magnetohydrodynamic flow of casson fluid over a stretching cylinder

[6] explored viscous dissipation 54 impact in Casson nanofluid flow in presence of variable conductiv-55 ity and mixed convection.. 225 Conclusions 226 Here Joule heating and magnetohyd

4

5

6 M Tamoora, M Waqasb, M Ijaz Khanb,⇑ , Ahmed Alsaedic, T Hayatb,c

Department of Basic Sciences, University of Engineering and Technology, Taxila 47050, Pakistan

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

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O Box 80257, Jeddah

10 21589, Saudi Arabia

11

14 Article history:

15 Received 4 October 2016

16 Received in revised form 1 January 2017

17 Accepted 2 January 2017

18 Available online xxxx

19 Keywords:

20 Viscous dissipation

21 Casson liquid

22 Joule heating

23 Newtonian heating

24 Stretching cylinder

25

2 6

a b s t r a c t

27 Here the Newtonian heating characteristics in MHD flow of Casson liquid induced by stretched cylinder

28 moving with linear velocity is addressed Consideration of dissipation and Joule heating characterizes the

29 process of heat transfer Applied electric and induced magnetic fields are not considered Magnetic

30 Reynolds number is low The convenient transformation yields nonlinear problems which are solved

31 for the convergent solutions Convergence region is determined for the acquired solutions Parameters

32 highlighting for velocity and temperature are graphyically discussed Numerical values of skin friction

33 and Nusselt number are also presented in the tabular form Present analysis reveals that velocity and

34 thermal fields have reverse behavior for larger Hartman number Moreover curvature parameter has

sim-35 ilar influence on the velocity, temperature, skin friction and Nusselt number

36

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

37

creativecommons.org/licenses/by-nc-nd/4.0/)

38 39

40 Introduction

41 Several physiological materials can be identified as viscoplastic

42 materials which are the sub-classes of rheological materials having

43 both elastic and viscous features Several investigators have

devel-44 oped models in this direction [1–3] In this regard Casson liquid

45 model has shown relatively advantageous This model behaves as

46 solid when the shear stress is less than the yield stress and it starts

47 to deform when shear stress becomes greater than the yield stress.

48 Several researchers studied this model under distinct physical

49 aspects For instance Dash et al [4] reported the characteristics

50 of Casson liquid under yield stress via homogeneous porous

med-51 ium bounded by circular tube Effectiveness of variable properties

52 in non-Darcy dissipative flow of Casson liquid is presented by

Ani-53 masaun et al [5] Hayat et al [6] explored viscous dissipation

54 impact in Casson nanofluid flow in presence of variable

conductiv-55 ity and mixed convection Characteristics of gyrotactic

microorgan-56 isms in magneto Casson nanomaterial is addressed by Raju et al.

57 [7] Raju and Sandeep [8] investigated the effects of ferrous

58 nanoparticles considering Casson model.

59 The magnetic field phenomenon in convection flows is topic of

60 advancement in technology and industry Specific examples

61 include collection of solar energy, insulation of nuclear reactor,

fur-62 naces, cooling of electronic chips and devices, crystal growth in

flu-63

ids, drying process and several others In electrically conducting

64

liquid, the Lorentz force overcomes the convective currents via

65

reduction in liquid velocities [9] Hence the involvement of

exter-66

nal magnetic field can be implemented to control the mechanism

67

in material industry [10–20] Furthermore the Joule heating can

68

generate enhancement in temperature and also yield temperature

69

gradient Moreover it has remarkable influence in nuclear

engi-70

neering and geophysical stream [21–25]

71

Viscous dissipation characterizes the degeneration of

mechani-72

cal energy into the thermal energy Such phenomenon transpires in

73

all the flow systems However for different flow configurations, the

74

characteristics of viscous dissipation is often neglected It is

mean-75

ingful just for the systems having larger velocity and velocity

gra-76

dients respectively It is for this reason that the viscous dissipation

77

is introduced in the present study Simultaneous characteristics of

78

viscous dissipation, Dufour-Soret and thermophoresis in radiative

79

viscous material flow towards isothermal wedge is explored by

80

Pal and Mondal [26] Zaib and Shafie [27] examined viscous

dissi-81

pation aspect in chemically reacting stratified stretched flow of

vis-82

cous liquid considering Hall current Hayat et al [28,29]

83

scrutinized the effectiveness of viscous dissipation and MHD

84

effects considering Jeffrey and second grade liquid models.

85

Three-dimensional dissipative stretched flow of radiative

Powell-86

Eyring nanomaterial is addressed by Mahanthesh et al [30]

87

Our prime theme here is to report the magnetohydrodynamic

88

(MHD) convective flow of Casson fluid Joule heating and

dissipa-89

tion are considered to predict the characteristics of heat transfer.

http://dx.doi.org/10.1016/j.rinp.2017.01.005

2211-3797/Ó 2017 Published by Elsevier B.V

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

⇑ Corresponding author

E-mail address:mikhan@math.qau.edu.pk(M.I Khan)

Results in Physics xxx (2017) xxx–xxx

Contents lists available at ScienceDirect

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

96 Here steady magnetohydrodynamic (MHD) flow of Casson

liq-97 uid is modeled Stretching of cylinder creates the flow Joule

heat-98 ing and dissipation effects are retained We imposed the

99 Newtonian heating condition at the stretching cylinder The flow

100 is anticipated in axial (x) direction whereas radial direction is

per-101 pendicular to x Entire treatment is deliberated via assumptions of

102 boundary layer Cylinder has linear stretching velocity Under

103 these hypothesis, 2D flow expressions are [6]

104

@ðruÞ

@x þ

@ðr v Þ

106

107

u @u

@x þ v @u @r ¼ m 1 þ 1 b

@2u

@r2þ 1 r

@u

@r

!

 r B2

109

110

u @T

@x þ v @T @r

¼ q k cp

@2T

@r2þ 1 r

@T

@r

!

þ lB 1 þ 1 b

@u

@r

2

þ r B20u2

112

113 The boundary conditions are

114

uðx; rÞ ¼ uwðxÞ ¼u0x

l ; v ðx; rÞ ¼ 0;@T

@r¼ hsTatr ¼ R;

116

117 In the aforestated expressions u and v are the velocity

compo-118 nents in the x and r directions, b the Casson fluid parameter,

119 uwðxÞ the stretching velocity, m the kinematic viscosity, q the

den-120 sity of fluid, cpthe specific heat, r the constant electrical

conduc-121 tivity, B0 the applied magnetic field, lB the plastic dynamic

122 viscosity, hs the heat transfer coefficient and concentration

expo-123 nent, T ð ; T1Þ the fluid and ambient temperatures respectively.

124 Utilizing suitable transformations

125

g ¼

ffiffiffiffiffi

u0

m l

r

r2 R2

2R

!

;u ¼ u0x

l f

0ð g Þ; v ¼  R

r

ffiffiffiffiffiffiffiffi

u0m

l

r

fð g Þ;hð g Þ ¼ T  T1

T1 ; ð5Þ

127

128 Eq (1) is identically satisfied while Eqs (2)–(4) have the forms:

129

1 þ 1 b

1 þ 2 ag

ð Þf000þ 2 c f00

þ ff00 f02 Ha2

f0¼ 0; ð6Þ

131

132

1 þ 2 ag

ð Þh00þ 2 a h0þ Prf h0þ PrEc 1 þ 2 ð ag Þ 1 þ 1 b

f002

þ PrEcHa2

134

135

f ¼ 0; f0¼ 1; h0ð0Þ ¼  c ð 1 þ h 0 ð Þ Þ at g ¼ 0;

137

138

140

141 where b the Casson fluid parameter, a the curvature parameter, Ha

142 the Hartman number, Pr the Prandtl number, Ec the Eckert number

143 and c the conjugate parameter of Newtonian heating These are

144 defined as

145

a ¼

ffiffiffiffiffiffiffiffiffiffi

m l

u0R2

s

;Ha2¼ r B20l

q u0;Pr ¼ l cp

k ;hs

ffiffiffiffiffi

v l

u0

s

;Ec ¼ u2w

cpT1; c ¼ hs

ffiffiffiffiffi

m l

u0

s 0

@

1 A:

ð9Þ 147 148

Skin friction 1CfRe1=2x 

and Nusselt number  NuRe1=2x 

are

149

expressed as

Fig 1.h-Curves for f and h

Fig 2 Impact of b on f0

Fig 3 Impact of Ha on f0

Please cite this article in press as: Tamoor M et al Magnetohydrodynamic flow of Casson fluid over a stretching cylinder Results Phys (2017),http://dx.doi

Cf¼ 2 sw

q u2

w

; Nux¼ xqw

152

154

sw¼ l 1 þ 1 b

@u

@r

 r¼R; qw¼ k @T @r



156

158

1

2 CfRe

1=2

x ¼ 1 þ 1 b

f00ð Þ; NuRe 0 1=2

161

where Rex¼u 0 x 2

vl is the local Reynolds number

162

Convergence analysis of obtained results

163

The solutions of the expressions comprising in Eqs (6)–(8) are

164

determined via homotopic approach The homotopic technique

Fig 4 Impact ofaon f0

Fig 5 Impact of Ha on h

Fig 6 Impact of Pr on h

Fig 7 Impact ofcon h

Fig 8 Impact of Ec on h

Fig 9 Impact ofaon h

165 yields a straight forward approach in order to control and adjust

166 the rate of convergence The parameters hf and hhneed the

appro-167 priate values for such objective No doubt these auxiliary

parame-168 ters play a vital role for the convergence of obtained solutions The

169 h-curves have been developed at 15thorder of approximations to

170 acquire valid ranges of these parameters (see Fig 1 ) It is seen that

171 the permissible values of these parameters are 1:45 6 hf 6 0:40

172 and 1:40 6 hh6 0:50.

173 Results and discussion

174 The behavior of physical variables on the velocity f0ð Þ and ther- g

175 mal fields h ð Þ is interpreted here g Figs 2–9 certify the roles of

176 dimensionless Casson fluid parameter b ð Þ, Hartman number Ha ð Þ,

177 curvature parameter ð Þ, Prandtl number Pr a ð Þ, conjugate parameter

178 ð Þ and Eckert number Ec c ð Þ Fig 2 elucidates the behavior of b on

179 f0ð Þ Clearly larger b reduced f g 0ð Þ and thickness of boundary layer g

180 Features of Ha on f0ð Þis interpreted in g Fig 3 It is of interest to note

181 that f0ð Þ decays via larger Ha whereas thickness of boundary layer g

182 reduces Since Lorentz force increments causes much resistance in

183 the fluid flow and consequently f0ð Þ decays The characteristics of g

184 a is illustrated in Fig 4 It is reported that velocity and thickness of

193

increment in h ð Þ Variation of Pr on h g ð Þ is disclosed in g Fig 6 It

194

is explored that h ð Þ decays via larger Pr As thermal diffusivity g

195

reduces with an increase in Prandtl number so h ð Þ decreases g

196

Fig 7 signifies the impact of c on h ð Þ As expected h g ð Þ and its g

197

associated thickness of thermal layer increment for larger c

Coef-198

ficient of heat transfer h ð Þ is enhanced via largers c Thus more heat

199

is transferred from heated surface of the cylinder to cooled surface

200

of liquid Consequently h ð Þ and corresponding thickness of bound- g

201

ary layer augment The role of Ec on h ð Þ is displayed through g

202

Fig 8 As expected both h ð Þ and related thickness of thermal layer g

203

augment through larger Ec Physically for larger Ec the liquid

par-204

ticles become more active because of storage of energy This fact

205

leads to higher temperature Fig 9 reveals the features of a on

206

h ð Þ Here h g ð Þ decays by enhancing g a close to the stretched

sur-207

face It shows augmenting behavior when one moves away from

208

the surface Physically higher a increase the thickness of thermal

209

layer due to which the heat transport rate decays and h ð Þ g

210

increases.

211

Table 1 reports the convergence of velocity f  0ð Þ g  and

temper-212

ature ð h ð Þ g Þ Presented values scrutinized that 15th-order of

213

approximations are enough for f00ð Þ and h 0 0ð Þ respectively 0 Table 2

214

is calculated to determine the coefficient of skin friction 1CfRe1=2x 

215

for distinct values of physical variables b ; a and Ha Here we

216

observed that skin friction 1CfRe1=2x 

shows increasing behavior

217

for larger a and Ha However opposite situation is noticed for

218

higher b Impacts of a ; Ha; Pr; Ec and c on Nusselt number

219

NuRe1=2x

are presented through Table 3 Clearly Nusselt number

220

NuRe1=2x

illustrates boosted behavior for increasing values of

221

a ; Pr and c whereas it decays when Ha and Ec are enhanced For

222

some limiting cases, comparison with previously available results

223

in the literature is made and an excellent agreement is achieved

224

(for detail see [40] and Table 4 ).

225

Conclusions

226

Here Joule heating and magnetohydrodynamic (MHD) effects in

227

dissipative stretched flow of Casson liquid are investigated The

228

following points are worthmentioning:

Table 3

Numerical values of Nusselt number NuRe1=2

x

for distinct values of the variablesa; Ha; Pr; Ec andcwhen b¼ 2:0

x

Table 2

Numerical values of skin friction 1CfRe1=2x

for distinct values of b;aand Ha when

Pr¼ 1:2; Ec ¼c¼ 0:3

Please cite this article in press as: Tamoor M et al Magnetohydrodynamic flow of Casson fluid over a stretching cylinder Results Phys (2017),http://dx.doi

229  Velocity field  f0ð Þ g  diminishes when b and Ha are

230 incremented.

231  Velocity and thermal fields describe insignificant behavior near

232 the cylinder surface and these quantities increment away from

233 the cylinder for larger curvature parameter ð Þ a

234  With the increment in Pr, the thermal field and related layer

235 thickness decay.

236  Larger Ec; c and Ha enlarge h ð Þ and associated thickness of g

237 boundary layer.

238  Skin friction 1CfRe1=2x 

decays via higher b however it

incre-239 ments when Ha and a are increased.

240  Nusselt number  NuRe1=2x 

and thermal field ð h ð Þ g Þ have

241 reverse behavior via larger Pr.

242  Presented model shows viscous fluid characteristics when

243 b ! 1.

244

245 Uncited reference

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Table 4

Comparative analysis forf00ð Þ with0 [40]for distinct values of Ha when b¼ 0 ¼a