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

Hall and Joule heating effects on peristaltic flow of Powell–Eyring liquid in an inclined symmetric channel... Hall and Joule heating effects on peristaltic flow of Powell–Eyring liquid

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5

6

7 T Hayata,b, Naseema Aslama, M Rafiqa,⇑, Fuad E Alsaadib

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

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1 3 a r t i c l e i n f o

14 Article history:

15 Received 12 November 2016

16 Received in revised form 24 December 2016

17 Accepted 3 January 2017

18 Available online xxxx

19 Keywords:

20 Peristalsis

21 Hall effects

22 Slip conditions

23 Joule heating

24 Eyring–Powell fluid and inclined channel

25

2 6

a b s t r a c t

27 This article is intended to investigate the influence of Hall current on peristaltic transport of conducting

28 Eyring–Powell fluid in an inclined symmetric channel Energy equation is modeled by taking Joule

heat-29 ing effect into consideration Velocity and thermal slip conditions are imposed Lubrication

approxima-30 tion is considered for the analysis Fundamental equations are non-linear due to fluid parameter A

31 Regular perturbation technique is employed to find the solution of systems of equations The key roles

32

of different embedded parameters on velocity, temperature and heat transfer coefficient in the problem

33 are discussed graphically Trapping phenomenon is analyzed carefully

34

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

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

36 37

39 Investigation regarding the flow of non-Newtonian fluid cannot

40 be overlooked due to its extensive applications in variety of field

41 like physiology, engineering and industry No doubt various

consti-42 tutive relations are suggested for the flow description of such fluids

43 diverse characteristics Some recent researchers are even now

44 engaged for the flow analysis of such fluids In connection with

45 peristalsis, the non-Newtonian fluids gained much attention due

46 to their various applications in physiological and industrial

pro-47 cesses Spontaneous compressing and relaxing movement along

48 the walls of tabular structures is termed as peristalsis Digestive

49 tract, blood flow in lymphatic transport are few examples that

50 can be observed within human body The phenomenon is also

51 involved in designing many devices like dialysis machine, heart

52 lung machine and blood pump machine to blood pump during

sur-53 gical processes Some worms also use this phenomenon for their

54 locomotion Pioneering studies in this direction were done by

55 Latham [1], Shapiro et at [2] and Lew et al [3] After these

56 attempts the investigators analyzed the peristaltic flow of

Newto-57 nian and non-Newtonian fluids under different flow situations[4–

58 10] Heat transfer also has a vital role in peristaltic flows especially

59 blood flows Heat conduction in tissues, convective heat transfer

60 during blood flow from pores of tissue, radiative heat transfer

61 between environment and surface, food processing and

vasodila-62 tion are some main applications of heat transfer Oxygenation

63 and hemodialysis are the processes involving heat transfer in

con-64 nection with peristalsis Recent attempts on peristaltic flow with

65 heat transfer effects can be visualized by Refs.[11–20]

66 Magnetic field has gained significance due to its variety of

appli-67 cations in biomedical engineering and industry Power generators,

68 electrostatic precipitation, purification of molten metal from

non-69 metallic inclusions etc are some processes that deals with

mag-70 netic field The shear rate of less than 100 s1 for blood flow

71 shows the model for MHD peristaltic flows in coronary arteries

72

[21] MHD may also be used to control the blood flow during

car-73 diac surgeries from stenosed arteries Hall effects cannot be

74 ignored when strong magnetic field is considered Representative

75 studies in this direction can be consulted by the Refs.[22–31]

76 The problems studying thin films, rarefied fluid, fluid motion

77 inside human body and polishing of artificial heart values etc do

78 not follow no-slip boundary condition Experimental investigations

79 show that slippage can occur in non-Newtonian fluids Moreover,

80 many physiological systems are neither horizontal nor vertical

81 but show inclination with axis (see Refs.[32–36]) Therefore, aim

82

of the present study is to investigate the peristaltic flow of

Pow-83 ell–Eyring liquid in an inclined symmetric channel Heat transfer

84

is studied in the presence of Joule heating Problem is formulated

85

by taking partial slip effects into account Nonlinear equations

86 are simplified by adopting lubrication approach Perturbation is

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/ ).

⇑ Corresponding author.

E-mail address: maimona_88@hotmail.com (M Rafiq).

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

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

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87 employed to find the solution of stream function, velocity,

temper-88 ature and heat transfer coefficient Results are analyzed via graphs

90 Here we assume the two-dimensional electrically conducting

91 non-Newtonian incompressible fluid in an inclined symmetric

92 channel having width 2d (seeFig 1) We consider Cartesian

coor-93 dinates ðx; yÞ in such a way that wave propagates in x-direction

94 and y-axis is taken transverse to it The walls of channel are

95 assumed compliant Also the channel is inclined at anglea Strong

96 magnetic field 0ð ; 0; B0Þ is applied Hall and Joule heating

contribu-97 tions are retained Peristaltic waves propagate with constant speed

98 c and wavelength k along channel walls The structure of wall

99 geometry is described as:

100

y¼ gðx; tÞ ¼ d þ asin2p

k ðx ctÞ

102

103 Here t is the time, a the wave amplitude andgthe

displace-104 ments of upper and lower walls respectively

105 The Cauchy stress tensor ðsÞ for Eyring–Powell fluid is Refs

107

109

110

S¼ lþb_1

csinh

1 c_

c1

112

113

_

c¼

ffiffiffiffiffiffiffiffi

1

2P

r

115

116

P¼ trðA2

1Þ; A1¼ grad V þ ðgrad VÞ: ð5Þ

118

119 Here S designates the extra stress tensor, I the identity tensor, b

120 and c1the material parameters of Powell–Eyring fluid andlthe

121 dynamic viscosity The term sinh1is

122

sinh1 c_

c1

¼ c_

c1

c_3 6c3;c_5

124

125 The generalized Ohms law with Hall effects is written as:

126

J¼r V B 1

enðJ BÞ

128

129

J B ¼ rB

2

1þ m2½ðu mvÞ;ðvþ muÞ; 0; ð8Þ

131

132 in which J characterizes the current density, V the velocity field, B

133 the applied magnetic field,rthe electrical conductivity, n the

num-134 ber density of electron, e the electric charge, u andvthe velocity

135 components in x and y directions respectively, B0the magnetic field

136 strength and m¼r B 0

en

the Hall parameter The fundamental flow

137 equations are

138

141

qdV

144

qCP dT

dt ¼ T:L þj r2TþJ:J

147

in whichqis the fluid density,jthe thermal conductivity and Cp

148 the specific heat

149 The two dimensional fundamental flow equations after using

150 Eqs.(2)–(8)in Eqs.(9)–(11)can be expressed as:

151

@u

@xþ

@v

154

q @v

@tþv@@yvþ u@v

@x

¼ @p

@yþ

@Syy

@y þ

@Syx

@x

þ rB

2 0

1þ m2ðvþ muÞ þqg cosa; ð13Þ 156

157

q @u

@tþv@u@yþ u@u

@x

¼ @p

@xþ

@Sxx

@x þ

@Sxy

@y

þ rB 2

1þ m2ðu mvÞ þqg sina; ð14Þ 159

160

qCp @

@tþv@y@ þ u@

@x

T¼j @2

T

@x2þ@

2 T

@y2

!

þ ðSyy SxxÞ@v

@y

þ Sxy @u

@yþ

@v

@x

þ rB

2 0

1þ m2ðu2þv2Þ; ð15Þ 162

163 where p; Sijði; j ¼ x; yÞ; g and T signify the pressure, the components

164

of extra stress tensor, the gravity and temperature respectively

165 The slip conditions for velocity and temperature at the walls

166 are:

167

170

T b1

@T

173 Flexible walls can be characterized by

174

s@3

@x3þ m1 @3

@x@t2þ d @

2

@t@x

g

¼@Sxx

@x þ

@Sxy

@y q

@u

@tþv@u@yþ u@u

@x

rB

2 0

1þ m2ðu mvÞ

177

In the above expressions T0is the temperature at the upper and

178 lower walls,sis the elastic tension, m1the mass per unit area and d

179 the coefficient of viscous damping

180 Dimensionless parameters are:

181

x¼x

k; y¼dy; u¼uc; v¼v

c; t¼ct

k;

g¼g

d; p¼d

2 p

ckl; c¼c

d; b

1¼b1

d;

h¼T T0

T0 ; S

ij¼dSij

cl; w¼w

cd:

ð19Þ

183 184

By using dimensionless variables, Eqs.(12)–(15)become:

Fig 1 Geometry of the problem.

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@v

@yþ d

@u

187

188

R1d d@v

@tþv@@yvþ du@v

@x

¼ @p

@yþ d

2@Syx

@x þ d

@Syy

@y

þH 2

dðvþ muÞ

1þ m2 þ R1

Fr

ð Þ2cosa; ð21Þ

190

191

R1 d@u

@tþ du

@u

@xþv@u@y

¼ @p

@xþ d

@Sxx

@x þ

@Sxy

@y

H2ðu mvÞ

1þ m2

þ R1 Fr

193

194

R1Prd d@

@tþv@y@ þ du@

@x

h¼ BrdðSxx SyyÞ@u

@x

þ SxyBr @u

@yþ d

2@v

@x

þ d2@2 h

@x2

þ@ 2 h

@y2þ H

2Br

1þ m2ðu2þv2Þ; ð23Þ

196

197 with the dimensionless boundary conditions

198

200

201

h b1

@h

203

204 wherecand b1represent the dimensionless velocity and thermal

205 slip parameters and for simplicity we omitted asterisk Now

206

E1@3

@x3þ E2 @3

@x@t2þ E3 @2

@t@x

g

¼ R1 d@u

@tþ du

@u

@xþv@u@y

þ d@Sxx

@x þ

@Sxy

@y

H2ðu mvÞ

1þ m2

þ R1

Fr

208

209

g¼ 1 þ½ sin 2pðx tÞ: ð27Þ

211

212 Here d is the wave number, E1; E2and E3the elasticity

parame-213 ters,the amplitude ratio, R1the Reynolds number, Pr the Prandtl

214 number, H the Hartman number, Ec the Eckert number; Br the

215 Brinkman number and Fr the Froude number These definitions are

216

d¼ d=k; E1¼ sd

3 1

k31lc; E2¼m1cd31

k31lc ; E3¼dd

3 1

k21l;

¼ a=d; R1¼qcd1

l ; Pr ¼lCp=j; H ¼ B0d ffiffiffiffiffiffiffiffiffi

r=l

p

;

Ec¼ c2=CpT0; Br ¼ EcPr; Fr ¼ cffiffiffiffiffiffi

gd

p :

218

219 Defining the stream function wðx; y; tÞ by

220

u¼@w

@y; v¼ d@w

222

223 the continuity equation(20)is identically satisfied Note that the

224 lubrication process remain useful for the chyme transport in small

225 intestine[36] Also Lew et al.[3]mentioned that Reynold number

226

in intestine is small Moreover the state of intrauterine fluid flow

227 due to myometrial contractions is a peristaltic type fluid motion

228

in a cavity The sagittal cross section of the uterus indicates a

nar-229 row channel bounded by two fairly parallel walls[37] Thus large

230 wavelength and small Reynolds number yield

231

@p@xþ@Sxy

@y

H2

1þ m2

@w

@y

þ R1 Fr

ð Þ2sina¼ 0; ð29Þ

233 234

@p

237

@2 h

@y2þ BrSxy@2

w

@y2þ H

2

1þ m2Br @w

@y

2

239 240

243

h b1

@h

246

E1 @3

@x3þ E2 @3

@x@t2þ E3 @2

@t@x

g

¼@Sxy

@y

H2

1þ m2

@w

@yþ

R1 Fr

ð Þ2sinaat y¼ g: ð34Þ

248 249 Combining the dimensionless equations(29)and(30) we obtain

250

resulting form

251

@2

Sxy

@y2 H

2

1þ m2

@2 w

@y2

!

253 254 Dimensionless form of extra stress tensor for Powell–Eyring

255 fluid is

256

Sxy¼a1wyyAða1 1Þ

3 wyy

3

259 witha1¼ 1 þ M; M ¼ 1

l bc 1and A¼1 c

1 d

2

and asterisks have been

260 suppressed for simplicity Viscous fluid model is obtained for

261

a1¼ 1

262 Heat transfer coefficient is defined as

263

266 Solution methodology

267 Here we used the perturbation technique for small parameter A

268

to solve the non-linear governing equations Expand the following

269 flow quantities as:

270

w¼w0þ Aw1þ ;

Syx¼S0yxþ AS1yxþ ;

h¼h0þ Ah1þ ;

273 Solving the resulting zeroth and first order systems through

274 Eqs.(31)–(37)we have the solutions as follows

275

w0¼ C3þ C4yþ

e

Hy

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þm2

ð Þa1

p

C2þ C1e

2Hy

1þm2

ð Þa1 p 0

@

1

A 1 þ m 2

a1

Trang 4

w1¼ e

3Hy

1þm2

ð Þa1

p

24H2a1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ m2

ð Þa1

4Hy

1þm2

ð Þa1

p

1þ m2

a1ðA13 A14ð2Hy þ A15ÞÞ þe

6Hy

1þm2

ð Þa1

p

A12 6e

2Hy

1þm2

ð Þa1

p

1þ m2

a1ðA16 A17ð2Hy þ A18ÞÞ

0 B

@

1 C

h0¼ L1þ L2y

Bre

2Hy

1þm2

ð Þa1

2

4H4y2e

2Hy

1þm2

ð Þa 1

p

þ C2

2þ C2

1e

4Hy

1þm2

ð Þa 1

p 0

@

1

AB11 þ4C4e

Hy

1þm2

ð Þa1

p

C2þ C1e

2Hy

1þm2

ð Þa1 p 0

@

1

AB12

0 B B B

@

1 C C C A

ð40Þ

h1¼ K1þ K2y Bra1

72H 1ðð þ m2Þa1Þ3

C11e

3Hy

1þm2

ð Þa1

p

C12e

3Hy

1þm2

ð Þa1 p

þC13e

4Hy

1þm2

ð Þa1

p

þ C14e

4Hy

1þm2

ð Þa1 p þ72y2C15þ 36e

Hy

1þm2

ð Þa1

p

1þ m2

yC16 C17

þ36e

Hy

1þm2

ð Þa1

p

1þ m2

a1ðyC18þ C19Þ þ3C1a11þ m22

e

2Hy

1þm2

ð Þa 1

p

yD11þD 12

H

3C2a11þ m22

e

2Hy

1þm2

ð Þa1

p

yD 13 þD 14

H

0 B B B B B B B B B B B

1 C C C C C C C C C C C

Expression of heat transfer coefficient is

Z0¼gx

L2þ

Bre

2H g ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þm2

ð Þa1

p E11 C22þ C2

1e

1þm2

ð Þa1 p 0

@

1

A þ C2

4H4g2e

1þm2

ð Þa1 p

þ4E12C4e

H g ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þm2

ð Þa1

p

C2þ C1e

1þm2

ð Þa1 p 0

@

1 A

0 B B B

@

1 C C C A

H 1ðþm 2Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið1 þm 2Þa 1

p

Bre

2H g

1þm2

ð Þa1

p E13e

1þm2

ð Þa1

p

þ 4E14C4e

1þm2

ð Þa1

p

C2þ C1e

1þm2

ð Þa1 p 0

@

1 A

þ2C2

4e

1þm2

ð Þa 1

p

H4gþ E15e

1þm2

ð Þa 1

p

þ e

1þm2

ð Þa 1

p

g2E16

0 B B

@

1 C C A 2H 2ð1þm 2Þ

0

B

@

1 C C C C C C C C C C C A

Z1¼gx K2 Bra1

72H 1ðð þ m2Þa1Þ3

F11e

1þm2

ð Þa1

p

F12e

1þm2

ð Þa1

p

þ F13e

1þm2

ð Þa1 p

þF14e

1þm2

ð Þa1

p

þ F15e

1þm2

ð Þa1

p

þ F16e

1þm2

ð Þa1 p

þF17e

1þm2

ð Þa1

p

þ F18e

1þm2

ð Þa1

p

þ F19g

þ36Hð1 þ m2Þe

1þm2

ð Þa1

p

g Gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi11 þG 12

1 þm 2

ð Þa 1

p

!

þ6C1a1ð1 þ m2Þ2

e

1þm2

ð Þa 1

p

g Gffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi13þG 14

1þm 2

ð Þa 1

p

!

þ6C2a1ð1 þ m2Þ2

e

1þm2

ð Þa1

p

1 þm 2

ð Þa 1

p

!

36 1 þ m 2

He

1þm2

ð Þa1

p

1 þm 2

ð Þa 1

p

!

0 B B B B B B B B B B B B B B B

@

1 C C C C C C C C C C C C C C C A

0

B

@

1 C C C C C C C C C C C C C C C A

Trang 5

Fig 2d Effect via m on u when ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E 1 ¼ 0:03; E 2 ¼ 0:02;

E 3 ¼ 0:01; Br ¼ 2:0;a¼ p

4 ; H ¼ 0:5; Fr ¼ 0:08; A ¼ 0:01; M ¼ 0:1; R 1 ¼ 0:2 and

c¼ 0:01.

Fig 2e Effect via Fr on u when ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E 1 ¼ 0:03; E 2 ¼ 0:02;

E 3 ¼ 0:01; Br ¼ 2:0;a¼ p

4 ; H ¼ 0:5; M ¼ 1:0; A ¼ 0:1; m ¼ 0:2; R 1 ¼ 0:2 andc¼ 0:01.

Fig 2g Effect via angle inclinationaon u when¼ 0:2; x ¼ 0:2; t ¼ 0:1; E 1 ¼ 0:03;

E 2 ¼ 0:02; E 3 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; R 1 ¼ 0:2 andc¼ 0:01.

Fig 2a Effect via wall parameters on u when¼ 0:2; x ¼ 0:2; t ¼ 0:1; Br ¼ 2:0;

a¼ p

4 ; H ¼ 0:5; Fr ¼ 0:8; A ¼ 0:1; m ¼ 0:2; M ¼ 1; R 1 ¼ 0:1 andc¼ 0:1.

Fig 2b Effect via a1 on u when ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E 1 ¼ 0:03; E 2 ¼ 0:02;

E 3 ¼ 0:01; Br ¼ 2:0;a¼ p

4 ; H ¼ 0:5; Fr ¼ 0:8; A ¼ 0:1; m ¼ 0:2; R 1 ¼ 0:2 andc¼ 0:01.

Fig 2c Effect via A on u when ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E 1 ¼ 0:03; E 2 ¼ 0:02;

E 3 ¼ 0:01; Br ¼ 2:0;a¼ p

6 ; H ¼ 0:5; Fr ¼ 0:8; M ¼ 1:0; m ¼ 0:2; R 1 ¼ 0:2 andc¼ 0:01.

Fig 2f Effect via H on u when ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E 1 ¼ 0:03; E 2 ¼ 0:02;

E 3 ¼ 0:01; Br ¼ 2:0;a¼ p

4 ; M ¼ 1:0; Fr ¼ 0:5; A ¼ 0:1; m ¼ 0:2; R 1 ¼ 0:2 andc¼ 0:01.

Fig 2h Effect via R 1 on u when ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E 1 ¼ 0:03; E 2 ¼ 0:02;

E 3 ¼ 0:01; Br ¼ 2:0;a¼ p

6 ; H ¼ 0:5; Fr ¼ 0:8; A ¼ 0:1; m ¼ 0:2; M ¼ 1:0 andc¼ 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

Trang 6

278 Here the values of Ai;s ði¼ 1; 2; ; 8Þ; Bi;s ði¼ 1; 2Þ; Ci;s

279

i¼ 1; 2; ; 7

ð Þ; Di;s ði¼ 1; 2; ; 4Þ; Ei;s ði¼ 1; 2; ; 6ÞFi;s

280

i¼ 1; 2; ; 9

ð Þ and Gi;s ði¼ 1; 2; ; 8Þ can be calculated

alge-281 braically using MATHEMATICA

282 Analysis

283 The purpose of this section is to analyze the behavior of

differ-284 ent embedded parameters on the velocity u, temperature h and

285 heat transfer coefficient Z Trapping phenomenon is also examined

286 via graphs

Fig 3b Effect via H on h when ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E 1 ¼ 0:3; E 2 ¼ 0:2;

E 3 ¼ 0:1; Br ¼ 2:0;a¼ p

4 ; M ¼ 1:0; Fr ¼ 0:8; A ¼ 0:1; m ¼ 0:2; b 1 ¼ 0:01; R 1 ¼ 0:1 and

c¼ 0:01.

Fig 3a Effect via wall parameters on h when ¼ 0:2; x ¼ 0:2; t ¼ 0:1; Br ¼ 2:0;

a¼ p

4 ; H ¼ 0:5; Fr ¼ 0:8; A ¼ 0:1; m ¼ 0:2; M ¼ 1; R 1 ¼ 0:1 b 1 ¼ 0:01 andc¼ 0:01.

Fig 3d Effect via angle inclinationaon h when¼ 0:2; x ¼ 0:2; t ¼ 0:1; E 1 ¼ 0:4;

E 2 ¼ 0:2; E 3 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; R 1 ¼ 0:1;

b 1 ¼ 0:01 andc¼ 0:01.

Fig 3e Effect via Br on h when¼ 0:2; x ¼ 0:2; t ¼ 0:1; E 1 ¼ 0:4; E 2 ¼ 0:2; E 3 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; R 1 ¼ 0:1; b 1 ¼ 0:01;a¼ p

4 and

c¼ 0:01.

Fig 3c Effect via m on h when¼ 0:2; x ¼ 0:2; t ¼ 0:1; E 1 ¼ 0:4; E 2 ¼ 0:2; E 3 ¼ 0:3;

Br ¼ 2:0;a¼ p

4 ; H ¼ 0:5; Fr ¼ 0:8; A ¼ 0:1; M ¼ 1:0; R 1 ¼ 0:1; b 1 ¼ 0:01 andc¼ 0:01.

Fig 3f Effect via Fr on h when ¼ 0:2; x ¼ 0:2; t ¼ 0:1; E 1 ¼ 0:03; E 2 ¼ 0:02;

E 3 ¼ 0:01; Br ¼ 4:0;a¼ p

4 ; H ¼ 0:2; M ¼ 1:0; A ¼ 0:01; m ¼ 0:2; b 1 ¼ 0:01; R 1 ¼ 0:1 andc¼ 0:01.

Fig 3g Effect via R 1 on h when¼ 0:2; x ¼ 0:2; t ¼ 0:1; E 1 ¼ 0:3; E 2 ¼ 0:2; E 3 ¼ 0:1;

Br ¼ 4:0;a¼ p

4 ; H ¼ 0:2; Fr ¼ 0:8; A ¼ 0:1; m ¼ 0:2; M ¼ 1:0; b 1 ¼ 0:01 andc¼ 0:01.

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Fig 4c Effect via Br on Z when¼ 0:2; t ¼ 0:1; E 1 ¼ 0:03; E 2 ¼ 0:02; E 3 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; R 1 ¼ 0:1; b 1 ¼ 0:01;a¼ p

6 and

c¼ 0:01.

Fig 3h Effect via A on h when¼ 0:2; x ¼ 0:2; t ¼ 0:1; E 1 ¼ 0:3; E 2 ¼ 0:2; E 3 ¼ 0:1;

Br ¼ 4:0;a¼ p

4 ; H ¼ 0:2; Fr ¼ 0:8; M ¼ 1:0; m ¼ 0:2; R 1 ¼ 0:2; b 1 ¼ 0:01 andc¼ 0:01.

Fig 4d Effect via wall parameters on Z when ¼ 0:2; t ¼ 0:1; Br ¼ 2:0;a¼ p

6 ;

H ¼ 0:5; Fr ¼ 0:8; A ¼ 0:2; m ¼ 0:2; M ¼ 1; R 1 ¼ 0:1 b 1 ¼ 0:01 andc¼ 0:01.

Fig 4g Effect via A on Z when ¼ 0:2; t ¼ 0:1; E 1 ¼ 0:03; E 2 ¼ 0:02; E 3 ¼ 0:01;

Br ¼ 4:0;a¼ p

3 ; H ¼ 0:5; Fr ¼ 0:8; M ¼ 1:0; m ¼ 0:2; R 1 ¼ 0:1; b 1 ¼ 0:01 andc¼ 0:01.

Fig 4e Effect via angle inclinationaon Z when¼ 0:2; t ¼ 0:1; E 1 ¼ 0:03; E 2 ¼ 0:02;

E 3 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; R 1 ¼ 0:1; b 1 ¼ 0:01 and

c¼ 0:01.

Fig 4a Effect via H on Z when ¼ 0:2; t ¼ 0:1; E 1 ¼ 0:03; E 2 ¼ 0:02; E 3 ¼ 0:01;

Br ¼ 2:0;a¼ p

6 ; M ¼ 1:0; Fr ¼ 0:8; A ¼ 0:2; m ¼ 0:2; b 1 ¼ 0:01; R 1 ¼ 0:1 andc¼ 0:01.

Fig 4b Effect via m on Z when¼ 0:2; t ¼ 0:1; E 1 ¼ 0:03; E 2 ¼ 0:02; E 3 ¼ 0:01;

Br ¼ 2:0;a¼ p

6 ; H ¼ 0:5; Fr ¼ 0:8; A ¼ 0:2; M ¼ 1:0; R 1 ¼ 0:1; b 1 ¼ 0:01 andc¼ 0:01. Fig 4f Effect via Fr on Z when ¼ 0:2; t ¼ 0:1; E 1 ¼ 0:03; E 2 ¼ 0:02; E 3 ¼ 0:01;

Br ¼ 2:0;a¼ p

3 ; H ¼ 0:5; M ¼ 1:0; A ¼ 0:2; m ¼ 0:2; b 1 ¼ 0:01; R 1 ¼ 0:1 andc¼ 0:01.

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287 Velocity profile

288 This subsection presents the effect of various parameters on

289 velocity distribution particularly the wall parameters Eð 1; E2; E3Þ,

290 material parameters of fluida1and A, the Hall parameter m, the

291 Froude number Fr, the Hartman number H, inclination angle a

292 and Reynolds number R1.Fig 2aexhibits that for large values of

293 E1and E2the velocity increases Physically increasing values of E1

294 and E2reduce the viscosity which yield more Wall parameter E3

295 shows decrease in velocity for increasing values of E3 Its reason

296 is that for large values of E3 viscous damping enhances due to

297 which velocity decreases Figs 2band 2cdepict the behavior of

298 fluid parameters on u Reverse results corresponding to larger

val-299 ues ofa1and A are observed i.e higher values ofa1gives reduction

300

in velocity while larger A favor the velocity u The fact behind this

301 behavior is that large A causes increase in kinetic energy of

parti-302 cles which results in increased velocity Fig 2d indicates the

303 increasing behavior when Hall parameter m increases Effect of

304 Froude number Fr on velocity profile shows decreasing impact

305 (see inFig 2e) It is noticed fromFig 2fthat larger values of

Hart-306 man number H decreases the velocity Physically this concept

307 holds because Lorentz force reduces the velocity Fig 2gshows

308 the increasing behavior of inclination anglea towards velocity

309 Increase in inclination angleacauses fluid to move with greater

Fig 5 Effect via H on w for E 1 ¼ 0:2; E 2 ¼ 0:2; E 3 ¼ 0:3;a¼ p

4 ; M ¼ 1:0;¼ 0:2; t ¼ 0:0; A ¼ 0:1;c¼ 0:01; Fr ¼ 0:8; m ¼ 0:2; R 1 ¼ 0:2 when ðaÞ :H ¼ 0:5 and ðbÞ :H ¼ 2:5.

Fig 6 Effect via A on w for E 1 ¼ 0:2; E 2 ¼ 0:2; E 3 ¼ 0:3;a¼ p

4 ; M ¼ 1:0;¼ 0:2; t ¼ 0:0; H ¼ 0:5;c¼ 0:01; Fr ¼ 0:8; m ¼ 0:2; R 1 ¼ 0:2 when ðaÞ :A ¼ 0:1 and ðbÞ :A ¼ 0:5.

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310 velocity due to increased effect of gravity Fig 2henhances the

311 velocity profile when R1is increased As R1is the ratio of inertial

312 forces to the viscous forces thus decrease in viscosity enhances

313 R1which in turn increases velocity

314 Temperature profile

315 Figs.3a–3hmanifest the impact of different emerging

parame-316 ters on temperature distribution h It is observed that h attains

317 maximum value near the centre of the channel.Fig 3adisplays

318 that temperature profile increases for increasing values of both

319 E1 and E2while it decreases for E3.Fig 3b discloses decrease in

320 temperature profile when Hartman number H is increased For

321 large values of Hall parameter m temperature profile enhances

322 (see in Fig 3c) The ascending values of inclination anglea on

323 temperature profile are depicted inFig 3d As growing values of

324 inclination angle acause increase in temperature profile.Fig 3e

325 illustrates that for higher values of Brinkman number Br the

tem-326 perature profile is enhanced The reason behind this effect is the

327 higher viscous dissipation which generates more heat and hence

328 causing rise in temperature occurs.Fig 3findicates that by

increas-329 ing Fr temperature profile decreases.Fig 3gensures that when we

330 increase Reynolds number R1then temperature enhances.Fig 3h

331 shows that for ascending values of A the temperature profile

Fig 7 Effect via m on w for E 1 ¼ 0:2; E 2 ¼ 0:2; E 3 ¼ 0:3;a¼ p

4 ; M ¼ 1:0;¼ 0:2; t ¼ 0:0; A ¼ 0:1;c¼ 0:01; Fr ¼ 0:8; H ¼ 0:5; R 1 ¼ 0:2 when ðaÞ :m ¼ 0:5 and ðbÞ :m ¼ 1:5.

Fig 8 Effect viaaon w for E 1 ¼ 0:7; E 2 ¼ 0:2; E 3 ¼ 0:1; m ¼ 0:2; M ¼ 1:0;¼ 0:2; t ¼ 0:0; A ¼ 0:1;c¼ 0:01; Fr ¼ 0:8; H ¼ 0:5; R 1 ¼ 0:2 when ðaÞ :a¼ p

6 and ðbÞ :a¼ p

3

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332 increases It is noticed that velocity and temperature show similar

333 behavior As temperature is defined as average kinetic energy of

334 molecules Hence increased/decreased velocity causes temperature

335 to get increased/decreased

336 Heat transfer coefficient

337 The purpose of this subsection is to investigate the behavior of

338 various parameters on the heat transfer coefficient Z From

339 Figs.4a–4git is observed that magnitude of heat transfer

coeffi-340 cient shows oscillatory behavior for the involved parameters due

341 to sinusoidal waves travelling along the walls Hartman number

342 H provides resistance to heat transfer and thus heat transfer rate

343

Z reduces due to the existence of magnetic field (see Fig 4a)

344

Fig 4bpotrays that for ascending values of m the heat transfer rate

345

Z decreases From Figs.4c and4d it is noticed that Z decreases

346 when Brinkman number Br and wall parameters Eð 1; E2; E3Þ are

347 increased Fig 4e displays that Z reduces when the inclination

348

alarger Froude number Fr enhances the heat transfer distribution

349

Z (seeFig 4f) The results inFig 4gillustrates that an increase of A

350 causes reduction in Z

351 Trapping

352 Formation of circular bolus by internally splitting of streamlines

353

is known as trapping The bolus moves forward through peristaltic

Fig 9 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;c¼ 0:01; Fr ¼ 0:8; H ¼ 0:5;a¼ p

4 ; R 1 ¼ 0:2 when ðaÞ :E 1 ¼ 0:1; E 2 ¼ 0:3; E 3 ¼ 0:1 ðbÞ :E 1 ¼ 0:4; E 2 ¼ 0:3; E 3 ¼ 0:1 ðcÞ : E 1 ¼ 0:1; E 2 ¼ 0:4; E 3 ¼ 0:1 ðdÞ :E 1 ¼ 0:1; E 2 ¼ 0:3; E 3 ¼ 0:02.

Tiêu đề	Hall and Joule Heating Effects on Peristaltic Flow of Powell–Eyring Liquid in an Inclined Symmetric Channel
Tác giả	T. Hayat, Naseema Aslam, M. Rafiq, Fuad E. Alsaadi
Trường học	Quaid-I-Azam University
Chuyên ngành	Physics
Thể loại	Research Article
Năm xuất bản	2017
Thành phố	Islamabad

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