid aturated porou medium. EL- Kabeir et al.( 7 olved magnetohydrodynamic ree convection low over inclined permeable urace embedded in porou medium in the preence o a uniorm magnetic ield. Rapti et al. (4 tudied the eect o thermal radiation on the magnetohydrodynamic low o a vicou luid pat emi-ininite tationary plate. When the low ield i o etreme ize or in high gravitational ield, vicou diipation i inevitable. Vajravelu (1 tudied ISBN:

2 Proceeding o ICFM 15 International Conerence on Frontier in Mathematic 15 the vicou low over a nonlinearly tretching heet. Cortell (7 analyzed the vicou low and heat traner over a nonlinearly tretching heet. Gebhart and Mollendor (1969 conidered the eect o vicou diipation or eternal natural convection low over a tretching urace. Soundalgekar (197 analyzed vicou diipative eect on the two-dimenional unteady ree convective low pat an ininite vertical porou plate. Duwairi (5 ha preented the eect o vicou diipation and Joule heating on the orced convection low in the preence o thermal radiation. Recently, Matin et al. (1 tudied the entropy analyi in mied MHD low o nanoluid over a non-linear tretching heet. Sreenivaulu et al. (13 preented the radiation and vicou diipation eect on teady MHD Marangoni convection low over a permeable lat urace with heat generation or aborption. With the above awarene, an eort i made to analyze the eect o variable vicoity on MHD low o nanoluid pat a moving vertical urace in the preence o vicou diipation. Etending the work o Pop et al.(199 and Mohamed Ali (6, thi problem i invetigated. The motive o thi paper i to tudy MHD thermal boundary-layer low over a moving vertical urace or dierent type o water-baed nanoluid, namely, Cu, Ag, CuO, Al O 3 and TiO. An eicient numerical hooting technique with a ourth order Runge-Kutta cheme wa ued to olve the normalized boundary layer equation and the eect o material parameter on the low ield and heat traner characteritic i dicued in detail.. Flow Analyi A teady two dimenional mied convection MHD boundary layer low o nanoluid along a moving vertical urace with variable magnetic ield and vicou diipation i conidered. The low i conidered along the ai, which i taken along the vertical lat plate in the upward direction and the y-ai i taken normal to it. The low i conirmed to y >. The urace o the plate i maintained at a contant temperature T w higher than the contant temperature T o the ambient nanoluid. A quiecent incompreible and electrically conducting in the preence o a magnetic ield B( perpendicular to the urace i taken into account in the preence o diipation eect. The tranvere applied magnetic ield and magnetic Reynold number are aumed to be very mall, o that the induced magnetic ield i negligible. The luid i a water baed nanoluid containing ive dierent type o nanoparticle namely Copper (Cu, Silver (Ag, Alumina (Al O 3, Copper Oide (CuO and Titanate (TiO. It i aumed that the bae luid and the nanoparticle are in thermal equilibrium and no lip occur between them. A chematic repreentation o the phyical model and coordinate ytem i depicted in Fig.1. Under the uual aumption, the governing boundary layer equation are given by: u v + (1 y u u 1 u u + v ( µ n + g( ρβ n ( T T σb ( u y ρn y y T T kn T µ n u u + v + (3 y ( ρcp n y ( ρcp n y The boundary condition or the velocity and temperature ield are u uw, v, T Tw at y (4 u, T T a y where u, v are the velocity component in the and y direction, repectively, T - the temperature o the nanoluid, T - the ambient luid temperature, σ - the electric conductivity, B( - the variable magnetic ield and g - the acceleration due to gravity. The eective denity o the nanoluid i given by Where φ i the olid volume ractiono nanoparticle. The eective dynamic vicoity o the nanoluid given by Brinkman [9] a Here µ i the coeicient o vicoity which i conidered to be vary a an invere unction o temperature can be written a Praad et al.( δ ( T T i.e., (7 1 µ µ at ( Tr µ where, ρ (1 φ ρ + φρ n µ µ n (1 φ.5 δ 1 a, Tr T µ δ The vicoity o a luid uually decreae with an increae in the temperature. To urther demontrate the appropriatene o equation (7 correlation between the vicoity and temperature oe air and water are given below becaue thee two are the mot commonly ued working luid in engineering (Vajravelu et al. 13. For water T 58.6 baed T 88K 15 C (8 ( µ ( and or air 1 baed (9 13. T 74.6 T 93K C ( µ ( The reerence temperature choen are o very practical in mot application. Here both a and T r are contant, and their value depend on the reerence tate and the thermal property o the luid. i.e. δ ( a contant in general, a > correpond to liquid and a < or gae. T i the temperature, T and are the contant value o the temperature and the coeicient o vicoity repectively, ar away rom the heet. Thermal diuivity o nanoluid i k n αn (1 ( ρcp n where the heat capacitance Cp o the nanoluid i obtained a ( ρc (1 φ( ρc + φ( ρc ( 11 p n p p µ (5 (6 ISBN:

3 Proceeding o ICFM 15 International Conerence on Frontier in Mathematic 15 The thermal epanion coeicient o the nanoluid can be determined by (1 and the eective thermal conductivity can be incorporated rom the ollowing epreion Rana and Bhargava (11 kn ( k + ( n 1 k ( n 1 φ( k k (13 k ( k + ( n 1 k + φ( k k where n i the nanoparticle empirical hape actor. In particular, n 3 tood or pherical haped nanoparticle and n 3/ or cylindrical one (nanotube. Here the ubcript n, and repreent the thermo phyical propertie o the nanoluid, bae luid and nano-olid particle repectively. and - the thermal conductivitie o the bae luid and nanoparticle, repectively. and - are the denitie o the bae luid and nanoparticle, repectively. Table 1: Model o nanoluid baed on dierent ormula or thermal conductivity. Model Shape o nanoparticle I Spherical II ( ρβ (1 φ( ρβ + φ( ρβ Cylindrical (nanotube n k k ρ Thermal conductivity Table Thermo phyical propertie o luid and nanoparticle given by (Oztop and Abu-Nada 8; Alloui 11 ρ kn ( k + k φ( k k k ( k + k + φ( k k kn ( k + (1/ k (1/ φ( k k k ( k + (1/ k + φ( k k ʹ ʹ ʹ φ ρ θ θ 1 φ φ ʹ ʹ ρ θr ( θ θr ʹ ʹ θʹ.5 r ( θ θ r ( ρβ (1 φ 1 φ+ φ λθ M ʹ θr ( ρβ (15 k n ( ρc p Pr Ecθ r θʹ ʹ + Pr 1 φ+ φ θʹ ( ʹ ʹ.5 k ( ρc p (1 φ ( θ θ r (16 The correponding boundary condition are ʹ ( 1, (, θ( 1 (17 ʹ (, θ( where prime denote the dierentiation with repect to η. θ r i the luid vicoity parameter, Pr i the Prandtl number, λ i the mied convection parameter, M i the magnetic ield parameter and Ec i the Eckert number. I θ r i large, in other word, i (T w T i mall, the eect o variable vicoity on the low can be neglected. Smaller value o θ r how that either the luid vicoity varie with temperature markedly or the operating temperature dierence i high. In either cae, the variable luid vicoity eect i taken to be mot important. Alo bearing in mind that the liquid vicoity varie wildly compared to that o ga, with temperature. θ r i negative or liquid and poitive or gae. The quantitie o practical interet in thi tudy are the kin riction or the hear tre coeicient C and the local Nuelt number C Nu µ, which are deined a u n ρ uw y y (18 To get imilarity olution o Eq.(1 - (3 ubject to the boundary condition (4, we introduce the ollowing imilarity tranormation. y ν 1/ 1/ η Re, ψ ν Re, u Re ʹ, 1/ ν Re T T σ B v [ η ʹ ], θ, M, T T u ρ gβ ( T T u λ Cp T T 3 Gr w w, Gr, Ec Re ν ( ( w 1 ν θr, Pr. α [ δ( T T ] w w w (14 where ν i the kinematic vicoity o the bae luid. η i the imilarity variable, and θ are the dimenionle velocity and temperature, repectively. The velocity component u and v in Eq. (14 automatically atiy the continuity Eq.(1. In term o (η and θ (η, the momentum Eq.( and energy Eq. (3 can be written a Phyical propertie Cp( J / kg K 3 ρ ( kg / m kw ( / mk β 5 1 (1/ K kn T Nu (19 k( Tw T y y Uing Eq.(6, the wall hear tre and local heat traner rate can be epreed a C Nu where Fluid phae (water (1 kn 1/ Re θʹ ( k 1/ 1 Re (.5 φ θr Cu Alo3 Tio Cuo Ag Re / i the local Reynold number. uν w ( (1 RESULTS AND DISCUSSION The coupled non linear dierential Eq. (15 and (16 along with the boundary condition (17 are olved numerically employing Runge - Kutta our order cheme along with hooting ʹ ʹ ISBN:

4 Proceeding o ICFM 15 International Conerence on Frontier in Mathematic 15 technique. In order to bring out the alient eature o the low and the heat traner characteritic, the numerical value or dierent value o the governing parameter φλ,, M, θr and Ec are portrayed in Fig Here, ive dierent type o nanoparticle, namely, Copper (Cu, Alumina (Al O 3, Titanium oide (TiO, Copper oide (CuO and Silver (Ag, with water a the bae luid i taken. The Prandtl number o the bae luid (water i kept contant at 6.. It i noticed that thi tudy reveal the governing boundary layer equation (15 - (16 condene to thoe o vicou or regular luid, when φ and the reult obtained i compared with that o Pop et. al. (199 and Ali (6, and ound a good agreement, when φ (Table-3. Fig. and 3 repreent the velocity and temperature proile or dierent value o the Cu - nanotube volume raction parameter repectively. It i ound that the nanoluid φ witne decreae with an increae in ize o the nanotube. It i oberved that the velocity proile tart with the plate velocity near the wall and an up thrut in the luid velocity i oberved, and gradually travel downward and reache zero atiying the boundary condition. The eect o increaing value o the nanotube volume raction i to decreae the velocity proile. Thi i due to the act that the preence o nanotube volume raction lead to urther thinning o the boundary layer. Thee igure illutrate thi agreement with the phyical behavior that the thermal conductivity increae and then the thermal boundary layer thickne increae a increae in volume o nanotube. Analyi o the graph (Fig.3 how that the eect o the increaing value o the nanoparticle volume raction parameter i to enhance the temperature. Alo the thermal boundary layer or nano-olid tube, i greater than that o pure water. Thi i becaue nano-olid tube volume raction parameter (o Cu ha high thermal conductivity, o the thickne o the thermal boundary layer increae. Fig. 4 and 5 illutrate the inluence o the magnetic parameter on the velocity and temperature o the low. It i oberved that the luid velocity decreae a the magnetic parameter increae (Fig.4. Thi i becaue o the Lorentz orce which tend to reit the luid low and thu reducing it velocity and enhancing the temperature o the nanoluid (Fig. 5. The eect o mied convection parameter λ on the velocity and the temperature are portrayed in Fig. 6 and 7, repectively. The Richardon number λ relate the meaure o the eect o the buoyancy in comparion with that o the inertia o the eternal λ orced or ree tream low on the heat and luid low. I, then orced convection play dominant role wherea ree convection heat traner i dominant when λ. A λ increae rapidly, ree convection pronounce more cauing a decreae in the momentum boundary layer and a continuou rie in the thermal boundary layer. From igure, it i oberved that velocity o the nanoluid decend (Fig. 6, while luid temperature rie (Fig. 7. Fig. 8 and 9 depict the uphot o the luid vicoity parameter θ r on the velocity and temperature. Here negative value are taken or θ r which repreent the liquid vicoity. It i een that the velocity o the luid decreae near the plate and a revere behavior i oberved ar away rom the plate. It i hown clearly that the hydrodynamic boundary layer thickne diminihe and the velocity proile become linear or large value o the Prandtl number. Thi i due to the act that or a given luid, maller θ r implie higher temperature dierence between the wall and the ambient luid. The reult preented in thi paper demontrate quite clearly that the vicoity variation parameter ha a trong eect on the velocity proile within the boundary layer and hence een on the kin riction characteritic. From the graphical repreentation we ee that the eect o increaing value o the luid vicoity parameter θ r i to enhance the temperature. Thi i due to the act that an increae in the luid vicoity parameter mean the temperature dierence i negligible; thereby the vicoity remain contant which increae the thermal boundary layer thickne reulting in temperature enhancement (Fig.9. The inluence o Ec on the velocity and the temperature i depicted in Fig.1 and 11. From Fig.1, it i clear that the velocity o nanoluid increae near the urace and it decreae or away rom the plate. The Eckert number Ec deine the relationhip between the kinetic energy in the low and the enthalpy. It embodie the conervation o kinetic energy into internal energy by work done againt the vicou luid tre. The poitive Eckert number implie cooling o the plate i.e., lo o heat rom the heet to the luid. From Fig. 11, it i noticed that the heat get tranerred rom the heet to the Cu-water nanoluid, thereby raiing the temperature o the luid. Fig. 1 and 13 decribe the eect o dierent type o nanoluid with nanotube (Ag, Cu, CuO, TiO, Al O 3 volume raction on velocity and temperature are hown. The momentum boundary layer increae rom Ag to Al O 3. The thermal boundary layer increae rom TiO to Ag. The eect o dierent nanoluid on the local kin riction and the local Nuelt number are hown in Fig. 1 and 13. Local Nuelt number increae rom TiO to Ag becaue Ag ha highet thermal conductivity. The local kin riction coeicient in term o againt φ i drawn in Fig.14. The value o ʹ ʹ ( ʹ ʹ ( i poitive which mean that the urace riction ha marked inluence on the luid. The nanoluid a it contain the nanotube act a a good riction reducer. It i oberved that the riction at the plate decreae with a an increae in the volume raction φ. A the liquid vicoity parameterθ r varie, the wall hear tre decreae. The heat θʹ ( traner rate in term o againt r i depicted in Fig.15. It i noticed that Nuelt number decreae or increaing φ. Alo, it i een that heat traner rate increae gradually a the liquid vicoity θ r increae. Fig. 16 and 17 eplain the coeicient o urace kin riction and Nuelt number veru the magnetic parameter M or ied value o vicoity parameter θ r. It i noticed that the plate kin riction and the Nuelt number (Fig. 17 decend or acending M. Alo it i oberved that the coeicient o kin riction drop up to M 1.5 and oppoite trend i oberved ater that, a the volume raction parameterφ, increae. The eect o mied convection parameter on the urace kin riction or dierent value o vicoity parameter θ r i hown in Fig. 18. It i noticed that the kin riction increae with an increae in λ. The rate o heat traner coeicient decreae with an increae in both the Eckert number Ec, thi i evident rom Fig. 19. The dierent model o nanoluid baed on dierent ormula or thermal conductivity i taken here. Model I or pherical haped nanoparticle and Model II or nanotube. Nanotube traner the heat more eiciently than that o pherical haped nanoparticle and thi i evident rom Fig.. Thi i becaue o the act that nanotube have higher thermal conductivity θ ISBN:

5 Proceeding o ICFM 15 International Conerence on Frontier in Mathematic 15 compared to nanoparticle. Fig. 1 how the graph o Nuelt number plotted againt the volume raction o dierent type o nanotube (Ag, Cu, CuO, Al O 3, and TiO. Nuelt number decreae more harply or Ag a compared to other nanoparticle becaue Ag ha highet thermal conductivity. Copper tood the net and the remaining marche. ACKNOWLEDGEMENTS Thi work i carried out by the inancial upport o Univerity Grant Commiion under Major Reearch Project (UGCF. No /1(SR, or which the author are highly grateul. 4. CONCLUSIONS The preent paper eamine the inluence o the dierent type o nanotube on MHD mied convective low o nanoluid along a moving vertical plate with variable vicoity and vicou diipation. The numerical analyi i carried out and the domino eect are ummarized a ollow: Variable vicoity play a predominant role on both velocity and temperature. It i a good riction reducer. Uage o nanotube i more advantageou than pherical haped nanoparticle. Becaue nanotube have high thermal conductivity and it traner heat more eiciently Silver ha highet cooling perormance due to it high thermal conductivity. It reduce the heat capably than any other nanoparticle. But economic aordability i concerned copper nanoparticle can be replaced a it alo a good conductor o heat. Both kin riction and Nuelt number decreae or variation in the volume raction parameter or Cu water nanoluid. Velocity o the nanoluid drop with an increae in buoyancy parameter (λ wherea the hear tre acend. The thermal boundary layer decreae and momentum boundary layer irtly decreae and then increae with an increae in the vicoity parameter θ r or Cu - water. The heat traner decreae with an increae in the Eckert number or ied value o vicoity parameter θ r or Cu - water. REFERENCES [1] Ali, M.E., The eect o variable vicoity on mied convection heat traner along a vertical moving urace, Int. 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6 [1] Rana, R. and R. Bhargav, Numerical tudy o heat traner enhancement in mied convection low along a vertical plate with heat ource/ink utilizing nanoluid, Commun. Nonlinear Sci. Numer. Simulat.,16, ,(11. [] Rapti, A., C.Perdiki and H.S.Takhar, Eect o thermal radiation on MHD low, Int. J. Heat Ma Traner, 153, ,(4. [3] Sedeek, M.A., Finite element method or the eect o chemical reaction, variable vicoity, thermophorei and heat generation / aborption on a boundary layer hydro magnetic low with heat and ma traner over a heat urace, Acta Mechanica,, 177 (1-4,1-18,(5. [4] Soundalgekar, V.M., Vicou diipation eect on unteady ree convective low pat an ininite vertical porou plate with contant uction, Int. J. Heat Ma traner,15, ,(197. [5] Sparrow, E.M. and R.D.Ce, Eect o magnetic ield on ree convection heat traner, Int. J. Heat Ma Traner,,3, 67 7,(1961. [6] Sreenivaulu,P.,N.Bhakar Reddy and M.Gnanewara Reddy, Radiation and vicou diipation eect on teady MHD Marangoni convection low over a permeable lat urace with heat generation or aborption, Int. J. o Mathematical Archive, 4(, , (13. [7] Takhar, H.S. and P.C.Ram, Magnetohydrodynamic ree convection low o water at 4 C through a porou wall, Int. Commun. Heat Ma Traner,1, , (1994. [8] Vajravelu, K., K.V. Praad and N.G.Chiu-on, The eect o variable vicoity on the low and heat traner o a vicou Ag-water and Cu-water nanoluid, Journal o Hydrodynamic, 5(1,1-9,(13. [9] Vajravelu, K., Vicou low over a nonlinearly tretching Sheet, Appl. Math. Comput., 14,81-88, (1. [3] Wen, D., Y.Ding, Natural convective heat traner o upenion o titanium dioide nanoparticle (nanoluid, IEEE Tran Nanotechnol., 5(3, 7,(6. [31] Wen, D.S., G.Lin, S.Vaaei, K.Zhang, Review o nanoluid or heat traner application,particuology, 7,141,(9. [3] Wen, D.S., Nanouel a a potential econdary energy carrier, Energy Environ. Sci., 3,591,(1. Proceeding o ICFM 15 International Conerence on Frontier in Mathematic 15 ISBN:

7 Proceeding o ICFM 15 International Conerence on Frontier in Mathematic 15 ISBN:

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