slip flow by a variable thickness rotating disk subject to magnetohydrodynamics

5 6 7 Maria Imtiaza,⇑, Tasawar Hayatb,c, Ahmed Alsaedic, Saleem Asghard Department of Mathematics, Mohi-Ud-Din Islamic University, Nerian Sharif AJ&K, Pakistan Department of Mathematics,

5

6

7 Maria Imtiaza,⇑, Tasawar Hayatb,c, Ahmed Alsaedic, Saleem Asghard

Department of Mathematics, Mohi-Ud-Din Islamic University, Nerian Sharif AJ&K, Pakistan

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

10 c

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University 80203, Jeddah 21589, Saudi Arabia

Department of Mathematics, CIIT, Chak Shahzad Park Road, Islamabad, Pakistan

12

1 4 a r t i c l e i n f o

15 Article history:

16 Received 5 November 2016

17 Received in revised form 13 December 2016

18 Accepted 19 December 2016

19 Available online xxxx

20 Keywords:

21 Variable thickness

22 Rotating disk

23 Slip flow

24 Magnetohydrodynamic (MHD)

25

2 6

a b s t r a c t

27

Objective of the present study is to determine the characteristics of magnetohydrodynamic flow by a

28

rotating disk having variable thickness At the fluid–solid interface we consider slip velocity The

govern-29

ing nonlinear partial differential equations of the problem are converted into a system of nonlinear

ordi-30

nary differential equations Obtained series solutions of velocity are convergent Impact of embedded

31

parameters on fluid flow and skin friction coefficient is graphically presented It is observed that axial

32

and radial velocities have an opposite impact on the thickness coefficient of disk Also surface drag force

33

has a direct relationship with Hartman number

34

Ó 2016 Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license (http://

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

36 37

38 Introduction

39 Engineering and industrial applications due to rotating surfaces

40 have attracted the attention of scientists and researchers These

41 applications include air cleaning machines, gas turbines, medical

42 equipment, food processing technology, aerodynamical

engineer-43 ing and in electric-power generating systems Initial work on

rotat-44 ing disk flow was undertaken by Karman[1] Ordinary differential

45 equations were obtained from Navier–Stokes equations by using

46 the Von Karman transformation Subsequently different physical

47 problems were discussed by various researchers In the internal

48 cooling-air systems of most gas turbines disks rotating at different

49 speeds are found Heat transfer and flow associated with an

air-50 cooled turbine disk and an adjacent stationary casing were

mod-51 eled using the rotor–stator system Bachok et al [2] examined

52 nanofluid flow due to rotation of a permeable disk Similarity

solu-53 tion for flow and heat convection from a porous rotating disk was

54 considered by Kendoush[3] MHD slip flow with variable

proper-55 ties and entropy generation due to rotation of a permeable disk

56 were investigated by Rashidi et al [4] Turkyilmazoglu [5]

57 described heat transfer and flow due to rotation of disk with

58 nanoparticles Sheikholeslami et al.[6]examined nanofluid

spray-59 ing on an inclined rotating disk for cooling process Hayat et al.[7]

60

studied partial slip effects in MHD flow due to rotation of a disk

61

with nanoparticles Mustafa et al [8] analyzed MHD stagnation

62

point flow of a ferrofluid past a stretchable rotating disk Xun

63

et al.[9]studied flow and heat transfer of Ostwald-de Waele fluid

64

over a variable thickness rotating disk Chemical reaction effects in

65

flow of ferrofluid due to a rotating disk were presented by Hayat

66

et al.[10]

67

There are promising applications in metallurgy, polymer

indus-68

try, chemistry, engineering and physics due to fluid flow in the

69

presence of a magnetic field Desired characteristics of the end

pro-70

duct are attained in such applications by controlling the rate of

71

heat cooling The rate of cooling is controlled by magnetic field

72

for an electrically conducting fluid Physiological fluid applications

73

like blood pump machines and blood plasma are of great

impor-74

tance for MHD flow Flow configurations under different conditions

75

for MHD flows were considered by numerous researchers Effects

76

of velocity slip and temperature jump in a porous medium by a

77

shrinking surface with magnetohydrodynamic were studied by

78

Zheng et al [11] Analytical and numerical solutions for MHD

79

Falkner-Skan Maxwell fluid flow were presented by Abbasbandy

80

et al.[12] Turkyilmazoglu[13]examined MHD flow of viscoelastic

81

fluid over a stretching/shrinking surface in three dimensional

anal-82

ysis Sheikholeslami et al.[14]analyzed radiative flow of nanofluid

83

with magnetohydrodynamics A numerical study of radiative MHD

84

flow of Al2O3–water nanofluid has been studied by Sheikholeslami

85

et al.[15] Hayat et al.[16]studied Cattaneo–Christov heat flux in

http://dx.doi.org/10.1016/j.rinp.2016.12.021

2211-3797/Ó 2016 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: mi_qau@yahoo.com (M Imtiaz).

Results in Physics xxx (2017) xxx–xxx

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: Imtiaz M et al Slip flow by a variable thickness rotating disk subject to magnetohydrodynamics Results Phys (2017),

86 MHD flow of Oldroyd-B fluid with chemical reaction Li et al.[17]

87 examined MHD viscoelastic fluid flow and heat transfer by a

verti-88 cal stretching sheet with Cattaneo–Christov heat flux Makinde

89 et al.[18]presented MHD Couette-Poiseuille flow of variable

vis-90 cosity nanofluids in a rotating permeable channel with Hall effects

91 Numerical study of MHD nanofluid flow and heat transfer past a

92 bidirectional exponentially stretching sheet was considered by

93 Ahmad et al [19] Buoyancy effects on the three dimensional

94 MHD stagnation-point flow of a Newtonian fluid were examined

95 by Borrellia et al.[20]

96 The formation and use of microdevices remain a hotly debated

97 and challenging topic of research by scientists The small size as

98 well as high efficiency of microdevices-such as microsensors,

99 microvalves and micropumps are some of the advantages of using

100 microelectromechanical systems (MEMS) and

nanoelectromechan-101 ical systems (NEMS) Wall slip readily occurs for an array of

com-102 plex fluids such as emulsions, suspensions, foams and polymer

103 solutions Also fluids that exhibit boundary slip have important

104 technological applications, such as polishing of artificial heart

105 valves and internal cavities Many attempts addressing slip flow

106 have been presented to guarantee the performance of such devices

107 A number of models have been proposed for describing slip that

108 occurs at solid boundaries A brief description of these models

109 may be found in the work of Rao and Rajagopal[21] Buscall[22]

110 reported that the importance of studying wall slip has grown

sub-111 stantially Akbarinia et al [23] used microchannels to study

112 nanofluids heat transfer enhancement in non-slip and slip flow

113 regimes Micropolar fluid flow with velocity slip and heat

genera-114 tion (absorption) has been examined by Mahmoud and Waheed

115 [24] Velocity and thermal slip effects in nanofluid flow have been

116 analyzed by Khan et al.[25] Mukhopadhyay[26]examined

radia-117 tive flow over a permeable exponentially stretching sheet with slip

118 effects Inside a circular microchannel slip flow of alumina/water

119 nanofluid has been considered by Malvandi and Ganji[27]

Nonlin-120 ear thermal radiation and slip velocity in MHD three-dimensional

121 nanofluid flow have been studied by Hayat et al.[28] Slip flow

122 in a microchannel for nanoparticles using lattice Boltzman method

123 has been analyzed by Karimipour et al.[29] Effect of mass transfer

124 induced velocity slip on heat transfer of viscous gas flows over

125 stretching/shrinking sheet has been presented by Wu[30]

126 In the past much attention has been given to flow due to a

rotat-127 ing disk with negligible thickness Our main focus of the present

128 analysis is to study MHD flow of viscous fluid due to a rotating disk

129 with variable thickness Formulation and analysis is presented

130 when no slip does not remain valid The technique used for solving

131 the present problem is homotopy analysis method (HAM)[31–38]

132 Convergent series solutions are obtained Impacts of pertinent

133 parameters on axial, radial and tangential velocity components

134 and surface drag force are examined

135 Formulation

136 Consider steady, laminar and axisymmetric flow due to a disk

137 rotating with angular velocityXabout the z-axis The disk is also

138 stretched with velocity uw¼ ra1 where a1 is the stretching rate

139 constant We assume that the disk at z¼ a r

R 0þ 1

m

is not flat

140 where a is the disk thickness coefficient, R0is the feature radius

141 and m is the disk thickness index Slip flow regime is considered

142 for viscous fluid A magnetic field of strength B0is applied in the

143 z-direction Magnetic Reynolds number is assumed small and thus

144 induced magnetic field is neglected Electric field is taken absent

145 The governing equations are as follows:

146

@u

@rþ

u

148

149

u@u

@u

@zv2

r ¼m @2u

@r2þ1r@u@rru2þ@

2u

@z2

!

rB2u

152

u@v

@v

@zþ

uv

r ¼m @2v

@r2þ1 r

@v

@rv

r2þ@

2v

@z2

!

rB20v

155

u@w

@w

@z¼m @2w

@r2 þ1 r

@w

@2w

@z2

!

157

158

with boundary conditions

159

u¼ ra1þ k1 @u

@z; v¼ rXþ k2 @v

@z; w ¼ 0 at z ¼ a r

R 0þ 1

;

162

where u;vand w are velocity components in the direction of r;H

163

and z respectively,m denotes kinematic viscosity,rthe electrical

164

conductivity,qthe density and k1; k2are slip velocity coefficients

165

Generalized Von Karman transformations are

166

u¼ rXFðgÞ;v¼ rXGðgÞ; w ¼ R0X 1þr

R 0

m XR2 q

l

nþ1

HðgÞ

g¼ z

R 0 1þ r

R 0

m XR2 q

l

þ1

:

ð6Þ

168

169

Mass conservation law is identically satisfied andEqs (2)–(5)

170

become

171

174

Re

1n 1þn

1þ r

177

Re

1n 1þn

1þ r

180

with boundary conditions

181

HðaÞ ¼ 0; FðaÞ ¼ A þc1ð1þ rÞm

F0ðaÞ; GðaÞ

¼ 1 þc2ð1þ rÞm

184

wheree¼ r

R 0 þris a dimensionless constant, Re¼XR 2

0

m is the Reynolds

185

number, A¼a 1

Xis scaled stretching parameter, r¼ r

R 0is the

dimen-186

sionless radius,a¼ a

R 0

X R 2

0 q l

1 nþ1

is the dimensionless disk thickness

187

coefficient, c1¼k 1

R 0

X R 2

0 q l

1 nþ1 and c2¼k 2

R 0

X R 2

0 q l

1 nþ1 are velocity slip

188

parameters and M¼rB 2

0

q X is the Hartman number

189

We now consider

190

193

and thusEqs (7)–(10) are reduced to

194

197

Re1n1þnð1þ rÞ2m

f00 f2

200

Re1n1þnð1þ rÞ2m

203

hð0Þ ¼ 0; f ð0Þ ¼ A þc1ð1þ rÞm

f0ð0Þ; gð0Þ

206

Here prime denotes the derivative with respect of n and h; f and

207

g are axial, radial and tangential velocity profiles respectively

208

At the disk the shear stress in radial and tangential directions is

209

szrandszh

210

2 M Imtiaz et al / Results in Physics xxx (2017) xxx–xxx

@z

z ¼0¼lrX1R01þr



ð Þ1 XR20 q l

þ1

~f 0 ð0Þ

szh¼l@^v

@z

z ¼0¼lrX1R01þr



ð Þ1 XR20 q l

þ1

~g 0 ð0Þ

ð16Þ

212

213 Total shear stresssw is defined as

214

sw¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2

zrþs2

zh

q

216

217 Skin friction coefficients Cfxin dimensionless form is

218

CfRen1nþ1¼ swjz ¼0

qð ÞrX2¼1

rð1þ rÞ1
~f0ð0Þ2þ ~gð 0ð0ÞÞ2

220

221 Solutions procedure

222 Linear operators L1; L2and L3and initial guesses h0ðnÞ; f0ðnÞ and

223 g0ðnÞ are taken in the forms

224

226

227

1þc1ð1þ rÞmen;

229

230 subject to the properties

231

L1½c1 ¼ 0;

233

234

L2½c2enþ c3en ¼ 0;

236

237

239

240 in which ciði ¼ 1  5Þ are the constants

241 If embedding parameter is denoted by p2 ½0; 1 and nonzero

242 auxiliary parameters byhh; hf andhgthen the zeroth order

defor-243 mation problems are as follows:

244

ð1  pÞL1½^hðn; pÞ  h0ðnÞ ¼ phhNh½^hðn; pÞ; ^fðn; pÞ; ð22Þ

246

247

ð1  pÞL2½^fðn; pÞ  f0ðnÞ ¼ phfNf½^fðn; pÞ; ^hðn; pÞ; ^gðn; pÞ; ð23Þ

249

250

ð1  pÞL3½^gðn; pÞ  g0ðnÞ ¼ phgNg½^fðn; pÞ; ^hðn; pÞ; ^gðn; pÞ; ð24Þ

252

253

^hð0;pÞ ¼ 0;^fð0;pÞ ¼ A þc1ð1þ rÞm^f0ð0; pÞ; ^fð1; pÞ ! 0;

255

256

258

259 Nonlinear operators are

260

Nh¼ 2^fðn; pÞ þ meðn þaÞ@^fðn; pÞ

@^hðn; pÞ

262

263

Nf ¼ Re1n1þnð1þ rÞ2m@2^fðn; pÞ

@n2 þ ^fðn; pÞ
2 meðn þaÞ^fðn; pÞ

@^fðn; pÞ@n þ ^gðn; pÞð Þ2

 ^hðn; pÞ@^fðn; pÞ@n  M^fðn; pÞ; ð27Þ

265

266

Ng¼ Re1n1þnð1þ rÞ2m@2^gðn; pÞ

@n2  2^fðn; pÞ^gðn; pÞ  meðn

þaÞ^fðn; pÞ@^gðn; pÞ

@n  ^hðn; pÞ

@^gðn; pÞ

268

269 The mth order deformation problems are

270

L1 hmvmhm 1

¼ hhRh

273

L2fmvmfm1

¼ hfRf

276

L3gmvmgm1

¼ hgRg

279

Rh

m¼ 2fm1þ meðn þaÞf0m 1þ h0m 1; ð32Þ 281

282

Rf

m¼ Re1n1þnð1þ rÞ2m

f00m1þXm1

k ¼0

fm 1kfk meðn

m 1

k ¼0

fm1kf0kþX

m 1

k ¼0

gm1kgkX

m 1

k ¼0

hm 1kf0k Mfm1; ð33Þ

284 285

Rg

m¼ Re1n1þnð1þ rÞ2m

g00m1 2X

m 1 k¼0

fm 1kgk meðn

þaÞXm1

k ¼0

fm 1kg0kXm1

k ¼0

287 288

hmjn¼0¼ fmjn¼0c1ð1þ rÞm@fm

@n



n!0¼ fmjn!1¼ 0;

290 291

gmjn¼0c2ð1þ rÞm@gm

@n





n!0

293 294

297

The general solutionsðhm; fm; gmÞ comprising the special

solu-298

tionsðh

m; f

m; g

mÞ are

299

302

305

gmðnÞ ¼ g

308

where the constants ci(i¼ 1  5) are

309

c1¼ h

mð0Þ; c2¼ c4¼ 0;

1 þ c1 ð 1 þr  Þ m c1ð1þ rÞm@f 

m ðnÞ

@n 

n¼0 f

mð0Þ

;

1þ c2 ð 1þr  Þ m c2ð1þ rÞm@g 

m ðnÞ

@n 

n¼0 g

mð0Þ

:

ð38Þ

311

312

Homotopy solutions

313

For the solution of linear and nonlinear problems the homotopy

314

analysis method (HAM) is a powerful technique An embedding

315

auxiliary parameterh is involved in HAM which enlarges the

con-316

vergence area Valid ranges of these parameters are obtained by

317

plottingh-curves (seeFig 1) Permissible ranges ofhh; hf andhg

318

are 2:5 6 hf6 2; 1 6 hh6 0:1 and 2:4 6 hU6 0:3 Also

319

HAM solutions converge whenhh¼ hf¼ hg¼ 0:5 (seeTable 1)

320

Results

321

In this section the impact of axial, radial and tangential

compo-322

nents of velocity for different dimensionless parameters is

consid-323

ered Similarly effect of these parameters on skin friction

324

coefficient is examined in this section Effects of disk thickness

325

index m, Reynolds number Re, disk thickness coefficient a and

326

Hartman number M on axial velocity hðnÞ is observed inFigs (2–

M Imtiaz et al / Results in Physics xxx (2017) xxx–xxx 3

Please cite this article in press as: Imtiaz M et al Slip flow by a variable thickness rotating disk subject to magnetohydrodynamics Results Phys (2017),

327 5) Fig 2exhibits that an increase in disk thickness index m causes

328 the magnitude of axial velocity to decrease.Fig 3depicts that

mag-329 nitude of axial velocity is an increasing function of Reynolds

num-330 ber Re As rotation of the disk reduces, the inertial effects so fluid

331 flow enhances when Re is increased Magnitude of axial velocity

332 reduces with increase in thickness coefficient of diska, due to that

333 with an increase inathe feature radius R0reduces and thus less

334 particles are in contact with the surface Consequently the velocity

335 decays (seeFig 4).Fig 5shows behavior of the axial velocity for

336 larger Hartman number M In fact for larger M, the magnetic field

337 increases due to the force of resistance called the Lorentz force

338 Lorentz force becomes more intense and as a result the velocity

339 decreases

340 Fig 6reveals that the radial velocity fðnÞ is an increasing

func-341 tion of disk thickness index m.Fig 7elucidates that radial velocity

342

increases for larger Reynolds number Re Impact of disk thickness

343

coefficientaon radial velocity is examined inFig 8 It is noted that

344

for higherathe radial velocity enhances Effect of increasing values

345

of Hartman number M on radial velocity is shown inFig 9 Results

346

shows that radial velocity decays for higher M For larger M the

347

Lorentz force enhances which produces resistance between the

348

particles and consequently both components of axial and radial

349

velocities are reduced.Fig 10shows that increasing velocity slip

350

parameterc1reduces the radial velocity fðnÞ As in radial direction

351

transport of momentum is less when slip velocity is enhanced

352

353

velocity The radial velocity enhances for larger stretching

param-354

eter A

Fig 1 The h-curves for h 0 ð0Þ; f 0 ð0Þ and g 0 ð0Þ whene¼ A ¼c1¼ 0:3; m ¼ n ¼ 1;c2¼

0:4; Re ¼ 0:9; r  ¼ 0:2;a¼ 1:2 and M ¼ 0:7.

Table 1

e¼ A ¼c1¼ 0:3; m ¼ n ¼ 1;c2¼ 0:4; Re ¼ 0:9; r  ¼ 0:2;a¼ 1:2 and M ¼ 0:7.

Order of approximations h 0 ð0Þ f 0 ð0Þ g 0 ð0Þ

Fig 2 Impact of m on axial velocity.

Fig 3 Impact of Re on axial velocity.

Fig 4 Impact ofaon axial velocity.

Fig 5 Impact of M on axial velocity.

4 M Imtiaz et al / Results in Physics xxx (2017) xxx–xxx

355 Effects of disk thickness index m, Reynolds number Re, disk

356 thickness coefficient a, Hartman number M and velocity slip

357 parameterc2on tangential velocity gðnÞ are observed inFigs (12–

358 16) Fig 12shows that tangential velocity is an increasing function

359 of m.Fig 13shows that tangential velocity increases for larger Re

360 Fig 14shows that tangential velocity is decreasing function ofa It

361 can be seen inFig 15that an increase in M reduces the tangential

362 velocity because resistive force enhances Fig 16 represents

363 decreasing behavior of tangential velocity withc2

364

Variation of Reynolds number Re on surface drag force CfRen1nþ1

365

via Hartman number M is shown inFig 17 Here surface drag force

366

decays for larger Re while it increases for larger M Influence of

367

thickness coefficient of diskaon surface drag force CfRen1nþ1via disk

368

thickness index m is depicted inFig 18 Here surface drag force

369

reduces whenaand mare increased.Fig 19shows the effect of slip

370

parameters c1andc2on surface drag force CfRen1nþ1 Here surface

371

drag force reduces for largerc1andc2

Fig 6 Impact of m on radial velocity.

Fig 7 Impact of Re on radial velocity.

Fig 8 Impact ofaon radial velocity.

Fig 9 Impact of M on radial velocity.

Fig 10 Impact ofc1on radial velocity.

Fig 11 Impact of A on radial velocity.

M Imtiaz et al / Results in Physics xxx (2017) xxx–xxx 5

Please cite this article in press as: Imtiaz M et al Slip flow by a variable thickness rotating disk subject to magnetohydrodynamics Results Phys (2017),

372 Concluding remarks

373 MHD flow of a rotating disk in the presence of velocity slip was

374 investigated Main findings are

375  Larger Hartman number decreases fluid flow due to larger

resis-376 tive force

377  Opposite behavior of the thickness coefficient of the disk is seen

378 on the axial and radial velocities

379

 Due to increment in stretching rate, radial velocity is an

increas-380

ing function of stretching parameter

381

 For higher velocity slip parameters, transport of momentum is

382

less which decays radial and tangential velocities

383

 Axial and radial velocities have opposite behavior for larger disk

384

thickness index

385

 Higher Hartman number enhances the surface drag force

386

Fig 12 Impact of m on tangential velocity.

Fig 13 Impact of Re on tangential velocity.

Fig 14 Impact ofaon tangential velocity.

Fig 15 Impact of M on tangential velocity.

Fig 16 Impact ofc2on tangential velocity.

Fig 17 Impact of Re on skin friction coefficient.

6 M Imtiaz et al / Results in Physics xxx (2017) xxx–xxx

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Fig 18 Impact ofaon skin friction coefficient.

Fig 19 Impact ofc2on skin friction coefficient.

M Imtiaz et al / Results in Physics xxx (2017) xxx–xxx 7

Please cite this article in press as: Imtiaz M et al Slip flow by a variable thickness rotating disk subject to magnetohydrodynamics Results Phys (2017),