simultaneous effects of single wall carbon nanotube and effective variable viscosity for peristaltic flow through annulus having permeable walls

Nadeem, Faranak Rabiei 9 Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan 10 Department of Mathematics, Faculty of Science, Universiti Putra Malaysia,

6

7

8 Iqra Shahzadi⇑, S Nadeem, Faranak Rabiei

9 Department of Mathematics, Quaid-i-Azam University 45320, Islamabad 44000, Pakistan

10 Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia

11

14 Article history:

15 Received 29 October 2016

16 Received in revised form 5 December 2016

17 Accepted 19 December 2016

18 Available online xxxx

19 Keywords:

20 Variable nanofluid viscosity

21 SWCNT

22 Annulus

23 Permeable walls

24 Exact solution

25

2 6

a b s t r a c t

27 The current article deals with the combine effects of single wall carbon nanotubes and effective viscosity

28 for the peristaltic flow of nanofluid through annulus The nature of the walls is assumed to be permeable

29 The present theoretical model can be considered as mathematical representation to the motion of

con-30 ductive physiological fluids in the existence of the endoscope tube which has many biomedical

applica-31 tions such as drug delivery system The outer tube has a wave of sinusoidal nature that is travelling along

32 its walls while the inner tube is rigid and uniform Lubrication approach is used for the considered

anal-33 ysis An empirical relation for the effective variable viscosity of nanofluid is proposed here interestingly

34 The viscosity of nanofluid is the function of radial distance and the concentration of nanoparticles Exact

35 solution for the resulting system of equations is displayed for various quantities of interest The outcomes

36 show that the maximum velocity of SWCNT-blood nanofluid enhances for larger values of viscosity

37 parameter The pressure gradient in the more extensive part of the annulus is likewise found to increase

38

as a function of variable viscosity parameter The size of the trapped bolus is also influenced by variable

39 viscosity parameter The present examination also revealed that the carbon nanotubes have many

appli-40 cations related to biomedicine

41

Ó 2017 The Authors Published by Elsevier B.V This is an open access article under the CC BY license (http://

42 creativecommons.org/licenses/by/4.0/)

43 44

46 Peristalsis is a conspicuous component in physiology for liquid

47 transport Peristaltic transport is totally vital among the most

48 recent researchers because of its application in physics, applied

49 mathematics, physiological world and engineering In this process,

50 sinusoidal waves move around the walls of tube as the organs of

51 human being pushing the fluid in the direction of their propagation

52 towards the tube Peristalsis has many applications in

medical/bio-53 logical employment where the traveling matter is not in the

imme-54 diate contact with another part prohibit the inside surface of tube

55 The witness of peristalsis is to transit food through oesophagus,

56 transport of urine from kidney to bladder, vasomotion of blood

57 vessels, transport of bile in bile duct, movement of chyme in

58 intestines and many others[1–3] Engineers have approved such

59 mechanism because of its utility in captivating different modern

60

apparatuses comprising of peristaltic pumps, finger pumps in

dial-61

ysis machines, roller and heart lung The phenomena peristalsis is

62

used in various hose pumps In the nuclear industry the transport

63

of destructive fluids is of peristaltic type Numerous theoretical

64

assessments are trucked out in physiology and industry because

65

of such wide existence of peristalsis[4–9]

66

In these days, the endoscope is a very significant tool utilize for

67

analyzing causes responsible for various complication in the

68

organs of human in which the fluid is carried by peristaltic

pump-69

ing like stomach and small intestine There is no difference

70

between catheter and an endoscope from dynamic point of view

71

Furthermore, the injection of a catheter will change the

distribu-72

tion and flow field in an artery [10] Numbers of investigations

73

are done to analyze the impact of endoscope on peristaltic

trans-74

port for Newtonian and non-Newtonian fluids[11–15]

75

Nanoparticle examination is in the blink of an eye a region of

76

effective experimental enthusiasm because of a gigantic scope of

77

potential applications in electronic, optical, biomedical field The

78

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/ ).

⇑ Corresponding author at: Department of Mathematics, Quaid-i-Azam

University 45320, Islamabad 44000, Pakistan (I Shahzadi).

E-mail address: iqrashahzadiwah@gmail.com (I Shahzadi).

Contents lists available atScienceDirect

Results in Physics

j o u r n a l h o m e p a g e : w w w j o u r n a l s e l s e v i e r c o m / r e s u l t s - i n - p h y s i c s

79 physical and chemical properties is defined as nanofluid and the

80 word ‘‘nanofluid” was firstly introduce by Choi[15] Buongiorno

81 [16]proposed that the thermophoresis and Brownian motion plays

82 a key role in the dynamics of nanofluids Nanofluid is basically the

83 liquid suspension that comprises of little particles having diameter

84 across lesser 100 nm These tiny particles are mostly found in the

85 metals such as nitrides, nitrides, Carbides or non-metals (Graphite,

86 Carbon nanotubes) In advanced examination miniaturized

compo-87 nents perform essential part in all types of utilizations Many

88 authors have considered the effect of nanoparticles[17–25] One

89 such development is carbon nanotube; Carbon nanotubes (CNTs)

90 are the hollow cylinders of carbon atoms Carbon nanotubes are

91 metal or semiconductor Their appearance is that of folded tubes

92 of graphite, such that the hexagonal carbon rings and their bundles

93 are formed Single-walled nanotubes (SWNTs) and multiwall

nan-94 otubes (MWNTs) are the two types of nanotubes because the

dif-95 ference in the arrangement of graphene cylinders SWNTs have

96 only one layer while MWNTs have more than one layer of graphene

97 cylinders [26] Carbon nanotube is stronger than steel per unit

98 weight while 50,000 times thinner than a human hair Recently,

99 it is determined by Murshed et al [27] that carbon nanotube

100 (CNT) have six times better thermal conductivity than other

mate-101 rials Iijima and Ichihashi[28]exposed that carbon nanotubes are

102 intriguing an extensive variety of industries as well as scientists

103 passion due to their charming chemical and physical properties

104 Peristaltic flow of carbon nanotubes in curved channel with heat

105 transfer was discussed by Akbar and Butt[29] Analysis of entropy

106 generation of CNT suspension in plumb duct is discussed by Akbar

107 [30] Effective viscosity and expressions for the nanofluids were

108 calculated by Li and Xuan[31] Brinkman’s[32]proposed the

effec-109 tive viscosity model for two phase flow Das and Tiwari [33]

110 designed a model for the study of nanofluids by using the results

111 of Li, Xuan[31]and Brinkman[32] For the perseverance of

nano-112 fluid dynamics, nanofluid model given by Das and Tiwari[33]was

113 used by various scientists

114 Darcy’s Law is used to drive the fluid flow through porous

med-115 ium while the fluid in free region is carried out by Navier Stokes

116 equations Beavers and Joseph suggested the boundary condition

117 at the permeable surface in the coupled flow motion in 1967

Dif-118 ferent pragmatic applications experience the flow through a

120 Limestone, Sandstone, gall bladder with stones in tiny blood

ves-121 sels, beach sand, bile duct and the human lung are the important

122 examples of natural porous media[34–38] Permeable wall

analy-123 sis for the nanofluid flow in stenosed arteries is examined by

124 Noreen et al [39] Nadeem and Ijaz [40] portray the impact of

125 metallic type nanoparticles on blood flow with permeable walls

126 through stenosed artery

127 In all of the mention citation, fluid viscosity was considered to

128 be constant The physical properties of the fluid may change

con-129 siderably with radius and temperature For lubricating fluids, heat

130 produced by the inner friction and the corresponding increase in

131 temperature affects the viscosity of the fluid and so the fluid

vis-132 cosity can not be considered constant anymore Therefore, to

133 examine the flow behavior accurately it is sufficient to consider

134 the incompressible fluids for viscosity variation[41,42]

135 The motivation behind the present examination is to inspect the

136 significance of nanoparticles infused in the annulus in the

exis-137 tence of variable effective viscosity which is not done before from

138 author’s knowledge Here we examine the impact of single wall

140 peristaltic flow in an annulus having permeable walls The aim of

141 this paper is to comprehend the fluid mechanics in a physiological

142 circumstance in the existence of concentrically set endoscope

143

Significant modeling is conferred with the aid of dimensionless

144

parameters and using approximation of low Reynolds number

145

and long wavelength Results acquired from this examination

pro-146

vide a useful understanding about the particular nature of SWCNT

147

which influence the peristalsis and provide new visions of

148

nanoparticles in the presence of variable viscosity

149

Formulation of the problem

150

Consider the unsteady, two-dimensional, incompressible flow

151

of a viscous fluid through the gap lies between the tubes with

vari-152

able effective viscosity The central tube is the endoscope while the

153

outer tube has a sinusoidal wave that is traveling down through its

154

wall The outer tube is maintained at a constant temperature T1

155

while the inner tube is rigid and retained at a temperature T0

156

R; Z

coordinates are preferred in such a way that the length of

157

the tube is along Z-axis whereas R-axis is normal to Z-axis The

158

two wall surface geometry is described by the equation:

159

R2¼ b sin2p

162

165

where a1and a2represents the radius of inner and outer tube, k is

166

the wavelength, c is the wave speed, b is the amplitude of the wave

167

and t is the time (SeeFig 1)

168

The two dimensional continuity equation for incompressible

169

fluid is defined below

170 U

Rþ@W

@Z þ

@U

173

In the laboratory frame, transverse and longitudinal

compo-174

nents of velocity are represented by W and U In the presence of

175

mixed convection, R and Z components of momentum equation are

176

qnf

@U

@ZWþ U

@U

@Rþ

@U

@t

!

¼ @

@Z

@U

@Zþ

@W

@R

!



lnfðRÞ

!

þ1 R

@

@R 2lnfðRÞ@U

@RR

!

 2lnfðRÞU

R2þð ÞqbnfðT  T0Þg @P

@R;

Fig 1 Geometry of the problem.

2 I Shahzadi et al / Results in Physics xxx (2017) xxx–xxx

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

qnf

@W

@ZWþ

@W

@RUþ

@W

@t

!

@W

@ZlnfðRÞ

!

þ1 R

@

@R lnfðRÞ @U

@Zþ

@W

@R

! R

!

þð ÞqbnfgðT  T0Þ @P

181

182 Energy equation in the presence of heat generation is given as,

183

qCp

nf

@T

@RUþ

@T

@ZWþ

@T

@t

¼ Knf @2T

@Z2þ@

2T

@R2þ1 R

@T

@R

!

þ Q0: ð6Þ 185

186 In the above equations, P is the pressure in laboratory frame, U

187 and W are the velocity components, T is the temperature of fluid,

188 Q0is the constant heat generation/absorption, Knf is the thermal

189 conductivity, bnf is the thermal expansion coefficient,qnf is the

190 density and
qCp

nf is the heat capacitance of the nanofluid with

191 thermophysical properties defined in[27,28]

192 For the proposed nanofluid model,lnfis the variable nanofluid

193 viscosity[5]and suppose the variation of viscosity following from

194 Brinkman[32]and Srivastava et al.[41]as follows:

195



lnfðRÞ ¼ lBðRÞ

1u

197

198 wherelBis the viscosity of the base fluid We further assume that

199 the viscosity of the base fluid varies according tothe following

200 relation:

201

lBðRÞ ¼l0eaR¼ l0

203

204 Herel0is the viscosity of blood,að 1Þ is the dimensional

vari-205 able viscosity parameter From Eqs.(6) and (7)the effective

viscos-206 ity of the nanofluid is reduced as follows:

207



209

211 model (i.e., the effective viscosity independent of R) canbe

recov-212 ered for a¼ 0 The viscosity of the fluid independent of the

213 nanoparticles can also be obtained by substitutingu¼ 0

214 Relations for effective density, thermal conductivity and specific

215 heat of nanofluid by Das and Tiwari[33]

216

qnf ¼uqSWCNTþqfð1uÞ;anf¼ knf

qCp

nf

; qcp

nf

¼u qcp

SWCNTþ 1 ð uÞqcp

218

219

qb

ð Þnf¼ð Þqbfð1uÞ þu qð ÞbSWCNT;Knf

Kf

¼ð1uÞ þ 2u

kSWCNT

kSWCNTk f lnkSWCNT þk f

2kf

1u

ð Þ þ 2u kf

kSWCNTk f lnkSWCNT þk f

2kf

221

222 Here effective thermal conductivity of nanofluid is given by

223 Maxwell-Gamett’s (MG-model) For the base fuid,lf is viscosity,

224 qfis density, bfis thermal expansion coefficient, Kfis thermal

con-225 ductivity and qcp

f is heat capacitance while for single wall

car-226 bon nanotubes bSWCNT is thermal expansion coefficient,qSWCNT is

227 density, kSWCNT is thermal conductivity,
qcp

SWCNT is heat

capaci-228 tance anduis nanoparticle volume fraction

229 The following transformation is used to swap fromðR; Z;tÞ fixed

230 frameðr; zÞ to wave frame,

231

z ¼ Z  ct; pðz;rÞ ¼ PðZ; R;tÞ; r ¼ R; w ¼ W  c; u ¼ U; ð11Þ 233

234

in whichu; w and p are the components of velocity and pressure in

235

wave frame Eqs.(3)–(5)through Eq.(11)gives,

236

@u

@rþ

@ w

@zþ

u

239

1u

@ru  c @@zuþ ðc þ wÞ@u

@z

¼ @

ð1 þarÞ 1 ð uÞ2 :5

@u

@zþ

@ w

@r

þ1r

 @

1u

ð Þ2 :5

2r

1þar

@u

@r

!

 2

ru2 l0

1u

ð Þ2 :5ðar þ 1Þ

!

þ 1 ð uÞð Þqbf

242

ðuqSWCNTþ 1 ð uÞqfÞ @ @rwu  c @@zwþ ðc þ wÞ@ w@z

¼ @

@z

2

1þar

l0

1u

ð Þ2 :5

@ w

@z

þ1r

 @

@r

r

1þar

l0

1u

ð Þ2 :5

@ w

@rþ

@u

@z

þð Þqbfð1uÞ

245

@T

@ru þ ðc þ wÞ @@zT c@T@z

1u

ð Þqcp

fþu qcp

SWCNT

þanf @2T

@r2þ1r@T@rþ@

2T

@z2

!

247 248

Bring out the following dimensionless quantities

249

w¼w

c; r ¼r

a2; u ¼k u

a2c; z ¼z

k; r2¼r 2

a2¼ 1 þ / sinð2pzÞ; ¼a1

a2; / ¼ b

2; t ¼ct

k; Gr¼a 2 ðT 1 T 0 Þ q f b f g

cl0 ; h ¼TT 0

T 1 T 0; r1¼r1

a 2¼;

c¼ a 2 Q 0

ðT 1 T 0 Þk f; Re¼a 2 c q f

l0 ; d ¼a 2

k; p ¼ a 2 p

ckl0:

252

and after applying the lubrication approach Eqs.(13)–(15)takes the

253

form:

254

@p

257

@p

@z¼

1 r

@

@r

r

ar þ 1@w@r

1

1u

ð Þ2:5þ Grð1 þuð ÞqbSWCNT

qb

ð Þf

260

2 kSWCNT

k SWCNT k fulnkf þk SWCNT

2k f þ 1 ð uÞ

2 kf

k SWCNT k fulnkf þk SWCNT

2k f þ 1 ð uÞ

0

@

1

A @2h

@r2þ@h

@r

1 r

!

262 263

Eq.(17)shows that p– pðrÞ In these equations, h is the

dimen-264

sionless temperature, Re the Reynolds number andcthe

dimen-265

sionless heat source parameter

266

In wave frame, the appropriate boundary conditions are defined

267

as[34–36]

268

273

274 where whis the slip velocity at r¼ r2

275

@w

@r ¼ fffiffiffiffiffiffi

Da

p whþ b1Da @p

@z

 

277

278 where f defines the dimensionless constant, Dadefines the Darcy’s

279 number and b1¼lf=lnf

280 The instantaneous volume flow rate in the fixed frame is given

281 by

282

Q ¼ 2pZ R2

R 1

284

285 where R1is a constant and R2is a function of Z and t On putting

286 (11)into(23), and then integrating, one obtains

287

289

291

q ¼ 2p

Z r2

r 1

293

294 is the volume flow rate in the moving coordinate system and is

295 independent of time Here, r2is a function of z only With the help

296 of dimensionless parameters, we find

297

F¼ q

2pca2¼

Z r 2

r1

299

300 The time-mean flow over a period T¼ k=c at a fixed Z-position

301 is given as

302

H¼1T

Z 1

0

304

305 By invoking Eq.(24)in(27)and integrating, we get

306

H¼ q þpc a2 a2þb

2

2

!

ð28Þ 308 309

which can be written as

310

H

2pca2¼ q

2pca2þ1

22

2 þ/

2

312 313

Defining the dimensionless time-mean flow as

314

316 317

we rewrite Eq.(29)as

318

q¼ F þ1

22

2þ/

2

321

Solution of the problem

322

The solutions of temperature and velocity profile are as follows

323

(SeeTable 1)

324

h ¼ cKKf

nf

r2

4þð4 c

Kf

K nfr2þcKf

K nfr2Þ ln r 4ðln r2 ln r1Þ

4 ln r2þc

K f

Knfr2ln r1þcKf

Knfr2ln r2

327

w¼ð1uÞ

2 :5

2

dp dz

r2

2þar3 3

ð ÞqbnfG 1ð uÞ2:5

qb

ð Þf

 161 ð4ðC1 C2Þr2



cKf

Knf

r4

449ð5C1 6C2Þr3

a

1

5r

5

cKKf

nfaþ4

3 1r

2ð3 þ 2raÞ ln rÞ



þ C3ðln r þarÞ þ C4; ð33Þ

329 330

Flow rate is given as

331

F¼

Z r2

r 1

rwdr:

333 334

The pressure gradient is defined as

335 dp

dz¼F l1

Table 1

Thermophysical parameters of SWCNT and blood.

Fig 2,3 Variation of velocity profile for different values of the (2) Darcy number D a , (3) heat source parameterc.

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Fig 4,5 Variation of velocity profile for different values of the (4) Grashoff number G r , (5) Viscosity parametera.

Fig 6,7 Variation of pressure gradient for (6) Darcy number D a , (7) heat source parameterc.

Fig 8,9 Variation of pressure gradient for (8) Grashoff number G r , (9) Viscosity parametera.

Fig 10,11 Variation of pressure rise for (10) Darcy number D a , (11) heat source or sink parameterc.

Fig 12,13 Variation of pressure rise for (12) Grashoff number G r (13) Viscosity parametera.

Fig 14 Streamlines for different values of (a) D a ¼ 0:1, (b) D a ¼ 0:2, (c) D a ¼ 0:3.

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Fig 15 Streamlines for different values of (a)c¼ 0:5, (b)c¼ 0:6, (c)c¼ 0:7.

Fig 16 Streamlines for different values of (a) G r ¼ 1:1, (b) G r ¼ 1:2, (c) G r ¼ 1:3.

Fig 17 Streamlines for different values of (a)a¼ 0:0, (b)a¼ 0:1, (c)a¼ 0:2.

338 where l1and l2are given inAppendix A.

339 Graphical results and discussion

340 In order to examine the implementation of the elongated set of

341 Navier–Stokes equations under the impact of radially varying

vis-342 cosity and nanoparticle contribution we have presented the graphs

343 of the velocity, pressure gradient, pressure rise and streamlines

346 ðu¼ 0:02;u¼ 0:05Þ SWCNT, Da¼ 0:01  0:3;c¼ 0:5  1:5; Gr¼

347 0:1  2; q ¼ 1:5  2;¼ 0:01  0:5;a¼ 0  0:2; b ¼ 1  3

Influ-348 ence of different embedded parameters like Darcy number Da, heat

349 source/sink parameterc, Grashoff number Grand viscosity

param-350 etera on velocity profile are exposed inFigs 2–5 These figures

351 demonstrate that velocity profile traces a curve like parabolic

tra-352 jectory Velocity enhances with the increasing values of Darcy

353 number Daas shown inFig 2 Velocity profile for distinct values

354 ofc(heat source) is plotted inFig 3 it is depicted that significance

355 of velocity enhances by increasingc Variation of fluid velocity for

356 Grashoff number is presented in Fig 4 It is observed that the

357 increasing values of Grashoff number increases the velocity of

359

increases Graph for different values of viscosity parameter like

360

ða¼ 0; 0:1; 0:2Þ cross ponds to constant viscosity and variable

vis-361

cosity, respectively is plotted inFig 5 It is analyzed that the

veloc-362

ity is higher for variable nanofluid viscosity as compared to

363

constant nanofluid viscosity Pressure gradient is investigated

364

throughFigs 6–9.Fig 6shows that the increasing values of Darcy

365

number Da increases the pressure gradient Effects of SWCNT

366

increases the pressure gradient more prominently in comparison

367

with pure blood Impact of heat source parametercon the pressure

368

gradient is presented inFig 7 It is noted that the rising values of

369

heat source parameterc increases the pressure gradient Fig 8

370

declared the influence of Grashoff number Gron pressure gradient

371

Observation shows that the increasing values of Gr increases the

372

pressure gradient.Fig 9is plotted for different values of viscosity

373

parametera It is indicated that the influence of pressure gradient

374

is more prominent for variable nanofluid viscosity as compared to

375

the constant viscosity Pressure rise per wavelength is necessary to

376

explain the pumping properly and represented here fromFigs.10–

377

13 One common observation from these figures is that pressure

378

rise decreases with the expansion of flow rate The free pumping

379

flux (value of q forDp¼ 0) increases with the inclusion of

nanopar-380

ticles On the other hand, pressure rise increases in the retrograde

381

pumping regionðq < 0;Dp> 0Þ by increasing the concentration of

382

nanoparticles Fig 10 shows that increasing values of Darcy

Fig 18 Streamlines for different values of (a)u¼ 0:00 (Pure blood), (b)u¼ 0:01, (c)u¼ 0:05.

Table 2

Variation of temperature profile for distinct values of heat source parameterc.

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383 number increases the pressure rise Figs 11 and 12declare that

384 raise in the values of heat source parameter and Grashoff number

385 increases the pressure rise per wavelength SWCNT enhances the

386 pressure rise more prominently in the retrograde pumping region

387 Fig 13show the impact of variable viscosityaon pressure rise and

388 it is elucidated that the pressure rise increases with the increasing

389 values of viscosity parameter when compared with constant

vis-390 cosity Streamlines by trapping describe an interesting

phe-391 nomenon for fluid flow of an inside flowing bolus and plotted

392 here throughFigs 14–18 Streamlines are plotted for variation in

393 Da;c; Gr;a and u Trapping phenomenon is investigate through

394 these plots The trapping phenomenon for Darcy number is given

395 inFig 14 The number of trapping bolus decreases with the closed

396 stream lines The trapping phenomena for heat sourcecand

Gra-397 shof number Gr are given inFigs 15 and 16 The size of the inner

398 bolus enhances with the increase ofcand Gr Effects of viscosity

399 parameter a on the streamlines is presented in Fig 17 It is

400 observed that the significantly large value of variable nanofluid

vis-401 cosityatend to increase the number of trapping bolus The

impor-402 tant significance of nanoparticles volume fraction is examined in

403 Fig 18 Size of the trapped bolus increases with the increase of

404 nanoparticle volume fraction when compared with pure blood case

405 ðu¼ 0:00Þ.Table 2is prepared for temperature profile for distinct

406 values of heat source parametercand it is observed from this table

407 that observed that the temperature profile increases with an

408 increase in the values of heat source parameter due to the increase

409 in the thermal state of the fluid i.e through metabolic process It is

410 also elucidated that the temperature is maximum near the wall of

411 inner tube and then start decreasing as we move towards the wall

415 this paper for the description of SWCNT analysis under the impact

416 of radially varying effective viscosity The present theoretical

anal-417 ysis was motivated by the applications in unique nanofluid drug

418 delivery systems Some main observations of the present

examina-419 tion acquired by the graphical demonstration are portrayed as

421  Temperature of the nanofluid decreases with the enhance of

423 conductivity plays an important role in dissipating heat

424  Velocity profile shows higher results for the SWCNT case than

426  The velocity profile grows with an increase in the Darcy number

427 Daand viscosity parametera

428  The pressure gradient increases due to the inclusion of SWCNT

429  Pressure rise enhances due to the addition of nanoparticle

432  The size of the inner bolus declines by the rise in the values of

436 nanoparticles as compared to the pure blood caseðu¼ 0Þ

437

439 [43]

440

Appendix A

441 442

C1¼ð4c

Kf Knfr2þ cKnfKfr 2 Þ 4ðlnr 1 lnr 2 Þ ; C2¼c

Kf Knfr2lnr1 þ4lnr 2 þ cKnfKfr 2 lnr2 4ðlnr 1 lnr 2 Þ ;

d1¼ 3r2þ2r3ar2ð3þ2r2aÞ; d2¼ 5r4þ4r5ar4ð5þ4r2a

d3¼ 9r2þ5r3ar2ð9þ5r2aÞ; d4¼ r2ð3þ2r1aÞlnr1;

d5¼ r2ð3þ2r1aÞlnr2; d6¼ð 1 u Þ 2:5

2880 ððr1r2Þaþlnr1lnr2Þ;

C3¼ d6ð240dp

dzd1þð ÞqbnfGd29cKf

Knfd380C1d4þ240ð ÞqbnfGC1;

d7¼ 2880wh 1 u

ð Þ 2:5Þ;d8¼ 720dp

dzþ45ð ÞqbnfG 16C116C2þcKf

Knf
r2þr1r2þr2

d9¼ 4ðr1þr2Þ 120dp

dzþð ÞqbnfG 100C1120C2þ9cKf

Knfðr2þr2Þa

d10¼ 240dp

dzr2ð3þ2r2aÞ; d11¼ 240C2r2ð3þ2r1aÞþ9cKf

Knfr3ð5þ4r2a

d12¼ 80C1ð9r2þ5r2aþ3r2aðð3þ2r1aÞÞ; d13¼ 240dp

dzr1ð3þ2r1aÞ

d14¼ 240C2r1ð3þ2r1aÞþ9cKf

Knfr3ð5þ4r1a

d15¼ 80C1ð9r1þ5r2aþ3r2að3þ2r1aÞÞ;

d16¼ 240ð ÞqbnfC1Gð3r2þ2r3

ar2ð3þ2r1aÞÞlnr1;

d17¼ lnr2þ 2880

1 u

ð Þ 2:5ðr1aþlnr1Þwh;

C4¼ d7ððr1r2Þa 2800

1 u

ð Þ 2:5þr1r2ðd8þd9Þa

 2880 1 u

ð Þ 2:5þðr2d10þð ÞqbnfGðd11þd12ÞÞ

þlnr1ð 12800u Þ 2:5þr1ðd13þð ÞqbnfGðd14þd15ÞÞd16lnr1Þd17Þ;

d18¼ 2800 1 u

ð Þ 2:5ð ffiffiffiffiffiffiDa

p þðr1r2Þak þ klnr1klnr2Þ;

wh¼ d18ð ffiffiffiffiffiffiDa

p

ðð ÞqbnfGð240C2ð3r2þ2r3ar2ð3þ2r1aÞÞ

9cKf

Knfð5r4þ4r5ar4ð5þ4r2aÞÞ80C1ð9r2þ5r3ar2ð9þ5r2aÞÞ

240dp

dzðr1r2Þð2r2

aþr1ð3þ2r1aÞþ12 ffiffiffiffiffiffiDa

p

að1þr2aÞkÞ þ240ððð ÞqbnfC1Gr2þr2ð3þ2r1aÞð3þ2r1aÞ

12 ffiffiffiffiffiffiDa

p dp

dzð1þr2aÞkÞlnr1þðð ÞqbnfC1Gr2ð3þ2r1aÞ

12 ffiffiffiffiffiffiDa

p dp

dzð1þr2aÞkÞlnr2ÞÞÞ;

l1¼ d18ð3 ffiffiffiffiffiffiDa

p ð15r430r2r2þ12r5a20r3r2aþr4ð15þ8r2aÞÞ þðr1r2Þð60 ffiffiffiffiffiffiDa

p ð1þ2r2aÞþðr1r2Þð45 rð1þr2Þ2

15ðr1þr2Þðr2þ4r1r2þr2Þaþ4 rð1r2Þ2

ð4r2þ7r1r2þ4r2Þa2ÞÞk þ9ð5r4þ4r5ar2ð40 ffiffiffiffiffiffiDa

p ð1þr2aÞþr2ð5þ4r2aÞÞÞkðlnr1lnr2ÞÞÞ;

l2¼ d18ð60400 1 u

ð Þ 2:5ðr1r2Þðr1þr2Þð ffiffiffiffiffiffiDa

p þðr1r2ÞakÞ

þð ÞqbnfGð90 ffiffiffiffiffiffiDa

p ð56C2ð15r430r2r2þ12r5

a20r3r2

a

þr4ð15þ8r2aÞÞcKf

Knfð70r6105r4r2þ60r7a84r5r2a

þr6ð35þ24r2aÞÞÞþ15 rð1r2Þ2ð315 rð1þr2Þ2ð16C2þcKf

Knfðr2þr2ÞÞ

42ðr1þr2Þð40C2ðr2þ4r1r2þr2Þ

þcKf

Knfðr4þ11r3r2þ6r2r2þ11r1r3þr4ÞÞa þ8 rð1r2Þ2

ð56C2ð4r2þ7r1r2þ4r2Þ þ3cKf

Knfð8r4þ17r3r2þ20r2r2þ17r1r3þ8r4ÞÞa2Þk þ28C1ðr1r2Þð3 ffiffiffiffiffiffiDa

p ð225ðr1þr2Þð3r25r2Þ þ8ð63r4þ63r3r262r2r262r1r362r4Þa2Þk þ12ð1260ð ÞqbnfGC1r4ð5þ4r1aÞklnr2

þlnr2ð50400 1 u

ð Þ 2:5ðr1r2Þðr1þr2Þk

ð ÞqbnfGð15ð84C2ð5r4þ4r5

ar4ð5þ4r2aÞÞ þ5cKf

Knfð7r6þ6r7

ar6ð7þ6r2aÞÞkþ7C1ð6 ffiffiffiffiffiffiDa

p

r4ð15þ8r2aÞ þð675r4þ504r5aþ300r2r2ð3þ2r2aÞ180r1r4að5þ4r2aÞ

þr4ð1575þ4r2að201þ80r2aÞÞÞkÞÞ

1260ð ÞqbnfGC1r4ð5þ4r2aÞklnr2Þþlnr1ð50400

1 u

ð Þ 2:5ðr1r2Þðr1þr2Þk

þð ÞqbnfGð420C1

ffiffiffiffiffiffi

Da

p

r2ð15r230r2þ12r3a20r1r2aÞ

7C1ð320r6a2þ300r2r2ð3þ2r2aÞþ200r3r2að3þ2r2aÞ

225r4ð7þ4r2aÞþ9r4ð75þ56r2aÞ12r5

að70þ60r2aÞÞk þ15ð84C2ð5r4þ4r5ar4ð5þ4r2aÞþ5cKf

Knfð7r6þ6r7ar6ð7þ6r2aÞÞÞk þ1260ð ÞqbnfGC1ð5ðr4þr4Þþ4ðr5þr5ÞaÞklnr2ÞÞÞ 444

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10 I Shahzadi et al / Results in Physics xxx (2017) xxx–xxx

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