flow and heat transfer of nanofluids over a rotating disk with uniform stretching rate in the radial direction

10:0:1465=WUnicodeDec222011ị 6ỵ model JPPR : 122 Prod:Type:FTP pp:026col:fig::NILị ED: PAGN: SCAN: Propulsion and Power Research ]]]];](]):]]]–]]] 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 HOSTED BY http://ppr.buaa.edu.cn/ Propulsion and Power Research www.sciencedirect.com Q2 Flow and heat transfer of nanofluids over a rotating disk with uniform stretching rate in the radial direction Q1 Chenguang Yina, Liancun Zhenga,n, Chaoli Zhanga,b, Xinxin Zhangb a School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China School of Mechanical Engineering, University of Science and Technology Beijing, Beijing 100083, China b Received 24 August 2015; accepted 23 October 2015 KEYWORDS Nanofluid; Rotating disk; Stretching; Heat transfer Abstract This paper studies flow and heat transfer of nanofluids over a rotating disk with uniform stretching rate Three types of nanoparticles-Cu, Al2O3 and CuO-with water-based nanofluids are considered The governing equations are reduced by Von Karman transformation and then solved by the homotopy analysis method (HAM), which is in close agreement with numerical results Results indicate that with increasing in stretching strength parameter, the skin friction and the local Nusselt number, the velocity in radial and axial directions increase, whereas the velocity in tangential direction and the thermal boundary layer thickness decrease, respectively Moreover, the effects of volume fraction and types of nanofluids on velocity and temperature fields are also analyzed & 2017 National Laboratory for Aeronautics and Astronautics Production and hosting by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Introduction The problem of fluid flow over a rotating disk is one of the classical problems in fluid mechanics, which has both theoretical and practical values Many researches have been n Corresponding author E-mail address: liancunzheng@ustb.edu.cn (Liancun Zheng) Peer review under responsibility of National Laboratory for Aeronautics and Astronautics, China carried out on flow over a rotating disk in theoretical disciplines and due to numerous practical applications in some areas such as computer storage devices, rotating machinery, electronic devices and medical equipment, such flow is also very important in the engineering processes Von Karman [1] originally investigated the hydrodynamic flow over an infinite rotating disk in 1921 In his work, Von Karman introduced his famous similarity transformations, which reduced the governing partial differential equations into ordinary differential equations In recent years, http://dx.doi.org/10.1016/j.jppr.2017.01.004 2212-540X & 2017 National Laboratory for Aeronautics and Astronautics Production and hosting by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Please cite this article as: Chenguang Yin, et al., (2017), http://dx.doi.org/10.1016/j.jppr.2017.01.004 57 58 59 60 61 62 63 64 65 66 67 68 69 70 2 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 Chenguang Yin et al Griffiths [2] considered the boundary-layer flow due to a rotating disk for a number of generalized Newtonian fluid models Dandapat and Singh [3] studied the two-layer film flow over a non-uniformly rotating disk in the presence of uniform transverse magnetic field under the assumption of planar interface The flow due to stretching surfaces is important in the extrusion processes in plastic and metal industries [4–6] The steady flow over a rotating and stretching disk was first studied by Fang [7] Recently, Fang and Zhang [8] investigated the flow between two stretching disks More recently, Turkyilmazoglu [9] studied the steady magnetohydrodynamic (MHD) laminar flow of an electrically conducting fluid on a radially stretchable rotating disk in the presence of a uniform vertical magnetic field Fang and Tao [10] investigated the laminar unsteady flow over a stretchable rotating disk with deceleration Rashidi et al [11] considered the first and second law analyzes of an electrically conducting fluid past a rotating disk in the presence of a uniform vertical magnetic field Asghar et al [12] studied steady three dimensional flow and heat transfer of viscous fluid on a rotating disk stretching in radial direction Turkyilmazoglu [13] investigated the traditional Bödewadt boundary layer of an incompressible viscous fluid flow and heat transfer over a stationary disk provided that the disk is allowed to radially stretch The term “nanofluids” was coined by Choi [14] in 1995 at the ASME Winter Annual Meeting Nanofluid is a colloidal mixture by adding nanoparticles (o100 nm) in a base fluid, which can considerably change the transport and thermal properties of the base fluid and thus may improve thermal conductivity A list of review papers on nanofluids can be found in Refs [15–17] Bachok et al [18] studied the flow and heat transfer over a rotating porous disk in a nanofluid Rashidi et al [19] considered the entropy generation in steady MHD flow due to a rotating porous disk in a nanofluid Turkyilmazoglu [20] investigated the flow and heat transfer characteristics over a rotating disk immersed in five distinct nanofluids The homotopy analysis method (HAM) introduced by Liao in 1992 [21–26], is an effective mathematical method which has been successfully employed to solve different types of nonlinear problems Many studies have verified the validity and effectiveness of this method In this work, we obtain the analytical solutions by using the homotopy analysis method Although the problem of fluid flow over a rotating disk that is stretching in the radial direction are already involved in some works as cited above, they have not yet been considered for nanofluids In this paper we investigate the flow and heat transfer of nanofluid over a stretching rotating disk, three types of nanoparticles: Cu, CuO and Al2O3 are considered Results show that with increasing in stretching strength parameter, the skin friction and the local Nusselt number, the velocity in radial and axial directions increase, whereas the velocity in tangential direction and the thermal boundary layer thickness decrease, respectively Formulation of the problem Consider an incompressible, steady and axially symmetric nanofluid flow past a rotating disk that is placed at z ¼ and rotates with an angular velocity Ω The disk is further stretching at a uniform rate s in the radial direction r Physical model of rotating disk is shown in Fig [13] The governing equations of the nanofluid motion and energy in cylindrical coordinates can be presented, respectively, as follows u u w ỵ ỵ ¼0 ð1Þ ∂r r ∂z u ∂u v2 ∂u À þw ∂r ∂z r   μnf ∂2 u u u u p ẳ ỵ ỵ þ ρnf ∂r ρnf ∂r r ∂r r z2 nf v uv v u ỵ ỵw ẳ r r ∂z ρnf  ∂2 v ∂ v ∂2 v þ þ ∂r ∂r r ∂z μnf ∂w w p ỵw ỵ ẳ u r z nf ∂z ρnf ð2Þ    ∂ w w w ỵ ỵ r r r ∂z   ∂T ∂T ∂ T ∂T T ẳ nf ỵ ỵ u ỵw r z ∂r r ∂r ∂z2 ð3Þ ð4Þ ð5Þ The boundary conditions are given by z ¼ : u ¼ sr; v ¼ Ωr; w ¼ 0; T ¼ T w ð6Þ z-1 : ð7Þ u-0; v-0; T-T ; P-P1 where T is the temperature of the nanofluid, T is the temperature of the ambient nanofluid, the pressure is P and the pressure of the ambient nanofluid is P1 , μnf and αnf are the dynamic viscosity and thermal diffusivity of Fig Physical model of rotating disk Please cite this article as: Chenguang Yin, et al., (2017), http://dx.doi.org/10.1016/j.jppr.2017.01.004 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 the nanofluid, ρnf is the density of the nanofluid, which are respectively defined as μf knf Á ; nf ẳ ịf ỵ s nf ẳ ; nf ẳ 2:5 Cp nf ị À Á À Á À Á ρC p nf ¼ ị C p f ỵ C p s ; ks ỵ 2kf 2φ k f À ks knf Á À Á ¼ 8ị kf ks ỵ 2k f ỵ kf À ks In which, φ is the nanoparticle volume fraction, the range of φ is to μf is the viscosity of the fluid fraction, ρf and ρs are the densities of the fluid and of the solid fractions, respectively À Á The heat capacitance of the nanofluid is given by ρC p nf and knf is the effective thermal conductivity of the nanofluid approximated by the model given by Oztop [27], which is confined to spherical nanoparticles only The thermophysical properties of water and different nanoparticles are given in Table [27] Nonlinear boundary value problem By means of the Von Karman's transformations, À Á1=2 À Á1=2 ẳ =f z; u ẳ rF ị; v ẳ rG ị; w ẳ f H ị; 9ị p p1 ẳ 2f p ị; ị ẳ T T Þ=ðT w À T Þ: The system (1)–(5) can be reduced into the following ordinary differential equations H ỵ 2F ẳ 10ị ! ị 2:5 À Á F″ À HF À F ỵ G2 ẳ ỵ s =f ð11Þ ! ð1 À φÞ 2:5 À Á G″ HG0 2FG ẳ ỵ s =ρf K =K À Ánf À f Á θ″ À Hθ0 ¼ Pr ρcp nf = ρcp f where C ¼ s=Ω denotes a stretching strength parameter measuring the ratio of radial stretch to swirl such that C ¼ corresponds to the classical non-stretching case, the range of C is to Pr is the Prandtl number The skin friction coefficient Cf and the Nusselt number Nu are physical quantities which are given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi τwr ỵ w rqw 15ị Cf ẳ ; Nu ¼ k f ðT w À T Þ ρf ðΩr Þ2 where τwr and τwϕ are the radial and transversal shear stress at the surface of the disk, respectively, and qw is the surface heat flux, which are introduced as  !  À Áà τwr ¼ μnf uz ỵ w z ẳ ; w ẳ nf vz ỵ w ; r zẳ0 qw ẳ k nf T z ịz ẳ Substituting Eq (8) in Eq (16) and using Eq (15), we have qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 0ị2 ỵ G0 0ị2 k nf 1=2 Re Cf ẳ ; Re1=2 Nu ẳ 0ị 2:5 kf ị 17ị Re ẳ r =υf is the local Reynolds number Results and discussion The nonlinear ordinary differential Eqs (10)–(13) subjected to the boundary conditions (14) are solved by the Homotopy analysis method [21–26] We obtain H ị ẳ 2Ce ỵ e ịh F ị ẳ ỵ e ịh ỵ C2 e ỵ C2 e e ! 3Ce ị2:5 ỵ s =f G1 ị ẳ ỵ e ịh 2C ỵ 2Ce e 12ị 13ị 16ị the transformed boundary conditions become À Á À 3e 2:5 ị ỵ s =f ! F 0ị ẳ C; G 0ị ẳ 1; H 0ị ¼ 0; θ ð0Þ ¼ F ð1Þ ¼ G 1ị ẳ 1ị ẳ P 1ị ẳ 14ị Table Thermophysical properties of water and different nanoparticles [27] Physical properties Pure water Cu CuO Al2O3 Cp/(J/(kg k)) ρ/(kg/m3) k/(W/(m k)) 4179 997.1 0.613 385 8933 400 531.8 6320 76.5 765 3970 40 ị ẳ e ỵ e ịhK nf =K f À Á À Á 2Pr ρcp nf = ρcp f We then evaluate the errors of Eqs (10)–(13) using the following error estimation functions [28,29] Z EH ¼ H ỵ 2F ị2 d 18ị Z EF ¼ !2 ! 2 À F HF F ỵ G d 1ị2:5 ỵ s =f Please cite this article as: Chenguang Yin, et al., (2017), http://dx.doi.org/10.1016/j.jppr.2017.01.004 ð19Þ 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 4 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 Chenguang Yin et al Table Computational errors for various of C for Cu-water nanofluid with φ ¼ 0:1 and Pr ¼ 6:2 in the case of h ¼ À1 Computational order C¼0 C ¼ 0:1 C ¼ 0:2 11 0:724620 8:37862  10 À 4:57449  10 À 2:55245  10 À 0:295475 7:79151  10 À 3:62427  10 À 1:78800  10 À 0:333653 7:66126  10 À 3:71540  10 À 2:33839  10 À Table Comparison of the numerical solutions for F ð0Þ, À G0 ð0Þ, À H ð1Þ and À θ0 0ị, when ẳ 0, C ẳ and Pr ¼ 6:2 F ð0Þ À G0 ð0Þ À H ð1Þ À θ0 ð0Þ Ref [19] Ref [20] Present 0.510186 0.61589 0.51023262 0.61592201 0.88447411 0.93387794 0.51022941 0.61591990 0.88446912 0.93387285 Fig Effects of C on tangential velocity profiles G ðηÞ for Cu-water nanofluid with φ ¼ 0:1 and Pr ¼ 6:2 Fig Effects of C on radial velocity profiles F ị for Cu-water nanouid with ẳ 0:1 and Pr ẳ 6:2 Z EG ẳ 1ị 2:5 ! !2 Á G″ ÀHG0 À 2FG dη ỵ s =f 20ị Z E ẳ 1 K =K À Ánf À f Á θ″ À Hθ0 Pr ρcp nf = ρcp f Err ¼ max ½EH ; EF ; E G ; Eθ Š !2 dη ð21Þ ð22Þ Substituting a certain order HAM solutions to Eq (22), we can obtain the corresponding error at that order, which are shown in Table The reliability of analytical results are verified with numerical ones obtained by finite difference technique with Richardson extrapolation [30,31] and results published in literatures [19] and [20], which are shown in Table Figs 2–5 show the effects of stretching strength parameter C on the velocity components in radial, tangential and axial directions and temperature distribution It can be seen that the velocity profiles in the radial and axial directions increase, whereas the velocity profiles in the tangential direction and the thermal boundary layer thickness decrease with the increasing C In order Fig Effects of C on axial velocity profiles H ðηÞ for Cu-water nanofluid with φ ¼ 0:1 and Pr ¼ 6:2 to validate the analytical results obtained by HAM, the numerical solutions are presented in Figs 2–4, the results are in very good agreement Figs 6–9 show the effects of the solid volume fraction of nanoparticles φ for a Cu-water nanofluid on the radial, tangential and axial velocity components and temperature distribution It is indicated that all velocity components decrease, respectively, with the increase in the value of φ The thermal conductivity of nanofluid increases and the thickness for thermal boundary layer increases as well, as the value of φ increases The analytical results for the skin friction coefficient Re1=2 Cf and the local Nusselt number Re À 1=2 Nu, for a wide range of the nanoparticle volume fraction and three different types of nanoparticles are presented in Figs 10 and 11 It is seen that the values of the skin friction coefficient and the local Nusselt number are both increase nearly linearly with the nanoparticle volume fraction Cu has the largest skin friction coefficient and heat transfer rate and Al2O3 has the lowest ones This is because of the largest thermal conductivity value of the Cu compared with other Please cite this article as: Chenguang Yin, et al., (2017), http://dx.doi.org/10.1016/j.jppr.2017.01.004 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 Fig Effects of C on temperature proles ị for Cu-water nanouid with ẳ 0:1 and Pr ¼ 6:2 Fig Effects of φ on temperature proles ị for Cu-water nanouid with ẳ 0:1, Pr ¼ 6:2 and C ¼ 0:1 Fig Effects of φ on radial velocity profiles F ðηÞ for Cu-water nanofluid with φ ¼ 0:1, Pr ¼ 6:2 and C ¼ 0:1 Fig 10 Variation of the skin friction coefficient with φ for different nanoparticles and C with Pr ¼ 6:2 Fig Effects of φ on tangential velocity proles G ị for Cu-water nanouid with ẳ 0:1, Pr ¼ 6:2 and C ¼ 0:1 Fig 11 Variation of the Nusselt number with φ for different nanoparticles and C with Pr ¼ 6:2 Nusselt number increases with the increasing stretching strength parameter C Conclusions Fig Effects of φ on axial velocity profiles H ðηÞ for Cu-water nanofluid with φ ¼ 0:1, Pr ¼ 6:2 and C ¼ 0:1 nanoparticles Fig 10 displays that the increase of stretching strength parameter C leads to increase the values of the skin friction coefficient It also can be seen from Fig 11 that the local In this paper we investigate the flow and heat transfer of nanofluid over a stretching rotating disk with three types of nanoparticles: Cu, CuO and Al2O3 The nonlinear governing equations are transformed into ordinary differential equations by Von Karman transformations and then solved by using homotopy analysis method (HAM) The effects of the stretching strength parameter, the solid volume fraction Please cite this article as: Chenguang Yin, et al., (2017), http://dx.doi.org/10.1016/j.jppr.2017.01.004 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 6 10 11 12 13 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75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 ... investigate the flow and heat transfer of nanofluid over a stretching rotating disk with three types of nanoparticles: Cu, CuO and Al2O3 The nonlinear governing equations are transformed into ordinary... Ω The disk is further stretching at a uniform rate s in the radial direction r Physical model of rotating disk is shown in Fig [13] The governing equations of the nanofluid motion and energy in. .. profiles in the radial and axial directions increase, whereas the velocity profiles in the tangential direction and the thermal boundary layer thickness decrease with the increasing C In order