Heat transfer analysis in carbon nanotube-water between rotating disks under thermal radiation conditions

Accepted Manuscript Heat transfer analysis in carbon nanotube-water between rotating disks under thermal radiation conditions

S. Mosayebidorcheh, M. Hatami PII: DOI: Reference:

S0167-7322(17)30615-3 doi: 10.1016/j.molliq.2017.05.085 MOLLIQ 7377

To appear in:

Journal of Molecular Liquids

Received date: Revised date: Accepted date:

10 February 2017 16 May 2017 19 May 2017

Please cite this article as: S. Mosayebidorcheh, M. Hatami , Heat transfer analysis in carbon nanotube-water between rotating disks under thermal radiation conditions, Journal of Molecular Liquids (2017), doi: 10.1016/j.molliq.2017.05.085

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ACCEPTED MANUSCRIPT ``

Heat Transfer Analysis in Carbon Nanotube-Water between Rotating Disks under Thermal Radiation Conditions S. Mosayebidorcheh*1, M. Hatami2* 1

Department of Mechanical Engineering, Esfarayen University of Technology, Esfarayen, North Khorasan, Iran

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2

Young Researchers and Elite Club, Najafabad Branch, Islamic Azad University, Najafabad, Iran.

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Abstract

In this paper, heat transfer in a carbon nanotube based fluid between two parallel rotating

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disks is investigated. The governing equations are solved by Least Square Method (LSM) using a

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semi-analytical code. Water is considered as the base fluid and two kind of carbon nanotube (CNTs), single-walled carbon nanotube (SWCNT) and Multi-walled carbon nanotube (MWCNT)

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are added as the second phase or additives to the base fluid. The problem is solved for different

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nanoparticles volume fraction and Reynolds numbers to study the effect of some parameters such as scaled stretching parameters (A1, A2), Rotation parameter (Ω), Radiation parameter (Rd) and

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thermal Biot numbers (γ1,γ2) on the skin friction factors and Nusselt numbers. Results show that in

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most cases, SWCNTs leads to more heat transfer and temperature profiles compared to MWCNTs.

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Keywords: Heat transfer; Thermal radiation; Carbon nanotube; Nanofluid; LSM; Nusselt number. Introduction

One of the applicable ways for increasing the heat transfer is using the nanofluids as a working fluid for conveying the heat from sources. Nanofluids are made from a base fluid such as water for first phase and solid nanoparticles suspended in it as second phase. Haghshenas Fard et al. 1

* Corresponding author, Tel/Fax:+98-919-743-0343 2

E-mails: [email protected] [email protected], [email protected]

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[1] compared the results of two-phase and single phase fluids in a circular tube, numerically. Also, Göktepe et al. [2] compared the single phase and two phase nanofluid modeling at the entrance of a uniformly heated tube and found higher accuracy for two-phase modeling. Mohyud-Din et al. [3] in an analytical study, considered the three dimensional heat and mass transfer with magnetic

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effects for the flow of a nanofluid between two parallel plates in a rotating system.

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Another three-dimensional flow of nanofluids study under the radiation has been analyzed by Hayat

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et al. [4] and Khan et al. [5]. They also computed and examined the effects of different parameters on the velocity, temperature, skin friction coefficient and Nusselt number of nanofluid flow.

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Hatami and Ganji [6] modeled the natural convection heat transfer of a non-Newtonian nanofluid

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flow between the parallel plates and Kefayati [7-10] simulated the natural and mixed convection of nanofluid in enclosures using an efficient numerical method called finite difference based Lattice

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Boltzmann Method (FDLBM). Also, Hatami et al. [11] analyzed the natural convection heat

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transfer of nanofluids in a circular-wavy cavity and optimized the cavity geometry using response surface methodology (RSM). Domairry and Hatami [12] analyzed the nanofluid treatment between

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the parallel plates by using an analytical method. More studies in the heat transfer and nanofluids

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heat transfers can be found in the literature [13-20]. Imtiaz et al. [21] modeled mathematically the heat transfer of nanofluids between rotating disks and discussed on different parameters such as

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rotation parameter on the results. In addition to numerical methods, there are some powerful mathematical or analytical methods for solving the nonlinear problems. Weighted Residual Methods or WRMs are some of user friendly methods among analytical methods while Least Square Method (LSM) is one of the accurate WRMs used in many applications [22-25]. Hatami and Ganji [26] used LSM to improve the thermal efficiency of circular convective–radiative porous fins by defining different section

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shapes. Hatami et al. [27] studied the thermal performance of longitudinal and compared the results for two different materials, Si3N4 and Al by LSM and based on their study, Ghasemi et al. [28] improved the fins efficiency by a realistic temperature-dependent thermal conductivity and heat generation. As an application of fins in numerical studies, Hatami and Ganji [29] used LSM to

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increase the heat rejected from the channel to nanofluids in a micro-channel heat sink and Ghasemi

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et al. [30] made a validation of this study by other analytical methods presented by Hatami et al.

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[31-34] for heat transfer and fluid flow modeling. In the present study, LSM is used to find the thermal treatment of carbon nanotube or CNT-water nanofluid treatment between rotating and

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stretching disks considering radiation effects. Also, effect of some physical parameters such as

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rotation parameter or radiation parameter on the temperature and velocity profiles of nanofluids is discussed.

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Problem description

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Consider two parallel infinite disks as shown in Fig. 1 which between them is filled by

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incompressible carbon nanotube-water nanofluid. Single walled and Multi walled Carbon nanotube or SWCNTs and MWCNTs are used as their physical properties are presented in Table 1. Distance

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between disks is h while lower disk is located at z=0. Disks are rotating with different angular

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velocities Ω1 and Ω2. Furthermore disks are stretching in radial direction with different rats of a1 and a2. It is assumed that lower disk is heated in T0 temperature and upper disk is in T1 temperature. The governing equation in cylindrical coordinate system (r, θ, z) will be:

u u w   0 r r z

u

(1)

  2u 1 u  2u u  u u  2 1 p w    nf  2     r z r nf r r r z 2 r 2   r

3

(2)

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u

  u w    nf r z r

u

  2 w 1 w  2 w  w w 1 p w   nf  2    r z nf z r r z 2   r

(4)

  2T 1 T  2T  16 *T13   2T 1 T  2T      2     r r z 2  3k *  r 2 r r z 2   r

(5)

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p nf

T   T  w   knf u z   r

(3)

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 c 

  2 1   2       2 r r z 2 r 2   r

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Where p is pressure and T is temperature,  * is Stefan Boltzmann constant and k * is the mean

at at

z0

(6)

zh

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T  h1 T0  T  z T u  ra2 ,  r2 , w  0, knf  h2 T  T1  z u  ra1 ,  r1 , w  0, knf

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absorption coefficient. For described problem, the boundary conditions are:

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In this study based on Imtiaz et al. [21] report, following questions are used to obtain the CNT-

f

1   

2.5

(8)

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nf   f 1     CNT

(7)

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nf 

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water nanofluid properties:

(  c p )nf  (  c p ) f 1     (  c p )CNT 

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(9)

k  kf kCNT ln CNT knf kCNT  k f 2k f  kf k  kf kf ln CNT 1     2 kCNT  k f 2k f

(10)

1     2

By defining the following transformation function:

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u  r1 f    ,  r1 g   , w  2h1 f  

   

T  T1 , p   f  f T0  T1

(11)

 1 r2  z P     ,     2 2h  h 

Now Eqs. (2)-(6) will be changed to

1

1   

2.5

   1    CNT   f  

1

 0 g   Re  2 fg   2 f g

2

P  4 Re ff  

1   

2.5

T (13)

f 

(14)

   1    CNT   f  

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   1    CNT   f  

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And the boundary conditions

 1 1   (0)  , (1)  

kf knf

(16)

 2 (1), P(0)  0

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knf

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f (0)  0, f (1)  0, f (0)  A1 , f (1)  A2 , g (0)  1, g (1)   kf

(15)

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  c p CNT    1  knf    Rd    2 Re 1      f  0    Pr  k f  c  p  f   

 (0)  

(12)

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   1    CNT   f  

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2.5

 0  1    CNT  f

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1   

f   Re  2 ff   f 2  g 2  

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1

Where Reynolds (Re), Prandtl (Pr), Scaled stretching parameters (A1, A2), Rotation parameter (Ω), Radiation parameter (Rd) and thermal Biot numbers (γ1,γ2) are:

Re 

1h 2

f

, Pr 

 c  p

kf

f

f

(17)

a a , A1  1 , A1  2 1 1

2 16 *T13 hh hh  , Rd  , 1  1 ,  2  2 * 1 3k f k kf kf

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To make more simplified equation of Eq. (12) and eliminate pressure parameter (ε), it can be differentiated respect to η 1 2.5

   1    CNT   f  

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1   

(18)

f iv  Re  2 ff   2 gg    0

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Now pressure parameter (ε) can be determined by Eqs. (12) and (16), also pressure term (P) can be

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obtained by integrating Eq. (14) respect to η as Imtiaz et al. [21] reported the results.

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Least Square Method (LSM)

For conception the main idea of LSM, a differential operator D is acted on a function u to produce a

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function p [27]:

(19)

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D (u (x ))  p (x )

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where u is approximated by a function u , which is a linear combination of basic functions chosen from a linearly independent set. That is,

i 1

(20)

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n

u  u   c i i

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Now, when substituted into the differential operator, D, the result of the operations generally isn’t

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p(x). Hence an error or residual will exist: R (x )  D (u (x ))  p (x )  0

(21)

The main concept in WRMs is to force the residual to zero in some average sense over the domain. That is:



X

R (x ) W i (x )  0

i  1, 2,..., n

(22)

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Where Wi , the number of weight functions, is exactly equal the number of unknown constants ci in

u . If the continuous summation of all the squared residuals is minimized, the rationale behind the name can be seen. In other words, a minimum of S   R (x )R (x )dx   R 2 (x )dx

(23)

X

T

X

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In order to achieve a minimum of this scalar function, the derivatives of S with respect to all the

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unknown parameters must be zero. That is,

(24)

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S R  2  R (x ) dx 0 c i c i X

Comparing with Eq. (22), the weight functions are seen to be R c i

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Wi  2

(25)

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However, the “2” coefficient can be dropped, since it cancels out in the equation. Therefore the

to the unknown constants [27],

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R c i

(26)

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Wi 

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weight functions for the Least Squares Method are just the derivatives of the residual with respect

Results and discussions

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To have a comparable discussion on the results, two main parameters are reported; skin friction coefficients and Nusselt number. Skin friction coefficients can be calculated when shear stresses are known. For the lower rotating disk, shear stress in radial (τzr) and tangential directions (τzθ) are:

 zr  nf  z  nf

u z  z

 z 0

 z 0

 f r1 f (0)

1   

2.5

(27)

h

 f r1 g (0)

1   

2.5

(28)

h

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And the total shear stress will be (29)

 w   zr2   z2 And the skin friction factors for lower and upper disks (C1 and C2)

C2 

2

 w z h

 f  r 2 

2

1 Rer 1   

2.5

1



(30)

1/ 2

 f (0) 2   g (0) 2   

1/ 2

Rer 1   

2.5

 f (1) 2   g (1) 2   

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 f  r1 



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 w z 0

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C1 

(31)

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z 0

hqw k f T0  T1 

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Nux 2 

hqw k f T0  T1 

z h

(32)

(33)

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Nux1 

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Where local Reynolds number (Rer=rΩh/υf) and the Nusselts numbers for lower and upper disks are

z 0



T z

 qr z 0

, qw z 0

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qr

z 0

 knf

16 *T13 T 3k * z

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qw

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Wall heat flux (qw) and radiative heat flux (qr) will be obtained by

, qr

z 0

z h

z h

 knf



T z

 qr z h

16 *T13 T 3k * z

(34) z h

(35) z h

Finally the dimensionless form of Nusselt numbers are  knf  Nu1     Rd   (0) k   f 

(36)

 knf  Nu2     Rd   (1) k   f 

(37)

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To solve the governing equations by LSM, the trial functions are:

f  x   A1x   2A1  A2  x 2   A1  A 2  x 3  c1x 2  x  1  c 2x 3  x  1 2

(38)

2

g  x   1  x  x  c 3x  x  1  c 4x 2  x  1

(39)

  x   c 5  c 6  x  0.5     1c 5  0.5 1c 6   1  c 6  x  x  1

(40)

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  2 1c 5  2 2c 5   1c 6   2c 6  2 1  4c 6  x 2  x  0.5 x  1

 x  , g  x  and

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By minimization of residuals the values of c i and finally the following distributions of f

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  x  can be obtained for   0.2, Rd  0.3, A1  0.7, A2  0.8,   0.8,  1  0.4,  2  0.5 . f  x   0.7x  2.2x 2  1.5x 3  0.0070x 2  x  1  0.0149x 3  x  1 2

(41)

2

(42)

  x   0.4696  0.0399x  0.0001x  x  1  0.0015x 2  x  0.5 x  1

(43)

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g  x   1  0.2x  0.0041x  x  1  0.0025x 2  x  1

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After solving the equations by LSM the results are presented here to show the effect of different parameters on the velocity and temperature profiles. Table 2 show the high accuracy of LSM

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compared to numerical methods in all profiles and the error of LSM is shown via Table 3 which

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confirm the accuracy of the trial functions. Results are presented for two kinds of carbon nano tubes (CNTs) called single walled (SWCNTs) and Multi walled (MWCNTs). Fig. 2 shows the effect of the number

on

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Reynolds

the

velocity

and

temperature

profiles

of

SWCNTs-water

when

  0.1, Rd  0.5, A1  0.9, A2  0.3,   0.5,  1  0.7,  2  0.3 . As seen in these figures, increasing the Re makes a decrease in all velocity profiles (radial velocity profile f’(η), axial velocity f(η) and tangential

velocity profile g(η)) near the lower disk while it has increase near the upper disk. This is due to increase in the inertial effects due to the rotation of lower plate in high Reynolds which causes the flow to be slow. It must be mentioned that negative values of velocity demonstrate that upper disk is moving

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faster than the lower disk. This figure also says that increasing the Re causes a reduction in nondimensional temperatures profile in whole domain between the disks. Fig. 3 compares the results of SWCNT and MWCNT which confirms that although these two kinds of CNTs have approximately close to each other tangential velocity profile, but SWCNTs have larger temperature profiles. Fig. 4

more

rapidly

for when φ is enhanced. It is due to the fact that by increasing the

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increases

temperature

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is depicted to show the effect of nanoparticles volume fraction of SWCNTs. Fluid

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volume fraction of nanoparticles, the thermal conductivity and thermal boundary layer are enhanced. Tangential velocity profile for lower nanoparticles volume fractions is larger.

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Figs. 5 and 6 demonstrate the effect of scaled stretching parameters, A1 and A2, respectively. As

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seen, A1 and A2 have a reverse effect on the profiles. By increasing the A1, temperature profile decreased while axial and tangential velocities profiles increased. The radial velocity profile, f’(η)

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have different treatment near the upper and lower disks as seen in the figure. Effect of Rotation

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parameter (Ω) on the tangential velocity profile is shown in Fig. 7. From the physical view point, when Ω < 0 means both disks rotate in opposite directions, Ω=0 stands for the stationary upper disk

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and Ω > 0 means disk rotation is in the same direction. Here, both disks rotation is in the same

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direction and by increasing this value, tangential velocity also increases. Effect of thermal Biot numbers (γ1 and γ2) on the temperature profiles are demonstrated via Fig. 8 and 9, respectively.

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These figures also confirm that γ1 and γ2 have different effect on the temperature profiles, i.e. increasing the γ1, increases the temperature profile, while for γ2 decreases. Radiation parameter (Rd) influence on temperature profile is depicted in Fig. 10. Fig. 11 shows the effect of Re and φ on the skin friction factors (C1 and C2 in Eqs. (30)-(31)) at the same time. For both skin friction contours larger values occur in larger nanoparticles volume fraction, but for C1 in the lower Re numbers and C2 in higher Re numbers. Fig. 12 is depicted for effects of A1 and A2 (at the same time) on the

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Nusselt numbers and skin friction factors. Both C1 and C2 are in maximum values when A1 and A2 are maximizing, while Nusselt numbers are in reverse treatments. Nu1 is in maximum value when the A1 is maximum and A2 is minimum while Nu2 is in maximum value when the A1 is minimum and A2 is maximum. 3D contours of Fig. 13 and 14 are depicted to show the effect of (  1 ,  2 ) and

T

(Re, Rd) on the Nusselt numbers, respectively. These figures confirm that to reach maximum heat

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transfer, or maximum Nusselt number, it is better to all these parameters be in their maximum

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possible values.

Conclusion

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In this paper, least square method or LSM code has been successfully applied to find the solution of CNT-water nanofluid flow and heat transfer between two rotating and stretching disks.

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Two different kinds of nanoparticles are considered, single walled (SWCNTs) and multi walled

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(MWCNTs) to add to water as base fluid. In this study, effects of nanoparticles type, nanoparticles

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volume fraction, Radiation parameter, stretching parameters, rotating parameter, etc. on the Nusselt numbers and skin friction factors are investigated and it is found that SWCNTs have more

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temperature profiles. Also, results show that for both lower and upper skin friction factors (C1 and

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C2) larger values occur in larger nanoparticles volume fraction, but for C1 in the lower Reynolds numbers and C2 in higher Reynolds numbers. References

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velocity slip and nonlinear thermal radiation. J. Magnetism Magnetic Mater. 396, 31–37 (2015).

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(2015).

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6. Hatami, M. & Ganji, D.D. Natural convection of sodium alginate (SA) non-Newtonian nanofluid flow between two vertical flat plates by analytical and numerical methods. Case Stud. Therm. Eng.

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2, 14-22 (2014).

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7. Kefayati, GH. R. FDLBM simulation of entropy generation in double diffusive natural convection of power-law fluids in an enclosure with Soret and Dufour effects. Int. J. Heat Mass

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Trans. 89, 267–290 (2015).

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8. Kefayati, GH. R. FDLBM simulation of mixed convection in a lid-driven cavity filled with nonNewtonian nanofluid in the presence of magnetic field. Int. J. Therm. Sci. 95, 29-46 (2015).

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9. Kefayati, GH. R. Mesoscopic simulation of mixed convection on non-Newtonian nanofluids in a two sided lid-driven enclosure. Adv. Powder Tech. 26, 576-588 (2015). 10. Kefayati, GH. R. FDLBM simulation of entropy generation due to natural convection in an enclosure filled with non-Newtonian nanofluid. Powder Tech. 273, 176 – 190 (2015). 11. Hatami, M., Song, D. & Jing, D. Optimization of a circular-wavy cavity filled by nanofluid under the natural convection heat transfer condition, Int. J. Heat Mass Trans. 98, 758-767 (2016).

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12. Domairry & G., Hatami, M. Squeezing Cu–water nanofluid flow analysis between parallel plates by DTM-Padé Method. J. Mol. Liq. 193, 37-44 (2014). 13. Ahmadi, A.R., Zahmatkesh, A., Hatami, M. & Ganji, D.D. A comprehensive analysis of the flow and heat transfer for a nanofluid over an unsteady stretching flat plate. Powder Tech. 258, 125-

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133 (2014).

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14. Fakour, M., Vahabzadeh, A., Ganji, D.D. & Hatami, M. Analytical study of micropolar fluid

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flow and heat transfer in a channel with permeable walls. J. Mol. Liq. 204, 198-204 (2015). 15. Ghasemi, S.E., Hatami, M., Sarokolaie, A.K., & Ganji, D.D. Study on blood flow containing

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nanoparticles through porous arteries in presence of magnetic field using analytical methods.

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Physica E, Low-dim. Sys. Nano. 70, 146-156 (2015).

16. Ghasemi, S.E., Hatami, M., Mehdizadeh GH.R. & Ganji D.D. Electrohydrodynamic flow

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analysis in a circular cylindrical conduit using least square method. J. Elect. 72, 47-52 (2014).

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17. Rahimi-Gorji M., Pourmehran O.,, Hatami, M. & Ganji, D. D. Statistical optimization of microchannel heat sink (MCHS) geometry cooled by different nanofluids using RSM analysis, Eur.

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Phys. J. Plus 130-22 (2015).

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18. Hatami, M. & Ganji, D.D. Thermal behavior of longitudinal convective–radiative porous fins

(2014).

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with different section shapes and ceramic materials (SiC and Si3N4), Ceramics Int. 40, 6765-6775

19. Hatami, M. & Ganji, D.D. Investigation of refrigeration efficiency for fully wet circular porous fins with variable sections by combined heat and mass transfer analysis, Int. J. Ref. 40, 140-151 (2014). 20. Hatami, M., Mehdizadeh GH.R., Ganji, D.D. & Boubaker K. Refrigeration efficiency analysis for fully wet semi-spherical porous fins, Energ. Convers. Manage. 84, 533-540 (2014).

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21. Imtiaz, M., Hayat, T., Alsaedi, A. & Ahmad, B. Convective flow of carbon nanotubes between rotating stretchable disks with thermal radiation effects. Int. J. Heat Mass Trans. 101, 948 (2016). 22. M Hatami, S Mosayebidorcheh, D Jing, Thermal performance evaluation of alumina-water nanofluid in an inclined direct absorption solar collector (IDASC) using numerical method, Journal

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of Molecular Liquids 231, 632-639 (2017).

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23. S Mosayebidorcheh, MA Tahavori, T Mosayebidorcheh, DD Ganji, Analysis of nano-

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bioconvection flow containing both nanoparticles and gyrotactic microorganisms in a horizontal channel using modified least square method (MLSM), Journal of Molecular Liquids 227, 356-365

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(2017).

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24. M Hatami, S Mosayebidorcheh, D Jing, Two-phase nanofluid condensation and heat transfer modeling using least square method (LSM) for industrial applications, Heat and Mass Transfer, 1-

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12 (2017).

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25. S Mosayebidorcheh, M Hatami, T Mosayebidorcheh, DD Ganji, Optimization analysis of convective–radiative longitudinal fins with temperature-dependent properties and different section

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shapes and materials, Energy Conversion and Management 106, 1286-1294 (2015).

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26. Hatami, M. & Ganji, D. D. Thermal performance of circular convective–radiative porous fins with different section shapes and materials. Energ. Convers. Manage. 76, 185-193 (2013).

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27. Hatami, M., Hasanpour, A. & Ganji, D. D. Heat transfer study through porous fins (Si 3 N 4 and AL) with temperature-dependent heat generation. Energ. Convers. Manage. 74, 9-16 (2013). 28. Ghasemi, S. E., Hatami, M. & Ganji, D. D. Thermal analysis of convective fin with temperaturedependent thermal conductivity and heat generation. Case Stud. Therm. Eng. 4 1-8, (2014).

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29. Hatami, M. & Ganji, D. D. Thermal and flow analysis of microchannel heat sink (MCHS) cooled by Cu–water nanofluid using porous media approach and least square method. Energ. Convers. Manage. 78, 347-358 (2014). 30. Ghasemi, S. E., Valipour, P., Hatami, M., & Ganji, D. D. Heat transfer study on solid and porous

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convective fins with temperature-dependent heat generation using efficient analytical method. J.

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Cent. South Uni., 21(12), 4592-4598 (2014).

31. Hatami, M. & Domairry, G. Transient vertically motion of a soluble particle in a Newtonian

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fluid media, Powder Tech. 253, 481-485 (2014).

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32. Hatami, M. & Ganji, D.D. Motion of a spherical particle on a rotating parabola using Lagrangian and high accuracy Multi-step Differential Transformation Method, Powder Tech. 258,

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94-98 (2014).

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33. Hatami, M. & Ganji, D.D. Motion of a spherical particle in a fluid forced vortex by DQM and DTM, Partic. 16, 206-212 (2014).

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34. Dogonchi, A.S., Hatami, M. & Domairry, G. Motion analysis of a spherical solid particle in

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plane Couette Newtonian fluid flow, Powder Tech. 274, 186-192 (2015).

Author Contribution statement

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S. Mosayebidorcheh obtained the governing equations and solved the problem. He also prepared all figures and tables. M. Hatami wrote the main manuscript text and the discussion of the results.

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Additional Information

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No financial support.

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Table 1. Thermal properties of base fluid (water) and nanoparticles SWCNT 425 2600 6600

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MWCNT 796 1600 3000

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Water 4179 997.1 0.613

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Unit Jkg−1∙K−1 kg⋅m−3 Wm−1⋅K−1

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Properties Heat capacitance Density Thermal conductivity

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Table 2. The comparison of the results of LSM (Eqs. (41) to (43)) and numerical solutions for SWCNTs-water when   0.2, Rd  0.3, A1  0.7, A2  0.8,   0.8,  1  0.4,  2  0.5 .

1.0 0.984446 0.973178 0.962134 0.948498 0.930656 0.908146 0.881607 0.852748 0.824307 0.80

ED PT CE AC

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LSM (Eq. (43)) 0.469676 0.465665 0.461652 0.457647 0.453654 0.449673 0.445703 0.441736 0.437762 0.433768 0.429735

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0.0 0.049514 0.064167 0.052924 0.024723 -0.011524 -0.046932 -0.072667 -0.079596 -0.058326 0.0

LSM (Eq. (42)) 1.0 0.979652 0.959421 0.93929 0.919252 0.899284 0.879373 0.859504 0.839662 0.819832 0.80

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

LSM (Eq. (41)) 0.0 0.049568 0.064256 0.053006 0.024748 -0.011594 -0.047079 -0.072729 -0.079514 -0.058334 0.0

 x 

g(x)

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Numerical 0.469547 0.465545 0.461575 0.457647 0.453754 0.449876 0.445991 0.442075 0.438111 0.434095 0.430041

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Table 3. Error of applied method compared with numerical outcomes of Table 2 data Error of f(x) (%)

Error of g(x) (%)

Error of   x  (%)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00

0.000 0.10906 0.1387 0.15494 0.10112 0.60743 0.31322 0.08532 0.10302 0.01372 0.000

0.000 0.486974 1.413616 2.374305 3.083401 3.370956 3.168323 2.507126 1.534568 0.54288 0.000

0.027473 0.025776 0.016682 0.000 0.02204 0.04512 0.06458 0.07668 0.07966 0.07533 0.07116

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Fig. 1 Schematic of the problem [21]

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Fig. 2. The effect of the Reynolds number on the velocity and temperature profiles of SWCNTs-water when

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  0.1, Rd  0.5, A1  0.9, A2  0.3,   0.5,  1  0.7,  2  0.3

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Fig 3. The velocity and temperature profiles of SWCNTs-water and the MWCNTs-water noanofluids when

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  0.2, Rd  0.5, A1  0.9, A2  0.1,   0.5,  1  0.7,  2  0.3

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Fig. 4. Effect of the nano-particle volume fraction on the g   and    for the SWCNTs-water when

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Re  10, Rd  0.5, A1  0.6, A2  0.3,   0.2,  1  0.1,  2  0.4

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Fig. 5. The effect of the parameter A1 on the velocity and temperature profiles of MWCNTS-water when Re  10,   0.2, Rd  0.5, A2  0.7,   0.2,  1  0.1,  2  0.4 .

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Fig.6. The effect of the parameter A2 on the velocity and temperature profiles of MWCNTs-water when

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Re  2,   0.2, Rd  0.3, A1  1,  1  0.5,  2  0.2

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Fig. 7. Effect of the rotation parameter on the velocity profiles for SWCNTs-water when Re  2,   0.2, Rd  0.3, A1  1, A2  2,  1  0.5,  2  0.2

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Fig. 8. Effect of the parameter  1 on the temperature profile of the SWCNTs-water when

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Re  8,   0.1, Rd  0.4, A1  0.1, A2  0.9,  2  1 .

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Fig. 9. Effect of the parameter  2 on the temperature profile of the MWCNTs-water when

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Re  6,   0.15, Rd  0.7, A1  0.1, A2  0.9,  1  1 .

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Fig. 10. Effect of the thermal parameter Rd on the temperature profile for the MWCNTs-water when Re  6,   0.2, A1  0.1, A2  0.9,  1  1,  1  0.4 .

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Fig. 11. Variation of the skin friction with the Reynolds number and nanoparticle volume fraction for MWCNTs-water when   0.8, A1  0.4, A2  0.9,  1  1,  1  0.4 .

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Fig. 12. Variation of the skin friction and the Nusselt number with the parameters A1 and A 2 for the

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SWCNTs-water when   0.6, Re  2,   0.15,  1  0.1,  2  0.5 .

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Fig. 13. Variation of the Nusselt number with the parameters  1 and  2 for SWCNTs-water when

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  0.6, Re  2,   0.15, A1  0.5, A2  0.9 .

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Fig. 14. . Variation of the Nusselt number with the parameters Re and Rd for MWCNTs-water when   0.5,  1  0.5,  1  1,   0.2, A1  0.4, A2  0.8 .

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Highlight

> Heat transfer of the carbon nano-tube (CNT)-water between the rotating disks is analyzed. > The governing equations are transformed to a set of nonlinear BVPs. > An analytical solution of the problem is obtained using the Least Square Method.

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> Effects of physical factors such as nanoparticle volume fraction and rotating velocity are discussed.

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