RINP 487 No of Pages 7, 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 Slip flow by a variable thickness rotating disk subject to magnetohydrodynamics Maria Imtiaz a,⇑, Tasawar Hayat b,c, Ahmed Alsaedi c, Saleem Asghar d 10 11 12 15 16 17 18 19 20 21 22 23 24 25 a Department of Mathematics, Mohi-Ud-Din Islamic University, Nerian Sharif AJ&K, Pakistan Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan c Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University 80203, Jeddah 21589, Saudi Arabia d Department of Mathematics, CIIT, Chak Shahzad Park Road, Islamabad, Pakistan b a r t i c l e i n f o Article history: Received November 2016 Received in revised form 13 December 2016 Accepted 19 December 2016 Available online xxxx Keywords: Variable thickness Rotating disk Slip flow Magnetohydrodynamic (MHD) a b s t r a c t Objective of the present study is to determine the characteristics of magnetohydrodynamic flow by a rotating disk having variable thickness At the fluid–solid interface we consider slip velocity The governing nonlinear partial differential equations of the problem are converted into a system of nonlinear ordinary differential equations Obtained series solutions of velocity are convergent Impact of embedded parameters on fluid flow and skin friction coefficient is graphically presented It is observed that axial and radial velocities have an opposite impact on the thickness coefficient of disk Also surface drag force has a direct relationship with Hartman number 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 Introduction 39 Engineering and industrial applications due to rotating surfaces have attracted the attention of scientists and researchers These applications include air cleaning machines, gas turbines, medical equipment, food processing technology, aerodynamical engineering and in electric-power generating systems Initial work on rotating disk flow was undertaken by Karman [1] Ordinary differential equations were obtained from Navier–Stokes equations by using the Von Karman transformation Subsequently different physical problems were discussed by various researchers In the internal cooling-air systems of most gas turbines disks rotating at different speeds are found Heat transfer and flow associated with an aircooled turbine disk and an adjacent stationary casing were modeled using the rotor–stator system Bachok et al [2] examined nanofluid flow due to rotation of a permeable disk Similarity solution for flow and heat convection from a porous rotating disk was considered by Kendoush [3] MHD slip flow with variable properties and entropy generation due to rotation of a permeable disk were investigated by Rashidi et al [4] Turkyilmazoglu [5] described heat transfer and flow due to rotation of disk with nanoparticles Sheikholeslami et al [6] examined nanofluid spraying on an inclined rotating disk for cooling process Hayat et al [7] 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 ⇑ Corresponding author E-mail address: mi_qau@yahoo.com (M Imtiaz) studied partial slip effects in MHD flow due to rotation of a disk with nanoparticles Mustafa et al [8] analyzed MHD stagnation point flow of a ferrofluid past a stretchable rotating disk Xun et al [9] studied flow and heat transfer of Ostwald-de Waele fluid over a variable thickness rotating disk Chemical reaction effects in flow of ferrofluid due to a rotating disk were presented by Hayat et al [10] There are promising applications in metallurgy, polymer industry, chemistry, engineering and physics due to fluid flow in the presence of a magnetic field Desired characteristics of the end product are attained in such applications by controlling the rate of heat cooling The rate of cooling is controlled by magnetic field for an electrically conducting fluid Physiological fluid applications like blood pump machines and blood plasma are of great importance for MHD flow Flow configurations under different conditions for MHD flows were considered by numerous researchers Effects of velocity slip and temperature jump in a porous medium by a shrinking surface with magnetohydrodynamic were studied by Zheng et al [11] Analytical and numerical solutions for MHD Falkner-Skan Maxwell fluid flow were presented by Abbasbandy et al [12] Turkyilmazoglu [13] examined MHD flow of viscoelastic fluid over a stretching/shrinking surface in three dimensional analysis Sheikholeslami et al [14] analyzed radiative flow of nanofluid with magnetohydrodynamics A numerical study of radiative MHD flow of Al2O3–water nanofluid has been studied by Sheikholeslami et al [15] Hayat et al [16] studied Cattaneo–Christov heat flux in http://dx.doi.org/10.1016/j.rinp.2016.12.021 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: Imtiaz M et al Slip flow by a variable thickness rotating disk subject to magnetohydrodynamics Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2016.12.021 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 RINP 487 No of Pages 7, Model 5G 11 January 2017 M Imtiaz et al / Results in Physics xxx (2017) xxx–xxx 134 MHD flow of Oldroyd-B fluid with chemical reaction Li et al [17] examined MHD viscoelastic fluid flow and heat transfer by a vertical stretching sheet with Cattaneo–Christov heat flux Makinde et al [18] presented MHD Couette-Poiseuille flow of variable viscosity nanofluids in a rotating permeable channel with Hall effects Numerical study of MHD nanofluid flow and heat transfer past a bidirectional exponentially stretching sheet was considered by Ahmad et al [19] Buoyancy effects on the three dimensional MHD stagnation-point flow of a Newtonian fluid were examined by Borrellia et al [20] The formation and use of microdevices remain a hotly debated and challenging topic of research by scientists The small size as well as high efficiency of microdevices-such as microsensors, microvalves and micropumps are some of the advantages of using microelectromechanical systems (MEMS) and nanoelectromechanical systems (NEMS) Wall slip readily occurs for an array of complex fluids such as emulsions, suspensions, foams and polymer solutions Also fluids that exhibit boundary slip have important technological applications, such as polishing of artificial heart valves and internal cavities Many attempts addressing slip flow have been presented to guarantee the performance of such devices A number of models have been proposed for describing slip that occurs at solid boundaries A brief description of these models may be found in the work of Rao and Rajagopal [21] Buscall [22] reported that the importance of studying wall slip has grown substantially Akbarinia et al [23] used microchannels to study nanofluids heat transfer enhancement in non-slip and slip flow regimes Micropolar fluid flow with velocity slip and heat generation (absorption) has been examined by Mahmoud and Waheed [24] Velocity and thermal slip effects in nanofluid flow have been analyzed by Khan et al [25] Mukhopadhyay [26] examined radiative flow over a permeable exponentially stretching sheet with slip effects Inside a circular microchannel slip flow of alumina/water nanofluid has been considered by Malvandi and Ganji [27] Nonlinear thermal radiation and slip velocity in MHD three-dimensional nanofluid flow have been studied by Hayat et al [28] Slip flow in a microchannel for nanoparticles using lattice Boltzman method has been analyzed by Karimipour et al [29] Effect of mass transfer induced velocity slip on heat transfer of viscous gas flows over stretching/shrinking sheet has been presented by Wu [30] In the past much attention has been given to flow due to a rotating disk with negligible thickness Our main focus of the present analysis is to study MHD flow of viscous fluid due to a rotating disk with variable thickness Formulation and analysis is presented when no slip does not remain valid The technique used for solving the present problem is homotopy analysis method (HAM) [31–38] Convergent series solutions are obtained Impacts of pertinent parameters on axial, radial and tangential velocity components and surface drag force are examined 135 Formulation 136 Consider steady, laminar and axisymmetric flow due to a disk rotating with angular velocity X about the z-axis The disk is also stretched with velocity uw ¼ ra1 where a1 is the stretching rate Àm constant We assume that the disk at z ẳ a Rr0 ỵ is not flat 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 149 ! @u @u v @ u @u u @ u rB20 u ỵw u þ þ ; ¼m @r @z @r r @r r2 @z2 r q @v @ v uv @2v @v v @2v u ỵ ỵw ỵ ẳm ỵ @r @z r @r r @r r @z2 ! ð2Þ 152 rB2 v À ; q ð3Þ 4ị with boundary conditions u ! 0; v ẳ rX þ k2 @@zv ; w ¼ at z ¼ a r R0 ỵ1 m 159 ; 5ị 161 where u; v and w are velocity components in the direction of r; H and z respectively, m denotes kinematic viscosity, r the electrical conductivity, q the density and k1 ; k2 are slip velocity coefficients Generalized Von Karman transformations are m nỵ1 XR0 q u ¼ r XFðgÞ; v ¼ r XGðgÞ; w ¼ R0 X ỵ Rr0 Hgị l m nỵ1 XR0 q g ẳ Rz0 ỵ Rr0 : l H ỵ 2F ỵ gemF ẳ 0; 138 139 140 141 142 143 144 145 146 148 @u u @w ỵ ỵ ẳ 0; @r r @z 7ị Re ỵ r ị F F gemFF þ G À HF À MF ¼ 0; à 2m 00 2 165 166 169 170 171 173 8ị 176 177 1n 1ỵn Re ỵ r à Þ2m G00 À 2FG À gemFG0 À HG0 À MG ẳ 0; 9ị with boundary conditions 181 Haị ẳ 0; Faị ẳ A ỵ c1 ỵ r ị F aị; Gaị ẳ ỵ c2 ỵ r à Þm G0 ðaÞ; Fð1Þ ! 0; Gð1Þ ! 0; where 179 180 m 10ị 183 e ẳ R0rỵr is a dimensionless constant, Re ẳ XmR0 is the Reynolds 184 number, A ¼ aX1 is scaled stretching parameter, r à ¼ Rr0 is the dimenÀ1 nỵ1 XR q sionless radius, a ẳ Ra0 l0 is the dimensionless disk thickness 1 nỵ1 nỵ1 XR q XR q coefficient, c1 ẳ Rk10 l0 and c2 ¼ Rk20 l0 are velocity slip parameters and M ¼ rB20 qX is the Hartman number 185 186 187 188 We now consider 189 Hðg À aÞ ¼ hðnÞ; Fðg À aÞ ¼ f ðnÞ; Gðg À aị ẳ gnị; 11ị 190 192 193 h ỵ 2f ỵ n ỵ aịemf ẳ 0; 1ị 164 174 1n 1ỵn where a is the disk thickness coefficient, R0 is the feature radius and m is the disk thickness index Slip flow regime is considered for viscous fluid A magnetic field of strength B0 is applied in the z-direction Magnetic Reynolds number is assumed small and thus induced magnetic field is neglected Electric field is taken absent The governing equations are as follows: 163 168 Mass conservation law is identically satisfied and Eqs (2)–(5) become 162 ð6Þ and thus Eqs (7)–(10) are reduced to 137 157 158 v ! as z ! 1; 154 155 ! @w @w @ w @w @ w þw ¼m þ u þ ; @r @z @r2 r @r @z2 u ẳ ra1 ỵ k1 @u ; @z 151 12ị 194 196 197 Re1ỵn ỵ r ị2m f f ỵ g hf emn ỵ aịff Mf ẳ 0; 13ị Re1ỵn ỵ r ị2m g 00 2fg hg emn ỵ aịfg Mg ẳ 0; ð14Þ 1Àn 00 199 200 1Àn 0 203 h0ị ẳ 0; f 0ị ẳ A ỵ c1 ỵ r ịm f 0ị; g0ị ẳ ỵ c2 ỵ r ịm g ð0Þ; f ð1Þ ! 0; gð1Þ ! 0: 202 ð15Þ 205 Here prime denotes the derivative with respect of n and h; f and g are axial, radial and tangential velocity profiles respectively At the disk the shear stress in radial and tangential directions is szr and szh 206 Please cite this article in press as: Imtiaz M et al Slip flow by a variable thickness rotating disk subject to magnetohydrodynamics Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2016.12.021 207 208 209 210 RINP 487 No of Pages 7, Model 5G 11 January 2017 M Imtiaz et al / Results in Physics xxx (2017) xxx–xxx szr ¼ l 212 szh ¼ l 217 218 ẳ sw XR2 q nỵ1 l R0 lr X1 R0 1ỵr ị1 XR2 q ~f 0ị nỵ1 l g~0 0ị R0 q ẳ s2zr ỵ s2zh : ; 16ị : sw is defined as C f Renỵ1 ẳ 17ị 2 sw jzẳ0 ẳ ỵ r ị1 ~f 0ị ỵ g~0 0ịị r qrXị 29ị  à L2 f m À vm f mÀ1 ¼ hf Rmf ; ð30Þ 275  à L3 g m vm g m1 ẳ hg Rmg ; 31ị 276 278 279 Rhm ẳ 2f m1 ỵ men ỵ aịf m1 ỵ hm1 ; 32ị 281 282 Rmf ẳ Re !1=2 272 273 Skin friction coefficients C fx in dimensionless form is nÀ1 220 @ v^ @z zẳ0 ẳ lr X1 R0 1ỵr ị1 Total shear stress 213 214 216 ^ @u @z z¼0  à L1 hm À vm hmÀ1 ¼ hh Rhm ; 1n 1ỵn ỵ 00 r ị2m f m1 þ mÀ1 X f mÀ1Àk f k À meðn k¼0 : 18ị ỵ aị m1 X f m1k f k ỵ kẳ0 m1 X g m1k g k k¼0 m À1 X hmÀ1Àk f k À MfmÀ1 ; 33ị kẳ0 284 285 221 Solutions procedure 222 Linear operators L1 ; L2 and L3 and initial guesses h0 ðnÞ; f ðnÞ and g ðnÞ are taken in the forms 223 224 226 Rmg 00 00 L1 ¼ h ; L2 ¼ f À f ; L3 ẳ g g; ẳ Re 229 ỵ aị ð19Þ 230 subject to the properties 231 233 234 236 237 239 L1 ẵc1 ẳ 0; n n n L2 ẵc2 e ỵ c3 e ẳ 0; L3 ẵc4 e ỵ c5 e ẳ 0; 21ị 243 in which ci i ẳ 5ị are the constants If embedding parameter is denoted by p ½0; 1 and nonzero h; auxiliary parameters by h hf and hg then the zeroth order deformation problems are as follows: 246 ^ pị h0 nị ẳ ph N ẵhn; ^ pị; ^f n; pị; pịL1 ½hðn; h h 240 241 242 244 247 249 250 252 253 ^ pị; g^n; pị; pịL2 ẵ^f n; pị f nị ẳ phf Nf ẵ^f ðn; pÞ; hðn; ð23Þ ^ pÞ; g^ðn; pÞ; ð1 À pịL3 ẵg^n; pị g nị ẳ phg Ng ẵ^f n; pị; hn; 24ị 255 ^ pị ẳ 0; ^f 0; pị ẳ A ỵ c ỵ r à Þm ^f ð0; pÞ; ^f ð1; pÞ ! 0; h0; 256 258 g^0; pị ẳ ỵ c2 ỵ r ịm g^0 0; pị; g^1; pÞ ! 0: 259 260 262 263 265 ð22Þ Nonlinear operators are ^ pị @^f n; pị @ hn; ỵ ; Nh ẳ 2^f n; pị ỵ men ỵ aị @n @n 2 1Àn @ f ðn; pÞ ^ Nf ẳ Re1ỵn ỵ r ị2m ỵ f n; pị men ỵ aị^f n; pị @n ^ ^ @ f ðn; pÞ ^ pÞ @ f n; pị M^f n; pị; ỵ g^n; pịị hðn;  @n @n ð26Þ 2^ ð27Þ 266 268 269 270 @ g^ðn; pÞ À 2^f ðn; pÞg^ðn; pÞ À meðn @n2 @ g^ðn; pÞ ^ @ g^ðn; pÞ hn; pị M g^n; pị: ỵ aị^f n; pị @n @n Ng ẳ Re 1n 1ỵn 2m ỵ r ị The mth order deformation problems are f mÀ1Àk g 0k À ð28Þ f mÀ1Àk g k À meðn m À1 X hmÀ1Àk g 0k À Mg mÀ1 ; n!0 0; m 1; m > ẳ f m jn!1 ẳ 0; 290 291 35ị hm nị ẳ : 36ị ỵ c1 ; 296 297 298 299 301 302 304 f m nị ẳ f m nị ỵ c2 en ỵ c3 en ; g m nị ẳ g m nị ỵ c4 en þ c5 eÀn ; ð37Þ where the constants ci (i ¼ À 5) are c1 ¼ 293 294 The general solutions ðhm ; f m ; g m Þ comprising the special soluà à tions ðhm ; f m ; g Ãm Þ are à hm ðnÞ 287 288 ẳ f m jnẳ0 c1 ỵ r Þ @n à m @f m & 34ị kẳ0 ẳ c4 ẳ 0; m nị c3 ẳ 1ỵc 11ỵr ịm c1 þ r à Þm@f @n À f m ð0Þ ; nẳ0 m nị c2 ỵ r ịm@g@n c5 ẳ 1ỵc 1ỵr À g Ãm ð0Þ : à Þm 305 307 308 309 à Àhm ð0Þ; c2 ð25Þ À2 mÀ1 X @g g m jnẳ0 c2 ỵ r ịm m ẳ g m jn!1 ẳ 0; @n n!0 vm ¼ n mÀ1 X k¼0 hm jnẳ0 20ị ỵ r ị2m g 00m1 kẳ0 227 A h0 nị ẳ 0; f nị ẳ en ; ỵ c1 ỵ r ịm en ; g nị ẳ ỵ c2 ỵ r ịm 1n 1ỵn 38ị nẳ0 311 Homotopy solutions 312 For the solution of linear and nonlinear problems the homotopy analysis method (HAM) is a powerful technique An embedding auxiliary parameter h is involved in HAM which enlarges the convergence area Valid ranges of these parameters are obtained by -curves (see Fig 1) Permissible ranges of plotting h hh ; hf and hg U À0:3 Also are À2:5 hf À2; À1 hh À0:1 and À2:4 h f ¼ HAM solutions converge when hh ¼ h hg ¼ À0:5 (see Table 1) 313 Results 320 In this section the impact of axial, radial and tangential components of velocity for different dimensionless parameters is considered Similarly effect of these parameters on skin friction coefficient is examined in this section Effects of disk thickness index m, Reynolds number Re, disk thickness coefficient a and Hartman number M on axial velocity hðnÞ is observed in Figs (2– 321 Please cite this article in press as: Imtiaz M et al Slip flow by a variable thickness rotating disk subject to magnetohydrodynamics Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2016.12.021 314 315 316 317 318 319 322 323 324 325 326 RINP 487 No of Pages 7, Model 5G 11 January 2017 M Imtiaz et al / Results in Physics xxx (2017) xxx–xxx Fig Impact of Re on axial velocity -curves for h ð0Þ; f ð0Þ and g ð0Þ when e ¼ A ¼ c1 ¼ 0:3; m ¼ n ¼ 1; c2 ¼ Fig The h 0:4; Re ¼ 0:9; rà ¼ 0:2; a ¼ 1:2 and M ¼ 0:7 0 Table HAM solutions convergence e ¼ A ¼ c1 ¼ 0:3; m ¼ n ¼ 1; c2 ¼ 0:4; Re ¼ 0:9; rà ¼ 0:2; a ¼ 1:2 and M ¼ 0:7 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 when Order of approximations Àh ð0Þ Àf ð0Þ Àg ð0Þ 10 16 20 25 30 35 0.1809 0.4832 0.4982 0.4978 0.4978 0.4978 0.4978 0.4978 0.1379 0.08556 0.09531 0.09456 0.09463 0.09463 0.09463 0.09463 0.6171 0.6299 0.6349 0.6344 0.6344 0.6344 0.6344 0.6344 5) Fig exhibits that an increase in disk thickness index m causes the magnitude of axial velocity to decrease Fig depicts that magnitude of axial velocity is an increasing function of Reynolds number Re As rotation of the disk reduces, the inertial effects so fluid flow enhances when Re is increased Magnitude of axial velocity reduces with increase in thickness coefficient of disk a, due to that with an increase in a the feature radius R0 reduces and thus less particles are in contact with the surface Consequently the velocity decays (see Fig 4) Fig shows behavior of the axial velocity for larger Hartman number M In fact for larger M, the magnetic field increases due to the force of resistance called the Lorentz force Lorentz force becomes more intense and as a result the velocity decreases Fig reveals that the radial velocity f ðnÞ is an increasing function of disk thickness index m Fig elucidates that radial velocity Fig Impact of a on axial velocity Fig Impact of M on axial velocity Fig Impact of m on axial velocity increases for larger Reynolds number Re Impact of disk thickness coefficient a on radial velocity is examined in Fig It is noted that for higher a the radial velocity enhances Effect of increasing values of Hartman number M on radial velocity is shown in Fig Results shows that radial velocity decays for higher M For larger M the Lorentz force enhances which produces resistance between the particles and consequently both components of axial and radial velocities are reduced Fig 10 shows that increasing velocity slip parameter c1 reduces the radial velocity f ðnÞ As in radial direction transport of momentum is less when slip velocity is enhanced Fig 11 shows influence of the stretching parameter A on radial velocity The radial velocity enhances for larger stretching parameter A Please cite this article in press as: Imtiaz M et al Slip flow by a variable thickness rotating disk subject to magnetohydrodynamics Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2016.12.021 342 343 344 345 346 347 348 349 350 351 352 353 354 RINP 487 No of Pages 7, Model 5G 11 January 2017 M Imtiaz et al / Results in Physics xxx (2017) xxx–xxx 355 356 357 358 359 360 361 362 363 Fig Impact of m on radial velocity Fig Impact of M on radial velocity Fig Impact of Re on radial velocity Fig 10 Impact of c1 on radial velocity Fig Impact of a on radial velocity Fig 11 Impact of A on radial velocity Effects of disk thickness index m, Reynolds number Re, disk thickness coefficient a, Hartman number M and velocity slip parameter c2 on tangential velocity gðnÞ are observed in Figs (12– 16) Fig 12 shows that tangential velocity is an increasing function of m Fig 13 shows that tangential velocity increases for larger Re Fig 14 shows that tangential velocity is decreasing function of a It can be seen in Fig 15 that an increase in M reduces the tangential velocity because resistive force enhances Fig 16 represents decreasing behavior of tangential velocity with c2 nÀ1 Variation of Reynolds number Re on surface drag force C f Renỵ1 via Hartman number M is shown in Fig 17 Here surface drag force decays for larger Re while it increases for larger M Influence of thickness coefficient of disk a on surface drag force C f Re via disk thickness index m is depicted in Fig 18 Here surface drag force reduces when a and mare increased Fig 19 shows the effect of slip n1 nỵ1 parameters c1 and c2 on surface drag force C f Renỵ1 Here surface drag force reduces for larger c1 and c2 nÀ1 Please cite this article in press as: Imtiaz M et al Slip flow by a variable thickness rotating disk subject to magnetohydrodynamics Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2016.12.021 364 365 366 367 368 369 370 371 RINP 487 No of Pages 7, Model 5G 11 January 2017 M Imtiaz et al / Results in Physics xxx (2017) xxx–xxx Fig 12 Impact of m on tangential velocity Fig 15 Impact of M on tangential velocity Fig 13 Impact of Re on tangential velocity Fig 16 Impact of c2 on tangential velocity Fig 14 Impact of a on tangential velocity Fig 17 Impact of Re on skin friction coefficient 372 373 374 375 376 377 378 Concluding remarks MHD flow of a rotating disk in the presence of velocity slip was investigated Main findings are Larger Hartman number decreases fluid flow due to larger resistive force Opposite behavior of the thickness coefficient of the disk is seen on the axial and radial velocities Due to increment in stretching rate, radial velocity is an increasing function of stretching parameter For higher velocity slip parameters, transport of momentum is less which decays radial and tangential velocities Axial and radial velocities have opposite behavior for larger disk thickness index Higher Hartman number enhances the surface drag force Please cite this article in press as: Imtiaz M et al Slip flow by a variable thickness rotating disk subject to magnetohydrodynamics Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2016.12.021 379 380 381 382 383 384 385 386 RINP 487 No of Pages 7, Model 5G 11 January 2017 M Imtiaz et al / Results in Physics xxx (2017) xxx–xxx Fig 18 Impact of a on skin friction coefficient Fig 19 Impact of c2 on skin friction coefficient 387 References 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 [1] Karman TV Uber laminare and turbulente Reibung Zeitschrift f ür Angewandte Mathematik und Mechanik 1921;1:233–52 [2] Bachok N, Ishak A, Pop I Flow and heat transfer over a rotating porous disk in a nanofluid Physica B 2001;406:1767–72 [3] Kendoush AA Similarity solution for heat convection from a porous rotating disk in a flow field J Heat Transfer 2013;135:1885–6 [4] Rashidi MM, Kavyani N, Abelman S Investigation of entropy generation in MHD and slip flow over a rotating porous disk with variable properties Int J Heat Mass Transfer 2014;70:892–917 [5] Turkyilmazoglu M Nanofluid flow and heat transfer due to a rotating disk Comput Fluids 2014;94:139–46 [6] Sheikholeslami M, Hatami M, Ganji DD Numerical investigation of nanofluid spraying on an inclined rotating disk for cooling process J Mol Liq 2015;211:577–83 [7] Hayat T, Rashid M, Imtiaz M, Alsaedi A Magnetohydrodynamic (MHD) flow of Cu-water nanofluid due to a rotating disk with partial slip AIP Adv 2015;5:067169 [8] Mustafa I, Javed T, Ghaffari A Heat transfer in MHD stagnation point flow of a ferrofluid over a stretchable rotating disk J Mol Liq 2016;219:526–32 [9] Xun S, Zhao J, Zheng L, Chen X, Zhang X Flow and heat transfer of Ostwald-de Waele fluid over a variable thickness rotating disk with index decreasing Int J Heat Mass Transfer 2016;103:1214–24 [10] Hayat T, Imtiaz M, Alsaedi A, Alzahrani F Effects of homogeneous– heterogeneous reactions in flow of magnetite-Fe3O4 nanoparticles by a rotating disk J Mol Liq 2016;216:845–55 [11] Zheng L, Niu J, Zhang X, Gao Y MHD flow and heat transfer over a porous shrinking surface with velocity slip and temperature jump Math Comput Model 2012;56(5–6):133–44 [12] Abbasbandy S, Naz R, Hayat T, Alsaedi A Numerical and analytical solutions for Falkner-Skan flow of MHD Maxwell fluid Appl Math Comput 2014;242:569–75 [13] Turkyilmazoglu M Three dimensional MHD flow and heat transfer over a stretching/shrinking surface in a viscoelastic fluid with various physical effects Int J Heat Mass Transfer 2014;78:150–5 [14] Sheikholeslami M, Ganji DD, Javed MY, Ellahi R Effect of thermal radiation on magnetohydrodynamics nanofluid flow and heat transfer by means of two phase model J Magn Magn Mater 2015;374:36–43 [15] Sheikholeslami M, Hayat T, Alsaedi A MHD free convection of Al2O3–water nanofluid considering thermal radiation: A numerical study Int J Heat Mass Transfer 2016;96:513–24 [16] Hayat T, Imtiaz M, Alsaedi A, Almezal S On Cattaneo–Christov heat flux in MHD flow of Oldroyd-B fluid with homogeneous–heterogeneous reactions J Magn Magn Mater 2016;401:296–303 [17] Li J, Zheng L, Liu L MHD viscoelastic flow and heat transfer over a vertical stretching sheet with Cattaneo–Christov heat flux effects J Mol Liq 2016;221:19–25 [18] Makinde OD, Iskander T, Mabood F, Khan WA, Tshehla MS MHD CouettePoiseuille flow of variable viscosity nanofluids in a rotating permeable channel with Hall effects J Mol Liq 2016;221:778–87 [19] Ahmad R, Mustafa M, Hayat T, Alsaedi A Numerical study of MHD nanofluid flow and heat transfer past a bidirectional exponentially stretching sheet J Magn Magn Mater 2016;407:69–74 [20] Borrellia A, Giantesio G, Patria MC, Rosca NC, Rosca AV, Pop I Buoyancy effects on the 3D MHD stagnation-point flow of a Newtonian fluid Commun Nonlinear Sci Numer Simul 2017;43:1–13 [21] Rao J, Rajagopal KR The effect of the slip boundary condition on the flow of fluids in a channel Acta Mech 1999;135:113–26 [22] Buscall R Letter to the Editor: Wall slip in dispersion rheometry J Rheol 2010;54:1177–84 [23] Akbarinia A, Abdolzadeh M, Laur R Critical investigation of heat transfer enhancement using nanofluids in microchannels with slip and non-slip flow regimes Appl Thermal Eng 2011;31:556–65 [24] Mahmoud MAA, Waheed SE MHD flow and heat transfer of a micropolar fluid over a stretching surface with heat generation (absorption) and slip velocity J Egypt Math Soc 2012;20:20–7 [25] Khan WA, Uddin MJ, Ismail AIM Hydrodynamic and thermal slip effect on double-diffusive free convective boundary layer flow of a nanofluid past a flat vertical plate in the moving free stream PLOS One 2013;8(3):e54024 [26] Mukhopadhyay S Slip effects on MHD boundary layer flow over an exponentially stretching sheet with suction/blowing and thermal radiation Ain Shams Eng J 2013;4:485–91 [27] Malvandi A, Ganji DD Brownian motion and thermophoresis effects on slip flow of alumina/water nanofluid inside a circular microchannel in the presence of a magnetic field Int J Thermal Sci 2014;84:196–206 [28] Hayat T, Imtiaz M, Alsaedi A, Kutbi MA MHD three-dimensional flow of nanofluid with velocity slip and nonlinear thermal radiation J Magn Magn Mater 2015;396:31–7 [29] Karimipour A, Nezhad AH, D’Orazia A, Esfe MH, Safaei MR, Shiran E Simulation of copper-water nanofluid in a microchannel in slip flow regime using the lattice Boltzman method Eur J Mech-B/Fluids 2015;49:89–99 [30] Wu L Effect of mass transfer induced velocity slip on heat transfer of viscous gas flows over stretching/shrinking sheets International Journal of Thermal Sciences 2017;112:165–73 [31] Hatami M, Nouri R, Gangi DD Forced convection analysis for MHD Al2O3water nannofluid flow over a horizontal plate J Mol Liq 2013;187:294–301 [32] Abbasbandy S, Yurusoy M, Gulluce H Analytical solutions of non-linear equations of power-law fluids of second grade over an infinite porous plate Math Comput Appl 2014;19(2):124 [33] Lin Y, Zheng L, Chen G Unsteady flow and heat transfer of pseudoplasticnano liquid in a finite thin film on a stretching surface with variable thermal conductivity and viscous dissipation Powder Technol 2015;274:324–32 [34] Farooq U, Hayat T, Alsaedi A, Liao SJ Effective computation of exact and analytic approximate solutions to singular nonlinear equations of Lane– Emden–Fowler type Numer Algorithms 2015;70:43–59 [35] Sui J, Zheng L, Zhang X, Chen G Mixed convection heat transfer in power law, fluids over a moving conveyor along an inclined plate Int J Heat Mass Transfer 2015;85:1023–33 [36] Shahzad SA, Hayat T, Alsaedi A, Chen B A useful model for Solar radiation Energy, Ecol Environ 2016;1:30–8 [37] Mustafa M, Mushtaq A, Hayat T, Alsaedi A Rotating flow of magnetite-water nanofluid over a stretching surface inspired by non-linear thermal radiation PLOS One 2016 0149304 [38] Hayat T, Qayyum S, Imtiaz M, Alsaedi A, Alzahrani F Partial slip effect in flow of magnetite-Fe3O4 nanoparticles between rotating stretchable disks J Magn Magn Mater 2016;413:39–48 Please cite this article in press as: Imtiaz M et al Slip flow by a variable thickness rotating disk subject to magnetohydrodynamics Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2016.12.021 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493