hall and joule heating effects on peristaltic flow of powell eyring liquid in an inclined symmetric channel

RINP 528 No of Pages 11, 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 Hall and Joule heating effects on peristaltic flow of Powell–Eyring liquid in an inclined symmetric channel T Hayat a,b, Naseema Aslam a, M Rafiq a,⇑, Fuad E Alsaadi b 10 11 14 15 16 17 18 19 20 21 22 23 24 25 a b Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan Department of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah 21589, Saudi Arabia a r t i c l e i n f o Article history: Received 12 November 2016 Received in revised form 24 December 2016 Accepted January 2017 Available online xxxx Keywords: Peristalsis Hall effects Slip conditions Joule heating Eyring–Powell fluid and inclined channel a b s t r a c t This article is intended to investigate the influence of Hall current on peristaltic transport of conducting Eyring–Powell fluid in an inclined symmetric channel Energy equation is modeled by taking Joule heating effect into consideration Velocity and thermal slip conditions are imposed Lubrication approximation is considered for the analysis Fundamental equations are non-linear due to fluid parameter A Regular perturbation technique is employed to find the solution of systems of equations The key roles of different embedded parameters on velocity, temperature and heat transfer coefficient in the problem are discussed graphically Trapping phenomenon is analyzed carefully Ó 2017 The Authors Published by Elsevier B.V This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/) 27 28 29 30 31 32 33 34 35 36 37 38 Introduction 39 Investigation regarding the flow of non-Newtonian fluid cannot be overlooked due to its extensive applications in variety of field like physiology, engineering and industry No doubt various constitutive relations are suggested for the flow description of such fluids diverse characteristics Some recent researchers are even now engaged for the flow analysis of such fluids In connection with peristalsis, the non-Newtonian fluids gained much attention due to their various applications in physiological and industrial processes Spontaneous compressing and relaxing movement along the walls of tabular structures is termed as peristalsis Digestive tract, blood flow in lymphatic transport are few examples that can be observed within human body The phenomenon is also involved in designing many devices like dialysis machine, heart lung machine and blood pump machine to blood pump during surgical processes Some worms also use this phenomenon for their locomotion Pioneering studies in this direction were done by Latham [1], Shapiro et at [2] and Lew et al [3] After these attempts the investigators analyzed the peristaltic flow of Newtonian and non-Newtonian fluids under different flow situations [4– 10] Heat transfer also has a vital role in peristaltic flows especially blood flows Heat conduction in tissues, convective heat transfer during blood flow from pores of tissue, radiative heat transfer 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 ⇑ Corresponding author between environment and surface, food processing and vasodilation are some main applications of heat transfer Oxygenation and hemodialysis are the processes involving heat transfer in connection with peristalsis Recent attempts on peristaltic flow with heat transfer effects can be visualized by Refs [11–20] Magnetic field has gained significance due to its variety of applications in biomedical engineering and industry Power generators, electrostatic precipitation, purification of molten metal from nonmetallic inclusions etc are some processes that deals with magnetic field The shear rate of less than 100 sÀ1 for blood flow shows the model for MHD peristaltic flows in coronary arteries [21] MHD may also be used to control the blood flow during cardiac surgeries from stenosed arteries Hall effects cannot be ignored when strong magnetic field is considered Representative studies in this direction can be consulted by the Refs.[22–31] The problems studying thin films, rarefied fluid, fluid motion inside human body and polishing of artificial heart values etc not follow no-slip boundary condition Experimental investigations show that slippage can occur in non-Newtonian fluids Moreover, many physiological systems are neither horizontal nor vertical but show inclination with axis (see Refs [32–36]) Therefore, aim of the present study is to investigate the peristaltic flow of Powell–Eyring liquid in an inclined symmetric channel Heat transfer is studied in the presence of Joule heating Problem is formulated by taking partial slip effects into account Nonlinear equations are simplified by adopting lubrication approach Perturbation is E-mail address: maimona_88@hotmail.com (M Rafiq) http://dx.doi.org/10.1016/j.rinp.2017.01.008 2211-3797/Ó 2017 The Authors Published by Elsevier B.V This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/) Please cite this article in press as: Hayat T et al Hall and Joule heating effects on peristaltic flow of Powell–Eyring liquid in an inclined symmetric channel Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.008 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 RINP 528 No of Pages 11, Model 5G 11 January 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx 88 employed to find the solution of stream function, velocity, temperature and heat transfer coefficient Results are analyzed via graphs 89 Formulation 87 90 91 92 93 94 95 96 97 98 99 100 102 103 104 105 106 107 109 110 112 Here we assume the two-dimensional electrically conducting non-Newtonian incompressible fluid in an inclined symmetric channel having width 2d (see Fig 1) We consider Cartesian coordinates ðx; yÞ in such a way that wave propagates in x-direction and y-axis is taken transverse to it The walls of channel are assumed compliant Also the channel is inclined at angle a Strong magnetic field ð0; 0; B0 Þ is applied Hall and Joule heating contributions are retained Peristaltic waves propagate with constant speed c and wavelength k along channel walls The structure of wall geometry is described as: ! y ẳ ặgx; tị ẳ ặ d ỵ asin ð1Þ Here t is the time, a the wave amplitude and Ỉg the displacements of upper and lower walls respectively The Cauchy stress tensor ðsÞ for Eyring–Powell fluid is Refs [19,23] s ẳ pI ỵ S; 2ị  ! _ 1 c A1 ; S ẳ l ỵ sinh bc_ c1 ð3Þ 113 115 2p ðx À ctÞ : k c_ ¼ rffiffiffiffiffiffiffiffi P; ber density of electron, e the electric charge, u and v the velocity components in x and y directions respectively, B0 the magnetic field À Á strength and m ¼ renB0 the Hall parameter The fundamental flow equations are div V ¼ 0; 118 Pẳ 119 ẳ grad V ỵ grad Vị : ð5Þ 120 Here S designates the extra stress tensor, I the identity tensor, b and c1 the material parameters of Powell–Eyring fluid and l the 121 dynamic viscosity The term sinh 124   _ c_ c_ c_ À1 c ¼ À ; ( 1: sinh c1 c1 6c1 c1 122 125 126 128 À1 The generalized Ohms law with Hall effects is written as: J¼r ! V  B À ðJ  BÞ ; en 7ị 129 131 132 133 JBẳ rB20 ẵu mv ị; v ỵ muị; 0; ỵ m2 8ị in which J characterizes the current density, V the velocity field, B the applied magnetic field, r the electrical conductivity, n the num- 136 137 138 140 143 144 ð11Þ in which q is the fluid density, j the thermal conductivity and C p the specific heat The two dimensional fundamental flow equations after using Eqs (2)–(8) in Eqs (9)–(11) can be expressed as: @u @ v ỵ ẳ 0; @x @y 12ị ỵ rB20 ỵ m2 v ỵ muị ỵ qg cos a; 13ị rB20 ỵ m2 u mv ị ỵ qg sin a; 147 148 149 150 151 153 156 157   @u @u @u @p @Sxx @Sxy ẳ ỵ ỵv ỵu q ỵ @t @y @x @x @x @y ỵ 146 154   @v @v @v @p @Syy @Syx q ỵv ỵu ẳ ỵ ỵ @y @t @y @x @y @x 14ị 159 !  @ @ @ @2T @2T @v ỵ Syy Sxx ị ỵv ỵu Tẳj ỵ @t @y @x @x2 @y2 @y   @u @ v rB0 2 ỵ Sxy ỵ u ỵ v ị; 15ị ỵ @y @x ỵ m2 160 where p; Sij i; j ẳ x; yị; g and T signify the pressure, the components of extra stress tensor, the gravity and temperature respectively The slip conditions for velocity and temperature at the walls are: 163  qC p is ð6Þ 10ị dT J:J qC P ẳ T:L ỵ jr2 T þ ; dt r ð4Þ Ã 135 141 dV q ẳ div T ỵ J B ỵ qg; dt 116 trA21 ị; A1 9ị 134 u ặ cSxy ẳ at y ẳ ặg; 16ị 164 165 166 167 169 170 @T ẳ T at y ẳ ặg: T ặ b1 @y " 162 17ị Flexible walls can be characterized by 172 173 174 # @3 @3 @2 s ỵ m1 ỵd g @x @t@x @x@t   @Sxx @Sxy @u @u @u rB20 ỵv ỵu ẳ u mv ị ỵ q @t @y @x @x @y ỵ m2 ỵ qg sin a at r ẳ ặg: 18ị 176 In the above expressions T is the temperature at the upper and lower walls, s is the elastic tension, m1 the mass per unit area and d the coefficient of viscous damping Dimensionless parameters are: 177 y u v ct yà ¼ ; uà ¼ ; v à ¼ ; tà ¼ ; d c k c g d p c b gà ¼ ; pà ¼ ; cà ¼ ; bÃ1 ¼ ; ckl d d d T À T0 dSij w à ; Sij ¼ ; wà ¼ : h¼ cd T0 cl 178 179 180 181 x xà ¼ ; k Fig Geometry of the problem ð19Þ By using dimensionless variables, Eqs (12)–(15) become: Please cite this article in press as: Hayat T et al Hall and Joule heating effects on peristaltic flow of Powell–Eyring liquid in an inclined symmetric channel Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.008 183 184 RINP 528 No of Pages 11, Model 5G 11 January 2017 T Hayat et al / Results in Physics xxx (2017) xxxxxx 185 187 @v @u ẳ 0; ỵd @x @y ð20Þ 188 ! R1 d d @v @v @v @p @Syx @Syy ỵv ỵ du ẳ ỵ d2 ỵd @y @t @y @x @x @y H dv þ muÞ R1 þ cos a; þ m2 ðFr ị2 ỵ 190 21ị 191 ! R1 d 193 194 @u @u @u @p @Sxx @Sxy H ðu À mv ị ỵ du ỵv ẳ ỵd ỵ @t @x @y @x ỵ m2 @x @y R1 ỵ sin a; ðFr Þ2 197 200 201 203 204 205 206 208 ð22Þ ð23Þ where c and b1 represent the dimensionless velocity and thermal slip parameters and for simplicity we omitted asterisk Now 212 213 214 215 216 d ¼ d=k; E1 ¼  ¼ a=d; R1 ¼ Ec ¼ c =CpT ; 218 219 220 222 223 224 s l 26ị 27ị E2 ẳ m1 cd1 k1 c l ; E3 ¼ dd1 k1 l ; pffiffiffiffiffiffiffiffiffi qcd1 ; Pr ¼ lC p =j; H ¼ B0 d r=l; l Br ¼ EcPr; v @w ; ẳ d @x 231 236 wy ặ cSxy ẳ at y ẳ ặg; 32ị 239 240 242 @h ẳ at y ẳ ặg; @y 33ị 245 246 " # @3 @3 @2 E1 ỵ E2 ỵ E g @x @t@x @x@t2 ẳ @Sxy H2 @w R1 ỵ sin a at y ẳ ặg: @y ỵ m2 @y Frị2 34ị Combining the dimensionless equations (29) and (30) we obtain resulting form @ Sxy H @y2 ỵ m2 ! @ w ẳ 0: @y2 248 249 250 251 35ị 36ị  2 with a1 ẳ ỵ M; M ¼ lbc and A ¼ 12 c1cd and asterisks have been suppressed for simplicity Viscous fluid model is obtained for a1 ¼ Heat transfer coefficient is defined as Z ẳ gx hy gị: 37ị 253 254 255 256 258 259 260 261 262 263 265 Solution methodology 266 Here we used the perturbation technique for small parameter A to solve the non-linear governing equations Expand the following flow quantities as: 267 268 269 270 w ẳw0 ỵ Aw1 ỵ ; Syx ẳS0yx ỵ AS1yx þ Á Á Á ; h ¼h0 þ Ah1 þ Á Á Á ; c Fr ¼ pffiffiffiffiffiffi : gd Z ẳZ ỵ AZ ỵ : 272 Solving the resulting zeroth and first order systems through Eqs (31)–(37) we have the solutions as follows Defining the stream function wx; y; tị by @w ; uẳ @y ð31Þ Aða1 À 1Þ À Á3 wyy ; Sxy ¼ a1 wyy À Here d is the wave number, E1 ; E2 and E3 the elasticity parameters,  the amplitude ratio, R1 the Reynolds number, Pr the Prandtl number, H the Hartman number, Ec the Eckert number; Br the Brinkman number and Fr the Froude number These definitions are d À ; k1 c 230 Dimensionless form of extra stress tensor for Powell–Eyring fluid is " 209 211  2 @ h @ w H @w ỵ BrS ỵ Br ẳ 0; xy @y2 @y2 ỵ m2 @y g ẳ ặẵ1 ỵ  sin 2pðx À tފ: 229 233 ð30Þ h ặ b1 25ị # @3 @3 @2 E1 ỵ E2 ỵ E g @x @t@x @x@t ! @u @u @u @Sxx @Sxy H2 ðu À mv Þ þ du þv þd ¼ ÀR1 d þ À @t @x @y ỵ m2 @x @y R1 ỵ sin a at y ẳ ặg: Frị2 228 234 24ị @h ẳ at y ẳ ặg; h ặ b1 @y 227 243 with the dimensionless boundary conditions u Ỉ cSxy ẳ at y ẳ ặg; 226 29ị @ h H Br ỵ 2ỵ u2 ỵ v ị; @y ỵ m2 225 237 ! @ @ @ @u ỵ du h ẳ BrdSxx Syy ị R1 Prd d ỵ v @t @y @x @x   @u @v @2h ỵ d2 ỵ Sxy Br þ d2 @y @x @x 198   @p @Sxy H2 @w R1 ỵ ỵ sin a ẳ 0; @x @y ỵ m2 @y Fr ị2 @p ¼ 0; @y 196 intestine [36] Also Lew et al [3]mentioned that Reynold number in intestine is small Moreover the state of intrauterine fluid flow due to myometrial contractions is a peristaltic type fluid motion in a cavity The sagittal cross section of the uterus indicates a narrow channel bounded by two fairly parallel walls [37] Thus large wavelength and small Reynolds number yield pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi À 1ỵm2 ịa1 1ỵm2 ịa1 @C ỵ C e A ỵ m2 a1 e Hy ð28Þ the continuity equation (20) is identically satisfied Note that the lubrication process remain useful for the chyme transport in small w0 ẳ C ỵ C y ỵ 273 274 275 2Hy H2 ; ð38Þ Please cite this article in press as: Hayat T et al Hall and Joule heating effects on peristaltic flow of Powell–Eyring liquid in an inclined symmetric channel Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.008 277 RINP 528 No of Pages 11, Model 5G 11 January 2017 T Hayat et al / Results in Physics xxx (2017) xxxxxx 3Hy p 1ỵm2 ịa1 B e p B w1 ẳ 24H2 a1 ỵ m2 ịa1 @ 4Hy p 1ỵm2 ịa1 C A11 ỵ 6e ỵ m a1 A13 A14 2Hy ỵ A15 ịị C ỵ B3 ỵ B4 y; 6Hy 2Hy A p p 1ỵm2 ịa1 1ỵm2 ịa1 ỵe A12 6e þ m2 a1 ðA16 À A17 ð2Hy þ A18 ÞÞ 0 1 2Hy 4Hy p p 1ỵm2 ịa1 þ @C þ C e ð1þm2 Þa1 AB C 11 C C C C Hy 2Hy p p C a a 1ỵm 1ỵm ị ị 1@ 1A B12 A ỵ4C e C ỵ C e 39ị B C 24 H4 y2 e B 2Hy p B 1ỵm2 a Bre ð Þ B B B @ h0 ẳ L1 ỵ L2 y 2H2 ỵ m2 ị 40ị ; 3Hy 3Hy p p 1ỵm2 ịa1 1ỵm2 ịa1 C B C 11 e C 12 e C B À4Hy 4Hy C B pffiffiffiffiffiffiffiffiffiffiffiffiffiffi p C B a a 1ỵm 1ỵm ị ị 1 C B e e ỵ C ỵC 13 14 C B C B Hy p C B 1ỵm a B ỵ72y2 C þ 36e ð Þ 1 þ m2 ðyC À C Þ C 15 17 C B 16 Br a1 C B Hy h1 ẳ K ỵ K y C: p B C 1ỵm2 ịa1 72H1 ỵ m2 ịa1 ị2 B C B ỵ36e a yC ỵ C ị ỵ m 19 18 C B C B 2Hy p   C B 1ỵm ịa1 yD11 ỵD12 C B 2 ỵ3C a1 ỵ m e C B H C B C B À2Hy p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi   A @ À Á 1ỵm2 a 3C a1 ỵ m2 e ị yD13HỵD14 Expression of heat transfer coefficient is 0 1 4Hg 2Hg pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi a 1ỵm ị A ỵ C H4 g2 e 1ỵm2 ịa1 C ỵ C C C C Hg 2Hg p p C 1ỵm2 ịa1 1ỵm2 ịa1 A @ A ỵ4E12 C e C ỵ C e 41ị C C C C C C C C C C p C L2 ỵ C; H1ỵm2 ị 1ỵm2 ịa1 11C C 3Hg Hg 2Hg pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffi C 1ỵm2 ịa1 1ỵm2 ịa1 1ỵm2 ịa1 @ A 2Hg CC ỵ 4E C E p B 13 e 14 C e ỵ C1e CC 1ỵm2 ịa1 B CC B Bre CC B AC @ 2Hg 4Hg 2Hg C p p p C 1ỵm2 ịa1 1ỵm2 ịa1 1ỵm2 ịa1 2 A g E16 ỵ2C e H g ỵ E15 e ỵe 2H2 1ỵm2 ị 42ị 11 4Hg 2Hg 2Hg p p p 1ỵm2 ịa1 1ỵm2 ịa1 1ỵm2 ịa1 ð ð ð B CC B ÀF 11 e À F 12 e ỵ F 13 e B CC B 4Hg À3Hg 3Hg B CC B p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p B B 1ỵm2 ịa1 1ỵm2 ịa1 1ỵm2 ịa1 CC B CC B ỵF 14 e e e ỵ F ỵ F 15 16 B CC B ÀHg Hg B CC B p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi B CC B a a 1ỵm 1ỵm ị ị 1 B CC B ỵF 17 e ỵ F 18 e ỵ F 19 g B CC B B CC B ! Hg p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi B CC B 1ỵm2 ịa1 B CC B g G11 ỵG12 p ỵ36H1 ỵ m ịe B CC B a B CC B 1ỵm ị1 Br a CC: B Z ¼ gx B K À B B CC ! 2H g pffiffiffiffiffiffiffiffiffiffiffiffiffiffi B CC 72H1 ỵ m2 ịa1 ị2 B a B CC B 1ỵm g G13 ỵG14 ị p B CC B ỵ6C a1 ỵ m ị e B CC B 1ỵm2 ịa1 B CC B B CC B ! ÀHg pffiffiffiffiffiffiffiffiffiffiffiffiffiffi B CC B B CC B 1ỵm2 ịa1 g G15 ỵG16 2 p B CC B ỵ6C a1 þ m Þ e B CC B 1þm Þa1 ð B CC B B CC B ! ÀHg pffiffiffiffiffiffiffiffiffiffiffiffiffiffi B CC B @ AA @ 1ỵm ịa1 g G17 ỵG18 p 36 ỵ m He 1ỵm2 ịa1 43ị B B B B B B B B B B B Z ¼ gx B B B B B B B B B B B @ B E11 @C 22 B À2Hg pffiffiffiffiffiffiffiffiffiffiffiffiffiffi B 1ỵm2 ịa1 B Bre B B @ C 21 e Please cite this article in press as: Hayat T et al Hall and Joule heating effects on peristaltic flow of Powell–Eyring liquid in an inclined symmetric channel Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.008 RINP 528 No of Pages 11, Model 5G 11 January 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx Fig 2a Effect via wall parameters on u when  ¼ 0:2; x ¼ 0:2; t ¼ 0:1; Br ¼ 2:0; a ¼ p4 ; H ¼ 0:5; Fr ¼ 0:8; A ¼ 0:1; m ¼ 0:2; M ¼ 1; R1 ¼ 0:1 and c ¼ 0:1 Fig 2b Effect via a1 on u when  ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E1 ¼ 0:03; E2 ¼ 0:02; E3 ¼ 0:01; Br ¼ 2:0; a ¼ p4 ; H ¼ 0:5; Fr ¼ 0:8; A ¼ 0:1; m ¼ 0:2; R1 ¼ 0:2 and c ¼ 0:01 Fig 2c Effect via A on u when  ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E1 ¼ 0:03; E2 ¼ 0:02; E3 ¼ 0:01; Br ¼ 2:0; a ¼ p6 ; H ¼ 0:5; Fr ¼ 0:8; M ¼ 1:0; m ¼ 0:2; R1 ¼ 0:2 and c ¼ 0:01 Fig 2d Effect via m on u when  ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E1 ¼ 0:03; E2 ¼ 0:02; E3 ¼ 0:01; Br ¼ 2:0; a ¼ p4 ; H ¼ 0:5; Fr ¼ 0:08; A ¼ 0:01; M ¼ 0:1; R1 ¼ 0:2 and c ¼ 0:01 Fig 2e Effect via Fr on u when  ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E1 ¼ 0:03; E2 ¼ 0:02; E3 ¼ 0:01; Br ¼ 2:0; a ¼ p4 ; H ¼ 0:5; M ¼ 1:0; A ¼ 0:1; m ¼ 0:2; R1 ¼ 0:2 and c ¼ 0:01 Fig 2f Effect via H on u when  ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E1 ¼ 0:03; E2 ¼ 0:02; E3 ¼ 0:01; Br ¼ 2:0; a ¼ p4 ; M ¼ 1:0; Fr ¼ 0:5; A ¼ 0:1; m ¼ 0:2; R1 ¼ 0:2 and c ¼ 0:01 Fig 2g Effect via angle inclination a on u when  ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E1 ¼ 0:03; E2 ¼ 0:02; E3 ¼ 0:01; Br ¼ 2:0; M ¼ 1:0; H ¼ 0:5; Fr ¼ 0:8; A ¼ 0:1; m ¼ 0:2; R1 ¼ 0:2 and c ¼ 0:01 Fig 2h Effect via R1 on u when  ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E1 ¼ 0:03; E2 ¼ 0:02; E3 ¼ 0:01; Br ¼ 2:0; a ¼ p6 ; H ¼ 0:5; Fr ¼ 0:8; A ¼ 0:1; m ¼ 0:2; M ¼ 1:0 and c ¼ 0:01 Please cite this article in press as: Hayat T et al Hall and Joule heating effects on peristaltic flow of Powell–Eyring liquid in an inclined symmetric channel Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.008 RINP 528 No of Pages 11, Model 5G 11 January 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx Fig 3a Effect via wall parameters on h when  ¼ 0:2; x ¼ 0:2; t ¼ 0:1; Br ¼ 2:0; a ¼ p4 ; H ¼ 0:5; Fr ¼ 0:8; A ¼ 0:1; m ¼ 0:2; M ¼ 1; R1 ¼ 0:1 b1 ¼ 0:01 and c ¼ 0:01 Fig 3b Effect via H on h when  ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E1 ¼ 0:3; E2 ¼ 0:2; E3 ¼ 0:1; Br ¼ 2:0; a ¼ p4 ; M ¼ 1:0; Fr ¼ 0:8; A ¼ 0:1; m ¼ 0:2; b1 ¼ 0:01; R1 ¼ 0:1 and c ¼ 0:01 Fig 3e Effect via Br on h when  ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E1 ¼ 0:4; E2 ¼ 0:2; E3 ¼ 0:3; Br ¼ 2:0; M ¼ 1:0; H ¼ 0:2; Fr ¼ 0:8; A ¼ 0:1; m ¼ 0:2; R1 ¼ 0:1; b1 ¼ 0:01; a ¼ p4 and c ¼ 0:01 Fig 3f Effect via Fr on h when  ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E1 ¼ 0:03; E2 ¼ 0:02; E3 ¼ 0:01; Br ¼ 4:0; a ¼ p4 ; H ¼ 0:2; M ¼ 1:0; A ¼ 0:01; m ¼ 0:2; b1 ¼ 0:01; R1 ¼ 0:1 and c ¼ 0:01 Fig 3c Effect via m on h when  ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E1 ¼ 0:4; E2 ¼ 0:2; E3 ¼ 0:3; Br ¼ 2:0; a ¼ p4 ; H ¼ 0:5; Fr ¼ 0:8; A ¼ 0:1; M ¼ 1:0; R1 ¼ 0:1; b1 ¼ 0:01 and c ¼ 0:01 Fig 3g Effect via R1 on h when  ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E1 ¼ 0:3; E2 ¼ 0:2; E3 ¼ 0:1; Br ¼ 4:0; a ¼ p4 ; H ¼ 0:2; Fr ¼ 0:8; A ¼ 0:1; m ¼ 0:2; M ¼ 1:0; b1 ¼ 0:01 and c ¼ 0:01 Fig 3d Effect via angle inclination a on h when  ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E1 ¼ 0:4; E2 ¼ 0:2; E3 ¼ 0:3; Br ¼ 2:0; M ¼ 1:0; H ¼ 0:2; Fr ¼ 0:08; A ¼ 0:01; m ¼ 0:2; R1 ¼ 0:1; b1 ¼ 0:01 and c ¼ 0:01 Here the values of Ai;s i ẳ 1; 2; ; 8ị; Bi;s i ẳ 1; 2ị; C i;s i ẳ 1; 2; ; 7ị; Di;s i ẳ 1; 2; ; 4ị; Ei;s i ẳ 1; 2; ; 6ịF i;s i ẳ 1; 2; ; 9ị and Gi;s i ẳ 1; 2; ; 8Þ can be calculated algebraically using MATHEMATICA 278 Analysis 282 The purpose of this section is to analyze the behavior of different embedded parameters on the velocity u, temperature h and heat transfer coefficient Z Trapping phenomenon is also examined via graphs 283 Please cite this article in press as: Hayat T et al Hall and Joule heating effects on peristaltic flow of Powell–Eyring liquid in an inclined symmetric channel Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.008 279 280 281 284 285 286 RINP 528 No of Pages 11, Model 5G 11 January 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx Fig 3h Effect via A on h when  ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E1 ¼ 0:3; E2 ¼ 0:2; E3 ¼ 0:1; Br ¼ 4:0; a ¼ p4 ; H ¼ 0:2; Fr ¼ 0:8; M ¼ 1:0; m ¼ 0:2; R1 ¼ 0:2; b1 ¼ 0:01 and c ¼ 0:01 Fig 4d Effect via wall parameters on Z when  ¼ 0:2; t ¼ 0:1; Br ¼ 2:0; a ¼ p6 ; H ¼ 0:5; Fr ¼ 0:8; A ¼ 0:2; m ¼ 0:2; M ¼ 1; R1 ¼ 0:1 b1 ¼ 0:01 and c ¼ 0:01 Fig 4a Effect via H on Z when  ¼ 0:2; t ¼ 0:1; E1 ¼ 0:03; E2 ¼ 0:02; E3 ¼ 0:01; Br ¼ 2:0; a ¼ p6 ; M ¼ 1:0; Fr ¼ 0:8; A ¼ 0:2; m ¼ 0:2; b1 ¼ 0:01; R1 ¼ 0:1 and c ¼ 0:01 Fig 4e Effect via angle inclination a on Z when  ¼ 0:2; t ¼ 0:1; E1 ¼ 0:03; E2 ¼ 0:02; E3 ¼ 0:01; Br ¼ 2:0; M ¼ 1:0; H ¼ 0:5; Fr ¼ 0:8; A ¼ 0:2; m ¼ 0:2; R1 ¼ 0:1; b1 ¼ 0:01 and c ¼ 0:01 Fig 4b Effect via m on Z when  ¼ 0:2; t ¼ 0:1; E1 ¼ 0:03; E2 ¼ 0:02; E3 ¼ 0:01; Br ¼ 2:0; a ¼ p6 ; H ¼ 0:5; Fr ¼ 0:8; A ¼ 0:2; M ¼ 1:0; R1 ¼ 0:1; b1 ¼ 0:01 and c ¼ 0:01 Fig 4f Effect via Fr on Z when  ¼ 0:2; t ¼ 0:1; E1 ¼ 0:03; E2 ¼ 0:02; E3 ¼ 0:01; Br ¼ 2:0; a ¼ p3 ; H ¼ 0:5; M ¼ 1:0; A ¼ 0:2; m ¼ 0:2; b1 ¼ 0:01; R1 ¼ 0:1 and c ¼ 0:01 Fig 4c Effect via Br on Z when  ¼ 0:2; t ¼ 0:1; E1 ¼ 0:03; E2 ¼ 0:02; E3 ¼ 0:01; Br ¼ 2:0; M ¼ 1:0; H ¼ 0:5; Fr ¼ 0:8; A ¼ 0:2; m ¼ 0:2; R1 ¼ 0:1; b1 ¼ 0:01; a ¼ p6 and c ¼ 0:01 Fig 4g Effect via A on Z when  ¼ 0:2; t ¼ 0:1; E1 ¼ 0:03; E2 ¼ 0:02; E3 ¼ 0:01; Br ¼ 4:0; a ¼ p3 ; H ¼ 0:5; Fr ¼ 0:8; M ¼ 1:0; m ¼ 0:2; R1 ¼ 0:1; b1 ¼ 0:01 and c ¼ 0:01 Please cite this article in press as: Hayat T et al Hall and Joule heating effects on peristaltic flow of Powell–Eyring liquid in an inclined symmetric channel Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.008 RINP 528 No of Pages 11, Model 5G 11 January 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx Fig Effect via H on w for E1 ¼ 0:2; E2 ¼ 0:2; E3 ¼ 0:3; a ¼ p4 ; M ¼ 1:0;  ¼ 0:2; t ¼ 0:0; A ¼ 0:1; c ¼ 0:01; Fr ¼ 0:8; m ¼ 0:2; R1 ¼ 0:2 when ðaÞ :H ¼ 0:5 and bị :H ẳ 2:5 Fig Effect via A on w for E1 ¼ 0:2; E2 ¼ 0:2; E3 ¼ 0:3; a ¼ p4 ; M ¼ 1:0;  ¼ 0:2; t ¼ 0:0; H ¼ 0:5; c ¼ 0:01; Fr ¼ 0:8; m ¼ 0:2; R1 ¼ 0:2 when aị :A ẳ 0:1 and bị :A ẳ 0:5 287 Velocity profile 288 This subsection presents the effect of various parameters on velocity distribution particularly the wall parameters ðE1 ; E2 ; E3 Þ, material parameters of fluid a1 and A, the Hall parameter m, the Froude number Fr, the Hartman number H, inclination angle a and Reynolds number R1 Fig 2a exhibits that for large values of E1 and E2 the velocity increases Physically increasing values of E1 and E2 reduce the viscosity which yield more Wall parameter E3 shows decrease in velocity for increasing values of E3 Its reason is that for large values of E3 viscous damping enhances due to which velocity decreases Figs 2b and 2c depict the behavior of 289 290 291 292 293 294 295 296 297 fluid parameters on u Reverse results corresponding to larger values of a1 and A are observed i.e higher values of a1 gives reduction in velocity while larger A favor the velocity u The fact behind this behavior is that large A causes increase in kinetic energy of particles which results in increased velocity Fig 2d indicates the increasing behavior when Hall parameter m increases Effect of Froude number Fr on velocity profile shows decreasing impact (see in Fig 2e) It is noticed from Fig 2f that larger values of Hartman number H decreases the velocity Physically this concept holds because Lorentz force reduces the velocity Fig 2g shows the increasing behavior of inclination angle a towards velocity Increase in inclination angle a causes fluid to move with greater Please cite this article in press as: Hayat T et al Hall and Joule heating effects on peristaltic flow of Powell–Eyring liquid in an inclined symmetric channel Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.008 298 299 300 301 302 303 304 305 306 307 308 309 RINP 528 No of Pages 11, Model 5G 11 January 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx Fig Effect via m on w for E1 ¼ 0:2; E2 ¼ 0:2; E3 ¼ 0:3; a ¼ p4 ; M ¼ 1:0;  ¼ 0:2; t ¼ 0:0; A ¼ 0:1; c ¼ 0:01; Fr ¼ 0:8; H ¼ 0:5; R1 ¼ 0:2 when ðaÞ :m ¼ 0:5 and bị :m ẳ 1:5 Fig Effect via a on w for E1 ¼ 0:7; E2 ¼ 0:2; E3 ¼ 0:1; m ¼ 0:2; M ¼ 1:0;  ¼ 0:2; t ¼ 0:0; A ¼ 0:1; c ¼ 0:01; Fr ¼ 0:8; H ¼ 0:5; R1 ¼ 0:2 when aị :a ẳ p6 and bị :a ẳ p3 313 velocity due to increased effect of gravity Fig 2h enhances the velocity profile when R1 is increased As R1 is the ratio of inertial forces to the viscous forces thus decrease in viscosity enhances R1 which in turn increases velocity 314 Temperature profile 315 Figs 3a–3h manifest the impact of different emerging parameters on temperature distribution h It is observed that h attains maximum value near the centre of the channel Fig 3a displays that temperature profile increases for increasing values of both E1 and E2 while it decreases for E3 Fig 3b discloses decrease in 310 311 312 316 317 318 319 temperature profile when Hartman number H is increased For large values of Hall parameter m temperature profile enhances (see in Fig 3c) The ascending values of inclination angle a on temperature profile are depicted in Fig 3d As growing values of inclination angle a cause increase in temperature profile Fig 3e illustrates that for higher values of Brinkman number Br the temperature profile is enhanced The reason behind this effect is the higher viscous dissipation which generates more heat and hence causing rise in temperature occurs Fig 3f indicates that by increasing Fr temperature profile decreases Fig 3g ensures that when we increase Reynolds number R1 then temperature enhances Fig 3h shows that for ascending values of A the temperature profile Please cite this article in press as: Hayat T et al Hall and Joule heating effects on peristaltic flow of Powell–Eyring liquid in an inclined symmetric channel Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.008 320 321 322 323 324 325 326 327 328 329 330 331 RINP 528 No of Pages 11, Model 5G 11 January 2017 10 T Hayat et al / Results in Physics xxx (2017) xxx–xxx Fig Effect via wall properties on w for m ¼ 0:2; M ¼ 1:0;  ¼ 0:2; t ¼ 0:0; A ¼ 0:1; c ¼ 0:01; Fr ¼ 0:8; H ¼ 0:5; a ¼ p4 ; R1 ¼ 0:2 when ðaÞ :E1 ¼ 0:1; E2 ¼ 0:3; E3 ẳ 0:1 bị :E1 ẳ 0:4; E2 ẳ 0:3; E3 ẳ 0:1 cị : E1 ẳ 0:1; E2 ẳ 0:4; E3 ẳ 0:1 dị :E1 ẳ 0:1; E2 ẳ 0:3; E3 ¼ 0:02 335 increases It is noticed that velocity and temperature show similar behavior As temperature is defined as average kinetic energy of molecules Hence increased/decreased velocity causes temperature to get increased/decreased 336 Heat transfer coefficient 337 The purpose of this subsection is to investigate the behavior of various parameters on the heat transfer coefficient Z From Figs 4a–4g it is observed that magnitude of heat transfer coefficient shows oscillatory behavior for the involved parameters due to sinusoidal waves travelling along the walls Hartman number H provides resistance to heat transfer and thus heat transfer rate 332 333 334 338 339 340 341 342 Z reduces due to the existence of magnetic field (see Fig 4a) Fig 4b potrays that for ascending values of m the heat transfer rate Z decreases From Figs 4c and 4d it is noticed that Z decreases when Brinkman number Br and wall parameters ðE1 ; E2 ; E3 Þ are increased Fig 4e displays that Z reduces when the inclination alarger Froude number Fr enhances the heat transfer distribution Z (see Fig 4f) The results in Fig 4g illustrates that an increase of A causes reduction in Z 343 Trapping 351 Formation of circular bolus by internally splitting of streamlines is known as trapping The bolus moves forward through peristaltic 352 Please cite this article in press as: Hayat T et al Hall and Joule heating effects on peristaltic flow of Powell–Eyring liquid in an inclined symmetric channel Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.008 344 345 346 347 348 349 350 353 RINP 528 No of Pages 11, Model 5G 11 January 2017 T Hayat et al / Results in Physics xxx (2017) xxx–xxx 364 wave with the same speed Fig 5(a) and (b) indicate that for large value of Hartman number H the size of trapped bolus increases Fig 6(a) and (b) depict that size of trapped bolus enhances when fluid parameter A increased Moreover, number of streamline are enhanced For ascending values of Hall parameter m, the size of trapped bolus becomes larger (see Fig 7(a) and (b)) From Figs (a) and (b) there is reduction in the size of trapped bolus However number of streamlines remain same when angle inclination a of channel is increased We noticed that for increasing values of elastic parameters E1 and E2 size of trapped bolus is enhanced However it decreases for E3 (See Fig 9) 365 Conclusions 366 The peristaltic transport of Eyring–Powell fluid in an inclined channel with Hall and slip effects is addressed Key observations are 354 355 356 357 358 359 360 361 362 363 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381  Increase in Hartman number H shows decay in velocity profile  Fluid parameters a1 and A show opposite behavior for velocity profile  The velocity distribution exhibits decreasing behavior as Froude number Fr increases  Increasing values of inclination aenhances temperature profile  Temperature profile decreases as Froude number Fr increases  Magnitude of heat transfer coefficient Z gives oscillatory behavior for all embedded parameters  Trapped bolus size enhances with increase in the Hartman number H, Hall parameter m and fluid parameter A  For a1 ¼ Powell–Eyring fluid model reduces to viscous fluid model 382 383 References 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 [1] Latham TW Fluid motion in a peristaltic pump (MS Thesis) 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http://dx.doi.org/10.1016/j.rinp.2017.01.008 RINP 528 No of. .. article in press as: Hayat T et al Hall and Joule heating effects on peristaltic flow of Powell? ? ?Eyring liquid in an inclined symmetric channel Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.008... article in press as: Hayat T et al Hall and Joule heating effects on peristaltic flow of Powell? ? ?Eyring liquid in an inclined symmetric channel Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2017.01.008