magnetohydrodynamic flow of casson fluid over a stretching cylinder

RINP 525 No of Pages 5, Model 5G 10 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 Magnetohydrodynamic flow of Casson fluid over a stretching cylinder M Tamoor a, M Waqas b, M Ijaz Khan b,⇑, Ahmed Alsaedi c, T Hayat b,c a 10 Department of Basic Sciences, University of Engineering and Technology, Taxila 47050, 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, P.O Box 80257, Jeddah 21589, Saudi Arabia 11 a r t i c l e 14 15 16 17 18 19 20 21 22 23 24 25 b i n f o Article history: Received October 2016 Received in revised form January 2017 Accepted January 2017 Available online xxxx Keywords: Viscous dissipation Casson liquid Joule heating Newtonian heating Stretching cylinder a b s t r a c t Here the Newtonian heating characteristics in MHD flow of Casson liquid induced by stretched cylinder moving with linear velocity is addressed Consideration of dissipation and Joule heating characterizes the process of heat transfer Applied electric and induced magnetic fields are not considered Magnetic Reynolds number is low The convenient transformation yields nonlinear problems which are solved for the convergent solutions Convergence region is determined for the acquired solutions Parameters highlighting for velocity and temperature are graphyically discussed Numerical values of skin friction and Nusselt number are also presented in the tabular form Present analysis reveals that velocity and thermal fields have reverse behavior for larger Hartman number Moreover curvature parameter has similar influence on the velocity, temperature, skin friction and Nusselt number Ó 2017 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 Several physiological materials can be identified as viscoplastic materials which are the sub-classes of rheological materials having both elastic and viscous features Several investigators have developed models in this direction [1–3] In this regard Casson liquid model has shown relatively advantageous This model behaves as solid when the shear stress is less than the yield stress and it starts to deform when shear stress becomes greater than the yield stress Several researchers studied this model under distinct physical aspects For instance Dash et al [4] reported the characteristics of Casson liquid under yield stress via homogeneous porous medium bounded by circular tube Effectiveness of variable properties in non-Darcy dissipative flow of Casson liquid is presented by Animasaun et al [5] Hayat et al [6] explored viscous dissipation impact in Casson nanofluid flow in presence of variable conductivity and mixed convection Characteristics of gyrotactic microorganisms in magneto Casson nanomaterial is addressed by Raju et al [7] Raju and Sandeep [8] investigated the effects of ferrous nanoparticles considering Casson model The magnetic field phenomenon in convection flows is topic of advancement in technology and industry Specific examples include collection of solar energy, insulation of nuclear reactor, furnaces, cooling of electronic chips and devices, crystal growth in flu- 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 ⇑ Corresponding author E-mail address: mikhan@math.qau.edu.pk (M.I Khan) ids, drying process and several others In electrically conducting liquid, the Lorentz force overcomes the convective currents via reduction in liquid velocities [9] Hence the involvement of external magnetic field can be implemented to control the mechanism in material industry [10–20] Furthermore the Joule heating can generate enhancement in temperature and also yield temperature gradient Moreover it has remarkable influence in nuclear engineering and geophysical stream [21–25] Viscous dissipation characterizes the degeneration of mechanical energy into the thermal energy Such phenomenon transpires in all the flow systems However for different flow configurations, the characteristics of viscous dissipation is often neglected It is meaningful just for the systems having larger velocity and velocity gradients respectively It is for this reason that the viscous dissipation is introduced in the present study Simultaneous characteristics of viscous dissipation, Dufour-Soret and thermophoresis in radiative viscous material flow towards isothermal wedge is explored by Pal and Mondal [26] Zaib and Shafie [27] examined viscous dissipation aspect in chemically reacting stratified stretched flow of viscous liquid considering Hall current Hayat et al [28,29] scrutinized the effectiveness of viscous dissipation and MHD effects considering Jeffrey and second grade liquid models Three-dimensional dissipative stretched flow of radiative PowellEyring nanomaterial is addressed by Mahanthesh et al [30] Our prime theme here is to report the magnetohydrodynamic (MHD) convective flow of Casson fluid Joule heating and dissipation are considered to predict the characteristics of heat transfer http://dx.doi.org/10.1016/j.rinp.2017.01.005 2211-3797/Ó 2017 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: Tamoor M et al Magnetohydrodynamic flow of Casson fluid over a stretching cylinder Results Phys (2017), http://dx.doi org/10.1016/j.rinp.2017.01.005 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 525 No of Pages 5, Model 5G 10 January 2017 M Tamoor et al / Results in Physics xxx (2017) xxx–xxx 94 Implementation of HAM [32–39] leads to convergent series solution The emerging diverse parameters are plotted for distinct values to explore their impact on the velocity and thermal fields The values of surface drag force and heat transfer rate are numerically elaborated 95 Governing problems 96 103 Here steady magnetohydrodynamic (MHD) flow of Casson liquid is modeled Stretching of cylinder creates the flow Joule heating and dissipation effects are retained We imposed the Newtonian heating condition at the stretching cylinder The flow is anticipated in axial (x) direction whereas radial direction is perpendicular to x Entire treatment is deliberated via assumptions of boundary layer Cylinder has linear stretching velocity Under these hypothesis, 2D flow expressions are [6] 106 @ruị @r v ị ỵ ¼ 0; @x @r 90 91 92 93 97 98 99 100 101 102 104 107 u 109 110 !   @u @u @ u @u rB2 0u ỵ ỵv ẳm 1ỵ @x @r b @r r @r q ỵ rB20 u2 ; qcp ux; rị ẳ uw xị ẳ u0l x ; v x; rị ẳ 0; @T ẳ hs Tatr ẳ R; @r 118 119 120 121 122 123 124 125 u ! 0; T ! T asr ! 1: ð5Þ Eq (1) is identically satisfied while Eqs (2)–(4) have the forms: 128 132   À 000 00 Á 00 02 1ỵ ỵ 2agịf ỵ 2cf ỵ ff f Ha2 f ẳ 0; b ỵ PrEcHa2 f 02 ¼ 0; 135 137 f ¼ 0; f ẳ 1; h0 0ị ẳ c1 ỵ h0ịị at g ¼ 0; 138 140 f ! 0; h ! as g ! 1; 142 143 144 ð6Þ   002 f ỵ 2agịh00 ỵ 2ah0 ỵ Prf h0 ỵ PrEc1 ỵ 2agị ỵ b 134 141 Fig Impact of b on f ! rffiffiffiffiffi rffiffiffiffiffiffiffiffi u0 r À R2 u0 x R u0 m T À T1 ;u ¼ f gị; v ẳ ; f gị; hgị ẳ l r ml 2R l T1 127 131 ð4Þ In the aforestated expressions u and v are the velocity components in the x and r directions, b the Casson fluid parameter, uw ðxÞ the stretching velocity, m the kinematic viscosity, q the density of fluid, cp the specific heat, r the constant electrical conductivity, B0 the applied magnetic field, lB the plastic dynamic viscosity, hs the heat transfer coefficient and concentration exponent, ðT; T Þ the fluid and ambient temperatures respectively Utilizing suitable transformations gẳ 129 3ị The boundary conditions are 113 114 117 ð2Þ !     2 @T @T k @ T @T @u ỵ ỵv ẳ u ỵ l ỵ B @x @r b @r qcp @r2 r @r 112 116 Fig  h-Curves for f and h ð1Þ ð7Þ sffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffi ml 0 Fig Impact of Ha on f ð8Þ where b the Casson fluid parameter, a the curvature parameter, Ha the Hartman number, Pr the Prandtl number, Ec the Eckert number and c the conjugate parameter of Newtonian heating These are defined as 145 0sffiffiffiffiffi1 rB2 l lc p vl u2 ml A ; Ha2 ¼ ; Pr ¼ ; Ec ¼ w ; c ¼ hs @ : a¼ ;hs u0 u0 qu0 k cp T u0 R Skin friction  C Re1=2 x f  ð9Þ   and Nusselt number NuRexÀ1=2 are expressed as Please cite this article in press as: Tamoor M et al Magnetohydrodynamic flow of Casson fluid over a stretching cylinder Results Phys (2017), http://dx.doi org/10.1016/j.rinp.2017.01.005 147 148 149 RINP 525 No of Pages 5, Model 5G 10 January 2017 M Tamoor et al / Results in Physics xxx (2017) xxx–xxx Fig Impact of a on f Fig Impact of c on h Fig Impact of Ha on h Fig Impact of Ec on h Fig Impact of a on h Fig Impact of Pr on h 150 152 153 154 156 157 Cf ¼ 2sw xqw ; ; Nux ¼ qu2w kðT w À T Þ ð10Þ C f Re1=2 x   00 f 0ị; NuRe1=2 ẳ 1ỵ ¼ Àh0 ð0Þ; x b 158 ð12Þ x2 160 where Rex ¼ u0v l is the local Reynolds number 161 Convergence analysis of obtained results 162 The solutions of the expressions comprising in Eqs (6)–(8) are determined via homotopic approach The homotopic technique 163 with  sw ¼ l þ or b     @u @T ; qw ẳ k ; @r rẳR @r rẳR 11ị Please cite this article in press as: Tamoor M et al Magnetohydrodynamic flow of Casson fluid over a stretching cylinder Results Phys (2017), http://dx.doi org/10.1016/j.rinp.2017.01.005 164 RINP 525 No of Pages 5, Model 5G 10 January 2017 M Tamoor et al / Results in Physics xxx (2017) xxx–xxx Table Convergent values at distinct order b ¼ 2:0; a ¼ Ha ¼ 0:1; Ec ¼ c ¼ 0:3 and Pr ¼ 1:2 of approximations when Order of approximations Àf ð0Þ 00 Àh0 ð0Þ 10 12 15 20 0.8810 0.8586 0.8576 0.8575 0.8575 0.8575 0.5187 0.6270 0.6423 0.6425 0.6425 0.6425 Table   Numerical values of skin friction 12 C f Re1=2 for distinct values of b; a and Ha when x Pr ¼ 1:2; Ec ¼ c ¼ 0:3 b a Ha 1=2 C f Rex 1.0 1.2 1.4 2.0 1.6 0.2 -1.4949 -1.4283 -1.3790 -1.2309 -1.2860 -1.3404 -1.2797 -1.3360 -1.4900 0.0 0.1 0.2 0.1 0.0 0.3 0.6 172 yields a straight forward approach in order to control and adjust the rate of convergence The parameters  hf and  hh need the appropriate values for such objective No doubt these auxiliary parameters play a vital role for the convergence of obtained solutions The  h-curves have been developed at 15th order of approximations to acquire valid ranges of these parameters (see Fig 1) It is seen that the permissible values of these parameters are À1:45  hf À0:40 and À1:40  hh À0:50 173 Results and discussion 165 166 167 168 169 170 171 174 175 176 177 178 179 180 181 182 183 184 The behavior of physical variables on the velocity f ðgÞ and thermal fields hðgÞ is interpreted here Figs 2–9 certify the roles of dimensionless Casson fluid parameter ðbÞ, Hartman number ðHaÞ, curvature parameter ðaÞ, Prandtl number ðPrÞ, conjugate parameter ðcÞ and Eckert number ðEcÞ Fig elucidates the behavior of b on 0 f ðgÞ Clearly larger b reduced f ðgÞ and thickness of boundary layer Features of Ha on f ðgÞis interpreted in Fig It is of interest to note that f ðgÞ decays via larger Ha whereas thickness of boundary layer reduces Since Lorentz force increments causes much resistance in the fluid flow and consequently f ðgÞ decays The characteristics of a is illustrated in Fig It is reported that velocity and thickness of boundary layer are enhanced when a increments Physically for larger a, the radius of curvature decays which reduces the contact area of the cylinder with the fluid Therefore resistance offered by the surface decays and velocity of the fluid enhances Fig demonstrates that larger Ha correspond to higher hðgÞ and thickness of layer In fact Lorentz force strengthens due to higher Ha which provides resistance to fluid motion and therefore some valuable energy is transferred into heat This mechanism is responsible for increment in hðgÞ Variation of Pr on hðgÞ is disclosed in Fig It is explored that hðgÞ decays via larger Pr As thermal diffusivity reduces with an increase in Prandtl number so hðgÞ decreases Fig signifies the impact of c on hðgÞ As expected hðgÞ and its associated thickness of thermal layer increment for larger c Coefficient of heat transfer ðhs Þ is enhanced via larger c Thus more heat is transferred from heated surface of the cylinder to cooled surface of liquid Consequently hðgÞ and corresponding thickness of boundary layer augment The role of Ec on hðgÞ is displayed through Fig As expected both hðgÞ and related thickness of thermal layer augment through larger Ec Physically for larger Ec the liquid particles become more active because of storage of energy This fact leads to higher temperature Fig reveals the features of a on hðgÞ Here hðgÞ decays by enhancing a close to the stretched surface It shows augmenting behavior when one moves away from the surface Physically higher a increase the thickness of thermal layer due to which the heat transport rate decays and hðgÞ increases À Á Table reports the convergence of velocity f ðgÞ and temperature ðhðgÞÞ Presented values scrutinized that 15th-order of 00 approximations are enough for f ð0Þ and h0 ð0Þ respectively Table   is calculated to determine the coefficient of skin friction 12 C f Re1=2 x for distinct values of physical variables b; a and Ha Here we   shows increasing behavior observed that skin friction 12 C f Re1=2 x for larger a and Ha However opposite situation is noticed for higher b Impacts of a; Ha; Pr; Ec and c on Nusselt number   are presented through Table Clearly Nusselt number NuReÀ1=2 x   À1=2 NuRex illustrates boosted behavior for increasing values of a; Pr and c whereas it decays when Ha and Ec are enhanced For 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 Conclusions 225 Here Joule heating and magnetohydrodynamic (MHD) effects in dissipative stretched flow of Casson liquid are investigated The following points are worthmentioning: 226 a Ha Pr Ec c NuRexÀ1=2 0.0 0.1 0.2 0.1 0.1 1.2 0.3 0.3 0.5504 0.5630 0.5753 0.5649 0.5565 0.5341 0.5777 0.5909 0.6028 0.7257 0.5330 0.4604 0.6225 0.6649 0.6969 1.3 1.4 1.5 1.2 186 some limiting cases, comparison with previously available results in the literature is made and an excellent agreement is achieved (for detail see [40] and Table 4) Table   Numerical values of Nusselt number NuReÀ1=2 for distinct values of the variables a; Ha; Pr; Ec and c when b ¼ 2:0 x 0.0 0.2 0.4 0.1 185 0.0 0.4 0.8 0.3 0.4 0.5 0.6 Please cite this article in press as: Tamoor M et al Magnetohydrodynamic flow of Casson fluid over a stretching cylinder Results Phys (2017), http://dx.doi org/10.1016/j.rinp.2017.01.005 223 224 227 228 RINP 525 No of Pages 5, Model 5G 10 January 2017 M Tamoor et al / Results in Physics xxx (2017) xxx–xxx Table 00 Comparative analysis for Àf ð0Þ with [40] for distinct values of Ha when b ¼ ¼ a 00 Ha 0.0 0.2 0.5 0.8 1.0 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 00 Àf ð0Þ Àf ð0Þ Fang et al [37] Present 1.00000 1.01980 1.11803 1.28063 1.41421 1.1180 À Á  Velocity field f ðgÞ diminishes when b and Ha are incremented  Velocity and thermal fields describe insignificant behavior near the cylinder surface and these quantities increment away from the cylinder for larger curvature parameter ðaÞ  With the increment in Pr, the thermal field and related layer thickness decay  Larger Ec; c and Ha enlarge hðgÞ and associated thickness of boundary layer   decays via higher b however it incre Skin friction 12 C f Re1=2 x ments when Ha and a are increased   and thermal field ðhðgÞÞ have  Nusselt number NuReÀ1=2 x reverse behavior via larger Pr  Presented model shows viscous fluid characteristics when b ! 244 245 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this article in press as: Tamoor M et al Magnetohydrodynamic flow of Casson fluid over a stretching cylinder Results Phys (2017), http://dx.doi org/10.1016/j.rinp.2017.01.005 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371