RINP 585 No of Pages 9, Model 5G 24 February 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 Influence of radial magnetic field on the peristaltic flow of Williamson fluid in a curved complaint walls channel Tasawar Hayat a,b, Sadaf Nawaz a,⇑, Ahmed Alsaedi b, Maimona Rafiq a 10 11 14 15 16 17 18 19 20 21 22 23 24 a b 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, Jeddah 21589, Saudi Arabia a r t i c l e i n f o Article history: Received 20 November 2016 Received in revised form 12 February 2017 Accepted 15 February 2017 Available online xxxx Keywords: Williamson fluid Curved channel Radial magnetic field Complaint walls a b s t r a c t Peristaltic transport of Williamson fluid in a curved geometry is modeled Problem formulation is completed by complaint walls of channel Radial magnetic field in the analysis is taken into account Resulting problem formulation is simplified using long wavelength and low Reynolds number approximations Series solution is obtained for small Weissenberg number Influences of different embedded parameters on the axial velocity and stream function are examined As expected the velocity in curved channel is not symmetric Axial velocity is noticed decreasing for Hartman number Trapped bolus increases for Hartman and curvature parameters 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/) 26 27 28 29 30 31 32 33 34 35 36 37 Introduction 38 The mechanism of peristalsis has prime importance in biomechanics This mechanism occurs due to traveling wave along flexible walls of channel This travelling wave forces the contained fluid to flow in the same direction even in absence of external involved pressure gradient Various physiological materials are transported through such mechanism Mention may be made to the transport of urine from kidney to bladder, food from oesophagus to gastrointestinal tract, chyme through intestine, vasomotion of small blood vessels, embro transport in the uterus, spermatic fluid in the ductus effertenes of the male reproductive tract, ovum movement in female fallopian tube etc Further peristalsis is well suited to design many machines including dialysis machine, open heart bypass machine, finger, roller and hose pumps, infusion pumps etc There is widespread applications of peristalsis in the pump industry for transport of corrosive and sterile materials A vast use of peristalsis is due to the fact that the transported fluid does not have direct contact with any moving part Because of all these applications there are intensive attempts on peristalsis through theoretical and experimental approaches Latham [1] and Shapiro et al [2] were the first who presented pioneering research on peristalsis of viscous fluids Afterwards many researchers worked on peristalsis by considering both viscous and non-Newtonian fluids 39 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: sadafnawaz26@gmail.com (S Nawaz) via diverse aspects and assumptions Some literature on peristalsis can be estimated through the references [3–23] There is no doubt that magnetohydrodynamics (MHD) deals with the motion of highly conducting fluid via a magnetic field Specifically peristalsis in presence of magnetic field has relevance with processes regarding motion of conducting materials for example the geothermal sources analysis, MHD power generators design, hyperthermia, drug delivery, removal of arteries blockage, MHD compressors, nuclear fuel debris treatment, the control of underground spreading of chemical wastes and pollution Having all this in mind, extensive attempts in the aforementioned quoted literature have been mostly for peristalsis in presence of constant applied magnetic field In other words the peristalsis subject to radial magnetic field is examined scarcely Further in these attempts the channel is taken straight This flow configuration is not realistic since most of the geometries involved in industry and physiological processes are curved Thus very recent the researchers in the field analyzed the impact of curvature on peristalsis in a channel (see [24–43]) The objective of present communication is to venture further in this regime Thus our intention here is to model the effect of radial magnetic field on peristaltic transport of Williamson fluid in a curved channel with flexible walls Note that Williamson fluid is a shear thinning viscoelastic material It also represents the behavior of diverse variety of pseudoplastic liquids accurately i-e a decrease in viscosity through increase in rate of shear stress is exhibited Many biophysical and physiological materials posses shear-thinning behavior for example blood suspension and gastro-intestinal fluids [9] Here http://dx.doi.org/10.1016/j.rinp.2017.02.022 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: Hayat T et al Influence of radial magnetic field on the peristaltic flow of Williamson fluid in a curved complaint walls channel Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.02.022 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 86 87 RINP 585 No of Pages 9, Model 5G 24 February 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx 91 stream function formulation and lubrication approach are employed Solution for small Weissenberg number is constructed Impacts of sundry variables on physical quantities of interest are addressed in detail 92 Modeling 88 89 90 93 94 95 96 97 98 99 100 102 103 104 105 106 107 109 110 111 112 113 Consider a curved channel of width 2d coiled in a circle of radius R⁄ and centre at O Here x -axis lies in the length of the channel and r -axis normal to it The axial flow direction is along x -axis while radial direction along the r -axis In the axial and radial directions the components of velocity are denoted by u and v respectively An incompressible electrically conducting Williamson fluid fills the channel The walls shape is defined as follows: 2p r ẳ ặgx; tị ẳ ặ d ỵ a sin x ctị ; k R B0 Bẳ ; 0; ; r ỵ R J B ¼ 0; À r 115 116 117 118 119 121 122 ð2Þ : ð4Þ @v @v uRà @ v u2 q ỵv ỵ @t @r r ỵ R @x r ỵ R @u @u uR @u uv þv þ þ @t @r r þ Rà @x r þ Rà ð5Þ Ã The boundary conditions for the present flow considerations are 157 u ¼ at r ¼ Æg; ¼ 135 136 À1 qffiffiffiffiffiffiffi P; T A1 ẳ gradV ỵ gradVị ; 138 139 140 141 ð7Þ where C denotes the time constant, l0 and l1 are the zero shear rate and infinite shear rate viscosities and c_ and A1 are defined below c_ ¼ P¼ 164 Here p represents the pressure, R the curvature parameter and 8ị trA21 ị: Here we have taken l1 ẳ and Cc_ < 1: Thus extra stress tensor becomes 165 q the density In above equation s, m and d1 describe the elastic 166 tension, mass per unit area and the coefficient of viscous damping respectively Further Srr, Srx and Sxx are the components of extra stress tensor S Considering 167 à x ¼ x ; k dSij cl0 à r ¼ à r ; d u ¼ à ; k ¼ Rd ; u ; c v à ¼ vc ; à t ¼ ct ; k à g g ¼ d; c_ à ¼ c_ dc ; pà ¼ ckd lp0 ; We ¼ C dc ; @v @v dku @ v u2 Red d ỵv ỵ @t @r r ỵ k @x r ỵ k @p @ kd @Sxr Sxx ; ẳ ỵd fr ỵ kịSrr g ỵ @r r ỵ k @r r ỵ k @x rỵk S ẳ ẵl1 ỵ l0 ỵ l1 ị1 Cc_ Þ A1 ; 134 159 161 ð15Þ ⁄ 132 133 ; @ @Sxx fr ỵ R ị Srx g ỵ R qr r ỵ R @r @x ! à @u uR @u uv rðRà Þ2 B20 u @u ỵR ị ỵv ỵ ỵ ; r ẳ ặg: @t @r r ỵ R @x r ỵ R r ỵ R ị 130 158 162 # @3 @3 @2 R s ỵ m þ d1 g @x @t@x @x@t2 in which relation for an extra stress tensor S for Williamson fluid is [13]: 14ị " 129 128 r ỵ R ị 154 156 ẳ 127 153 13ị 168 169 170 171 16ị 173 174 175 ð17Þ 177 178 @u @u dku @u uv ỵv ỵ ỵ Re d @t @r r ỵ k @x r ỵ k R @p @ R @Sxx ỵ fr ỵ R ị Srx g ỵ r ỵ R @x r ỵ R ị2 @r r ỵ R @x rR ị2 B20 u 150 151 ð12Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 à 2 2 @v v Rà @u R @v u @u _c ¼ ỵ2 ỵ : ỵ ỵ @r r ỵ R r ỵ R @x r ỵ R @x r ỵ R @r 6ị 147 148 11ị the non-dimensional problems are ẳ 10ị v R @u ; Sxx ẳ 2l0 ỵ Cc_ ị þ r þ Rà r þ Rà @x SÃij ¼ @p @ R @Sxr Sxx fr ỵ R ịSrr g ỵ ; ẳ ỵ @r r þ R @r r þ Rà @x r þ Rà 145 3ị @ẵr ỵ R ịv @u þ Rà ¼ 0; @r @x q 144 @v Srr ẳ 2l0 ỵ Cc_ ị ; @r R @v u @u ; Srx ¼ l0 ð1 þ Cc_ Þ þ À r þ Rà @x r þ Rà @r # Here r denotes the electrical conductivity and J(=V  B) is the current density of fluid Present flow is governed by the following expressions: 124 125 r ỵ R ị The components of extra stress tensor are 143 à where B0 denotes the strength of applied magnetic field The Lorentz force (F = J  B) in absence of electric field takes the form as follows: ðRÃ Þ B20 u ; à ð9Þ ð1Þ in which a, c and k denote the amplitude, speed and wavelength of the wave respectively and t stands for time The definition of magnetic field B applied in the radial direction is " S ẳ l0 ẵ1 ỵ Cc_ ịA1 : k @p @ kd @Sxx ỵ fr ỵ kị Srx g ỵ r ỵ k @x r ỵ kị2 @r r ỵ k @x k 2 r ỵ kị M u; 18ị @v ; @r @v u @u ; ỵ Srx ẳ ỵ Wec_ ị d @x r ỵ k @r Srr ẳ 21 ỵ Wec_ ị k @u d ; Sxx ẳ 21 ỵ Wec_ ị ỵ r ỵ k r ỵ k @x v s 2 2 2 @v v k @u k @v u @u d d ỵ ỵ c_ ẳ ỵ2 ỵ ; r ỵ k @x r ỵ k @r @r r ỵ k r ỵ k @x 180 181 ð19Þ 183 184 ð20Þ 186 187 ð21Þ 189 190 ð22Þ Please cite this article in press as: Hayat T et al Influence of radial magnetic field on the peristaltic flow of Williamson fluid in a curved complaint walls channel Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.02.022 192 RINP 585 No of Pages 9, Model 5G 24 February 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx u ẳ 0; at r ẳ ặg ẳ ặ1 ỵ e sin 2pðx À tÞÞ; ð23Þ 193 195 # @3 @3 @2 k E1 ỵ E2 ỵ E3 gẳ @x @t@x @x@t2 ! @S @ @u @u ukd @u uv c ỵ rỵk @r fr ỵ kị Srx g ỵ dk @xxx Rer ỵ kị d ỵ v ỵ ỵ @t @r r ỵ k @x r ỵ k k M2 u; atr ẳ ặg; 196 24ị 198 where e( = a/d) denotes the amplitude ratio, k the dimensionless curvature parameter, Re( = qcd/l0) the Reynolds number, dẳ d=kị p the wave number, M ¼ r=l0 B0 d the Hartman number, We the Williamson fluid parameter known as Weissenberg number and 199 " rỵk Fig Geometry of the problem 3 E1 ẳ sd =k3 cl0 ị; E2 ẳ mcd =k3 l0 ị and E3 ẳ d1 d =k2 l0 ị depict the non-dimensional form of elastance parameters respectively Writing the velocity components ðu; v Þ in terms of stream function (w) by [26,32,34]: Fig u via change in M when E1 = 0.003, E2 = 0.003, E3 = 0.01, t = 0.1, x = 0.2, e = 0.2, We = 0.01.(a)k = 5.0.(b)k ? Fig u via change in We when E1 = 0.003, E2 = 0.003, E3 = 0.01, t = 0.1, x = 0.2, e = 0.2, M = 0.8.(a)k = 5.0.(b)k ? Please cite this article in press as: Hayat T et al Influence of radial magnetic field on the peristaltic flow of Williamson fluid in a curved complaint walls channel Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.02.022 200 201 202 203 204 205 206 207 RINP 585 No of Pages 9, Model 5G 24 February 2017 209 210 211 212 213 215 T Hayat et al / Results in Physics xxx (2017) xxx–xxx u¼À @w ; @r vẳ dk @w : r ỵ k @x Zeroth order system and stream function 250 222 ð34Þ @p ¼ 0; @r w0r ¼ 0; at r ¼ Æg; ð35Þ À ð25Þ k @p @ k @w ỵ fr ỵ kị Srx g ỵ ẳ 0; M2 r ỵ k @x r ỵ kị2 @r @r r ỵ kị wr ẳ at r ẳ ặg ẳ ặ1 ỵ e sin 2px tịị; " ð26Þ 225 227 228 230 @ k @w fr ỵ kị Srx g ỵ ; M2 r ỵ k @r @r rỵk @ k @w fr þ kÞ S0rx g þ M ; at r ẳ ặg; r ỵ k @r rỵk @r 36ị ð28Þ ð37Þ ð29Þ ! @w @ w Srx ẳ ỵ Wec_ ị ; r ỵ k @r @r 30ị ỵ 263 1ỵk2 M2 1ỵk2 M2 C k ỵ rị p : ỵ ỵ k M2 38ị 231 233 234 235 237 c_ ¼ @w @ w À : r ỵ k @r @r2 31ị By cross differentiation of Eqs (25) and (26) we obtain: @ @ k @w fr ỵ kị Srx g ỵ M2 ẳ 0: @r kr ỵ kị @r rỵk @r ð32Þ 239 Eq (25) indicates that p does not depend upon r Note that the results for planar channel case are recovered when k ? 1.; 240 Solution methodology 241 It is difficult to find the closed form solution for the Eq (32) Thus we intend to compute the solution for small Weissenberg number (0 < We < 1) For this purpose we write 238 242 243 244 246 w ẳ w0 ỵ Wew1 ỵ OWe2 ị: 247 The corresponding zeroth and first order systems and respective stream functions are: 248 262 r C k ỵ rị p þ 2 À þ k M2 pffiffiffiffiffiffiffiffiffiffiffiffiffi w0 ẳ C ỵ C kr ỵ C 1ỵ Srr ẳ ẳ Sxx ; 259 260 @w0 @ w0 ẳ ; r ỵ k @r @r2 p at r ẳ ặg; 254 256 S0rx ¼ # @3 @3 @2 k E1 ỵ E2 ỵ E3 g @x @t@x @x@t 27ị @3 @3 @2 ỵ E2 ỵ E3 g @x3 @t@x @x@t 253 257 " ¼ # k E1 224 251 @ @ k @w fðr ỵ kị S0rx g ỵ M2 ẳ 0; @r kr ỵ kị @r rỵk @r 219 221 The continuity equation is identically satisfied Using above expressions in Eqs (17)(24) and then implementing the long wavelength and low Reynolds number approximations we obtain 216 218 249 ð33Þ Fig u via change in k when E1 = 0.001, E2 = 0.001, E3 = 0.01, t = 0.1, x = 0.2, e = 0.2, We = 0.01, M = 0.8 Fig u via change in E1, E2, E3when t = 0.1, x = 0.2, e = 0.2, We = 0.01, M = 2.0.(a)k = 2.0.(b)k ? Please cite this article in press as: Hayat T et al Influence of radial magnetic field on the peristaltic flow of Williamson fluid in a curved complaint walls channel Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.02.022 265 RINP 585 No of Pages 9, Model 5G 24 February 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx Fig w via change in M when E1 = 0.003, E2 = 0.003, E3 = 0.01, t = 0, k = 5.0, e = 0.1, We = 0.03.(a)M = 5.0.(b)M = 7.0.(c)M = 9.0 266 First order system and stream function S1rx ¼ 267 268 270 @ @ k @w fr ỵ kị S1rx g ỵ M ẳ 0; @r kr ỵ kị @r rỵk @r 39ị 271 273 w1r ẳ 0; at r ẳ ặg; ð40Þ 274 276 @ k @w fðr ỵ kị S1rx g ỵ M ẳ 0; at r ẳ ặg; r ỵ k @r rỵk @r ð41Þ @w1 @ w1 @w0 @ w0 þ À À r þ k @r r þ k @r @r @r2 277 !2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 2 2 1ỵk2 M C 22 21 ỵ ỵ k M2 ị ỵ k M 2 ỵ ỵ k M ị k ỵ rị p w1 ẳ 2 þ k MIn2 ð8 þ 9k M Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi p p 2 2 1ỵk2 M C 21 21 ỵ ỵ k M2 ị ỵ k M 2 ỵ ỵ k M2 ịk ỵ rị p ỵ 2 ỵ k M2 ỵ 9k M ị p p 1ỵ 1ỵk2 M2 1ỵk2 M2 B1 k ỵ rị B2 k ỵ rị r2 p ỵ p ỵ krB3 ỵ B3 ỵ B4 ; ỵ 2 2 1ỵ 1ỵk M 1ỵk M 42ị 279 280 43ị Please cite this article in press as: Hayat T et al Influence of radial magnetic field on the peristaltic flow of Williamson fluid in a curved complaint walls channel Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.02.022 282 RINP 585 No of Pages 9, Model 5G 24 February 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx Fig w via change in k when E1 = 0.003, E2 = 0.003, E3 = 0.01, t = 0, M = 4.0, e = 0.1, We = 0.03.(a)k = 3.0.(b)k = 5.0.(c)k ? 283 284 where the values for Ci,s (i = 1–4)and Bi,s(i = 1–4) are obtained with the help of MATHEMATICA (Fig 1) 285 Results and discussion 286 First of all the velocity is analyzed here Impacts of different parameters namely the wall properties (E1, E2, E3), magnetic field (M), Weissenberg number (We) and curvature parameter k on the 287 288 axial velocity u and stream function w have been examined Fig 2(a) and 2(b) are plotted to see the Hartman number effect on the axial velocity in the curved and straight channels respectively It can be seen that larger magnetic field decreases axial velocity in view of the resistive nature of Lorentz force It is also noted that axial velocity u in case of curved channel is tilted towards left Fig 3(a) and (b) elucidate the influence of Weissenberg number on the axial velocity It is noted that the axial velocity shows dual behavior for Weissenberg number By enhanc- Please cite this article in press as: Hayat T et al Influence of radial magnetic field on the peristaltic flow of Williamson fluid in a curved complaint walls channel Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.02.022 289 290 291 292 293 294 295 296 297 RINP 585 No of Pages 9, Model 5G 24 February 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx Fig w via change in We when E1 = 0.003, E2 = 0.003, E3 = 0.01, t = 0, k = 5.0, e = 0.1, M = 7.0.(a)We = 0.01.(b)We = 0.02.(c)We = 0.03 298 299 300 301 302 303 304 305 306 ing Weissenberg number the velocity decreases in the region ½À1; 0 whereas it increases in the region ½0; 1 Fig 4(a) and (b) represent the influence of wall properties on the axial velocity We can see that with the increase in E1 and E2 the velocity enhances The fact behind this is the less resistance offered to the flow due to the elastance of wall which cause an enhancement in the fluid velocity Decrease in the axial velocity is noticed when E3 increases Here damping is responsible for decrease in the axial velocity Fig represents the behavior of curvature parameter k on the axial velocity Mixed behavior of curvature parameter on the axial velocity is observed Axial velocity decreases near the lower wall while it increases in rest part of channel It is also noted that for very large values of curvature parameter k the curve becomes symmetric Moreover in the straight channel the velocity is more than in the curved channel Fig 6(a)–(c) have been displayed for the behavior of Hartman number on the stream function It can be seen that the size of trapped bolus increases for larger Hartman number Fig 7(a)–(c) Please cite this article in press as: Hayat T et al Influence of radial magnetic field on the peristaltic flow of Williamson fluid in a curved complaint walls channel Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.02.022 307 308 309 310 311 312 313 314 315 RINP 585 No of Pages 9, Model 5G 24 February 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx Fig w via change in E1, E2, E3when t = 0, k = 5.0, e = 0.1, We = 0.03, M = 7.0.(a)E1 = 0.7, E2 = 0.4, E3 = 0.2.(b)E1 = 0.9, E2 = 0.4, E3 = 0.2.(c)E1 = 0.7, E2 = 0.6, E3 = 0.2.(d)E1 = 0.7, E2 = 0.4, E3 = 0.5 316 317 318 319 320 321 322 323 324 325 elucidate the impact of curvature parameter on the stream function Increase in the size of trapped bolus is noticed when the curvature parameter attains larger values Effect of different values of Weissenberg number on the stream function is shown through Fig 8(a)–(c) It is observed that by enhancing the values of Weissenberg parameter the size of trapped bolus decreases The effect of wall properties on the stream function is examined through the Fig 9(a)–(d) We can see that by increasing the values of E1 and E2 the size of the trapped bolus increases whereas it decreases for an increase in E3 Conclusions 326 Influences of wall properties and radial magnetic field on the peristaltic flow of Williamson fluid in a curved channel is modeled and analyzed Main points of presented analysis are listed below 327 Axial velocity decreases with increase via Hartman number and E3 whereas it enhances for E1 and E2 Dual behavior of axial velocity is noticed for Weissenberg number and curvature parameter 330 Please cite this article in press as: Hayat T et al Influence of radial magnetic field on the peristaltic flow of Williamson fluid in a curved complaint walls channel Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.02.022 328 329 331 332 333 RINP 585 No of Pages 9, Model 5G 24 February 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx 334 335 336 337 338 339 Axial velocity in the curved channel is slanted towards left when compared with the straight channel However the axial velocity in straight channel is symmetric Size of trapped bolus increases for larger curvature parameter k, Hartman number M, E1 and E2 whereas it decreases against E3 Viscous fluid results are obtained when We = 340 341 Uncited references 342 [44–46] 343 References 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 [1] Latham TW Fluid motion in a peristaltic pump [MS Thesis] Cambridge, MA: MIT; 1966 [2] Shapiro AH, Jafrin MY, Weinberg SL Peristaltic pumping with long wavelengths 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