RINP 506 No of Pages 9, Model 5G 11 January 2017 Results in Physics xxx (2017) xxx–xxx Contents lists available at ScienceDirect Results in Physics journal homepage: www.journals.elsevier.com/results-in-physics On MHD nonlinear stretching flow of Powell–Eyring nanomaterial Tasawar Hayat a,b, Rai Sajjad c, Taseer Muhammad a,⇑, Ahmed Alsaedi b, Rahmat Ellahi c,d 10 11 12 15 16 17 18 19 20 21 22 23 24 25 a Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia c Department of Mathematics and Statistics, Faculty of Basic & Applied Sciences, International Islamic University, Islamabad 44000, Pakistan d Department of Mechanical Engineering, Bourns Hall A373, University of California, Riverside, CA 92521, USA b a r t i c l e i n f o Article history: Received 24 November 2016 Received in revised form 18 December 2016 Accepted 24 December 2016 Available online xxxx Keywords: Powell–Eyring fluid Magnetohydrodynamics Nanomaterial Nonlinear stretching surface a b s t r a c t This communication addresses the magnetohydrodynamic (MHD) flow of Powell–Eyring nanomaterial bounded by a nonlinear stretching sheet Novel features regarding thermophoresis and Brownian motion are taken into consideration Powell–Eyring fluid is electrically conducted subject to non-uniform applied magnetic field Assumptions of small magnetic Reynolds number and boundary layer approximation are employed in the mathematical development Zero nanoparticles mass flux condition at the sheet is selected Adequate transformation yield nonlinear ordinary differential systems The developed nonlinear systems have been computed through the homotopic approach Effects of different pertinent parameters on velocity, temperature and concentration fields are studied and analyzed Further numerical data of skin friction and heat transfer rate is also tabulated and interpreted 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/) 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Introduction 41 Alternative form of fluids that are composition of nanometer sized particles and convectional base liquids are termed as nanofluids Nanoparticles utilized in the nanomaterials are basically made of metals (Ag, Cu, Al) or nonmetals (carbon nanotubes, graphite) and the base liquids include ethylene glycol, water or oil Suspension of nanoparticles in the base liquids greatly varies the heat transfer characteristics and transport property To obtain prominent thermal conductivity enhancement in the nanofluids, the studies have been processed both theoretically and experimentally Applications of nanofluids in technology and engineering are nuclear reactor, vehicle cooling, vehicle thermal management, heat exchanger, cooling of electronic devices and many others Moreover magneto nanofluids (MNFs) are helpful in removal of blockage in arteries, wound treatment, cancer therapy, hyperthermia and resonance visualization Further the nanomaterials enhances the heat transfer rate of microchips in microelectronics, computers, fuel cells, transportation, biomedicine, food processing etc The pioneer investigation regarding enhancement of thermal properties of base liquid through the suspension of nanoparticles was presented by Choi [1] Later the development of mathematical relationship of nanofluid with Brownian diffusion and thermophoresis is presented by Buongiorno [2] Turkyilmazoglu [3] derived the 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 ⇑ Corresponding author exact analytical solutions for MHD slip flow of nanofluids by considering heat and mass transfer characteristics Further relevant attempts on nanofluid flows can be quoted through the analysis [4–27] and various studies therein Non-Newtonian fluids are regarded very prominent for applications in chemical and petroleum industries, biological sciences and geophysics The flow of non-Newtonian fluids due to stretching surface occurs in several industrial processes, for example, drawing of plastic films, polymer extrusion, oil recovery, food processing, paper production and numerous others The well-known Navier– Stokes expressions are not appropriate to describe the flow behavior of non-Newtonian materials However various constitutive relations of non-Newtonian materials are proposed in the literature due to their versatile nature Such materials are categorized as differential, integral and rate types The Powell–Eyring fluid model [28–33] is derived from kinetic theory of gases instead of empirical relation as in the case of the power-law model Further it appropriately recovers Newtonian behavior at low and high shear rates Ketchup, human blood, toothpaste, etc are the examples of Powell–Eyring fluid There is no doubt that much attention in the past has been devoted to the flow caused by linear stretching velocity However this consideration is not realistic in plastic industry Hence some researchers studied the flow problem of nonlinear stretching surface Gupta and Gupta [34] initially studied the flow by nonlinear stretching velocity Heat transfer in flow of viscous fluid generated by nonlinear stretching velocity is analyzed by Vajravelu [35] Cor- E-mail address: taseer@math.qau.edu.pk (T Muhammad) http://dx.doi.org/10.1016/j.rinp.2016.12.039 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/) Please cite this article in press as: Hayat T et al On MHD nonlinear stretching flow of Powell–Eyring nanomaterial Results Phys (2017), http://dx.doi.org/ 10.1016/j.rinp.2016.12.039 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 RINP 506 No of Pages 9, Model 5G 11 January 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx Nomenclature u; v l m T T1 r a ðqcÞp DB uw a f h K; K Le Nb sw sij Nux x; y qf b; C à 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 velocity components dynamic viscosity kinematic viscosity temperature ambient fluid temperature electrical conductivity thermal diffusivity effective heat capacity of nanoparticles Brownian diffusion coefficient surface velocity positive constant similarity variable dimensionless temperature fluid parameters Lewis number Brownian motion parameter wall shear stress extra stress tensor local Nusselt number coordinate axes density of base fluid material constants tell [36] extended the work of [35] through prescribed surface temperature and constant surface temperature cases Prasad et al [37] investigated mixed convection flow over a nonlinear stretching surface Mustafa et al [38] analyzed axisymmetric nanoliquid flow past a nonlinear stretching surface Magnetohydrodynamic (MHD) flow of second grade nanomaterial induced due to a nonlinear stretching sheet is studied by Hayat et al [39] This communication addresses the magnetohydrodynamic (MHD) flow of Powell–Eyring nanomaterial over a nonlinear stretching surface Powell–Eyring fluid’s constitutive relations are used in the problem formulation Novel features regarding thermophoresis and Brownian motion are taken into consideration Recently proposed condition for zero mass flux of nanoparticles at stretching sheet is taken into account This condition describes that the nanofluid particle fraction on the boundary is passively rather than actively controlled, i.e., it is no longer assumed that one can control the value of the nanoparticle fraction at the wall but rather that the nanoparticle flux at the wall is zero This change necessitates a rescaling of the parameters that are involved The governing nonlinear systems have been computed through the homotopic approach [40–47] Effectiveness of influential parameters on velocity, temperature and concentration fields have been studied in detail Further numerical data of skin friction and heat transfer rate are explained 114 Statement 115 Let us consider two dimensional (2D) magnetohydrodynamic flow of an incompressible Powell–Eyring nanomaterial The flow is caused by a nonlinear stretching surface Features of thermophoresis and Brownian motion are taken into consideration The x- and y-axes are taken parallel and transverse to the stretching surface The sheet at y ¼ is stretching along the x-direction with velocity uw xị ẳ axn where a and n are positive constants Powell–Eyring fluid is electrically conducted subject to nonuniform magnetic field applied in the y-direction (see Fig 1) Here the induced magnetic field is neglected for low magnetic Reynolds number [48–50] Assumptions of low magnetic Reynolds number and boundary layer approximation are employed in the mathemat- 116 117 118 119 120 121 122 123 124 125 126 C C1 B0 k ðqcÞf DB DT uw Tw a n f / M Pr Nt qw C fx Rex concentration ambient fluid concentration magnetic field strength thermal conductivity heat capacity of fluid Brownian diffusion coefficient thermophoretic diffusion coefficient surface velocity surface temperature positive constant power-law index dimensionless velocity dimensionless concentration magnetic parameter Prandtl number thermophoresis parameter wall heat flux skin friction coefficient local Reynolds number ical development The extra stress tensor for Powell–Eyring fluid is [31]: sij ẳ l @ui 1 ỵ sinh @xj b sinh 128 129 @ui ; C à @xj ð1Þ in which l stands for dynamic viscosity and b and C à for material constants Considering À1 127 3 @ui @ui @ui 1 @ui ( 1: ¼ ~ à À ; à à à C @xj C @xj C @xj C @xj 131 132 133 134 ð2Þ The boundary layer expressions for two-dimensional (2D) magnetohydrodynamic flow of Powell–Eyring nanofluid are [31,39]: 136 137 138 139 @u @ v ỵ ẳ 0; @x @y ð3Þ 142 ! 2 @2u @u @ u mỵ qf bC @y 2q bC @y @y2 @u @u ỵv ẳ u @x @y 141 f rB2 ðxÞ À u; qf ð4Þ 2 ! @T @T @ T qcịp DT @T @T @C ỵv ẳa 2ỵ ; u þ DB @x @y @y @y @y ðqcÞf T @y @C @C @2C ỵv ẳ DB u @x @y @y2 Here u and ! ! DT @ T þ : T @y2 144 145 ð5Þ 147 148 ð6Þ v show the velocity components along the horizonnÀ1 tal and vertical directions, Bxị ẳ B0 x represents the non-uniform magnetic field, m ¼ ql stands for kinematic viscosity, qf for denf sity of base liquid, r for electrical conductivity, T for temperature, C for concentration, ðqcÞp for effective heat capacity of nanoparticles, a ẳ k=qcịf for thermal diffusivity of fluid, k for thermal conductivity, ðqcÞf for heat capacity of fluid, DT for thermophoretic Please cite this article in press as: Hayat T et al On MHD nonlinear stretching flow of Powell–Eyring nanomaterial Results Phys (2017), http://dx.doi.org/ 10.1016/j.rinp.2016.12.039 150 151 152 153 154 155 156 157 RINP 506 No of Pages 9, Model 5G 11 January 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx Fig Geometry of the problem 160 diffusion coefficient, DB for Brownian diffusivity and T for ambient fluid temperature The associated boundary conditions are [8,39]: 163 u ¼ uw ðxÞ ¼ ax ; 164 166 u ! 0; T ! T ; C ! C as y ! 1; 158 159 161 n v @C DT @T ỵ ẳ at y ẳ 0; ẳ 0; T ¼ T w ; DB @y T @y 7ị Cf x ẳ sw xqw ; Nux ẳ ; qf u2w kðT w À T Þ ð15Þ in which wall shear stress ðsw Þ and heat flux qw ị are given by sw ẳ þ bC1 à @u À @y 3 @u ;> > = Ã3 @y 6bC y¼0 > > ; : qw ¼ Àk @T @y ð8Þ 168 169 170 171 where a shows the rate of the stretching surface, T w the temperature of the stretching surface, n the power-law index and C the ambient fluid concentration Introducing the suitable transformations 173 174 175 000 178 180 nÀ1 = f ỵ n1 ff ; > nỵ1 9ị Expression (3) is vanishes identically while (4)(8) yield 00 ỵ K ịf ỵ ff M2 f nỵ1 177 1=2 u ẳ ax f fị; v ẳ amnỵ1ị x 1=2 n1 > 1 ; f ẳ anỵ1ị x y; hfị ẳ TTT ; /fị ẳ CC C1 2m w ÀT n nỵ1 2n 002 000 02 K Kf f f nỵ1 10ị h000 ỵ Pr f h0 ỵ Nb h0 /0 ỵ Nt h0 ẳ 0; 11ị 201 Nt 00 h ẳ 0; Nb 12ị 0 f ẳ 0; f ẳ 1; h ẳ 1; Nb / ỵ Nt h ẳ at f ẳ 0; 189 190 191 194 195 13ị Here K and K stand for fluid parameters, Pr for Prandtl number, M for magnetic parameter, N t for thermophoresis parameter, N b for Brownian motion parameter and Le for Lewis number These parameters are defined by K¼ 193 ) f ! 0; h ! 0; / ! as f ! 186 188 qffiffiffiffiffiffiffi 00 003 nỵ1 nỵ1 = Re1=2 C ẳ 0ị ịK K f 0ị ;> ỵ Kịf fx x qffiffiffiffiffiffiffi > ; ReÀ1=2 Nux ¼ À nỵ1 h0 0ị; x Nt ẳ lbC ; Kẳ 202 203 17ị 205 where Rex ẳ uw x=m stands for local Reynolds number 206 Homotopic solutions 207 The appropriate initial approximations ðf ; h0 ; /0 Þ in homotopic solutions are defined as 208 ¼À u3w ; M ẳ 2mxC qcịp DT T w T ị ; Nb qcịf mT ẳ rB20 qf a ; ðqcÞp DB C ðqcÞf m Pr ¼ m a; a ; Le ¼ DB > = ; :> ð14Þ Nt expðÀfÞ; Nb ð18Þ À Á and auxiliary linear operators Lf ; Lh ; L/ are 184 187 In dimensionless variables 209 210 212 213 214 df d h d / Lf ¼ À ; Lh ¼ À h; L/ ¼ À /: df df df df d f 181 183 199 f fị ẳ expfị; h0 fị ẳ expfị; /0 fị ẳ 0; /00 ỵ LePrf /0 ỵ 198 16ị yẳ0 167 197 19ị 216 The above auxiliary linear operators satisfy the following characteristics: )  Lf E1 ỵ E2 expfị ỵ E3 expfị ẳ 0; Lh E4 expfị ỵ E5 expfị ẳ 0; L/ E6 expfị ỵ E7 expfị ¼ 0; 217 ð20Þ 221 where ðr ¼ À 7Þ elucidate the arbitrary constants Deformation problems at zeroth-order are 222 Er h i ị f fị ẳ p hN f ẵ^f f; p ị; p ÞLf ^f ðf; p ð21Þ h i ^ p ^ p ị h0 fị ẳ p ịLh hf; hN h ẵ^f f; p ị; ^hf; p ị; /f; ị; ð1 À p ð22Þ 218 219 223 224 226 227 Skin friction coefficient and local Nusselt number are Please cite this article in press as: Hayat T et al On MHD nonlinear stretching flow of Powell–Eyring nanomaterial Results Phys (2017), http://dx.doi.org/ 10.1016/j.rinp.2016.12.039 229 RINP 506 No of Pages 9, Model 5G 11 January 2017 230 232 233 235 236 238 239 T Hayat et al / Results in Physics xxx (2017) xxx–xxx h i ^ p ^ p Þ /0 fị ẳ p ịL/ /f; hN / ẵ^f ðf; p Þ; ^hðf; p Þ; /ðf; Þ; ð1 À p ^f 0; p > ị ẳ 0; ^f 0; p ị ẳ 1; ^h0; p ị ẳ 1; > = 0 ^ ^ Nb / 0; pị ỵ Nt h 0; pị ẳ 0; > > ^f 1; p ^ ^ ị ẳ 0; h1; ị ẳ 0; /1; ị ẳ 0; ; p p ð24Þ !2 @ 2^f @ ^f h i @ ^f @ 2^f nỵ1 ị ẳ ỵ K ị ỵ f KK N f ^f ðf; p @f @f @f2 ! ^ 2n @f @ ^f M2 ; n ỵ @f nỵ1 @f h i @ ^h ^ p ị; ^hf; p ị; /f; ị ẳ N h ^f ðf; p @f @f3 ð25Þ ^ @ ^h @ ^h @ / @ ^h ỵ Nb ỵ Nt ỵ Pr@^f @f @f @f @f 244 !2 A; h i @2/ ^ ^ Nt @ ^h @/ ^ p ị; ^hf; p ị; /f; ị ẳ ỵ LePr^f ỵ N / ^f f; p : @f Nb @f2 @f ð38Þ Â Ã e m^ ðfÞ; L/ /m^ fị vm^ /m1 fị ẳ h R ^ / 39ị 27ị vm^ ẳ & ^hf; 0ị ẳ h0 fị; ^hf; 1ị ẳ hfị; 29ị f m^ fị ẳ ^ 0ị ẳ / fị; /f; ^ 1ị ẳ /fị: /f; ð30Þ 255 256 257 258 259 260 262 263 p p changes from to then f ðf; p Þ, hðf; Þ and /ðf; Þ disWhen p play alteration from primary approximations f ðfÞ; h0 ðfÞ and /0 ðfÞ to desired ultimate solutions f ðfÞ; hðfÞ and /ðfÞ The following expressions are derived via Taylor’s series expansion: X Þ @ m ^f f; p m ^f f; p ị ẳ f fị ỵ ; f m fị ẳ f m fịp m m! @ p mẳ1 265 Þ @ hðf; p m ^hðf; p Þ ẳ h0 fị ỵ ; hm fị ẳ hm fịp m m! @ p mẳ1 266 ^ p ị ẳ /0 fị ỵ /f; 268 269 270 271 272 274 275 277 278 280 281 282 284 ; m^ X m ; /m fị ẳ /m fịp mẳ1 31ị ; 32ị f fị ẳ f fị ỵ f m fị; ^ p ị @ m /ðf; m m! @ p : 33ị ẳ0 p 34ị mẳ1 X hfị ẳ h0 fị ỵ hm fị; 35ị mẳ1 X /fị ẳ /0 fị ỵ /m fị: 296 297 h0m1k h0k ; ^ f m1k /0k ^ Nt 00 ỵ h^ ; Nb mÀ1 ð43Þ 299 302 303 ^ 1; m ^ > 1; 1; m ð44Þ Ã ðf m^ ; hÃm^ ; /Ãm^ Þ, In terms of special solutions the general solutions ðf m^ ; hm^ ; /m^ Þ of the Eqs (37)–(39) are defined by the following expressions: 305 306 307 308 45ị 309 311 hm^ fị ẳ hm^ fị ỵ E4 expfị ỵ E5 expfị; 46ị 312 314 /m^ fị ẳ /m^ fị ỵ E6 expfị ỵ E7 expfị; 47ị 315 317 f m^ fị ỵ E1 þ Ễ2 expðfÞ þ Ễ3 expðÀfÞ; in which Ễr ðr ¼ À 7Þ through the boundary conditions (40) are given by E2 E5 ẳ E4 ẳ hm^ 0ị; ẳ EÃ6 ¼ 0; EÃ7 EÃ3 ¼ à @f m^ fị ; @f fẳ0 E1 ẳ E3 f m^ ð0Þ; ! @/Ãm^ ðfÞ Nt @hÃm^ ðfÞ : ẳ ỵ E ỵ @f fẳ0 Nb @f fẳ0 48ị 318 319 320 322 323 49ị 325 Convergence analysis 326 Here the homotopic solutions ð34Þ À ð36Þcontain the nonzero h and auxiliary parameters hf ; h h/ Such auxiliary parameters play a significant role to control and adjust the region of convergence To get the suitable values of auxiliary parameters, the hÀcurves are sketched at 25th order of deformations Fig displays that the convergence zone lies within the ranges À1:8 hf À0:1; À1:75 hh À0:15 and À1:7 h/ À0:2 The residual errors for velocity, temperature and concentration distributions are calculated through the following expressions: 327 Z Dmf Dhm D/m h Z ¼ h The mth-order deformation problems are presented as follows: 37ị ẳ 36ị mẳ1 e m^ ðfÞ; Lf f m^ ðfÞ À vm^ f mÀ1 fị ẳ h R ^ f kẳ0 h0m1k /0k ỵ N t ^ ! 294 ¼0 p The convergence regarding Eqs (31)–(33) is strongly based upon the suitable choices of hf ; hh and h/ Choosing suitable values ¼ then of hf ; hh and h/ so that Eqs (31)–(33) converge at p X k¼0 0; ¼0 p X kẳ0 kẳ0 28ị 253 ^ m1 X ^ m1 X ^f f; 0ị ẳ f fị; ^f f; 1ị ẳ f ðfÞ; 252 ^ mÀ1 X e / ðfÞ ẳ /00^ ỵ LePr R ^ m1 m 249 250 293 ð42Þ 246 247 ð40Þ ^ mÀ1 X f mÀ1Àk h0k ỵ Nb ^ 290 291 X ^ ^ k m1 ml X X nỵ1 00 000 00 00 e f fị ẳ ỵ K ịf 000^ ỵ K K f m1k f m1k fk À f kÀl f l R ^ ^ ^ mÀ1 m l¼0 k¼0 k¼0 X ^ À 2n mÀ1 0Á M2 f mÀ1 À f mÀ1Àk fk À ; ð41Þ ^ ^ n þ k¼0 nþ1 Here N f ; N h and N / are nonlinear operators and hf ; hh and h/ ¼ and p ¼ one obtains the nonzero auxiliary parameters For p 245 288 f m^ 0ị ẳ f m^ 0ị ẳ hm^ 0ị ẳ 0; = > Nb /0m^ 0ị ỵ Nt h0m^ 0ị ẳ 0; > ; f m^ 1ị ẳ hm^ 1ị ẳ /m^ 1ị ẳ 0; e h^ fị ẳ h00^ ỵ Pr R mÀ1 m 287 300 ð26Þ 241 242 ð23Þ 285  à e m^ ðfÞ; R Lh hm^ ðfÞ À vm^ hm1 fị ẳ h ^ h Z ẳ h ef R ^ m À f; hf Á i2 df; i2 e h^ ðf; hh Þ df; R m À Á i2 e / f; h/ df: R ^ m ð50Þ 328 329 330 331 332 333 334 335 336 338 339 ð51Þ 341 342 ð52Þ Please cite this article in press as: Hayat T et al On MHD nonlinear stretching flow of Powell–Eyring nanomaterial Results Phys (2017), http://dx.doi.org/ 10.1016/j.rinp.2016.12.039 344 RINP 506 No of Pages 9, Model 5G 11 January 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx Fig hh -curve for the residual error Dhm Fig The h-curves for f ðfÞ, hðfÞ and /ðfÞ 350 , the h -curves for the residual To get the suitable range for h errors of velocity, temperature and concentration distributions are plotted in Figs 3–5 It is observed that the correct results up to fifth decimal place are obtained when we choose the values of hfrom this range HAM solutions convergence via Table is satisfactorily achieved by considering 20th orders of approximation 351 Discussion 352 This portion has been arranged to explore the impacts of several effective parameters including fluid parameter K, magnetic parameter M, thermophoresis parameter N t , Brownian motion parameter N b , Lewis number Le, Prandtl number Pr and power-law index n on velocity f ðfÞ, temperature hðfÞ and concentration /ðfÞ distributions Fig shows the impact of fluid parameter K on velocity dis0 tribution f ðfÞ Both velocity field and momentum boundary layer thickness are increased for larger K Behavior of M on velocity dis0 tribution f ðfÞis presented in Fig Here both velocity and momentum boundary layer thickness decay for M Fig shows influence of power-law index n for velocity f ðfÞ By increasing n, both the velocity and momentum boundary layer thickness have been reduced Here n ¼ corresponds to linear stretching surface case and n – for nonlinear stretching surface The impacts of fluid parameter K, magnetic parameter M, thermophoresis parameter 345 346 347 348 349 353 354 355 356 357 358 359 360 361 362 363 364 365 366 Fig hf -curve for the residual error Dmf Fig h/ -curve for the residual error D/m Table Homotopic solutions convergence when K ¼ K ¼ M ¼ 0:1; Pr ¼ n ¼ 1:2; N t ¼ 0:2; N b ¼ 0:3 and Le ¼ 1:0 Order of approximations Àf ð0Þ 00 Àh0 ð0Þ /0 ð0Þ 10 15 20 25 30 35 0.98485 0.98476 0.98475 0.98475 0.98475 0.98475 0.98475 0.98475 0.70000 0.64522 0.64301 0.64292 0.64294 0.64294 0.64294 0.64294 0.46667 0.43015 0.42867 0.42862 0.42863 0.42863 0.42863 0.42863 N t , Prandtl number Pr and power-law index n for temperature hðfÞ have been displayed in the Figs 9–13 respectively It is observed that by increasing magnetic parameter M, thermophoresis parameter N t and power-law index n, both the temperature distribution and thermal boundary layer thickness are increased whereas opposite behavior is seen for fluid parameter K and Prandtl number Pr It is a valuable fact to mention here that the properties of liquid metals are characterized by small values of Prandtl number ðPr < 1Þ, which have larger thermal conductivity but smaller viscosity, whereas higher values of Prandtl number ðPr > 1Þ associate with high-viscosity oils Particularly Prandtl number Pr ¼ 0:72; 1:0 and 6:2 are associated to air, electrolyte Please cite this article in press as: Hayat T et al On MHD nonlinear stretching flow of Powell–Eyring nanomaterial Results Phys (2017), http://dx.doi.org/ 10.1016/j.rinp.2016.12.039 367 368 369 370 371 372 373 374 375 376 377 378 RINP 506 No of Pages 9, Model 5G 11 January 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx Fig Plots of hðfÞ for K Fig 10 Plots of hðfÞ for M Fig Plots of f ðfÞ for n Fig 11 Plots of hðfÞ for N t solution such as salt water and water respectively Moreover it is also observed that N t portrays the strength of thermophoresis effects Higher N t leads to more strength to thermophoresis The variations in concentration field /ðfÞfor various values of fluid parameter K, magnetic parameter M, thermophoresis parameter N t , Brownian motion parameter N b , Lewis number Le, Prandtl num- Fig Plots of f ðfÞ for K Fig Plots of f ðfÞ for M 379 380 381 Please cite this article in press as: Hayat T et al On MHD nonlinear stretching flow of Powell–Eyring nanomaterial Results Phys (2017), http://dx.doi.org/ 10.1016/j.rinp.2016.12.039 382 383 384 RINP 506 No of Pages 9, Model 5G 11 January 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx Fig 12 Plots of hðfÞ for Pr Fig 13 Plots of hðfÞ for n Fig 15 Plots of /ðfÞ for M Fig 16 Plots of /ðfÞ for N t Fig 14 Plots of /ðfÞ for K Fig 17 Plots of /ðfÞ for N b Please cite this article in press as: Hayat T et al On MHD nonlinear stretching flow of Powell–Eyring nanomaterial Results Phys (2017), http://dx.doi.org/ 10.1016/j.rinp.2016.12.039 RINP 506 No of Pages 9, Model 5G 11 January 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx Table Numerical data of skin friction coefficient for K; K; M and n K K M n ÀRex1=2 C fx 0.0 0.1 0.2 0.1 0.1 0.1 1.2 0.1 1.2 0.1 0.0 0.2 0.4 0.1 1.2 0.1 0.1 0.0 0.1 0.2 0.1 1.0832 1.1324 1.1802 1.1361 1.1288 1.1214 1.1276 1.1324 1.1468 0.9629 1.0511 1.1324 0.8 1.0 1.2 Table Numerical data of local Nusselt number for K; M; N t ; N b ; Le; Pr and n when K ¼ 0:1 Fig 18 Plots of /ðfÞ for Le Fig 19 Plots of /ðfÞ for Pr Fig 20 Plots of /ðfÞ for n K M Nt Nb Le Pr n Nux ÀReÀ1=2 x 0.0 0.1 0.2 0.1 0.1 0.2 0.3 1.0 1.2 1.2 0.2 0.3 1.0 1.2 1.2 0.1 0.0 0.1 0.2 0.1 0.3 1.0 1.2 1.2 0.1 0.1 0.1 0.2 0.3 0.2 1.0 1.2 1.2 0.1 0.1 0.2 0.1 0.2 0.3 0.3 1.2 1.2 0.1 0.1 0.2 0.3 0.0 0.5 1.0 1.0 1.2 0.1 0.1 0.2 0.3 1.0 0.8 1.0 1.2 1.2 0.6638 0.6742 0.6836 0.6753 0.6743 0.6715 0.6848 0.6743 0.6639 0.6743 0.6743 0.6743 0.6951 0.6824 0.6743 0.5177 0.6001 0.6743 0.6172 0.6463 0.6743 0.8 1.0 1.2 ber Pr and power-law index n are displayed in the Figs 14–20 Concentration field through these sketches enhances for larger magnetic parameter M, thermophoresis parameter N t and powerlaw index n whereas reverse trend is observed for fluid parameter K, Brownian motion parameter N b , Lewis number Le and Prandtl number Pr Table depicts the numerical data of skin friction coefficient for several effective parameters K; K; M and n Skin friction coefficient is higher for larger K; M and n while the reverse behavior is noticed through n Table is presented to analyze the numerical data of local Nusselt numbers via different parameters Here local Nusselt number increases for larger fluid parameter K, Prandtl number Pr and power-law index n whereas opposite result holds for magnetic parameter M, thermophoresis parameter N t and Lewis number Le There is no significant change of N b on local Nusselt number 385 Conclusions 400 Magnetohydrodynamic (MHD) flow of Powell–Eyring nanomaterial bounded by a nonlinear stretching surface is investigated Main observations of presented analysis are: 401 Larger values of fluid parameter K depict increasing behavior for velocity field while opposite behavior holds for temperature and concentration fields 404 Please cite this article in press as: Hayat T et al On MHD nonlinear stretching flow of Powell–Eyring nanomaterial Results Phys (2017), http://dx.doi.org/ 10.1016/j.rinp.2016.12.039 386 387 388 389 390 391 392 393 394 395 396 397 398 399 402 403 405 406 RINP 506 No of Pages 9, Model 5G 11 January 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 Impact of magnetic parameter M on temperature and concentration fields is quite opposite to that of velocity field Temperature and concentration fields through Prandtl number Pr are qualitatively similar An increase in Lewis number Le causes a decay in the concentration field and related 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535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 ... change of N b on local Nusselt number 385 Conclusions 400 Magnetohydrodynamic (MHD) flow of Powell? ? ?Eyring nanomaterial bounded by a nonlinear stretching surface is investigated Main observations of. .. a nonlinear stretching sheet is studied by Hayat et al [39] This communication addresses the magnetohydrodynamic (MHD) flow of Powell? ? ?Eyring nanomaterial over a nonlinear stretching surface Powell? ? ?Eyring. .. convection flow over a nonlinear stretching surface Mustafa et al [38] analyzed axisymmetric nanoliquid flow past a nonlinear stretching surface Magnetohydrodynamic (MHD) flow of second grade nanomaterial