RINP 490 No of Pages 10, Model 5G 25 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 Simultaneous effects of single wall carbon nanotube and effective variable viscosity for peristaltic flow through annulus having permeable walls Iqra Shahzadi ⇑, S Nadeem, Faranak Rabiei 10 Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia 11 a r t i c l e 14 15 16 17 18 19 20 21 22 23 24 25 i n f o Article history: Received 29 October 2016 Received in revised form December 2016 Accepted 19 December 2016 Available online xxxx Keywords: Variable nanofluid viscosity SWCNT Annulus Permeable walls Exact solution a b s t r a c t The current article deals with the combine effects of single wall carbon nanotubes and effective viscosity for the peristaltic flow of nanofluid through annulus The nature of the walls is assumed to be permeable The present theoretical model can be considered as mathematical representation to the motion of conductive physiological fluids in the existence of the endoscope tube which has many biomedical applications such as drug delivery system The outer tube has a wave of sinusoidal nature that is travelling along its walls while the inner tube is rigid and uniform Lubrication approach is used for the considered analysis An empirical relation for the effective variable viscosity of nanofluid is proposed here interestingly The viscosity of nanofluid is the function of radial distance and the concentration of nanoparticles Exact solution for the resulting system of equations is displayed for various quantities of interest The outcomes show that the maximum velocity of SWCNT-blood nanofluid enhances for larger values of viscosity parameter The pressure gradient in the more extensive part of the annulus is likewise found to increase as a function of variable viscosity parameter The size of the trapped bolus is also influenced by variable viscosity parameter The present examination also revealed that the carbon nanotubes have many applications related to biomedicine 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 39 40 41 42 43 44 45 Introduction 46 Peristalsis is a conspicuous component in physiology for liquid transport Peristaltic transport is totally vital among the most recent researchers because of its application in physics, applied mathematics, physiological world and engineering In this process, sinusoidal waves move around the walls of tube as the organs of human being pushing the fluid in the direction of their propagation towards the tube Peristalsis has many applications in medical/biological employment where the traveling matter is not in the immediate contact with another part prohibit the inside surface of tube The witness of peristalsis is to transit food through oesophagus, transport of urine from kidney to bladder, vasomotion of blood vessels, transport of bile in bile duct, movement of chyme in intestines and many others [1–3] Engineers have approved such mechanism because of its utility in captivating different modern 47 48 49 50 51 52 53 54 55 56 57 58 59 ⇑ Corresponding author at: Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan (I Shahzadi) E-mail address: iqrashahzadiwah@gmail.com (I Shahzadi) apparatuses comprising of peristaltic pumps, finger pumps in dialysis machines, roller and heart lung The phenomena peristalsis is used in various hose pumps In the nuclear industry the transport of destructive fluids is of peristaltic type Numerous theoretical assessments are trucked out in physiology and industry because of such wide existence of peristalsis [4–9] In these days, the endoscope is a very significant tool utilize for analyzing causes responsible for various complication in the organs of human in which the fluid is carried by peristaltic pumping like stomach and small intestine There is no difference between catheter and an endoscope from dynamic point of view Furthermore, the injection of a catheter will change the distribution and flow field in an artery [10] Numbers of investigations are done to analyze the impact of endoscope on peristaltic transport for Newtonian and non-Newtonian fluids [11–15] Nanoparticle examination is in the blink of an eye a region of effective experimental enthusiasm because of a gigantic scope of potential applications in electronic, optical, biomedical field The combination of the base fluid with nanoparticles that have unique http://dx.doi.org/10.1016/j.rinp.2016.12.024 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: Shahzadi I et al Simultaneous effects of single wall carbon nanotube and effective variable viscosity for peristaltic flow through annulus having permeable walls Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2016.12.024 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 RINP 490 No of Pages 10, Model 5G 25 January 2017 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 I Shahzadi et al / Results in Physics xxx (2017) xxx–xxx physical and chemical properties is defined as nanofluid and the word ‘‘nanofluid” was firstly introduce by Choi [15] Buongiorno [16] proposed that the thermophoresis and Brownian motion plays a key role in the dynamics of nanofluids Nanofluid is basically the liquid suspension that comprises of little particles having diameter across lesser 100 nm These tiny particles are mostly found in the metals such as nitrides, nitrides, Carbides or non-metals (Graphite, Carbon nanotubes) In advanced examination miniaturized components perform essential part in all types of utilizations Many authors have considered the effect of nanoparticles [17–25] One such development is carbon nanotube; Carbon nanotubes (CNTs) are the hollow cylinders of carbon atoms Carbon nanotubes are metal or semiconductor Their appearance is that of folded tubes of graphite, such that the hexagonal carbon rings and their bundles are formed Single-walled nanotubes (SWNTs) and multiwall nanotubes (MWNTs) are the two types of nanotubes because the difference in the arrangement of graphene cylinders SWNTs have only one layer while MWNTs have more than one layer of graphene cylinders [26] Carbon nanotube is stronger than steel per unit weight while 50,000 times thinner than a human hair Recently, it is determined by Murshed et al [27] that carbon nanotube (CNT) have six times better thermal conductivity than other materials Iijima and Ichihashi [28] exposed that carbon nanotubes are intriguing an extensive variety of industries as well as scientists passion due to their charming chemical and physical properties Peristaltic flow of carbon nanotubes in curved channel with heat transfer was discussed by Akbar and Butt [29] Analysis of entropy generation of CNT suspension in plumb duct is discussed by Akbar [30] Effective viscosity and expressions for the nanofluids were calculated by Li and Xuan [31] Brinkman’s [32] proposed the effective viscosity model for two phase flow Das and Tiwari [33] designed a model for the study of nanofluids by using the results of Li, Xuan [31] and Brinkman [32] For the perseverance of nanofluid dynamics, nanofluid model given by Das and Tiwari [33] was used by various scientists Darcy’s Law is used to drive the fluid flow through porous medium while the fluid in free region is carried out by Navier Stokes equations Beavers and Joseph suggested the boundary condition at the permeable surface in the coupled flow motion in 1967 Different pragmatic applications experience the flow through a permeable medium especially in geophysical fluid dynamics Limestone, Sandstone, gall bladder with stones in tiny blood vessels, beach sand, bile duct and the human lung are the important examples of natural porous media [34–38] Permeable wall analysis for the nanofluid flow in stenosed arteries is examined by Noreen et al [39] Nadeem and Ijaz [40] portray the impact of metallic type nanoparticles on blood flow with permeable walls through stenosed artery In all of the mention citation, fluid viscosity was considered to be constant The physical properties of the fluid may change considerably with radius and temperature For lubricating fluids, heat produced by the inner friction and the corresponding increase in temperature affects the viscosity of the fluid and so the fluid viscosity can not be considered constant anymore Therefore, to examine the flow behavior accurately it is sufficient to consider the incompressible fluids for viscosity variation [41,42] The motivation behind the present examination is to inspect the significance of nanoparticles infused in the annulus in the existence of variable effective viscosity which is not done before from author’s knowledge Here we examine the impact of single wall carbon nanotubes on blood (as the base flood) to depict the peristaltic flow in an annulus having permeable walls The aim of this paper is to comprehend the fluid mechanics in a physiological circumstance in the existence of concentrically set endoscope Significant modeling is conferred with the aid of dimensionless parameters and using approximation of low Reynolds number and long wavelength Results acquired from this examination provide a useful understanding about the particular nature of SWCNT which influence the peristalsis and provide new visions of nanoparticles in the presence of variable viscosity 143 Formulation of the problem 149 Consider the unsteady, two-dimensional, incompressible flow of a viscous fluid through the gap lies between the tubes with variable effective viscosity The central tube is the endoscope while the outer tube has a sinusoidal wave that is traveling down through its wall The outer tube is maintained at a constant temperature T while the inner tube is rigid and retained at a temperature T À Á R; Z coordinates are preferred in such a way that the length of the tube is along Z-axis whereas R-axis is normal to Z-axis The two wall surface geometry is described by the equation: 150 144 145 146 147 148 151 152 153 154 155 156 157 158 159 2p R2 ẳ b sin Z ctị ỵ a2 ; k 1ị 161 R1 ẳ a1 ; ð2Þ 162 164 where a1 and a2 represents the radius of inner and outer tube, k is the wavelength, c is the wave speed, b is the amplitude of the wave and t is the time (See Fig 1) The two dimensional continuity equation for incompressible fluid is defined below U @W @U ỵ ỵ ẳ 0: R @Z @R ð3Þ In the laboratory frame, transverse and longitudinal components of velocity are represented by W and U In the presence of mixed convection, R and Z components of momentum equation are qnf @U @U @U W ỵU ỵ @Z @R @ t ! ẳ @ @U @Z @Z ỵ @W @R ! ! 166 167 168 169 170 172 173 174 175 176 l nf ðRÞ @ @U nf Rị 2l R ỵ R @R @R nf Rị 2l 165 ! U @P ; ỵ ðqbÞnf ðT À T Þg À R @R ð4Þ Fig Geometry of the problem Please cite this article in press as: Shahzadi I et al Simultaneous effects of single wall carbon nanotube and effective variable viscosity for peristaltic flow through annulus having permeable walls Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2016.12.024 178 RINP 490 No of Pages 10, Model 5G 25 January 2017 I Shahzadi et al / Results in Physics xxx (2017) xxxxxx 179 @W @W @W Wỵ Uỵ @t @Z @R qnf ! ! ẳ @ ỵ R @R 187 188 189 190 191 192 193 194 195 197 198 199 200 201 203 204 205 206 207 209 210 211 212 213 214 215 216 qC p @T @T Uỵ Wỵ @t @R @Z nf @T @2T ¼ K nf @Z ỵ @2T @R2 5ị ỵ @T R @R ! ỵ Q 0: 6ị In the above equations, P is the pressure in laboratory frame, U and W are the velocity components, T is the temperature of fluid, Q is the constant heat generation/absorption, K nf is the thermal conductivity, bnf is the thermal expansion coefficient, qnf is the À Á density and qC p nf is the heat capacitance of the nanofluid with thermophysical properties defined in [27,28] For the proposed nanofluid model, lnf is the variable nanofluid viscosity [5] and suppose the variation of viscosity following from Brinkman [32] and Srivastava et al [41] as follows: l nf Rị ẳ lB Rị À uÞ2:5 ; ð7Þ where lB is the viscosity of the base fluid We further assume that the viscosity of the base fluid varies according tothe following relation: lB ðRÞ ẳ l0 eaR ẳ l0 : ỵ aRị ð8Þ Here l0 is the viscosity of blood, að( 1Þ is the dimensional variable viscosity parameter From Eqs (6) and (7) the effective viscosity of the nanofluid is reduced as follows: l nf Rị ẳ l0 ỵ aRị1 uÞ2:5 : ð9Þ We note from the above equation that Brinkman’s viscosity model (i.e., the effective viscosity independent of R) canbe recovered for a ¼ The viscosity of the fluid independent of the nanoparticles can also be obtained by substituting u ¼ Relations for effective density, thermal conductivity and specific heat of nanofluid by Das and Tiwari [33] k qnf ẳ uqSWCNT ỵ qf uị; anf ẳ nf ; qcp qC p nf 218 @P : @Z Energy equation in the presence of heat generation is given as, 182 183 186 ! ! @U @W ỵ l nf Rị R @Z @R ỵ ðqbÞnf gðT À T Þ À 181 185 z ¼ Z À ct; p ðz; r Þ ¼ PðZ; R; tị; r ẳ R; w ẳ W c; u ¼ U; @ @W l nf Rị @Z @Z ẳ u qcp SWCNT ỵ uị qcp nf 219 qbịnf ẳ qbịf uị ỵ uqbịSWCNT ; ẳ 221 222 223 224 225 226 227 228 229 230 231 kSWCNT uị ỵ 2u kSWCNT ln k f k uị ỵ 2u kSWCNTf k ln f K nf Kf kSWCNT ỵkf 2kf kSWCNT ỵkf 2kf ; 233 ; w and p are the components of velocity and pressure in in which u wave frame Eqs (3)(5) through Eq (11) gives, 234 12ị ỵ uqbịSWCNT gT À T Þ À @p ; @r ð13Þ @p ; @z ð14Þ ð10Þ Here effective thermal conductivity of nanofluid is given by Maxwell-Gamett’s (MG-model) For the base fuid, lf is viscosity, qf is density, bf is thermal expansion coefficient, K f is thermal conÀ Á ductivity and qcp f is heat capacitance while for single wall carbon nanotubes bSWCNT is thermal expansion coefficient, qSWCNT is À Á density, kSWCNT is thermal conductivity, qcp SWCNT is heat capacitance and u is nanoparticle volume fraction The following transformation is used to swap from ðR; Z; tÞ fixed frame ðr; zÞ to wave frame, ð15Þ Bring out the following dimensionless quantities w¼ r¼ ku ; a2 c ; u¼ / ¼ ab2 ; t ¼ ckt ; Gr ¼ z¼ z ; k r2 ¼ a22 ðT T ịqf bf g ; cl0 r2 a2 ẳ ỵ / sin2pzị; 244 245 @T @T @T Q ỵ c ỵ wị À Á ¼ u Àc @r @z @z ð1 uị qcp f ỵ u qcp SWCNT ! @ T @T @ T ỵ ỵ anf þ : @r2 r @r @ z2 r a2 241 242 @w @w @w Àc u ỵ c ỵ wị uqSWCNT ỵ uÞqf Þ @r @z @z ! @ l0 @w ỵ ẳ r @z ỵ ar ð1 À uÞ2:5 @z ! @u r @ l0 @w ỵ ỵ qbịf uị @r ỵ ar uị2:5 @r @z ỵ uðqbÞSWCNT gðT À T Þ À 238 239 @u @u @u Àc u þ ðc þ wÞ ð1 À uÞqf þ uqSWCNT Þ @r @z @ z ! @w @ l0 @u ỵ ẳ ỵ r @z ỵ arÞð1 À uÞ2:5 @ z @r ! @ l0 2r @ u À2  @r ð1 À uÞ2:5 þ ar @r ! l0 u þ ð1 À uịqbịf r uị2:5 ar ỵ 1ị w ; c 235 236 @w u @u ỵ ỵ ẳ 0; @ r @ z r 247 248 ¼ 249 a1 a2 ; h ¼ TTÀT ; r ¼ ar12 ¼ ; ÀT 2 Q a p a cq c ¼ ðT 1aÀT ; Re ¼ 2l f ; d ¼ ak2 ; p ¼ ck2l : Þkf 0 ð16Þ 251 and after applying the lubrication approach Eqs (13)–(15) takes the form: 252 9ị f 11ị @p ẳ 0; @r r @p @ @w qbịSWCNT ẳ ỵ Gr ỵ u 2:5 @z r @r ar ỵ @r ðqbÞf ð1 À uÞ À uÞh; 259 ð19Þ 262 f at r ¼ r ¼ ; 256 260 Eq (17) shows that p – pðrÞ In these equations, h is the dimensionless temperature, Re the Reynolds number and cthe dimensionless heat source parameter In wave frame, the appropriate boundary conditions are defined as [34–36] w ¼ À1; h ẳ 1; 254 257 18ị ! kSWCNT 2 kSWCNT u ln kf ỵk2kSWCNT ỵ uị @ h @h kf f @ A ỵ c ẳ 0: ỵ k k ỵk @r @r r kSWCNTf k u ln f 2kSWCNT ỵ À uÞ f ð17Þ 253 ð20Þ Please cite this article in press as: Shahzadi I et al Simultaneous effects of single wall carbon nanotube and effective variable viscosity for peristaltic flow through annulus having permeable walls Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2016.12.024 263 264 265 266 267 268 270 RINP 490 No of Pages 10, Model 5G 25 January 2017 I Shahzadi et al / Results in Physics xxx (2017) xxx–xxx Fig 2,3 Variation of velocity profile for different values of the (2) Darcy number Da , (3) heat source parameter c 271 273 w ¼ wh À 1; h ¼ 0; 274 where wh is the slip velocity at r ¼ r 275 277 278 279 280 281 282 284 at r ¼ r2 ; @w f @p ; ẳ p wh ỵ b1 Da @r @z Da 21ị at r ẳ r ; ð22Þ where f defines the dimensionless constant, Da defines the Darcy’s number and b1 ¼ lf =lnf The instantaneous volume flow rate in the fixed frame is given by ẳ 2p Q Z R2 23ị RWdR R1 H ẳ q ỵ pc a22 a21 b ỵ 306 ! ð28Þ which can be written as 309 310 q 2 /2 ẳ ỵ ỵ 2pca22 2pca22 2 H ð29Þ Defining the dimensionless time-mean flow as qẳ 308 H 313 314 30ị 2pca22 312 we rewrite Eq (29) as 316 317 318 286 where R1 is a constant and R2 is a function of Z and t On putting (11) into (23), and then integrating, one obtains 2 / qẳFỵ ỵ 2 289 ẳq ỵ pcr 22 À r 21 Þ Q Solution of the problem 321 290 where The solutions of temperature and velocity profile are as follows (See Table 1) 322 285 287 291 293 294 295 296 297 299 ¼ 2p q Z r r1 ð24Þ rwd r ð25Þ is the volume flow rate in the moving coordinate system and is independent of time Here, r is a function of z only With the help of dimensionless parameters, we find q Fẳ ẳ 2pca22 Z r2 26ị rwdr r1 301 The time-mean flow over a period T ¼ k=c at a fixed Z-position is given as 304 H¼ 300 302 305 T Z dt Q ð27Þ By invoking Eq (24) in (27) and integrating, we get h ẳ c Kf r ỵ K nf ð31Þ ðÀ4 À c Kf K nf r 21 ỵc nf nf 4ln r ln r2 Þ ; ð32Þ ðqbÞnf Gð1 À uÞ2:5 ð1 uị2:5 dp r ar wẳ ỵ dz 2 ðqbÞf K r f ðÀ4ðC À C Þr Àc  À ð5C À 6C Þr3 a 16 K nf Kf À r5 c a ỵ C r2 ỵ 2raị ln rị ỵ C ln r ỵ arị ỵ C ; 33ị K nf Z Fẳ Physical properties SWCNT Blood K (W = mK) cp (J = kgK) q ðkg=m3 Þ 6600 425 2600 0.492 3594 1063 b  10À5 ð1=KÞ 1.5 0.18 324 K ln r ỵ c K f r22 ln r þ c K f r21 ln r Flow rate is given as Table Thermophysical parameters of SWCNT and blood 323 r 22 Þ ln r 4ðln r À ln r1 Þ K À Kf K nf 320 326 327 329 330 331 r2 rwdr: 333 r1 The pressure gradient is defined as dp F À l1 ¼ ; dz l2 334 335 ð34Þ Please cite this article in press as: Shahzadi I et al Simultaneous effects of single wall carbon nanotube and effective variable viscosity for peristaltic flow through annulus having permeable walls Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2016.12.024 337 RINP 490 No of Pages 10, Model 5G 25 January 2017 I Shahzadi et al / Results in Physics xxx (2017) xxx–xxx Fig 4,5 Variation of velocity profile for different values of the (4) Grashoff number Gr , (5) Viscosity parameter a Fig 6,7 Variation of pressure gradient for (6) Darcy number Da , (7) heat source parameter c Fig 8,9 Variation of pressure gradient for (8) Grashoff number Gr , (9) Viscosity parameter a Please cite this article in press as: Shahzadi I et al Simultaneous effects of single wall carbon nanotube and effective variable viscosity for peristaltic flow through annulus having permeable walls Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2016.12.024 RINP 490 No of Pages 10, Model 5G 25 January 2017 I Shahzadi et al / Results in Physics xxx (2017) xxx–xxx Fig 10,11 Variation of pressure rise for (10) Darcy number Da , (11) heat source or sink parameter c Fig 12,13 Variation of pressure rise for (12) Grashoff number Gr (13) Viscosity parameter a Fig 14 Streamlines for different values of (a) Da ¼ 0:1, (b) Da ¼ 0:2, (c) Da ¼ 0:3 Please cite this article in press as: Shahzadi I et al Simultaneous effects of single wall carbon nanotube and effective variable viscosity for peristaltic flow through annulus having permeable walls Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2016.12.024 RINP 490 No of Pages 10, Model 5G 25 January 2017 I Shahzadi et al / Results in Physics xxx (2017) xxx–xxx Fig 15 Streamlines for different values of (a) c ¼ 0:5, (b) c ¼ 0:6, (c) c ¼ 0:7 Fig 16 Streamlines for different values of (a) Gr ¼ 1:1, (b) Gr ¼ 1:2, (c) Gr ¼ 1:3 Fig 17 Streamlines for different values of (a) a ¼ 0:0, (b) a¼ 0:1, (c) a ¼ 0:2 Please cite this article in press as: Shahzadi I et al Simultaneous effects of single wall carbon nanotube and effective variable viscosity for peristaltic flow through annulus having permeable walls Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2016.12.024 RINP 490 No of Pages 10, Model 5G 25 January 2017 I Shahzadi et al / Results in Physics xxx (2017) xxx–xxx Fig 18 Streamlines for different values of (a) u ¼ 0:00 (Pure blood), (b) u ¼ 0:01, (c) u ¼ 0:05 Table Variation of temperature profile for distinct values of heat source parameter c 338 h Pure blood u ẳ 0:00ị r c ¼ 0:4 c ¼ 0:6 c ¼ 0:8 SWCNT u ẳ 0:03ị c ẳ 0:4 c ẳ 0:6 c ¼ 0:8 c ¼ 0:4 c ¼ 0:6 c ¼ 0:8 a1 ¼ 0:1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 h 1.0000 0.7004 0.5249 0.4003 0.3035 0.2241 0.1568 0.0983 0.0465 0.0000 1.0000 0.7011 0.5260 0.4015 0.3047 0.2252 0.1578 0.0990 0.0469 0.0000 1.0000 0.7019 0.5271 0.4027 0.3059 0.2264 0.1587 0.0997 0.0473 0.0000 1.0000 0.7004 0.5249 0.4003 0.3034 0.2240 0.1568 0.0983 0.0465 0.0000 1.0000 0.7011 0.5259 0.4014 0.3046 0.2251 0.1577 0.0989 0.0469 0.0000 1.0000 0.7018 0.5269 0.4026 0.3058 0.2263 0.1586 0.0997 0.0473 0.0000 1.0000 0.7003 0.5249 0.4002 0.3033 0.2239 0.1567 0.0982 0.0465 0.0000 1.0000 0.7010 0.5259 0.4013 0.3045 0.2250 0.1576 0.0989 0.0469 0.0000 1.0000 0.7017 0.5269 0.4025 0.3056 0.226 0.1585 0.0996 0.0472 0.0000 where l1 and l2 are given in Appendix A 339 Graphical results and discussion 340 In order to examine the implementation of the elongated set of Navier–Stokes equations under the impact of radially varying viscosity and nanoparticle contribution we have presented the graphs of the velocity, pressure gradient, pressure rise and streamlines These graphs are prepared by controlling the emerging flow parameters restricted such as u ¼ 0:00 (pure blood) and u ẳ 0:02; u ẳ 0:05ị SWCNT, Da ¼ 0:01 À 0:3; c ¼ 0:5 À 1:5; Gr ¼ 0:1 À 2; q ¼ À1:5 À 2; ¼ 0:01 À 0:5; a ¼ À 0:2; b ¼ À Influence of different embedded parameters like Darcy number Da , heat source/sink parameter c, Grashoff number Gr and viscosity parameter a on velocity profile are exposed in Figs 2–5 These figures demonstrate that velocity profile traces a curve like parabolic trajectory Velocity enhances with the increasing values of Darcy number Da as shown in Fig Velocity profile for distinct values of c (heat source) is plotted in Fig it is depicted that significance of velocity enhances by increasing c Variation of fluid velocity for Grashoff number is presented in Fig It is observed that the increasing values of Grashoff number increases the velocity of the base fluid and curve become steeper as the value of Gr 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 SWCNT u ẳ 0:06ị increases Graph for different values of viscosity parameter like ða ¼ 0; 0:1; 0:2Þ cross ponds to constant viscosity and variable viscosity, respectively is plotted in Fig It is analyzed that the velocity is higher for variable nanofluid viscosity as compared to constant nanofluid viscosity Pressure gradient is investigated through Figs 6–9 Fig shows that the increasing values of Darcy number Da increases the pressure gradient Effects of SWCNT increases the pressure gradient more prominently in comparison with pure blood Impact of heat source parameter c on the pressure gradient is presented in Fig It is noted that the rising values of heat source parameter c increases the pressure gradient Fig declared the influence of Grashoff number Gr on pressure gradient Observation shows that the increasing values of Gr increases the pressure gradient Fig is plotted for different values of viscosity parameter a It is indicated that the influence of pressure gradient is more prominent for variable nanofluid viscosity as compared to the constant viscosity Pressure rise per wavelength is necessary to explain the pumping properly and represented here from Figs 10– 13 One common observation from these figures is that pressure rise decreases with the expansion of flow rate The free pumping flux (value of q for Dp ¼ 0) increases with the inclusion of nanoparticles On the other hand, pressure rise increases in the retrograde pumping region ðq < 0; Dp > 0Þ by increasing the concentration of nanoparticles Fig 10 shows that increasing values of Darcy Please cite this article in press as: Shahzadi I et al Simultaneous effects of single wall carbon nanotube and effective variable viscosity for peristaltic flow through annulus having permeable walls Results Phys (2017), http://dx.doi.org/10.1016/j.rinp.2016.12.024 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 RINP 490 No of Pages 10, Model 5G 25 January 2017 I Shahzadi et al / Results in Physics xxx (2017) xxx–xxx 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 number increases the pressure rise Figs 11 and 12 declare that raise in the values of heat source parameter and Grashoff number increases the pressure rise per wavelength SWCNT enhances the pressure rise more prominently in the retrograde pumping region Fig 13 show the impact of variable viscosity a on pressure rise and it is elucidated that the pressure rise increases with the increasing values of viscosity parameter when compared with constant viscosity Streamlines by trapping describe an interesting phenomenon for fluid flow of an inside flowing bolus and plotted here through Figs 14–18 Streamlines are plotted for variation in Da ; c; Gr ; a and u Trapping phenomenon is investigate through these plots The trapping phenomenon for Darcy number is given in Fig 14 The number of trapping bolus decreases with the closed stream lines The trapping phenomena for heat source c and Grashof number Gr are given in Figs 15 and 16 The size of the inner bolus enhances with the increase of c and Gr Effects of viscosity parameter a on the streamlines is presented in Fig 17 It is observed that the significantly large value of variable nanofluid viscosity atend to increase the number of trapping bolus The important significance of nanoparticles volume fraction is examined in Fig 18 Size of the trapped bolus increases with the increase of nanoparticle volume fraction when compared with pure blood case u ẳ 0:00ị Table is prepared for temperature profile for distinct values of heat source parameter c and it is observed from this table that observed that the temperature profile increases with an increase in the values of heat source parameter due to the increase in the thermal state of the fluid i.e through metabolic process It is also elucidated that the temperature is maximum near the wall of inner tube and then start decreasing as we move towards the wall of outer tube 413 Conclusions 414 A comprehensive mathematical investigation has been done in this paper for the description of SWCNT analysis under the impact of radially varying effective viscosity The present theoretical analysis was motivated by the applications in unique nanofluid drug delivery systems Some main observations of the present examination acquired by the graphical demonstration are portrayed as follows, 415 416 417 418 419 420 Appendix A 440 441 442 C1 ẳ Kf Kf nf nf 4cK r 21 ỵcK r 22 Þ 4ðlnr Àlnr Þ ; C2 ẳ Kf Kf nf nf cK r22 lnr1 ỵ4lnr2 ỵcK r21 lnr2 4lnr1 lnr2 ị ; d1 ẳ 3r 21 þ 2r 31 a À r 22 ð3 þ 2r aị; d2 ẳ 5r 41 ỵ 4r 51 a r 42 ỵ 4r aị; d3 ẳ 9r 21 ỵ 5r 31 a r 22 ỵ 5r aị; d4 ẳ r 21 ỵ 2r aÞlnr ; 2:5 uÞ ððr À r ịa ỵ lnr lnr ị; d5 ẳ r 22 ỵ 2r aịlnr ; d6 ¼ ð1À 2880 K C ¼ d6 240 dp d ỵ qbịnf Gd2 9c K nff d3 80C d4 ỵ 240qbịnf GC ; dz Á K À h d7 ¼ À 2880w ị;d8 ẳ 720 dp ỵ 45qbịnf G 16C 16C ỵ c K f r 21 ỵ r r ỵ r 22 dz 1uị2:5 nf K d9 ẳ 4r ỵ r ị 120 dp ỵ qbịnf G 100C 120C ỵ 9c K f r 21 ỵ r 22 ịa dz nf K r ỵ 2r aị; d11 ẳ 240C r ỵ 2r aị ỵ 9c K f r 32 ỵ 4r aị; d10 ẳ 240 dp dz nf d12 ẳ 80C 9r2 ỵ 5r22 a ỵ 3r 21 a3 ỵ 2r aịị; d13 ẳ 240 dp r ỵ 2r aị dz K d14 ẳ 240C r ỵ 2r aị þ 9c K f r31 ð5 þ 4r1 aÞ; nf d15 ẳ 80C 9r1 ỵ 5r21 a ỵ 3r 22 a3 ỵ 2r aịị; d16 ẳ 240qbịnf C G3r 21 ỵ 2r 31 a r 22 ỵ 2r aịịlnr ; 2880 r a ỵ lnr ịwh ; d17 ẳ lnr ỵ uị2:5 2800 ỵ r r2 d8 ỵ d9 ịa C ẳ d7 r r ịa uị2:5 2880 ỵ r d10 ỵ qbịnf Gd11 ỵ d12 ịị uị2:5 2800 ỵ r d13 ỵ qbịnf Gd14 ỵ d15 ịị d16 lnr ịd17 ị; ỵlnr 1 uị2:5 p 2800 Da ỵ r r ịak ỵ klnr klnr ị; d18 ẳ uị2:5 p wh ẳ d18 Da qbịnf G240C 3r 21 ỵ 2r 31 a r 22 ỵ 2r aịị K 9c K f 5r 41 ỵ 4r 51 a r 42 ỵ 4r aịị 80C 9r 21 ỵ 5r 31 a r 22 ỵ 5r aịị nf p 240 dp r r ị2r21 a ỵ r ỵ 2r aị ỵ 12 Da a1 ỵ r aịkị dz ỵ240qbịnf C Gr21 ỵ r ỵ 2r aị3 ỵ 2r aị p 12 Da dp ỵ r aịkịlnr ỵ qbịnf C Gr 22 ỵ 2r aị dz p dp 12 Da dz ỵ r aịkịlnr ịịị; p l1 ẳ d18 Da 15r 41 30r 21 r 22 ỵ 12r 51 a 20r 31 r 22 a ỵ r42 15 ỵ 8r aịị p ỵr r ị60 Da ỵ 2r aị ỵ r r ị45r ỵ r ị2 15r þ r Þðr 21 þ 4r r þ r 22 Þa þ 4ðr À r ị2 4r 21 ỵ 7r r ỵ 4r 22 ịa2 ịịk p ỵ95r 41 ỵ 4r 51 a r 22 40 Da ỵ r aị þ r 22 ð5 þ 4r2 aÞÞÞkðlnr À lnr ịịị; p 60400 l2 ẳ d18 r r ị r ỵ r ị Da ỵ r r ịakị uị2:5 p ỵqbịnf G90 Da 56C 15r 41 30r 21 r 22 ỵ 12r 51 a 20r 31 r22 a K ỵr 42 15 ỵ 8r aịị À c K f ð70r 61 À 105r 41 r 22 ỵ 60r 71 a 84r 51 r 22 a nf K 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 Temperature of the nanofluid decreases with the enhance of nanoparticle volume fraction for SWCNT because high thermal conductivity plays an important role in dissipating heat Velocity profile shows higher results for the SWCNT case than for the pure blood case The velocity profile grows with an increase in the Darcy number Da and viscosity parameter a The pressure gradient increases due to the inclusion of SWCNT Pressure rise enhances due to the addition of nanoparticle volume fraction when compared to that of the pure blood ðu ¼ 0:00Þ The size of the inner bolus declines by the rise in the values of the Darcy number Da The trapping bolus phenomenon shows that the size of the bolus increases with an increase in the concentration of nanoparticles as compared to the pure blood case u ẳ 0ị 437 438 439 Uncited reference [43] þr 62 ð35 þ 24r aÞÞÞ þ 15ðr r ị2 315r ỵ r ị2 16C ỵ c K f r 21 ỵ r 22 ịị nf 42r ỵ r ị40C r 21 ỵ 4r r ỵ r 22 ị K ỵ K f r 41 ỵ 11r 31 r ỵ 6r 21 r 22 ỵ 11r r32 ỵ r 42 ịị nf c a ỵ8r r ị 56C 4r 21 ỵ 7r r ỵ 4r 22 ị K ỵ3c K f 8r 41 ỵ 17r 31 r ỵ 20r 21 r 22 ỵ 17r r 32 ỵ 8r 42 ịịa2 ịk nf p ỵ28C r r ị3 Da 225r ỵ r ị3r 21 5r 22 ị ỵ863r þ 63r r À 62r 21 r 22 À 62r r 32 À 62r 42 Þa2 Þk þ12ðÀ1260ðqbÞnf GC r 41 ð5 þ 4r aÞklnr 21 ỵlnr 50400 r r ịr þ r Þk 1ÀuÞ2:5 ÀðqbÞnf Gð15ðÀ84C ð5r 41 ỵ 4r 51 a r 42 ỵ 4r aịị p K ỵ5c K f 7r 61 þ 6r 71 a À r 62 ð7 þ 6r aịịk ỵ 7C Da r 42 15 þ 8r aÞ nf þð675r 41 þ 504r 51 a ỵ 300r 21 r 22 ỵ 2r aị 180r r 42 a5 ỵ 4r aị ỵr 42 1575 ỵ 4r a201 ỵ 80r aịịịkịị 50400 r r ịr ỵ r ịk 1260qbịnf GC r 42 ỵ 4r aịklnr ị ỵ lnr 1 uị2:5 p 2 ỵqbịnf G420C Da r 15r 30r ỵ 12r a À 20r r 22 aÞ À7C ð320r 61 a2 ỵ 300r 21 r 22 ỵ 2r aị ỵ 200r 31 r 22 a3 ỵ 2r aị 225r 41 ỵ 4r aị ỵ 9r 42 75 ỵ 56r aị 12r 51 a70 ỵ 60r aịịk K ỵ1584C 5r41 ỵ 4r51 ar 42 ỵ 4r2 aị ỵ 5c K nff 7r 61 ỵ 6r 71 a r 62 ỵ 6r aịịịk ỵ1260q bịnf GC 5r 41 ỵ r42 ị ỵ 4r 51 ỵ r 52 Þ aÞklnr2 ÞÞÞ Please cite this 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