newtonian heating effect in nanofluid flow by a permeable cylinder

Results in Physics (2017) 256–262 Contents lists available at ScienceDirect Results in Physics journal homepage: www.journals.elsevier.com/results-in-physics Newtonian heating effect in nanofluid flow by a permeable cylinder T Hayat a,b, M Ijaz Khan a,⇑, M Waqas a, A Alsaedi b a Department of Mathematics, Quaid-i-Azam University 45320 Islamabad 44000, Pakistan Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P O Box 80257, Jeddah 21589, Saudi Arabia b a r t i c l e i n f o Article history: Received 22 August 2016 Received in revised form 19 November 2016 Accepted 20 November 2016 Available online 23 November 2016 Keywords: Stretched cylinder Nanofluid Newtonian heating Brownian motion Thermophoresis a b s t r a c t Here characteristics of Newtonian heating in permeable stretched flow of viscous nanomaterial are investigated Adopted nanomaterial model incorporates the phenomena of Brownian motion and thermophoresis Concept of boundary layer is employed for the formulation procedure Convergent homotopic solutions are established for the nonlinear systems Velocity, thermal and nanoparticles fields for nonlinear boundary value problems are computed and discussed The velocity, temperature and concentration gradients are also evaluated It is noticed that impacts of curvature and suction/injection parameters on skin friction coefficient are qualitatively similar Moreover temperature distribution enhances for larger thermophoresis and Brownian motion parameters Ó 2016 Published 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 It is well known fact that improvement of human culture enormously depends the energy sources Therefore specialists and researchers are attempting to build up the new energy assets and energy innovations in order to utilize the sun powered/solar energy which is very suitable, easily available and friendly source for the various heating processes in technological and industrial processes (which when it reaches the earth is about  1015 MW) This solar energy is 2000 times bigger than the worldwide energy utilization The term nanofluid is coined by Choi [1] This pioneering experimental research witnessed thermal conductivity enhancement of nanofluid He concluded that addition of very small amount of nanoparticles to traditional heat transfer liquids enhanced the thermal conductivity of liquid up to two times Nanoparticles have various shapes for example spherical, rod-like or tubular Impact of Soret and Dufour on MHD convective radiative heat and mass transfer is presented by Pal et al [2] Turkyilmazoglu [3] examined boundary layer unsteady flow of nanofluid with heat transfer in the presence of vertical flat plate Hussain et al [4] explored magnetohydrodynamic flow of Jeffrey nanomaterial induced by exponentially stretched surface in the presence of Joule heating and thermal radiation Behavior of magnetohydrodynamic nanofluid flow over a permeable exponentially stretched ⇑ Corresponding author E-mail addresses: mikhan@math.qau.edu.pk (M.I Khan), mw_qau88@yahoo com (M Waqas) surface was presented by Bhattacharyya and Layek [5] Rashidi et al [6] analyzed MHD flow of nanofluid with entropy generation due to rotating porous disk Influences of viscous dissipation and melting heat transport in MHD stretched flow of viscous nanoliquid is reported by Mabood and Mastroberardino [7] Hayat et al [8] addressed Joule heating and radiation characteristics in magneto thixotropic nanofluid Analysis of power law nanoliquid towards stretched surface with mixed convection is presented by Si et al [9] Hayat et al [10] analyzed hydromagnetic mixed convective flow of viscous nanomaterial subject to curved stretching surface Further relevant studies on nanofluids can be seen in [11–15] and many studies therein Fluid flow over a flat plate or stretching cylinder has promising uses in engineering and industrial processes such as polymer expulsion, in a melt turning forms, streamlined expulsion of plastic sheets, glass fiber creation, the cooling and drying of paper and materials, water funnels, sewer funnels, watering system channels, veins and so forth Flow caused by stretching of a sheet is initially investigated by Crane [16] Hsiao [17] examined heat and mass transfer in mixed convection MHD flow of viscoelastic fluid past a stretched surface Lin et al [18] analyzed stretched flow of pseudo-plastic nanofluid Rosca and Pop [19] analyzed fluid flow over an unsteady curved stretching/shrinking surface Abbas et al [20] examined radiative flow of nanofluid by a curved stretched surface with partial slip Mixed convection and variable thermal conductivity effects in viscoelastic nanofluid flow over a stretched cylinder is studied by Hayat et al [21] Simultaneous influences of magnetic dipole and homogeneous/heterogeneous http://dx.doi.org/10.1016/j.rinp.2016.11.047 2211-3797/Ó 2016 Published 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/) 257 T Hayat et al / Results in Physics (2017) 256–262 reactions in stretched flows of ferrofluid is explored by Hayat et al [22,23] Alsaedi et al [24] established homotopic solutions for viscous nanomaterial considering gyrotactic microorganisms Mixed convection and viscous dissipation characteristics in nonlinear stretching flow of micropolar material with convective condition is addressed by Waqas et al [25] Hayat et al [26] examined the impacts of variable properties in three-dimensional mixed convective flow induced by exponential surface Waqas et al [27] and Hayat et al [28] explored the characteristics of non-Fourier flux theory in flows of generalized Burgers and Jeffrey fluids Farooq et al [29] examined nonlinear radiation and magnetohydrodynamics (MHD) T in convective flow of viscoelastic nanoliquid In many realistic cases the heat transfer from the surface is proportional to the local surface temperature Such an effect is known as Newtonian heating effect Researchers utilized the Newtonian heating process in their practical life applications such as conjugate heat transfer around fins, to design heat exchanger and also in convection flows setup where the surrounding bounding surfaces absorb heat by solar radiations Merkin [30] presented the boundary layer natural convection flow by a vertical surface in the presence of Newtonian heating Effect of Newtonian heating on MHD unsteady flow with Navier slip is analyzed by Makinde [31] Shehzad et al [32] studied Jeffrey fluid flow in three dimensional with Newtonian heating Hayat et al [33] examined stagnation region flow of carbon nanotubes with chemical reactions in the presence of Newtonian heating Newtonian heating and heat generation/ absorption characteristics in flows of micropolar, Jeffrey and Eyring-Powell materials are discussed by Hayat et al [34–36] From above mentioned investigations, it has been noticed that the heat transfer analysis in the past has been mostly dealt with the boundary condition either through prescribed temperature or heat flux at the surface Few studies in this direction are made using temperature Newtonian heating condition at the surface instead of prescribed surface temperature or heat flux However, no attempt is yet presented for the Newtonian heating mass condition at the surface Theme of this study is to introduce Newtonian mass flux condition in the literature Even such condition has not been utilized yet in flow analysis without nanoparticles We thus attempt here the flow by a permeable stretching cylinder with Newtonian heating and mass flux conditions Series solutions are computed by homotopy analysis method (HAM) [37–50] The behaviors of different parameters on the physical quantities have been examined u @C @C @ C @C ỵv ẳ DB þ @x @r @r r @r DT @ T @T : ỵ T @r r @r ux; rị ẳ uw xị ẳ 1ị u ! 0; T ! T ; C ! C as r ! 1: ð5Þ where the velocity components in (x; r) direction are (u; v ) respectively, m denotes kinematic viscosity, q the density, cp the specific heat, s ¼ ðqcÞp ðqcÞf the heat capacity ratio, ðqcÞp the heat capacity of fluid, ðqcÞf the effective heat capacity of nanoparticles, DB the Brownian diffusion coefficient, DT the thermophoresis diffusion coefficient, T the ambient temperature, T the temperature, C the nanoparticles concentration, ðT ; C Þ the ambient fluid temperature and concentration respectively, uw ðxÞ the velocity of stretching surface, v w the suction/injection velocity, h1 the heat transfer coefficient and h2 the mass diffusion coefficient Using u¼ u0 x R f gị; v ẳ l r /gị ẳ C C1 ;g ẳ C1 r u0 m T T1 ; f gị; hgị ẳ l T1 ! rffiffiffiffiffi u0 r À R2 ; ml 2R ð6Þ continuity condition is indistinguishably fulfilled and Eqs (2)–(5) can be reduced as follows: 000 00 00 ð1 ỵ 2agịf ỵ 2af ỵ ff f 02 ẳ 0; 7ị ỵ 2agịh00 ỵ 2ah0 ỵ Prf h0 ỵ Pr1 ỵ 2agị N b h0 /0 ỵ Nt h02 ẳ 0; ỵ 2agị/00 ỵ 2a/0 þ Scf /0 þ f ¼ S; Nt ðð1 ỵ 2agịh00 ỵ 2ah0 ị ẳ 0; Nb 2ị !    2 ! @T @T k @ T @T @C @T DT @T ỵ s DB u ; ỵ ỵv ẳ ỵ @x @r @r @r T @r qcp @r2 r @r ð3Þ ð8Þ ð9Þ h0 0ị ẳ c1 ỵ h0ịị; f ẳ 1; /0 0ị ẳ c2 ỵ /0ịị at g ẳ 0; ð10Þ qffiffiffiffiffiffiffi À Á is the curvature parameter, N b ¼ sDm B the À lc Á Brownian motion parameter, Pr ¼ k p the Prandtl number,     Nt ¼ Ts1DTm the thermophoretic parameter, Sc ¼ DmB the Lewis where a ¼ ml u0 R2 number, S the suction (S > 0) and S the injection (S < 0) and   qffiffiffiffi qffiffiffiffi c1 ¼ h1 uv0l and c2 ¼ h2 uv0l are the conjugate heat and mass parameters respectively À Á Skin friction coefficient C f Nusselt number ðNux Þ and Sherwood number ðShx Þ are Cf ¼ ð4Þ u0 x @T @C ; v x; rị ẳ v w ; ẳ h1 T; ¼ Àh2 C at r ¼ R; l @r @r Here we assume the steady axisymmetric flow of viscous nanomaterial along a stretched cylinder of radius R Heat and mass transfer analysis is reported in presence of Brownian motion, Newtonian heating and thermophoresis The flow here is being assumed in the axial ðxÞ direction Radial direction is normal to x Fluid is assumed in compression Whole treatment is considered via boundary layer assumptions Suction through permeable cylinder is considered Stretching velocity for cylinder is linear The governing equations are @u @u ỵv ẳm u @x @r ỵ The boundary condition for present flow are designed in the form:  ! @ u @u ; ỵ @r2 r @r ! f ! 0; h ! 0; / ! as g ! 1; Governing problems @ðruÞ @ðr v ị ỵ ẳ 0; @x @r ! 2sw xqw xqm ; Nux ¼ ; Shx ¼ ; qu2w kðT w À T Þ DB ðC w À C Þ ð11Þ where sw ¼ l   @u ; @r r¼R qw ¼ Àk In dimensional form   @T ; @r r¼R qm ¼ ÀDB   @C : @r rẳR 12ị 258 T Hayat et al / Results in Physics (2017) 256–262 00 ¼ c1 C f Rex1=2 ẳ f 0ị; Nux Re1=2 x   ẳ c2 ỵ ; Shx Re1=2 x /0ị   1ỵ ; h0ị 13ị where Rex ẳ uwvxị ẳ u0vxl as the Reynolds number Homotopic solutions and convergence In 1992, Liao [37] initiated the concept of homotopy analysis method (HAM) This method is used to solve highly nonlinear equations the initial guesses and auxiliary linear operators along with associate characteristic are f ðgÞ ẳ S ỵ eg ; ẳ c2 c2 000 h0 gị ẳ /0 gị 14ị Lf C ỵ C eg ỵ C eg ị ẳ 0; g expgị; expgị; Lh ẳ h00 À h; Lf ¼ f À f ; c1 c1 L/ ẳ /00 /; 15ị Lh C eg ỵ C eg ị ẳ 0; g L/ C e ỵ C e ị ẳ 0; 16ị where C i i ẳ 7ị indicate the arbitrary constants Employing HAM and solving the corresponding zeroth order and mth deformation problems one obtains f m gị ẳ f m gị ỵ C ỵ C eg ỵ C eg ; 17ị hm gị ẳ hm gị ỵ C eg ỵ C eg ; 18ị /m gị ẳ /m gị ỵ C eg ỵ C eg ; 19ị f m ; hÃm in which the and of C i ẳ i ẳ 7ị are /m indicate the special solutions The values  à  @f m gị ; @g gẳ0   @hm gị ỵ c1 hm gị gẳ0 @g C ẳ C ¼ C ¼ and C ¼ à C ¼ ÀC À f m 0ị; C ẳ   @/m gị ỵ c2 /m gị gẳ0 @g gẳ0 C7 ẳ c2 gẳ0 c1 ; 20ị Fig  h-curve for f ; h and / are addressed through graphs and Tables To achieve this motto Figs 2–13 along with Tables 2–4 are presented Fig is displayed for the effect of S on f ðgÞ For higher S ð> 0Þ some of the fluid particles are sucked through the cylinder which provides a resistance to the fluid flow and therefore the f ðgÞ decreases Fig disclosed the features of a on f ðgÞ It is noticed that f ðgÞ decays near the stretching surface while it augments as one moves away from the stretching surface Physically an increase in a reduces the radius of cylinder due to which the contact area of the cylinder with fluid is reduced Therefore less resistance is pro0 vided by the surface and consequently f ðgÞ enhances The analysis of Pr on hðgÞ is described in Fig It is found that an increase in Pr reduces hðgÞ The increase in Pr causes the thinning of thermal boundary layer which enhances the heat transfer rate As a result the temperature of fluid reduces Hence Pr can be utilized to control the relative thickness of momentum and associated boundary layer in heat transfer phenomenon Behavior of c1 on hðgÞ is analyzed in Fig Clearly hðgÞ and its associated boundary layer thickness enhances Heat transfer coefficient increases for larger c1 Therefore more heat transfers from the heated surface of cylinder to the cooled surface of the fluid and as a whole temperature of the fluid increases which transfers more heat from the cylinder to the fluid It is noticed that c1 ¼ relates to insulated wall while c ! represents the constant wall temperature Subsequently c1 can be utilized as a cooling operator as a part of the progressed innovative procedure Fig displays the variation of N t on hðgÞ It is inspected that temperature of the fluid Convergence analysis Our motto here is to ensure the convergence of developed series solutions through homotopy analysis method (HAM) Auxiliary À Á variables  hf ;  hh ;  h/ in the developed series solutions have crucial role for such motto Therefore Fig outlines the  h-plots for 14th order of approximations It is noticed that the permissible values   hh and h/ are of the auxiliary parameters hf ;  hh À0:35 and À1:70  h/ À0:30 À1:35  hf À0:30; À1:50  Moreover the series converge in the entire region of g when  hh ¼  h/ ¼ À0:6 hf ¼  Discussion Here impacts of distinct variables on velocity f ðgÞ, temperature   hðgÞ, nanoparticles concentration /ðgÞ, skin friction 12 C f Rex1=2 ,     À1=2 Nusselt number NuRexÀ1=2 and Sherwood number ShRex Fig Effects of S on f 259 T Hayat et al / Results in Physics (2017) 256–262 Fig Effects of a on f Fig Effects of N t on h Fig Effects of Pr on h Fig Effects of N b on h Fig Effects of c1 on h Fig Effects of a on h rises as value of N t increases It is because for larger N t , the difference between wall temperature and reference temperature augments Fig pointed out the impact of N b on hðgÞ It is examined that hðgÞ and associated thickness of boundary layer are enhanced for larger N b Physically an increase in N b , the random movement of molecules increases which results in an enhancement of hðgÞ Characteristics of a on hðgÞ are addressed through Fig Clearly hðgÞ reduces by augmenting a near the stretching surface but it demonstrates increasing behavior as we move far away from the surface From physical point of view larger a enhance the thickness of thermal boundary layer due to which the heat transport rate reduces and thus hðgÞ increases Fig is sketched to see the impact of N t on /ðgÞ It is examined that an increase in N t enhances /ðgÞ Physically more nanoparticles are pressed away from the hot surface Consequently the volume fraction distribution ð/ðgÞÞ boosts Fig 10 shows that larger Schmidt number ðScÞ corresponds to a decrease in /ðgÞ Since weaker Brownian diffusion coefficient rises for higher Sc that retards /ðgÞ and thickness of boundary layer Influence of N b on /ðgÞ is portrayed in Fig 11 It is noticed for larger N b , the random movement 260 T Hayat et al / Results in Physics (2017) 256–262 Fig Effects of N t on / Fig 12 Effects of a on / Fig 10 Effects of Sc on / Fig 13 Effects of c2 on / Table HAM solutions Convergence when a ¼ N t ¼ N b ¼ 0:1; c1 ¼ 0:4; c2 ¼ 0:2; Pr ¼ S ¼ 1:0 and Sc ¼ 0:6 Fig 11 Effects of N b on / and also the collision of macroscopic particles of the liquid enhances which decays /ðgÞ Fig 12 describes the influence of a on /ðgÞ It is exposed that /ðgÞ decays near the stretching cylinder however it enhances as one moves away from the cylinder The thickness of concentration boundary layer also enhances by increasing the value of a Behavior of c2 on /ðgÞ is disclosed through Fig 13 As expected /ðgÞ and corresponding thickness of boundary layer are enhanced via larger c2 Table interprets the convergence analysis of series solutions It is depicted that 15th order of approximations are enough for Order of approximations Àf ð0Þ 00 Àh0 ð0Þ À/0 ð0Þ 10 15 20 25 30 35 40 1.3300 1.6337 1.6429 1.6429 1.6429 1.6429 1.6429 1.6429 1.6429 0.6176 0.5782 0.5804 0.5804 0.5804 0.5804 0.5804 0.5804 0.5804 0.2894 0.3406 0.3505 0.3536 0.3548 0.3554 0.3555 0.3555 0.3555 Table Skin friction coefficient ð12 C f Re1=2 x Þ via a and S when N t ¼ N b ¼ 0:1; c1 ¼ 0:4; c2 ¼ 0:2; Pr ¼ 1:0 and Sc ¼ 0:6 a 0.0 0.1 0.2 0.3 0.1 S 0.0 0.3 0.6 0.9 1=2 C f Rex À1.6180 À1.6429 À1.6690 À1.6952 À1.0368 À1.1952 À1.3745 À1.5728 261 T Hayat et al / Results in Physics (2017) 256–262 Table   via a; Pr; c1 ; c2 ; N t ; N b and Sc when S ¼ 1:0 Local Nusselt number Nux ReÀ1=2 x a Pr c1 c2 Nt Nb Sc Nux Rex 0.0 0.1 0.2 0.1 1.0 0.4 0.2 0.1 0.1 0.6 0.8 0.9 1.0 1.2706 1.3037 1.3414 1.5112 1.7306 1.9443 1.2681 1.2428 1.2043 1.2740 1.2576 1.2355 1.3311 1.2413 1.1465 1.2740 1.2488 1.2239 1.2913 1.2931 1.2944 1.2 1.4 1.6 1.0 0.5 0.6 0.7 0.4 0.3 0.4 0.5 0.2 0.0 0.2 0.4 0.1 0.2 0.4 0.6 0.1 À1=2 Table Sherwood number ðShx ReÀ1=2 Þ via a; Pr; c1 ; c2 ; N t ; N b , and Sc when S ¼ 0:6 x a Pr c1 c2 Nt Nb Sc Shx Rex 0.0 0.2 0.4 0.1 1.0 0.4 0.2 0.1 0.1 0.6 0.4456 0.4733 0.5076 0.4682 0.4731 0.4770 0.3989 0.3516 0.3114 0.5994 0.7071 0.7926 1.0813 0.3501 0.2805 0.5980 0.7473 0.8254 0.5495 0.5947 0.6392 1.2 1.4 1.6 1.0 0.5 0.6 0.7 0.4 0.3 0.4 0.5 0.2 the convergence of f and h while 35th order is sufficient for the convergence of / Table indicates the numerical values of skin friction coefficient for different values of S and a As expected the skin friction coefficient enhances for larger S and a Impacts of emerging variables on Nusselt and Sherwood numbers are demonstrated in Tables and It is examined that Nusselt number shows increasing behavior for higher a; Pr and Sc whereas reverse behavior is noticed for c1 ; c2 ; N t and N b (for detail see Table 3) We further analyzed that Sherwood number enhances for larger a; Pr; c2 ; Nb and Sc whereas it reduces via c1 and Nt (for detail see Table 4) Conclusions Newtonian heating characteristics in the boundary layer flow of viscous nanofluid are explored Major highlights of presented analysis are given below  Larger suction parameter (S > 0) reduces velocity distribution f ðgÞ 0.0 0.2 0.4 0.1 0.2 0.4 0.6 0.1 0.8 0.9 1.0 À1=2  Temperature distribution hðgÞ boosts through larger N t and N b   are  Influences of a and S on skin friction coefficient 12 C f Re1=2 x qualitatively similar    Nusselt number NuReÀ1=2 diminishes via larger N t and N b x   À1=2 has reverse effects for N t while Sherwood number ShRex and N b  Effects of Schmidt number Sc on heat and mass transfer rates are similar References [1] Choi SUS Enhancing thermal conductivity of fluids with nanoparticles ASME Int Mech Eng 1995;66:99–105 [2] Pal D, Mandal G, Vajravalu K Soret and Dufour effects on MHD convective– radiative heat and mass transfer of nanofluids over a vertical non-linear stretching/shrinking sheet Appl Math Comput 2016;288:184–200 [3] Turkyilmazoglu M Unsteady convection flow of some nanofluids past a moving vertical flat plate with heat transfer ASME J Heat Transfer 2013;136:031704 262 T Hayat et al / Results in Physics (2017) 256–262 [4] Hussain T, Shehzad SA, Hayat T, Alsaedi A, Al-Solamy F, Ramzan M Radiative hydromagnetic flow of Jeffrey nanofluid by an exponentially stretching sheet Plos One 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