Radiative flow due to stretchable rotating disk with

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Dec 21, 2016 - Maria Imtiaz a, Ahmed Alsaedi b a Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan b Nonlinear ...
Results in Physics 7 (2017) 156–165

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Radiative flow due to stretchable rotating disk with variable thickness Tasawar Hayat a,b, Sumaira Qayyum a,⇑, Maria Imtiaz a, Ahmed Alsaedi b a b

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, Jeddah 21589, Saudi Arabia

a r t i c l e

i n f o

Article history: Received 9 November 2016 Received in revised form 24 November 2016 Accepted 9 December 2016 Available online 21 December 2016 Keywords: Variable thickness Stretchable rotating disk MHD Thermal radiation

a b s t r a c t Present article concerns with MHD flow of viscous fluid by a rotating disk with variable thickness. Heat transfer is examined in the presence of thermal radiation. Boundary layer approximation is applied to the partial differential equations. Governing equations are then transformed into ordinary differential equations by utilizing Von Karman transformations. Impact of physical parameters on velocity, temperature, skin friction coefficient and Nusselt number is presented and examined. It is observed that with an increase in disk thickness and stretching parameter the radial and axial velocities are enhanced. Prandtl number and radiation parameter have opposite behavior for temperature field. Skin friction decays for larger disk thickness index. Magnitude of Nusselt number enhances for larger Prandtl number. Ó 2016 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/).

Introduction Presently the flow by rotating surfaces is very popular area of research. It is because of its relevance in engineering and industrial sectors including jet motors, food processing, electric power generating system and turbine system. Theoretical and experimental studies for this type of flow seem interesting. Pioneer work on flow due to rotating disk is done by Karman [1]. He provided transformations which help us to construct ordinary differential equation from Navier Stokes equations. Cochran [2] also used these transformations to examine rotating disk flow by numerical integration method. Rotating flow by two disks is firstly examined by Stewartson [3]. Chapple and Stokes [4] and Mellor et al. [5] studied flow between rotating disks. Heat transfer between two rotating disks is explored by Arora and Stokes [6]. Kumar et al. [7] described flow phenomenon between porous stationary disk and solid rotating disk. Hayat et al. [8] analyzed thermal stratification effects in rotating flow between two disks. Radiative flow of carbon nanotubes between rotating stretchable disks with convective conditions is studied by Hayat et al. [9]. Anderson et al. [10] and Ming et al. [11] examined the flow and heat transfer of power law fluid by a rotating disk for unsteady and steady cases respectively. Surfaces of variable thickness have applications in engineering particularly mechanical, architectural, civil, marine and aeronautical processes. It also helps to reduce the weight of structural elements and improve the utilization of material. However it is

⇑ Corresponding author.

noted that very little literature is present for flow due to stretching surfaces having variable thickness. Hayat et al. [12] examined flow over variable thicked surface with variable thermal conductivity and Cattaneo-Christov heat flux. Ramesh et al. [13] studied Casson fluid flow over variable stretching sheet with thermal radiation. Xun et al. [14] analyzed Ostwald-de Waele fluid flow by rotating disk of variable thickness with index decreasing. Hayat et al. [15] worked on stagnation point flow past a variable thicked surface with nonlinear stretching and carbon nanotube effects. Fang et al. [16] described the flow over stretching sheet with variable thickness. Flow of Williomson nanofluid over a stretching sheet with variable thickness has been examined by Hayat et al. [17]. Zhang et al. [18] worked on bending collapse of square tubes with variable thickness. Hayat et al. [19] discussed the homogeneousheterogeneous reactions and melting heat transfer effects in the flow by a stretching surface with variable thickness. Acharya et al. [20] analyzed the variable thickness on nanofluid flow by a slendering stretching sheet. Khader and Megahed [21] examined the partial slip effects on boundary layer flow due to a nonlinearly stretching sheet with variable thickness. Radiation has many applications in engineering as well as industrial sector such as nuclear reactor, glass production, furnace design, power plant and also in space technology. In radiation process the electromagnetic waves are responsible for transfer of energy which carry energy away from the emitting object. Radiative flow and heat transfer over a permeable stretching sheet with Hall current are analyzed by Pal [22]. Hayat et al. [23] studied stretched flow of nanofluid in presence of nonlinear thermal radiation and mixed convection. Sheikholeslami et al. [24] worked on

E-mail address: [email protected] (S. Qayyum). http://dx.doi.org/10.1016/j.rinp.2016.12.010 2211-3797/Ó 2016 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/).

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T. Hayat et al. / Results in Physics 7 (2017) 156–165

radiative flow of nanofluid. Flow due to porous wedge in presence of mixed convection and thermal radiation are examined by Rashidi et al. [25]. Hayat et al. [26] studied partial slip effects in radiative flow of nanofluid. MHD flow of Oldroyd-B nanofluid with radiative surface is analyzed by Shehzad et al. [27]. Battacharyya et al. [28] described radiative flow of micropolar fluid over a porous shrinking sheet. There are many methods to solve the nonlinear problems. Homotopy analysis method (HAM) is firstly developed by S. Liao in 1992 [29]. He further modified [30] with a non-zero auxiliary parameter which is also known as convergence control parameter  h. This parameter is a non-physical variable that provides a simple way to verify and ensure convergence of solution series. The HAM always helps no matter whether there exist small/large physical parameters or not in the problem statement. It provides a convenient way to guarantee the convergence of approximation series. It also provides great freedom to choose the equation type of linear sub-problems and the base functions of solutions.The capability of the HAM to naturally show convergence of the series solution is unusual in analytical and semi-analytic approaches to nonlinear partial differential equations. There are also some disadvantages that mathematical foundation of this method is not very well established and it will not work for cases with zero radius of convergence. The method has been used by several authors and proved to be very effective in deriving an analytic solution especially for nonlinear differential equation [31–42]. Much attention in past has been given to the flow due to rotating disk with negligible thickness. It is hard to find any such study in presence of radiation. Furthermore the radiative flow by stretchable rotating disk having variable thickness is not studied yet. Our main objective is to fill this void. MHD effects are also taken in account. Graphical technique is used to elaborate the impact of involved parameters on velocity, temperature, skin friction coefficient and Nusselt number.

Modeling We consider steady incompressible flow due to a stretchable rotating disk with angular velocity X and stretching rate c. Fluid occupies the semi infinite region over the disk with variable thick 1 ness and surface is considered at z ¼ a Rr0 þ 1 . Temperature at b w and ambient temperature is assumed the surface of the disk is T b T 1 . Magnetic field of strength B0 is applied parallel to z-axis. We are considering cylindrical coordinates (r; h; z) and physical model is presented in Fig. 1. Under the assumptions

^ Þ ¼ Oðv ^ Þ ¼ OðrÞ ¼ Oð1Þ and OðwÞ ^ ¼ OðzÞ ¼ OðdÞ the ¼ @@zp ¼ 0; Oðu equations for flow and heat transfer [14] are as follows: ^ @p @r

^

^ u ^ ^ @w @u þ þ ¼ 0; @r r @z

ð1Þ

^ u

^ ^ v^ 2 ^ r @u @u @2u ^ ^; þw  ¼ m 2  B20 u @r @z r q @z

ð2Þ

^ u

^ v^ @ v^ @ v^ u @ 2 v^ r ^ þw þ ¼ m 2  B20 v^ ; @r @z r @r q

ð3Þ

^ ðqcp Þ u

@ Tb @ Tb ^ þw @r @z

!

! 16r T 31 @ 2 Tb kþ ; @z2 3k

¼

ð4Þ

with boundary conditions

^ ¼ rc; u

v^ ¼ rX;

^ ¼ 0; u

v^ ¼ 0;

^ ¼ 0; w ^ ¼ 0; w

 1 r Tb ¼ Tb w at z ¼ a þ1 ; R0 Tb ¼ Tb 1 at z ! 1;

ð5Þ

where m denotes kinematic viscosity, r the electrical conductivity, q the density, cp the specific heat, k the thermal conductivity, r the stefan–Boltzmann constant, k the mean absorption coefficient, a is the thickness coefficient of the disk which is very small, R0 the feature radius and 1 the disk thickness index. Generalized Von Karman transformations are

^ ¼ r  R0 X e F ðgÞ; u Tb  Tb 1 ; #~ ¼ b T w  Tb 1

e gÞ; v^ ¼ r R0 X Gð

^ ¼ R0 Xð1 þ r Þ1 w

XR20 q

l

1 !nþ1 2 z  1 XR0 q g ¼ ð1 þ r Þ : l R0

1 !nþ1

e gÞ Hð ð6Þ

After using transformations Eqs. (1)–(5) take the form

e 0 þ g1 e F 0 ¼ 0; 2e FþH

ð7Þ

1n e2  H e ee F 00 ðReÞ1þn ð1 þ r  Þ21  e F2 þ G F0  e Fe F 0 1g  M e F ¼ 0;

ð8Þ

21 e 00 ðReÞ1n eH e0  e e 0 1g  M G e ¼ 0; eG 1þn ð1 þ r  Þ G  2e FG FG

ð9Þ

1n 1 e #~0 ¼ 0; F #~0 1g  H ð1 þ RÞðReÞ1þn ð1 þ r Þ21 #~00  e Pr

ð10Þ

with boundary conditions

e aÞ ¼ 0; Fe ðaÞ ¼ A; e Hð F ð1Þ ¼ 0; e ~ ~ Gð1Þ ¼ 0; #ðaÞ ¼ 1; #ð1Þ ¼ 0;

e aÞ ¼ 1; Gð ð11Þ

where

Re ¼ R¼

XR20

m

;

Pr ¼

16r T 31 ; 3kk

qc p m



k

;



r ; R0 þ r  1 !nþ1

a

XR20 q

R20

l



c

X

; ð12Þ

where  is a dimensionless constant, 1 is disk thickness index, Re denotes Reynolds number, Pr is Prandtl number, A is scaled stretching parameters, R is radiation parameter, r  is the dimensionless radius and M is magnetic parameter. We now introduce deformations

~ g  aÞ ¼ hðnÞ; ~ e ¼ hð H e ¼ g~ðg  aÞ ¼ g~ðnÞ; G Fig. 1. Flow geometry.

e F ¼ ~f ðg  aÞ ¼ ~f ðnÞ; #~ ¼ ~hðg  aÞ ¼ ~hðnÞ:

Eqs. (7)–(11) are reduced to the forms:

ð13Þ

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T. Hayat et al. / Results in Physics 7 (2017) 156–165

~0 þ ðn þ aÞ1~f 0 ¼ 0; 2~f þ h

ð14Þ

21 ~f 00 ðReÞ1n ~~f 0  ~f ~f 0 1ðn þ aÞ  M~f ¼ 0; 1þn ð1 þ r  Þ  ~f 2 þ g~2  h

ð15Þ

~g~0  ~f g~0 1ðn þ aÞ  Mg~ ¼ 0; g~00 ðReÞ1þn ð1 þ r  Þ21  2~f g~  h

ð16Þ

1n 1 ~~h0 ¼ 0; ð1 þ RÞðReÞ1þn ð1 þ r  Þ21 ~h00  ~f ~h0 1ðn þ aÞ  h Pr

ð17Þ

1n

~ hð0Þ ¼ 0;

~f ð0Þ ¼ A; ~f ð1Þ ¼ 0; g~ð0Þ ¼ 1; g~ð1Þ ¼ 0; ~hð0Þ ¼ 1; ~hð1Þ ¼ 0:

ð18Þ

~ ~f ; g~ Here prime denotes the derivative with respect of n and h; ~ are axial, radial, tangential velocities and temperature proand h files respectively. At lower disk the shear stress in radial and tangential directions are szr and szh



2  1 XR0 q

^ @u szr ¼ l  @z

¼

lr X1 R0 ð1 þ r Þ



@z

¼

lr X1 R0 ð1 þ r Þ1



~f 0 ð0Þ

1 nþ1

XR20 q

l

;

g~0 ð0Þ

R0

z¼0

Total shear stress

sw

1 nþ1

l

R0

z¼0

@ v^  szh ¼ l 



qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ s2zr þ s2zh :

:

ð19Þ

sw is defined as ð20Þ

C fx Re

1=2 2 sw jz¼0 1 2 ¼ ¼  ð1 þ r  Þ1 ½ð~f 0 ð0ÞÞ þ ðg~0 ð0ÞÞ  : 2 r qðrXÞ

Initial guesses and auxiliary linear operators are

~0 ðnÞ ¼ 0; h

ð27Þ

~f 0 ðnÞ ¼ A expðnÞ;

ð28Þ

g~0 ðnÞ ¼ expðnÞ;

ð29Þ

~h0 ðnÞ ¼ expðnÞ;

ð30Þ

~0 ; Lh~ ¼ h

L~f ¼ ~f 00  ~f ;

Lg~ ¼ g~00  g~0 ;

L~h ¼ ~h00  ~h;

ð31Þ

with

Lh~ ½c1  ¼ 0;  L~f c2 en þ c3 en ¼ 0;  n Lg~ c4 e þ c5 en ¼ 0;  L~h c6 en þ c7 en ¼ 0;

ð32Þ ð33Þ ð34Þ

where ci ði ¼ 1  7Þ are the constants.  h~ ;   g~ Denoting q 2 ½0; 1 as the embedding parameter and h h~f ; h

Skin friction coefficients C fx is n1 nþ1

Zeroth-order deformation problems

ð21Þ

and  h~h the non-zero auxiliary parameters then the zeroth order deformation problems are

h i ~0 ðnÞ ¼ qh~ N ~ ½Hðn; qÞ; Fðn; qÞ; ð1  qÞLh~ Hðn; qÞ  h h h

ð35Þ

h i ð1  qÞL~f Fðn; qÞ  ~f 0 ðnÞ ¼ qh~f N ~f ½Fðn; qÞ; Hðn; qÞ; Gðn; qÞ;

ð36Þ

ð1  qÞLg~ ½Gðn; qÞ  g~0 ðnÞ ¼ q hg~ N g~ ½Gðn; qÞ; Hðn; qÞ; Fðn; qÞ;

ð37Þ

h i ð1  qÞL~h #ðn; qÞ  ~h0 ðnÞ ¼ qh~h N ~h ½#ðn; qÞ; Fðn; qÞ; Gðn; qÞ;

ð38Þ

Hð0; qÞ ¼ 0;

ð39Þ

Nusselt number is defined as

  R0 qw  Nux ¼  kð Tb w  Tb 1 Þ

;

ð22Þ

Fð0; qÞ ¼ A;

Fð1; qÞ ¼ 0;

ð40Þ

where wall heat flux qw is given by

Gð0; qÞ ¼ 1;

Gð1; qÞ ¼ 0;

ð41Þ

qw jz¼0

#ð0; qÞ ¼ 1;

#ð1; qÞ ¼ 0:

ð42Þ

z¼0

  @ Tb  þ qr  ¼ k  @z

z¼0

¼ kð Tb w  Tb 1 Þð1 þ r Þ

 1

XR20 q

l

1 !nþ1

Nonlinear differential operators N h~ ; N ~f ; N g~ and N ~h are

ð1 þ RÞ~h0 ð0Þ:

ð23Þ

Nusselt number can be written as follows:

Nux Re

1 nþ1



1

~0

¼ ð1 þ r Þ ð1 þ RÞh ð0Þ:

N h~ ½Hðn; qÞ; Fðn; qÞ ¼ 2Fðn; qÞ þ

ð24Þ

¼

ð25Þ

such that for each xX.

Fðx; 1Þ ¼ f 2 ðxÞ

@ 2 Fðn; qÞ @n2  Hðn; qÞ

Concept of homotopy [42] comes from topology to find the convergent solutions of highly nonlinear problems. If one function can continously be deformed into another then these two functions are called homotopy. If there exist two continous maps f 1 and f 2 from topological spaces X into Y then f 1 is homotopic to f 2 if F is a continous map.

Fðx; 0Þ ¼ f 1 ðxÞ;

ð43Þ

N ~f ½Fðn; qÞ; Gðn; qÞ; Hðn; qÞ

Solution technique

F : X  ½0; 1 ! Y

@Hðn; qÞ @Fðn; qÞ þ ðn þ aÞ1 ; @n @n

ð26Þ

then between f 1 and f 2 the continous map F is called homotopy. Detail of HAM is as follows:

1n

ðReÞ1þn ð1 þ r Þ21  ðFðn; qÞÞ2 þ ðGðn; qÞÞ2 @Fðn; qÞ @Fðn; qÞ  Fðn; qÞ 1ðn þ aÞ  MFðn; qÞ; @n @n

ð44Þ

N g~ ½Gðn; qÞ; Fðn; qÞ; Hðn; qÞ ¼

@ 2 Gðn; qÞ 2



@n @ 2 Gðn; qÞ @n2

1n

ðReÞ1þn ð1 þ r  Þ21  2Fðn; qÞGðn; qÞ  Hðn; qÞ

1ðn þ aÞ  MGðn; qÞ;

@Gðn; qÞ @n ð45Þ

N ~h ½#ðn; qÞ; Fðn; qÞ; Hðn; qÞ ¼

1n 1 @ 2 #ðn; qÞ @#ðn; qÞ ð1 þ RÞðReÞ1þn ð1 þ r  Þ21  Fðn; qÞ 1ðn þ aÞ Pr @n @n2 @#ðn; qÞ : ð46Þ  Hðn; qÞ @n

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T. Hayat et al. / Results in Physics 7 (2017) 156–165

mth order deformation problems The mth order deformation problems are

h i ~m ðnÞ  v h ~ Lh~ h hh~ Rh;m ~ ðnÞ; m m1 ðnÞ ¼ 

ð47Þ

h i L~f ~f m ðnÞ  vm~f m1 ðnÞ ¼ h~f R~f ;m ðnÞ;

ð48Þ

 Lg~ g~m ðnÞ  vm g~m1 ðnÞ ¼ hg~ Rg~;m ðnÞ;

ð49Þ

h i L~h ~hm ðnÞ  vm ~hm1 ðnÞ ¼ h~h R~h;m ðnÞ;

ð50Þ

~f m ð0Þ ¼ ~f m ð1Þ ¼ 0;

~m ð0Þ ¼ 0; h

g~m ð0Þ ¼ g~m ð1Þ ¼ ~hm ð0Þ

¼ ~hm ð1Þ ¼ 0;

ð51Þ

00 0 Fig. 2.  h-curve for h ð0Þ; f ð0Þ; g 0 ð0Þ and h0 ð0Þ.

where Rh;m ~ ðnÞ; R~f ;m ðnÞ; Rg~;m ðnÞ and R~ h;m ðnÞ are

~0 ~ ~0 Rh;m ~ ðnÞ ¼ 2f m1 þ hm1 þ ðn þ aÞ1f m1 ; R~f ;m ðnÞ ¼ ~f 00m1 ðReÞ1þn ð1 þ r  Þ21  1n

m1 X



~ ~0 h m1k f k 

k¼0

m1 X

ð52Þ

m1 X

m1 X

k¼0

k¼0

~f ~ m1k f k þ

g~m1k g~k

1ðn þ aÞ  M~f m1 ;

0 f m1k f k

ð53Þ

k¼0

1n

Rg~;m ðnÞ ¼ g 00m1 ðReÞ1þn ð1 þ r  Þ21  2

m 1 X

~f ~k m1k g

k¼0



m1 X

~ ~0k  h m1k g

k¼0

R~h;m ðgÞ ¼

m1 X ~f ~0k 1ðn þ aÞ  Mg~m1 ; m1k g

ð54Þ

vm ¼

0;

m1 X

~ ~0 h m1k hk ;

m 6 1

1; m > 1

Order of approximations

~ 00 ð0Þ h

~f 0 ð0Þ

~0 ð0Þ g

~ h0 ð0Þ

1 10 20 21 23 25 30 40 50 60

0.4074 0.2018 0.1999 0.1999 0.1999 0.1999 0.1999 0.1999 0.1999 0.1999

0.04312 0.01080 0.01069 0.01069 0.01071 0.01071 0.01071 0.01071 0.01071 0.01071

0.8920 0.9471 0.9482 0.9482 0.9483 0.9483 0.9483 0.9483 0.9483 0.9483

0.7223 0.4289 0.4252 0.4253 0.4253 0.4253 0.4253 0.4253 0.4253 0.4253

Table 2 Comparison of present result n ¼ 1; 1 ¼ 0; Pr ¼ 1; M ¼ R ¼ A ¼ 0.

ð55Þ

k¼0



1 ¼ 1;  ¼ 0:3; a ¼ 1:2; Re ¼ 0:9; n ¼ 1; r ¼

k¼0

m1 X 1n 1 ~f ~0 ð1 þ RÞðReÞ1þn ð1 þ r Þ21 ~h00m1  m1k hk Pr k¼0

1ðn þ aÞ 

Table 1 Convergence of series solutions when 0:2; M ¼ 0:7; A ¼ 0:3; Pr ¼ 1:9 and R ¼ 0:1.

:

ð56Þ

with

Refs.

[10,11,14]

where

Authors

~f 0 ð0Þ

g~0 ð0Þ

~ h0 (0)

Present Xun et al. [14] Anderson et al. [10] Ming et al. [11]

0.5109 0.510231 0.510 0.51021

0.61598 0.615921 0.616 0.61591

0.3959 0.396271 – 0.39632

~m ; ~f m ; g~m ; ~ General solutions (h hm Þ comprising special solutions

~  ; ~f  ; g ~m ; ~ ðh hm Þ are m m

~m ðnÞ ¼ h ~  ðnÞ þ c1 ; h m

ð57Þ

~f m ðnÞ ¼ ~f  ðnÞ þ c2 en þ c3 en ; m

ð58Þ

g~m ðnÞ ¼ g~m ðnÞ þ c4 en þ c5 en ;

ð59Þ

~hm ðnÞ ¼ ~h ðnÞ þ c6 en þ c7 en : m

ð60Þ

Convergence analysis Convergence region is adjusted with the help of auxiliary  g~ and   h~ ;  h~f ; h h~h . Here we have plotted the  hcurves parameters h at 15th order of approximation (see Fig. 2). Ranges of admissible  are 1:2 6  h~f 6 0:3; 1:1 6 values for h hh~ 6 0:4; 1:1 6   ~h 6 0:5. Solution converges for whole  hg~ 6 0:25 and 0:9 6 h h~f ¼ 0:6;  hg~ ¼ 0:5 and region of nð0 6 n 6 1Þ when  hh~ ¼ 0:7;   ~h ¼ 0:6. Series solution convergence can be shown in Table 1. h ~ 00 ð0Þ, radial ~f 0 ð0Þ and tangential g ~0 ð0Þ velocities converge at Axial h 20th, 23rd, 23rd order of approximation respectively. Temperature

Fig. 3. Behavior of

~ 1 against h(n).

profile ~ h0 ð0Þ converges for 21st order of approximation. Table 2 is constructed to verify our numerical scheme. The present results are compared with the published results and the results are found in an excellent agreement.

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T. Hayat et al. / Results in Physics 7 (2017) 156–165

Fig. 4. Behavior of

1 against ~f (n).

~ Fig. 7. Behavior of M against h(n).

Fig. 5. Behavior of

~  against h(n).

Fig. 8. Behavior of M against ~f (n).

Fig. 6. Behavior of

 against ~f (n).

~ Fig. 9. Behavior of A against h(n).

Discussion This section elucidates the behavior of velocity, temperature, skin friction coefficient and Nusselt number for different involved parameters. Axial and radial velocity components Figs. 3 and 4 show the behavior of axial and radial velocities for larger disk thickness power law index 1. Here opposite behavior is observed for axial and radial velocities. Magnitude of axial velocity decays while radial velocity increases for larger 1. Opposite behav-

ior of axial and radial velocities is noticed for increasing value of constant number  (see Figs. 5 and 6). It is observed that radial velocity profile is increasing function of  due to fact that for larger  the radius R0 decreases so less surface of disk is in contact with fluid particles and less resistance produced lead to increase the velocity. Figs. 7 and 8 elucidate that magnitude of axial and radial velocity fields decays with an increase in Hartman number M. For larger M the Lorentz force enhances which produces resistance between the particles and consequently velocity reduces for both axial and radial components. Impact of stretching parameter A on radial and axial velocity is examined in Figs. 9 and 10. It is noted that for higher A the magnitude of both velocities enhances. It is

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T. Hayat et al. / Results in Physics 7 (2017) 156–165

~ Fig. 13. Behavior of a against h(n).

Fig. 10. Behavior of A against ~f (n).

~ Fig. 11. Behavior of n against h(n).

Fig. 14. Behavior of

1 against g~(n).

Fig. 15. Behavior of

 against g~(n).

Fig. 12. Behavior of n against ~f (n).

because of the fact that for increase in A stretching rate c at lower disk increases. Effect of increasing values of power law exponent of fluid n for radial and axial velocities is shown in Figs. 11 and 12. For larger values of n power of radius R0 decreases. Hence less resistance increases the velocity. Magnitude of axial velocity reduces with an increase in thickness coefficient of disk a (see Fig. 13). Tangential velocity Fig. 14 shows the impact of disk thickness power law index 1on ~ðnÞ enhances tangential velocity profile g~ðnÞ. Results depict that g

for larger 1. Fig. 15 characterizes the effect of constant number ~ðnÞ. Tangential velocity has direct relation with . Figs. 16 g and 17 show that for larger values of Hartman number M and stretching parameter A the tangential velocity reduces. With an increase in M resistance produces due to Lorentz force due to ~ðnÞ reduces. Tangential velocity depends on rotational which g velocity of disk X. When A increases then rotational velocity is decreased and so g~ðnÞ decays (see Fig. 17). Tangential velocity increases for larger values of power law exponent of fluid n. Infact for larger n the power of R0 decreases and consequently velocity increases (see Fig. 18).

on

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T. Hayat et al. / Results in Physics 7 (2017) 156–165

Fig. 16. Behavior of M against g~(n).

Fig. 17. Behavior of A against g~(n).

Fig. 18. Behavior of n against g~(n).

Temperature profile Influence of disk thickness power law index 1on temperature field is portrayed in Fig. 19. Results indicate that temperature of the fluid increases when there is enhancement in 1. Fig. 20 shows the impact of on temperature profile. It is noted that radius R0 is decreasing for larger  so less fluid particles are attached to the surface of disk which enhances the heat transfer by convection and as a result temperature increases. Fig. 21 portrays the effect of stretching parameter A on temperature field. With an increase in

Fig. 19. Behavior of

1 against ~h(n).

Fig. 20. Behavior of

 against ~h(n).

Fig. 21. Behavior of A against ~ h(n).

A the temperature profile decays because with an increase in A rotational velocity becomes less. Thus less resistance is produced between the particles and consequently temperature reduces. ~ Influence of Prandtl number Pr on hðnÞ is displayed in Fig. 22. Temperature of the fluid reduces for larger Pr because with an increase in Pr the thermal diffusivity reduces. In fact the particles are able to conduct less heat and consequently temperature decreases. Fig. 23 ~ displays the effect of radiation parameter R on hðnÞ. Here temperature profile enhances when radiation effects strengthen. As increase in radiation parameter corresponds to a decrease in mean

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Fig. 25. Behavior of

Fig. 22. Behavior of Pr against ~ h(n).

1 against C fx ðReÞn1=nþ1 .

Fig. 26. Behavior of a against C fx ðReÞn1=nþ1 .

Fig. 23. Behavior of Ragainst ~ h(n).

Fig. 27. Behavior of n against C fx ðReÞn1=nþ1 .

Fig. 24. Behavior of n against ~ h(n).

absorption coefficient. Hence the rate of radiative heat transfer to the fluid increases. Fig. 24 depicts the behavior of temperature field for larger physical power law exponent of fluid n. Decreasing behavior of ~ hðnÞ and thermal boundary layer thickness are

ber M on surface drag force against stretching parameter A is presented in Figs. 25–28. It is noted that skin friction coefficient reduces when 1; a; n and M are increased while it shows increasing behavior for larger stretching parameter A.

observed for larger n. It shows that efficiency of heat transfer is enhancing gradually from the shear-thinning fluid to shearthickening fluid.

Nusselt number

Surface drag force Impact of disk thickness power law exponent 1, thickness coefficient of disk a, power law exponent of fluid n and Hartman num-

Figs. 29–31 depict the influence of disk thickness power law exponent 1, thickness coefficient of disk a and radiation parameter R against stretching parameter A on Nusselt number. Heat transfer rate increases for larger 1; a; R and A.

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Fig. 28. Behavior of M against C fx ðReÞn1=nþ1 .

Fig. 31. Behavior of R against ð1 þ RÞð1 þ r Þ1 h0 ð0Þ:

 With an increase in values of Prandtl number Pr the temperature of fluid decreases while for larger values of radiation parameter R and power law exponent it increases.  Skin friction coefficient decreases for larger a.  Heat transfer rate enhances for larger Pr and A.

References

Fig. 29. Behavior of

1 against ð1 þ RÞð1 þ r  Þ1 h0 ð0Þ.

Fig. 30. Behavior of a against ð1 þ RÞð1 þ r  Þ1 h0 ð0Þ.

Conclusions Here we studied radiative flow by a stretchable rotating disk of variable thickness. Main points are as follows:  Magnitude of radial and axial velocities decreases for larger M.  With increase in power law exponent of fluid the magnitude of ~ hðnÞ and ~f ðnÞ are enhanced.  Tangential velocity profile decays for larger M while it shows increasing behavior for n.

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