Hydromagnetic flow of third grade nanofluid with viscous dissipation and flux conditions , T Hussain, S A Shehzad , T Hayat, and A Alsaedi Citation: AIP Advances 5, 087169 (2015); doi: 10.1063/1.4929725 View online: http://dx.doi.org/10.1063/1.4929725 View Table of Contents: http://aip.scitation.org/toc/adv/5/8 Published by the American Institute of Physics AIP ADVANCES 5, 087169 (2015) Hydromagnetic flow of third grade nanofluid with viscous dissipation and flux conditions T Hussain,1 S A Shehzad,2,a T Hayat,3,4 and A Alsaedi4 Faculty of Computing, Mohammad Ali Jinnah University, Islamabad 44000, Pakistan Department of Mathematics, Comsats Institute of Information Technology, Sahiwal 57000, Pakistan Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia (Received 21 June 2015; accepted 14 August 2015; published online 24 August 2015) This article investigates the magnetohydrodynamic flow of third grade nanofluid with thermophoresis and Brownian motion effects Energy equation is considered in the presence of thermal radiation and viscous dissipation Rosseland’s approximation is employed for thermal radiation The heat and concentration flux conditions are taken into account The governing nonlinear mathematical expressions of velocity, temperature and concentration are converted into dimensionless expressions via transformations Series solutions of the dimensionless velocity, temperature and concentration are developed Convergence of the constructed solutions is checked out both graphically and numerically Effects of interesting physical parameters on the temperature and concentration are plotted and discussed in detail Numerical values of skin-friction coefficient are computed for the hydrodynamic and hydromagnetic flow cases C 2015 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License [http://dx.doi.org/10.1063/1.4929725] INTRODUCTION The suspension of nanoparticles such as Al2O3, Cu or CuO in the base fluids like ethylene glycol, oil or water is known as the nanofluid The recent researchers have paid special attention to explore the characteristics of nanofluid It is because of higher thermal performance and their potential role for high heat exchange and with zero pressure drop A combination of nanofluid with biotechnical components has been extensively used in various agricultural, biological sensors and pharmaceuticals processes Different types of nanomaterials like nanofibers, nanostructures, nanowires and nanomachines are utilized in the biotechnological applications The nanofluids can also be involved in heat removal system and the standby safety systems The heat transfer enhancement characteristics of nanofluid are quite useful in the nuclear reactor processes and nuclear reactor safety systems In recent years, the sustainable energy generation is also a challenging issue globally The solar radiation is the most suitable candidate for the renewable energy with less environmental impact By the implementation of solar power, heat, water and electricity can be directly obtained from nature The scientists and researchers have explored that the solar collector processes and heat transfer rate can be improved through the addition of nanoparticles in the base fluids Nanomaterials are the new energy materials because their size is similar or smaller than the coherent or de Brogile waves The nanoparticles are most suitable to strongly absorb the incident radiation Due to such properties of nanomaterials, the nanofluid in solar thermal system is a new study area for the researchers and engineers Choi1 experimentally pointed out that the addition of nanoparticles enhances the thermal conductivity of the fluid twice a Corresponding author email address: ali_qau70@yahoo.com (S.A Shehzad) 2158-3226/2015/5(8)/087169/15 5, 087169-1 © Author(s) 2015 087169-2 Hussain et al AIP Advances 5, 087169 (2015) Buongiorno2 provided a mathematical model to investigate the effects of Brownian motion and thermophoresis on the flow of nanofluid After that the flow of nanofluids is investigated largely by the different researchers under various types of nanoparticles and flow geometries Some investigations on nanofluid can be seen in the Refs 3–10 and many therein Very recently, Sheikholeslami et al.11 numerically examined the unsteady two-phase nanofluid flow between parallel plates in presence of magnetic field Lin et al.12 reported the unsteady pseudo-plastic nanofluid flow and heat transfer over a thin liquid film in presence of variable thermal conductivity and viscous dissipation Analytical solutions of single and multi-phase models of nanofluid with heat transfer are presented by Turkyilmazoglu.13 In another study, Turkyilmazoglu14 discussed the convective heat transfer analysis of nanofluids in circular concentric pipes with partial slip conditions Hayat et al.15 analyzed the similarity solutions of three-dimensional viscoelastic nanofluid due to a bidirectional stretching surface Mixed convection peristaltic flow of water based nanofluids with two-different models is studied by Shehzad et al.16 Mustafa and Khan17 numerically analyzed the boundary layer flow of Casson nanofluid over a nonlinearly stretching sheet The fully developed mixed convection flow of nanofluid in a vertical channel is reported by Das et al.18 The magnetic nanofluid manufacturing applications are enhanced in the industrial processes The magnetic nanofluids are involved in the manufacturing process of biomaterials for wound treatment, gastric medications, sterilized devices etc The magneto nanoparticles significantly appeared in cancer therapy, tumor analysis, sink float separation, construction of loud speakers and many others Recently, Sheikholeslami et al.19 investigated the effects of applied magnetic field and viscous dissipation on nanofluid flow between two horizontal plates in a rotating system Here they calculated the viscosity and effective thermal conductivity of nanofluid through KKL correlation The radiative hydromagnetic flow of Jeffrey nanofluid induced by an exponentially stretching sheet is reported by Hussain et al.20 Mixed convection flow of MHD nanofluid over a linearly stretching sheet in the presence of thermal radiation is examined by Rashidi et al.21 The solutions are developed by employing the shooting technique together with fourth-order Runge-Kutta integration criteria Hayat et al.22 studied the MHD mixed convection peristaltic transport of nanofluid under Soret and Dufour effects They utilized the slip boundary conditions in the presence of Joule heating and viscous dissipation They presented the analysis via long wave length and low Reynolds number approximation Zhang et al.23 considered the MHD flow and radiation effects in nanofluid saturating through a porous medium in presence of chemical reaction They used the variable surface heat flux condition instead of prescribed surface flux condition in this investigation Magnetohydrodynamic free convection flow of Al2O3-water nanofluid under Brownian motion and thermophoresis effects is discussed by Sheikholeslami et al.24 Shehzad et al.25 carried out an analysis to examine the MHD mixed convection peristaltic flow of nanofluid in the presence of Joule heating and thermophoresis In present analysis, we considered the magnetohydrodynamic flow of third grade nanofluid in the presence of viscous dissipation over a stretching sheet Third grade fluid26–30 has an ability to describe the both shear thinning and shear thickening effects This fluid model is the generalization of second grade fluid which only exhibits the effects of normal stress We utilized the prescribed surface heat flux and prescribed surface mass flux conditions In the previous studies, the researchers used the constant surface temperature condition or prescribed heat flux condition There is not a single study in nanofluid literature that dealt with both the conditions Here we introduced the prescribed surface mass flux condition to explore the characteristics of third grade nanofluid with viscous dissipation and thermal radiation Mathematical formulation is made under boundary layer assumptions The expression of thermal radiation is invoked via Rosseland’s approximation Governing nonlinear problems along with corresponding boundary conditions are solved through homotopy analysis method (HAM).31–38 Results of interesting physical parameters on the dimensionless temperature and nanoparticles concentration fields are presented graphically and examined carefully GOVERNING PROBLEMS We consider the two-dimensional steady incompressible flow of third grade nanofluid with thermophoresis and Brownian motion effects The flow is considered to be thermally radiative An 087169-3 Hussain et al AIP Advances 5, 087169 (2015) applied magnetic field of strength B0 is applied normal to the flow direction Effects of induced magnetic field are neglected due to the low magnetic Reynolds number The prescribed surface heat flux and mass flux conditions are imposed at the boundary The governing equations of third grade nanofluid with viscous dissipation and thermal radiation after employing the boundary layer theory are expressed as follows: ∂u ∂v + = 0, ∂x ∂ y u ( ) ∂u ∂ 3u ∂u ∂ 2u ∂u ∂ 2u ∂u ∂ 2u α1 ∂ 3u + v + + +v = ν 2+ u ∂x ∂y ρ ∂ y ∂ x∂ y ∂y ∂ x∂ y ∂ y3 ∂ x ∂ y2 ( )2 2 α2 ∂u ∂ u α3 ∂u ∂ u σB0 +2 − +6 u, ρ ∂ y ∂ x∂ y ρ ∂ y ∂ y2 ρf ( )2 ∂C ∂T DT ∂T ∂T ∂T ∂ 2T u +v = α + τ DB + ∂x ∂y ∂ y ∂ y T∞ ∂ y ∂y )2 ) ( ( α1 ∂u ∂ 2u ∂u ∂u ∂ 2u ν + +v u + (ρc) f ∂ y (ρc) f ∂ y ∂ x∂ y ∂ y ∂ y2 ( )4 α3 ∂u ∂qr +2 − , (ρc) f ∂ y (ρc) f ∂ y u ∂C ∂C ∂ 2C DT ∂ 2T +v = DB + ∂x ∂y T∞ ∂ y ∂y (1) (2) (3) (4) The appropriate boundary conditions for the present flow problems are u = u w (x) = cx, v = 0, at y = 0; u → 0, v → as y → ∞ (5) The boundary conditions for the prescribed heat flux (PHF) and prescribed concentration flux (PCF) are imposed as follows: ( ) ∂T PHF : −k = Tw at y = and T → T∞ when y → ∞, (6) ∂y ( ) ∂C PCF : −D B = Cw at y = and C → C∞ when y → ∞, (7) ∂y in which u and v denote the velocity components parallel to the x− and y− directions, α1, α2 and α3 are the material parameters, ν = (µ/ρ) is the kinematic viscosity, µ is the dynamic viscosity, (ρc) ρ f is the density of fluid, T is the fluid temperature, α is the thermal diffusivity, τ = (ρc)p is the f ratio of nanoparticle heat capacity and the base fluid heat capacity, D B is the Brownian diffusion coefficient, DT is the thermophoretic diffusion coefficient, C is the nanoparticle concentration, Tw is the temperature of the fluid at the wall, Cw is the nanoparticle concentration of the fluid at the wall and T∞ and C∞ are the ambient temperature and nanoparticle concentration far away from the sheet The radiative heat flux qr via Rosseland’s approximation is given below qr = 4σ ∂T , 3k ∗ ∂ y (8) in which σ is the Stefan-Boltzmann constant and k ∗ is mean absorption coefficient It is assumed that the temperature difference within the flow is such that T can be written in the linear combination of temperature By expanding T about T∞ in terms of Taylor’s series and neglecting higher order terms we have T4 4T∞3 T − 3T∞4 , (9) and 16σT∞3 ∂ 2T ∂qr =− ∂y 3k ∗ ∂ y (10) 087169-4 Hussain et al AIP Advances 5, 087169 (2015) Eq (3) now becomes ( )2 ∂T ∂T ∂ 2T ∂C ∂T DT ∂T u +v = α + τ DB + ∂x ∂y ∂ y ∂ y T∞ ∂ y ∂y ( )2 ( ) α1 ∂u ∂ 2u ∂u ∂ 2u ν ∂u + u +v + Cp ∂ y ρCp ∂ y ∂ x∂ y ∂ y ∂ y2 ( ) α3 ∂u 16σT∞ ∂ T +2 + (ρc) f 3k ∗ ∂ y ρCp ∂ y (11) The transformations for the considered flow problems can be put in the forms √ c ν Tw ′ u = cx f (η), v = − cν f (η), η = y , T = T∞ + θ(η), ν c k ν Cw C = C∞ + φ(η) c DB (12) Now Eq (2) is satisfied automatically and the Eqs (3)-(6) and (11) can be expressed as follows: f ′′′ + ff ′′ − f ′2 + β1(2 f ′ f ′′′ − ff ′′′′) + (3 β1 + β2) f ′′2 + 6ε 1ε f ′′′ f ′′2 − M f ′ = 0, ) ( + Rd θ ′′ + Pr f θ ′ + Pr Nbθ ′φ ′ + Pr Ntθ ′2 + Pr Ec f ′′2 + Pr Ecε f ′ f ′′2 − Pr Ecε 1ff ′′ f ′′′ + Pr Ecφφ1 f ′′4 = 0, (13) φ ′′ + Pr Le f φ ′ + (Nt/Nb) θ ′′ = 0, f = 0, f ′ = 1, θ ′ = −1, φ ′ = −1 at η = 0, f ′ → 0, θ → 0, φ → as η → ∞ (15) (16) (17) (14) cα Here β1 = cαµ , β2 = cαµ , ε = µ are the material parameters for third grade fluid, ε = cνx is the local Reynolds number, M = σB02/ρ f c is the Hartman number, Pr = ν/α is the Prandtl number, u2 Rd = 4σT∞3 /k k ∗ is the radiation parameter, Ec = c pTw∞ is the Eckert number, Nb = (ρc) p D B(C f − C∞)/(ρc) f ν is the Brownian motion parameter, Nt = (ρc) p DT /(ρc) f ν is the thermophoresis parameter and Le = α/D B is the Lewis number The dimensionless expressions of skin-friction coefficient, local Nusselt and Sherwood numbers can be expressed as follows: ′′ ′ ′′ ′′′ ′′3 Cf x Re1/2 x = f (0) + β1(3 f (0) f (0) − f (0) f (0)) + 2ε 1ε f (0), ( ) ( ) 1 NuRe−1/2 = + Rd , ShRe−1/2 = x x θ(0) φ(0) (18) DEVELOPMENT OF SERIES SOLUTIONS To develop the solution expressions via homotopy analysis method, the initial guesses and auxiliary linear operators are expressed in the forms (Liao (2009)): f 0(η) = − exp(−η), θ 0(η) = exp(−η), φ0(η) = exp(−η), L f = f ′′′ − f ′, L θ = θ ′′ − θ, L φ = φ ′′ − φ (19) (20) The above initial guesses and auxiliary linear operators satisfy the following conditions L f (C1 + C2eη + C3e−η ) = 0, L θ (C4eη + C5e−η ) = 0, L φ (C6eη + C7e−η ), (21) where Ci (i = − 7) are the arbitrary constants The problems at zeroth order deformation are given by Hayat et al (2014b), Abbasbandy et al (2014) and Shehzad et al (2014b): (1 − p) L f fˆ(η; p) − f 0(η) = p f N f fˆ(η; p) , (22) 087169-5 Hussain et al AIP Advances 5, 087169 (2015) ˆ ˆ p), φ(η, ˆ p) , (1 − p) L θ θ(η; p) − θ 0(η) = p θ Nθ fˆ(η; p), θ(η, ˆ p), φ(η, ˆ ˆ p) , (1 − p) L φ φ(η; p) − φ0(η) = p φ Nφ fˆ(η; p), θ(η, (23) (24) fˆ(0; p) = 0, fˆ′(0; p) = 1, θˆ′(0, q) = −1, φˆ′(0, q) = −1, (25) ˆ ˆ fˆ′(∞; p) = 0, θ(∞, p) = 0, φ(∞, p) = 0, )2 ( ∂ fˆ(η, p) ∂ fˆ(η, p) ∂ fˆ(η, p) ∂ fˆ(η, p) ˆ ˆ N f [ f (η, p)] = − M2 + f (η, p) − ∂η ∂η ∂η( ∂η ) ˆ ˆ ∂ ∂ f (η, q) f (η, q) + β1 fˆ(η, q) − fˆ(η, q) ∂η ∂η ( ) ( )2 ∂ fˆ(η, p) ∂ fˆ(η, p) ∂ fˆ(η, p) + (3 β1 + β2) + 6ε ε , ∂η ∂η ∂η ( ) ˆ p) ˆ p) ∂ θ(η, ∂ θ(η, ˆ ˆ ˆ + Pr fˆ(η, p) Nθ [θ(η, p), f (η, p), φ(η, p)] = + Rd ∂η ∂η ( ) ˆ p) ∂ φ(η, ˆ p) ˆ p) ∂ θ(η, ∂ θ(η, + Pr Nb + Pr Nt ∂η ∂η ∂η ( )2 ( )2 ˆ ˆ ∂ f (η, p) ∂ f (η, p) ∂ fˆ(η, p) + Pr Ec + Pr Ec β ∂η ∂η ∂η ( )2 ∂ fˆ(η, p) ∂ fˆ(η, p) − Pr Ec β1 fˆ(η, p) ∂η ∂η ( )4 ∂ fˆ(η, p) + Pr Ecε 1ε , ∂η 2ˆ 2ˆ ˆ ˆ p)] = ∂ φ(η, p) + Pr Le fˆ(η, p) ∂ φ(η, p) + Nt ∂ θ(η, p) ˆ p), fˆ(η, p), θ(η, Nφ [φ(η, ∂η N b ∂η ∂η (26) (27) (28) In above equations, p is an embedding parameter, f , θ and φ are the non-zero auxiliary parameters and N f , Nθ and Nφ are the nonlinear operators For p = and p = we have (Rashidi et al (2014b): ˆ 0) = θ 0(η), φ(η, ˆ 0) = φ0(η), fˆ(η; 0) = f 0(η), θ(η, ˆ 1) = θ(η), φ(η, ˆ 1) = φ(η) fˆ(η; 1) = f (η), θ(η, (29) When variation of p is considered from to then f (η, p), θ(η, p) and φ(η, p) vary from f 0(η), θ 0(η), φ0(η) to f (η), θ(η) and φ(η) Using Taylor series we can write (Hayat et al (2014c) and Shehzad et al (2014c)): f (η, p) = f 0(η) + ∞ f m (η)pm , f m (η) = m=1 θ(η, p) = θ 0(η) + ∞ θ m (η)pm , θ m (η) = m=1 φ(η, p) = φ0(η) + ∞ φ m (η)pm , φ m (η) = m=1 ∂ m f (η; p) m! ∂η m p=0 ∂ m θ(η; p) m! ∂η m p=0 ∂ m φ(η; p) m! ∂η m p=0 , (30) , (31) (32) The convergence of Eqs (30)-(32) strongly depends upon the suitable values of auxiliary parameters f , θ and φ We select that f , θ and φ in such a manner that Eqs (30)-(32) converge for p = and thus one has f (η) = f 0(η) + ∞ m=1 f m (η), (33) Hussain et al 087169-6 AIP Advances 5, 087169 (2015) θ(η) = θ 0(η) + ∞ θ m (η), (34) φ m (η) (35) m=1 φ(η) = φ0(η) + ∞ m=1 The general solutions are represented by the following expressions: ∗ f m (η) = f m (η) + C1 + C2eη + C3e−η , θ m (η) = φ m (η) = in which ∗ fm θ ∗m and φ∗m θ ∗m (η) + C4eη + C5e−η , φ∗m (η) + C6eη + C7e−η , (36) (37) (38) are the special solutions ANALYSIS AND DISCUSSION The equations (30)-(33) depends upon the auxiliary parameters f , θ and φ which have vital role in controlling and adjusting the convergence regions for the development of homotopic solutions For this purpose, we plotted the − curves at 15th-order of approximations that give us the proper ranges of these auxiliary parameters Fig demonstrates that the admissible values of f , θ , φ are −0.90 ≤ f ≤ −0.20, −0.90 ≤ θ ≤ −0.40 and −0.90 ≤ φ ≤ −0.40 Here the series solutions are convergent in the whole region of η for f = −0.6 = θ = φ The effects of interesting physical parameters namely material parameters β1 and β2, Hartman number M, Prandtl number Pr, Brownian motion parameter N b, thermophoresis parameter Nt, Eckert number Ec and thermal radiation parameter Rd on the dimensionless temperature field θ(η) are investigated in the Figs 2-9 Figs and are plotted to see the behavior of the temperature profile for different values of material parameter β1 and β2 Here we examined that an increase in the material parameters β1 and β2 show a reduction in the temperature and thermal boundary layer thickness It is also seen that the temperature at the wall also decreases for the larger material parameters β1 and β2 Physically, an increase in material parameters β1 and β2 correspond to an enhancement in the normal stress differences due to which lower temperature and thinner thermal boundary layer thickness are achieved Fig presents the variations in temperature profile for different values of Hartman number This Fig depicts that the temperature field and thermal boundary layer thickness are stronger for hydromagnetic flow situation in comparison to hydrodynamic FIG – curves for functions f (η), θ(η) and φ(η) when β = 0.2 = ε 1, β = 0.1 = ε 2, Pr = 0.7 = Le, Nt = 0.1 = Nb, M = 0.5 = Ec and Rd = 0.3 087169-7 Hussain et al AIP Advances 5, 087169 (2015) FIG Influence of β on temperature θ(η) when ε = 0.2, β = 0.1 = ε 2, Pr = 0.7 = Le, Nt = 0.1 = Nb, M = 0.5 = Ec and Rd = 0.3 FIG Influence of β on temperature θ(η) when β = 0.2 = ε 1, ε = 0.1, Pr = 0.7 = Le, Nt = 0.1 = Nb, M = 0.5 = Ec and Rd = 0.3 FIG Influence of M on temperature θ(η) when β = 0.2 = ε 1, β = 0.1 = ε 2, Pr = 0.7 = Le, Nt = 0.1 = Nb, Ec = 0.5 and Rd = 0.3 087169-8 Hussain et al AIP Advances 5, 087169 (2015) FIG Influence of Pr on temperature θ(η) when β = 0.2 = ε 1, β = 0.1 = ε 2, Le = 0.7, Nt = 0.1 = Nb, M = 0.5 = Ec and Rd = 0.3 FIG Influence of Nb on temperature θ(η) when β = 0.2 = ε 1, β = 0.1 = ε 2, Pr = 0.7 = Le, Nt = 0.1, M = 0.5 = Ec and Rd = 0.3 FIG Influence of Nt on temperature θ(η) when β = 0.2 = ε 1, β = 0.1 = ε 2, Pr = 0.7 = Le, Nb = 0.1, M = 0.5 = Ec and Rd = 0.3 087169-9 Hussain et al AIP Advances 5, 087169 (2015) FIG Influence of Ec on temperature θ(η) when β = 0.2 = ε 1, β = 0.1 = ε 2, Pr = 0.7 = Le, Nt = 0.1 = Nb, M = 0.5 and Rd = 0.3 flow (M = 0) The larger values of Hartman number lead to higher temperature and thicker thermal boundary layer thickness Obviously Lorentz force appears in Hartman number Hence Lorentz force resists the flow that caused to en enhancement in the temperature and thermal boundary layer thickness From Fig we studied that an increase in the values of Prandtl number creates a reduction in the temperature field and its related boundary layer thickness An enhancement in Prandtl number corresponds to weaker thermal diffusivity It is well known fact that the fluids with weaker thermal diffusivity have lower temperature Due to such weaker thermal diffusivity, lower temperature and thinner thermal boundary layer thickness is obtained in Fig It is also seen that the temperature at the wall is decreased gradually for the larger Prandtl number Figs and are displayed to observe the influences of Brownian motion and thermophoresis parameters on the dimensionless temperature field θ(η) These Figs illustrate that an enhancement in the Brownian motion and thermophoresis parameters lead to larger temperature and thicker thermal boundary layer thickness Clearly the Brownian motion and thermophoresis parameters appeared due to the presence of nanoparticles The presence of nanoparticles enhances the thermal conductivity of fluid The fluids with stronger thermal conductivity have higher temperature Due to such reasons an enhancement in the temperature and thermal boundary layer thickness is observed corresponding to the larger values of Brownian motion and thermophoresis parameters From Fig 8, it is examined that the temperature and thermal boundary layer thickness are increased when we give rise to the values of Eckert number Ec Temperature is higher in the presence of viscous FIG Influence of Rd on temperature θ(η) when β = 0.2 = ε 1, β = 0.1 = ε 2, Pr = 0.7 = Le, Nt = 0.1 = Nb and M = 0.5 = Ec 087169-10 Hussain et al AIP Advances 5, 087169 (2015) FIG 10 Influence of β on nanoparticles concentration φ(η) when ε = 0.2, β = 0.1 = ε 2, Pr = 0.7 = Le, Nt = 0.1 = Nb, M = 0.5 = Ec and Rd = 0.3 dissipation in comparison to case without viscous dissipation In Fig 8, Ec = corresponds to no viscous dissipation effect case An increase in Eckert number corresponds to an increase in kinetic energy of the fluid due to which temperature and thermal boundary layer thickness are enhanced Fig shows the variations in temperature profile θ(η) for different values of thermal radiation parameter Rd From this Fig it is analyzed that temperature and its related boundary layer thickness are increasing functions of thermal radiation parameter Here Rd = corresponds to flow analysis without thermal radiation The temperature is weaker in absence of thermal radiation effects in comparison to temperature in the presence of thermal radiation Figs 10-15 are sketched to see the influences of material parameters β1 and β2, Hartman number M, Brownian motion parameter N b, thermophoresis parameter Nt and Lewis number Le on the dimensionless nanoparticles concentration φ(η) Figs 10 and 11 elucidate that both nanoparticles concentration and its related boundary layer thickness are reduced for the larger values of material parameters β1 and β2 A comparison of Figs 2, 3, 10, and 11 depicts that the effects of material parameters β1 and β2 are qualitatively similar for temperature and nanoparticles concentration profiles It is also observed that the nanoparticles concentration at the wall corresponding to different values of material parameters β1 and β2 is higher when compared with the temperature at the wall From Fig 12 it is analyzed that the nanoparticles concentration profile is higher for hydromagnetic flow (M > 0) and lower for hydrodynamic case (M = 0) It is also observed that the nanoparticles concentration is enhanced and going away from the wall of the sheet for hydromagnetic flow Fig 13 presents that an enhancement in Brownian motion parameter shows a reduction in the FIG 11 Influence of β on nanoparticles concentration φ(η) when β = 0.2 = ε 1, ε = 0.1, Pr = 0.7 = Le, Nt = 0.1 = Nb, M = 0.5 = Ec and Rd = 0.3 087169-11 Hussain et al AIP Advances 5, 087169 (2015) FIG 12 Influence of M on nanoparticles concentration φ(η) when β = 0.2 = ε 1, β = 0.1 = ε 2, Pr = 0.7 = Le, Nt = 0.1 = Nb, Ec = 0.5 and Rd = 0.3 FIG 13 Influence of Nb on nanoparticles concentration φ(η) when β = 0.2 = ε 1, β = 0.1 = ε 2, Pr = 0.7 = Le, Nt = 0.1, M = 0.5 = Ec and Rd = 0.3 FIG 14 Influence of Nt on nanoparticles concentration φ(η) when β = 0.2 = ε 1, β = 0.1 = ε 2, Pr = 0.7 = Le, Nb = 0.1, M = 0.5 = Ec and Rd = 0.3 087169-12 Hussain et al AIP Advances 5, 087169 (2015) FIG 15 Influence of Le on nanoparticles concentration φ(η) when β = 0.2 = ε 1, β = 0.1 = ε 2, Pr = 0.7, Nt = 0.1 = Nb, M = 0.5 = Ec and Rd = 0.3 nanoparticles concentration and its associated boundary layer thickness From Fig 14 we investigated that an increase in thermophoresis parameter leads to the higher nanoparticles concentration and boundary layer thickness Due to presence of nanoparticles, the nanoparticles concentration profile is enhanced rapidly in Fig 14 Fig 15 shows that the larger values of Lewis number correspond to a reduction in the nanoparticles concentration field and its associated boundary layer thickness Table I is given to examine the numerical values of f ′′(0), θ ′′(0) and φ ′′(0) at different order of HAM approximations when β1 = 0.2 = ε 1, β2 = 0.1 = ε 2, Pr = 0.7, M = 0.5, Le = 2.0, Nt = 0.2 = N b, Ec = 0.5, Rd = 0.3, Le = 2.0 and f = −0.6 = θ = φ This Table clearly shows that the convergent values of f ′′(0) are achieved at 13th order of HAM deformations On the other hand, the values of θ ′′(0) and φ ′′(0) start to repeat from 35th order of HAM deformations Hence we conclude that 35th order of HAM approximations give us the convergent solutions of velocity, temperature and nanoparticles concentration The values of skin-friction coefficient C f x Re1/2 x for different values of β1, β2, ε and ε when M = 0.0, 0.4 and M = 1.0 are investigated in Table II Here M = corresponds to hydrodynamic flow while M > represents the hydromagnetic fluid flow It is visualized that the values of skin-friction coefficient are larger for hydromagnetic flow in comparison to hydrodynamic case Such values are increased for larger Hartman number M The Lorentz force is stronger for the larger Hartman number which resists the fluid flow that leads to an enhancement in the values of skin-friction coefficient The larger values of skin-friction correspond to higher material parameter β1 An increase in the values of β1 corresponds to the larger material parameter α1 Such higher material parameter gives rise to the stress rate at the wall The values of C f x Re1/2 x are decreased when we give rise to β2, ε and ε The normal stress differences are increased for the larger values of β2, ε and ε that corresponds to smaller values of skin-friction TABLE I Convergence of homotopy solution for different order of approximations when β = 0.2 = ε 1, β = 0.1 = ε 2, Pr = 0.7, M = 0.5, Le = 2.0, Nt = 0.2 = Nb, Ec = 0.5, Rd = 0.3, Le = 2.0, f = −0.6 = θ = φ Order of approximation − f ′′(0) θ ′′(0) φ ′′(0) 13 20 30 35 45 50 0.79700 0.80851 0.80850 0.80850 0.80850 0.80850 0.80850 0.80850 0.50328 0.50369 0.51622 0.52269 0.52426 0.52491 0.52491 0.52491 0.82000 1.10803 1.10305 1.09887 1.09790 1.09716 1.09716 1.09716 087169-13 Hussain et al AIP Advances 5, 087169 (2015) TABLE II Numerical values of skin-friction coefficient C f x Re1/2 x for different values of β 1, β 2, ε and ε when M = 0.0, 0.4 and M = 1.0 −C f x Re1/2 x M = 0.0 −C f x Re1/2 x M = 0.4 −C f x Re1/2 x M = 1.0 β1 β2 ε1 ε2 0.0 0.3 0.5 0.1 0.2 0.1 0.92032 1.2673 1.5221 0.98735 1.3615 1.6240 1.2735 1.7747 2.0429 0.2 0.0 0.2 0.4 0.2 0.1 1.2125 1.1216 1.0436 1.3025 1.2049 1.1211 1.6950 1.5690 1.4610 0.2 0.1 0.0 0.2 0.4 0.1 1.1719 1.1622 1.1562 1.2600 1.2479 1.2405 1.6480 1.6211 1.6056 0.2 0.1 0.2 0.0 0.2 0.4 1.1719 1.1591 1.1448 1.2600 1.2442 1.2293 1.6480 1.6131 1.5851 and Sherwood numcoefficient Table III presents the values of local Nusselt number NuRe−1/2 x for various values of Rd, Le, N , N and Ec when β = 0.2 = ε ber ShRe−1/2 t b 1, β2 = 0.1, ε = 0.4, x M = 0.5 and Pr = 1.0 From this Table it is observed that the values of NuRe−1/2 are reduced while x the values of ShRe−1/2 increase when we enhance the values of Rd Physically, the larger values of x Rd give more heat to fluid due to which the heat transfer rate at the wall is decreased while mass transfer rate enhances An increase in Lewis number corresponds to larger NuRe−1/2 and ShRe−1/2 x x The Brownian diffusion coefficient is weaker for larger Lewis number Such weaker Brownian diffusion coefficient reduced the heat and mass transfer but enhanced the heat and mass transfer rates at the wall The larger values of thermophoresis parameter Nt create a reduction in the Nusselt and Sherwood numbers Physically, thermal conductivity of fluid is enhanced for larger Nt Such higher thermal conductivity reduce the heat and mass transfer rates at the wall The larger values of Eckert number Ec lead to a reduction in Nusselt number while these enhance the Sherwood number The TABLE III Numerical values of NuRe−1/2 and ShRe−1/2 for different values of Rd, Le, N t , N b and Ec when β = 0.2 = ε 1, x x β = 0.1, ε = 0.4, M = 0.5 and Pr = 1.0 Rd Le Nt Nb Ec NuRe−1/2 x ShRe−1/2 x 0.0 0.2 0.5 1.0 0.2 0.2 0.5 0.7519 0.6781 0.5938 0.6703 0.8540 1.1314 0.3 0.5 1.5 2.0 0.2 0.2 0.5 0.6395 0.6518 0.6548 0.5712 1.2643 1.5428 0.3 1.0 0.1 0.3 0.5 0.2 0.5 0.6716 0.6230 0.5766 1.1432 0.8098 0.6339 0.3 1.0 0.2 0.1 0.3 0.5 0.5 0.6651 0.6293 0.5956 0.6966 1.0745 1.2044 0.3 1.0 0.2 0.2 0.0 0.4 0.8 0.7350 0.6632 0.6034 0.9296 0.9427 0.9562 087169-14 Hussain et al AIP Advances 5, 087169 (2015) stronger viscous dissipation appeared due to larger Ec that corresponds to lower values of Nusselt number CONCLUSIONS In this analysis, we discussed the steady flow of third grade fluid with thermophoresis, Brownian motion and magnetic field effects Heat transfer characteristics are explored in the presence of thermal radiation and viscous dissipation Heat and mass flux conditions are imposed at the boundary The main points of this article are summarized below: • Temperature field θ(η) and nanoparticles concentration φ(η) are larger for smaller material parameters β1 and β2 and it starts to decrease when we enhance the values of β1 and β2 • The temperature θ(η) and nanoparticles concentration φ(η) are enhanced for the hydromagnetic flow M > when we compared it with the hydrodynamic case • The larger values of thermophoresis parameter Nt correspond to higher temperature and nanoparticles concentration The higher temperature is achieved due to higher thermal conductivity of fluid Such higher thermal conductivity is obtained due to presence of nanoparticles • Temperature field θ(η) and thermal boundary layer thickness is increased when we increase the values of thermal radiation parameter and Eckert number Here we also noticed that the temperature enhances rapidly for increasing values of thermal radiation parameter in comparison to Eckert number • Brownian motion parameter leads to an increase in temperature field but a reduction in the nanoparticles concentration In nanoparticles concentration equation, Brownian motion parameter is appeared in denominator due to which a decrease is shown • The values of skin-friction coefficient corresponding to increase in material parameter β1 and Hartman number M are larger These values are reduced when we increase the values of β2, ε and ε S.U.S Choi, in The Proceedings of the 1995 ASME International Mechanical Engineering Congress and Exposition (ASME, San Francisco, USA, 1995), FED 231/MD Vol 66, pp 99–105 J Buongiorno, Journal of Heat Transfer 128, 240-250 (2006) W Ibrahim and O.D Makinde, Computers and Fluids 86, 433-441 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