Flow between two stretchable rotating disks with

Results in Physics 7 (2017) 126–133

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Flow between two stretchable rotating disks with Cattaneo-Christov heat flux model 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 28 October 2016 Received in revised form 30 November 2016 Accepted 6 December 2016 Available online 11 December 2016 Keywords: Cattaneo-Christov heat flux Two stretchable rotating disks Porous medium

a b s t r a c t An analysis is performed to investigate flow between two stretchable rotating disks. Thermal equation is constructed by Cattaneo-Christov heat flux theory. Porous medium is also taken into account. The nonlinear partial differential equations are first converted to ordinary differential equations and then computed for the convergent series solutions. Discussion about impact of dimensionless parameters on velocities, temperature and skin friction coefficient is given. It is observed that the radial velocity at upper disk enhances for larger values of ratio of corresponding stretching rate to angular velocity. Velocity in y-direction decays with an increase in rotational parameter. Magnitude of temperature profile decays for larger Prandtl number and thermal relaxation parameter. Ó 2016 The Authors. 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/).

Introduction Flow by rotating surfaces is very popular area of research in view of its usage in engineering and industrial sectors including jet motors, food processing, electric power generating system and turbine system. Therefore 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] also 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]. Xun et al. [10] worked on Ostwald-de Waele fluid flow and heat transfer due to rotating disk with variable thickness and index decreasing. Rotating variable thickness disks for a new thickness profile is discussed by Erasalan and Ciftici [11]. Allam et al. [12] described the stresses of a rotating disk having a current and bearing a coaxial viscoelastic coating ⇑ Corresponding author. E-mail address: [email protected] (S. Qayyum).

with variable thickness. Flow and heat transfer induced by a stretched surface is a relevant problem in many industrial processes such as glass-fiber and paper production, manufacture and drawing of plastics and rubber sheets, cooling of metallic sheets in a cooling bath, metal and polymer extrusion processes, crystal growing and many others. Crane [13] was the first one who studied the stretching problem taking into account the fluid flow over a linearly stretched surface. After this various researchers worked on stretching with different norms (see Refs. [14–17]). There are very useful applications of heat transfer mechanism like nuclear reactor of cooling, medical applications for example heat conduction in tissues and drug targeting. Heat transfer mechanism was firstly described by Fourier law of heat conduction [18] such law leads to the argument that initial temperature is immediately sensed by the medium under consideration. To eliminate this issue Cattaneo [19] included thermal relaxation time in Fourier law of heat conduction (which is time required by the medium to transfer heat to its neighboring particles when temperature gradient is imposed across it). Christov [20] studied frame indifferent formulation of the Maxwell–Cattaneo model. Riesz fractional Cattaneo-Christov flux with an improved heat conduction model is investigated by Liu et al. [21]. Hayat et al. [22] studied Jeffrey fluid flow with homogeneous-heterogeneous reactions and Cattaneo-Christov heat flux. Impact of Cattaneo-Christov heat flux in flow of variable thermal conductivity fluid over a variable thicked surface is analyzed by Hayat et al. [23]. Mustafa [24] examined heat transfer in rotating flow of Maxwell fluid subject to Cattaneo-Christov heat flux.

http://dx.doi.org/10.1016/j.rinp.2016.12.007 2211-3797/Ó 2016 The Authors. 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/).

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Fluid flow through porous media is a subject of common interest. It has emerged a separate field of study. Many natural substances such as zeolites, biological tissues (e.g. wood, cork, bones), soil and rocks (e.g. aquifers, petroleum reservoirs) and man made materials like ceramics and cements can be considered as porous media. Many of their important properties can only be rationalized by considering as porous media. Some applications in engineering and applied science include filtration, mechanics (geomechanics, acoustics, rock mechanics, soil mechanics), engineering (bio-remediation, construction engineering, petroleum engineering), geosciences (petroleum geology, hydrogeology, geophysics), biophysics, material science and biology etc. Shehzad et al. [25] analyzed impact of heat generation in Casson fluid flow saturating porous medium. Moneim and Hassanin [26] studied flow with porous medium and oscillatory suction. Ellahi et al. [27] investigated peristaltic flow of Jeffrey fluid through porous medium in a rectangular duct. Mass and heat transfer in flow of micropolar fluid filling porous medium are examined by Sheikholeslami et al. [28]. Zhang et al. [29] worked on nanofluid flow through porous medium with radiation, chemical reaction and surface heat flux. Flow between two stretchable rotating disks with CattaneoChristov heat flux theory is not yet studied. The objective here is to provide such attempt. Porosity effects are also addressed. Convergent series solutions are developed through HAM [30–38]. Through graphs and tables, the impact of different parameters on velocity, temperature and skin friction coefficient is shown and analyzed. Modeling We consider flow between two stretchable rotating disks. The lower disk is situated at z ¼ 0 whereas upper disk is at distance h apart. We denote X1 and X2 as rotational velocities of lower and upper disks respectively and a1 and a2 their respective stretching rates (see Fig. 1). Porous medium between disks is also investigated where k0 is the permeability constant. Cattaneo-Christov model for heat conduction is used to analyze the heat transfer.

We have used the cylindrical coordinates (r; h; z) with velocity ^; v ^ ; w) ^ to construct the velocity and temperature equations as (u follows:

^ u ^ ^ @w @u þ þ ¼ 0; @r r @z ^ u

ð1Þ

^ 1 @u ^ u ^ ^ ^ v^ 2 ^ ^ @2u @u @u 1 @p @2u ^ þw  þm þ þ  ¼ @r2 r @r @z2 r2 @r @z r q @r

! 

l^

k0

u; ð2Þ

^ u

^ v^ @ v^ @ v^ u @ 2 v^ 1 @ v^ @ 2 v^ v^ ^ þ  þw þ ¼m þ @r @z r @r2 r @r @z2 r 2

^ w

! 

l^

k0

v;

! ^ ^ ^ ^ @2w ^ 1 @w ^ @w @w 1 @p @2w l^ ^  w; þu ¼ þm þ þ @r @r r @r @r 2 @z2 q @z k0

^ ðqcp Þ u

@ Tb @ Tb ^ þw @r @z

ð3Þ

ð4Þ

! ¼ $  q;

ð5Þ

with boundary conditions

^ ¼ 0; Tb ¼ Tb 1 at z ¼ 0; ^ ¼ ra1 ; v^ ¼ r X1 ; w u ^ ¼ 0; Tb ¼ Tb 2 at z ¼ h; ^ ¼ ra2 ; v^ ¼ r X2 ; w u

ð6Þ

b is the temperature, T b 1 and T b 2 the tem^ denotes pressure, T where p peratures at lower and upper disks respectively and the heat flux q satisfies

    @q þ V  $q  q  $V þ $  V q ¼ kr Tb ; qþc @t

ð7Þ

where c is thermal relaxation time and k is thermal conductivity. Now we omit q from Eq. (5) and Eq. (7) and get

^ u

@ Tb @ T^ ^ þw @r @z

!

¼

k @ 2 T^ 1 @ Tb @ 2 Tb þ þ qcp @r2 r @r @z2

!

@ 2 Tb @ 2 Tb @ 2 Tb ^ 2 2 þ 2u ^w ^ þw 2 @r @z @z@r   b   ! ^ ^ @T ^ ^ @ Tb @u @u @w @w ^ ^ ^ ^ þ u þw þ u þw : ð8Þ @r @z @r @r @z @z

^2 c u

We consider the Von Karman transformations [1]:

b b ^ ¼ r X1~f 0 ðnÞ; v^ ¼ r X1 g~ðnÞ; w ^ ¼ 2hX1~f ðnÞ; ~h ¼ T  T 2 ; u bT 1 bT 2   ^ ¼ qf X1 mf PðnÞ þ 12 r22  ; n ¼ hz : p h

ð9Þ

Mass conservation law is satisfied identically and Eqs. (2), (3), (4), (6) and (8) take the form

  ~f 000 þ Re 2~f ~f 00  ~f 02 þ g~2  1 ~f 0   ¼ 0; b

ð10Þ

  1 Re 2~f 0 g~  2~f g~0 þ g~  g~00 ¼ 0; b

ð11Þ

  2 P0 ¼ Re ~f  4~f ~f 0  2~f 00 ; b

ð12Þ

1 ~00 h þ 2Re~f ~h0  4kReð~f 2 ~h00 þ ~f ~f 0 ~h0 Þ ¼ 0; Pr

ð13Þ

with Fig. 1. Flow geometry.

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T. Hayat et al. / Results in Physics 7 (2017) 126–133

~f ð0Þ ¼ 0; ~f ð1Þ ¼ 0; ~f 0 ð0Þ ¼ A1 ; ~f 0 ð1Þ ¼ A2 ; g~ð0Þ ¼ 1; g~ð1Þ ¼ s; ~hð0Þ ¼ 1; ~hð1Þ ¼ 0; Pð0Þ ¼ 0;

ð14Þ

where 2

Re ¼ Xm1 h ; Pr ¼

ðqcp Þf mf kf

f

a1

a2

A1 ¼ X1 ; A2 ¼ X1 ;

; k ¼ cX1

ð15Þ

s ¼ XX21 ; b ¼ k0mX1

where Re denotes Reynolds number, Pr Prandtl number, A1 and A2 are scaled stretching parameters, k thermal relaxation parameter, s rotation number and b porosity parameter. For making more simpler form of Eq. (10) and removing 2 we differentiate it with respect to n

  ~f iv þ Re 2~f ~f 000 þ 2g~g~0  1 ~f 00 ¼ 0: b

ð16Þ

Also the pressure parameter 2 can be obtained by using Eqs. (10) and (14) as

  2 1 2 2¼ ~f 000 ð0Þ  Re ð~f 0 ð0ÞÞ  ðg~ð0ÞÞ þ ~f 0 ð0Þ : b

ð17Þ

We integrate Eq. (12) with respect to n to get pressure term and taking limit from 0 to n i.e.

    Z 1 n~ f dn  ~f 0 þ ~f 0 ð0Þ : P ¼ 2 Re ~f 2 þ b 0

ð18Þ

At lower disk the shear stress in radial and tangential directions are szr and szh



^ @u szr ¼ l  @z

¼

lrX1~f 00 ð0Þ h

z¼0

Total shear stress

sw

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ s2zr þ s2zh :



;

@ v^  szh ¼ l  @z

¼

z¼0

lrX1 g~0 ð0Þ h

:

ð19Þ

sw is defined as ð20Þ

L~h ½c7 þ c8 n ¼ 0;

ð29Þ

where ci ði ¼ 1  8Þ are the constants.  g~ and Denoting q 2 ½0; 1 as the embedding parameter and  h~f ; h  ~h the non-zero auxiliary parameters then the zeroth order deforh mation problems are

h i e qÞ; F ðn; qÞ  ~f 0 ðnÞ ¼ qh~f N ~f ½ e ð1  qÞL~f e F ðn; qÞ; Gðn;

ð30Þ

h i e qÞ  g~0 ðnÞ ¼ qhg~ N g~ ½ Gðn; e qÞ; e F ðn; qÞ; ð1  qÞLg~ Gðn;

ð31Þ

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

ð32Þ

e F 0 ð1; qÞ ¼ A2 ; F ð0; qÞ ¼ 0; e F ð1; qÞ ¼ 0; e F 0 ð0; qÞ ¼ A1 ; e

ð33Þ

e qÞ ¼ 1; Gð1; e qÞ ¼ X; Gð0;

ð34Þ

~ qÞ ¼ 1; #ð1; ~ qÞ ¼ 0: #ð0;

ð35Þ

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

 4e i e qÞ ¼ @ F ðn; qÞ þ Re 2 Gðn; e qÞ @ g^ðn; qÞ F ðn; qÞ; Gðn; N ~f e @n @n4 ! 3e 2e @ F ðn; qÞ 1 @ F ðn; qÞ e þ 2 F ðn; qÞ  ; b @n3 @n2 " e qÞ e qÞ @ 2 Gðn; @ Gðn; e e þ Re 2 e F ðn; qÞ N g~ ½ Gðn; qÞ; F ðn; qÞ ¼ 2 @n @n # @e F ðn; qÞ e 1e 2 Gðn; qÞ  Gðn; qÞ ; @n b h

Cf 2 ¼

1=2 2 sw jz¼0 1 ~00 2 ¼ ½ðf ð0ÞÞ þ ðg~0 ð0ÞÞ  ; 2 qðrX1 Þ Rer

ð21Þ

1=2 2 sw jz¼h 1 ~00 2 ¼ ½ðf ð1ÞÞ þ ðg~0 ð1ÞÞ  ; 2 qðrX1 Þ Rer

ð22Þ

ð37Þ

2~ h i e qÞ ¼ 1 @ #ðn; qÞ ~ g; qÞ; e N ~h #ð F ðg; qÞ; Gðn; Pr @n2

~ qÞ @ #ðn; þ 2Re e F ðn; qÞ @n 2~ @ #ðn; qÞ 4kReð e F 2 ðn; qÞ @n2 ! e ~ qÞ @ F ðn; qÞ @ #ðn; : þe F ðn; qÞ @n @n

Skin friction coefficients C f 1 and C f 2 at the lower and upper disks are

Cf 1 ¼

ð36Þ

ð38Þ

mthorder deformation problems

where local Reynolds number is Rer ¼ rXm1 h.

The mth order deformation problems are

Solution technique

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

ð39Þ

Zeroth-order deformation problems

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

ð40Þ

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

ð41Þ

~f m ð0Þ ¼ @~f m ð0Þ ¼ @~f m ð1Þ ¼ ~f m ð1Þ ¼ 0; @n @n

ð42Þ

Initial guesses and auxiliary linear operators are

~f 0 ðnÞ ¼ A1 n  2A1 n2  A2 n2 þ A1 n3 þ A2 n3 ;

ð23Þ

g~0 ðnÞ ¼ 1 þ ðX  1Þn;

ð24Þ

~h0 ðnÞ ¼ 1  n;

ð25Þ

0000 L~f ¼ ~f ; Lg~ ¼ g~00 ; L~h ¼ ~h00 ;

ð26Þ

with



2

L~f c1 þ c2 n þ c3 n þ c4 n Lg~ ½c5 þ c6 n ¼ 0;

3



g~m ð0Þ ¼ g~m ð1Þ ¼ ~hm ð0Þ ¼ ~hm ð1Þ ¼ 0; where R~f ;m ðnÞ; Rg~;m ðnÞ and R~h;m ðnÞare

R~f ;m ðnÞ ¼

1 ð1  /Þ2:5 ð1  / þ qqs /Þ

~f iv m1

f

¼ 0;

ð27Þ ð28Þ

þ Re 2

m1 X k¼0

~ ~0 ~ ð~f 000 m1k f k þ g m1k g k Þ 

M 1  / þ qqs / f

!

rnf ~00 f ; rf m1 ð43Þ

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T. Hayat et al. / Results in Physics 7 (2017) 126–133

Rg~;m ðnÞ ¼

1 ð1  /Þ2:5 ð1  / þ qqs /Þ

g~00m1

f

þ Re 2

m1 X

ð~f m1k g~0k  ~f 0m1k g~k Þ 

k¼0

! rnf ~ M g m1 ; 1  / þ qqs / rf f

ð44Þ R~h;m ðgÞ ¼

m1 X 1 ~00 hm1 þ 2Re ~f m1k ~h0k Pr k¼0

 4kRe

m 1 X

~f mk1

k¼0

vm ¼



0;

m61

1; m > 1

 ~h 6 0:1. Solution converges for whole region of 1:6 6 h  ~f ¼  nð0 6 n 6 1Þ when h hg~ ¼  h~h ¼ 1. Table 1 shows the convergence of series solutions. Velocity in x-direction ~f 00 ð0Þ converges

at 2nd order of approximation, tangential velocity g~0 ð0Þ converges for 3rd order of approximation is sufficient and temperature ~ h0 ð0Þ converges at 15th order of approximation. Discussion

! k X ð~f kl ~h00l þ ~f 0kl ~h0l Þ

ð45Þ

This section elucidates the behaviors of velocity, temperature and skin friction coefficient for different involved parameters.

l¼0

:

ð46Þ

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

~f m ðnÞ ¼ ~f  ðnÞ þ c1 þ c2 n þ c3 n2 þ c4 n3 ; m

ð47Þ

g~m ðnÞ ¼ g~m ðnÞ þ c5 þ c6 n;

ð48Þ

~hm ðnÞ ¼ ~h ðnÞ þ c7 þ c8 n: m

ð49Þ

Convergence analysis Fig. 3. Behavior of ~f ðnÞ against Re.

 g~ and  h~h have pivotal role in adjusting Auxiliary parameters  h~f ; h and controlling the convergence region. We have plotted  h-curves at 20th order of approximations (see Fig. 2). Range of admissible  is hg~ 6 0:1 and values for h 2 6  h~f 6 0:1; 2:1 6 

Fig. 4. Behavior of ~f 0 ðnÞ against Re.

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

Table 1 Convergence of series solutions b ¼ 0:9; Re ¼ 0:01; A1 ¼ 0:2; A2 ¼ 0:4; s ¼ 0:8; k ¼ 0:2 and Pr ¼ 0:7.

when

Order of approximations

~f 00 ð0Þ

g~00 ð0Þ

~ h0 ð0Þ

1 2 3 15 20 25 30 40 50 60

1.59936148 1.59936096 1.59936096 1.59936096 1.59936096 1.59936096 1.59936096 1.59936096 1.59936096 1.59936096

0.204878519 0.204875663 0.204875663 0.204875663 0.204875663 0.204875663 0.204875663 0.204875663 0.204875663 0.204875663

0.99993028 0.99996032 0.99994745 0.99995131 0.99995131 0.99995131 0.99995131 0.99995131 0.99995131 0.99995131

Fig. 5. Behavior of ~f ðnÞ against A1 .

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T. Hayat et al. / Results in Physics 7 (2017) 126–133

Axial and radial velocity components Behaviors of axial and radial velocities for Reynolds number Re is described in Figs. 3 and 4. Magnitude of axial and radial velocities decays at lower disk with an increase in Re. In fact inertial forces have direct relation with Re. Velocity of upper disk is more than lower one. That is why there are negative values at lower disk. Impact of stretching parameter of lower disk A1 for axial and radial velocities is presented in Figs. 5 and 6. With an increment in values of A1 the velocities ~f ðnÞ and ~f 0 ðnÞ enhance at lower disk and magni-

tude of these velocities decreases at upper disk because stretching rate of lower disk is increasing continuously. Negative values of ~f ðnÞ near the upper disk show that velocity at lower disk is more than upper disk. Figs. 7 and 8 portray the influence of stretching parameter A2 on ~f ðnÞ and ~f 0 ðnÞ. It shows that the velocities at lower disk decay with an increase in A2 while magnitude of these velocities increases near upper disk because stretching rate of upper disk is higher.

Fig. 6. Behavior of ~f 0 ðnÞ against A1 .

Fig. 9. Behavior of g~ðnÞ against Re.

Fig. 7. Behavior of ~f ðnÞ against A2 .

Fig. 10. Behavior of g~ðnÞ against A2 .

Fig. 8. Behavior of ~f 0 ðnÞ against A2 .

Fig. 11. Behavior of g~ðnÞ against s.

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T. Hayat et al. / Results in Physics 7 (2017) 126–133

Fig. 12. Behavior of g~ðnÞ against b.

Fig. 13. Behavior of ~ hðnÞ against Re.

Fig. 15. Behavior of ~ hðnÞ against b.

Fig. 16. Behavior of ~ hðnÞ against A1 .

Fig. 14. Behavior of ~ hðnÞ against A2 . Fig. 17. Behavior of ~ hðnÞ against Pr.

Tangential velocity Influence of Reynolds number on tangential velocity is presented in Fig. 9. For larger values of Re the velocity increases. Impact of stretching parameter of upper disk A2 on g~ðnÞ is shown in Fig. 10. It is found that tangential velocity is less for larger A2 . Fig. 11 is sketched to show the effect of rotational parameter s on g~ðnÞ. Here tangential velocity enhances for larger s. Fig. 12 is ~ðnÞ. As we increase the valsketched to show the influence of b on g

ues of b the tangential velocity decreases because porosity constant increases continuously. Temperature profile Figs. 13 and 14 show that temperature of the fluid has opposite hðnÞ is increasimpact for larger values of Re and A2 . It shows that ~ ing function of Re and it decays for larger values of A2 . Figs. 15 and

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T. Hayat et al. / Results in Physics 7 (2017) 126–133

16 are portrayed to show the impact of porosity parameter b and hðnÞ. We observed that temperature stretching parameter A1 on ~ of the fluid increases with an increase in values of b and A1 . Effect of Prandtl number Pr on ~ hðnÞ is presented in Fig. 17. Magnitude of temperature profile decays for larger Pr because thermal diffusivity decreases when Pr is increased. Fig. 18 is portrayed to show the behavior of temperature profile for larger thermal relaxation parameter k. It is noted that for larger k the temperature reduces. In fact thermal relaxation time increases for larger k. It means particles requires much more time to transfer heat to its neighboring particles and thus temperature decreases.

disks. Table 3 is presented to show the comparison of present results with previous published articles in limiting sense for different values of s. Good agreement is noted here. Here s 6 0 means both disks rotate in opposite directions, s ¼ 0 means upper disk is not rotating and s P 0 means both disks are rotating in the same direction.

Conclusions Here we studied flow between two stretchable rotating disks with heat transfer by Cattaneo-Christov heat flux theory. Main points are as follows:

Surface drag force Impact of different involved parameters on surface drag force at both disks is examined in Table 2. It is noted that skin friction coefficient enhances for larger values of Reynolds number Re, stretching parameters A1 and A2 while it shows decreasing behavior for porosity parameter b and rotation parameter s at lower and upper

 Radial and axial velocity profiles enhance at lower disk for larger A1 while for A2 the velocities increase near the upper disk.  Tangential velocity is increasing function of stretching and rotational parameters.  Temperature reduces for both Prandtl number and thermal relaxation parameter.  Surface drag force at both disks is less for larger rotational parameter.

References

Fig. 18. Behavior of ~ hðnÞ against k.

Table 2 Values of skin friction coefficient at lower and upper disks. b

A2

Re

A1

s

Cf 0

Cf 1

0.9 1 1.1 0.9

0.4

0.01

0.4

0.8

2.408261 2.408173 2.408101 2.607280 2.806356 2.408224 2.409185 2.807095 3.206236 2.401958 2.399810

2.409302 2.409297 2.409293 2.808393 3.207768 2.418446 2.429209 2.608926 2.808676 2.403112 2.401089

0.5 0.6 0.4

0.1 0.2 0.01

0.5 0.6 0.4

0.9 1

Table 3 Comparison of ~f 00 ð0Þ and g~0 ð0Þ with Stewartson [3] and Imtiaz et al. [9] when / ¼ A1 ¼ A2 ¼ 0 and Re ¼ 1.

s

~f 00 ð0Þ [3]

g~0 ð0Þ ½3

~f 00 ð0Þ ½9

g~0 ð0Þ ½9

~f 00 ð0Þ

g~0 ð0Þ

1 0:8 0:3 0 0:5

0.06666 0.08394 0.10395 0.09997 0.06663

2.00095 1.80259 1.30442 1.00428 0.50261

0.06666 0.08394 0.10395 0.09997 0.06663

2.00095 1.80259 1.30442 1.00428 0.50261

0.06666 0.08399 0.10395 0.09997 0.0667

2.00095 1.80259 1.30443 1.00428 0.50261

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