Computational approach on three-dimensional flow of couple-stress fluid with convective boundary conditions

Journal Pre-proof Computational approach on three-dimensional flow of couple-stress fluid with convective boundary conditions R. Ali, A. Farooq, A. Shahzad, A.C. Benim, A. Iqbal, M. Razzaq

PII: DOI: Reference:

S0378-4371(19)32242-3 https://doi.org/10.1016/j.physa.2019.124056 PHYSA 124056

To appear in:

Physica A

Received date : 7 May 2019 Revised date : 13 September 2019 Please cite this article as: R. Ali, A. Farooq, A. Shahzad et al., Computational approach on three-dimensional flow of couple-stress fluid with convective boundary conditions, Physica A (2020), doi: https://doi.org/10.1016/j.physa.2019.124056. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Journal Pre-proof 1   

Computational approach on three-dimensional flow of couple-stress fluid with convective boundary conditions

Technical University Dortmund, Department of Applied Mathematics, 40227 Dortmund, Germany b c

University of Central Asia, 310 Lenin Street, 722918, Naryn, Kyrgyz Republic

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R. Alia,b,1, A. Farooqc , A. Shahzadd , A. C. Benim e,f ,A. Iqbalg and M. Razzaqh

The Abdus Salam International Centre for Theoretical Physics ICTP, Strada Costiera 11, 34151 d

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

Center of Flow Simulation (CFS), Faculty of Mechanical and Process Engineering, Duesseldorf University of Applied Sciences, 156 Rather Street 23b, D-40476, Dusseldorf Germany f Institute of Thermal Power Engineering, Cracow University of Technology, Al. Jana Pawla II

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e

37, 31-864 Cracow, Poland g

Department of Mathematics, The National School and College System, I-10/1, Islamabad. Pakistan h

Syed Babar Ali School of Science and Engineering, Department of Mathematics, 54792 LUMS

Abstract: This investigation is concerned with the heat transfer analysis in a realistic three-

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dimensional flow of a couple-stress fluid. Convective boundary conditions have been adopted in the mathematical formulation keeping in view the physical consequences and detail examination

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of corresponding parameters. The boundary layer analysis has been invoked for simplification of the highly nonlinear system of differential equations. Afterward the governing system of partial differential equations are transformed into coupled nonlinear ordinary differential equations by appropriate set of transformations. The resulting system of differential equations are eventually

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solved for the series solutions employing HAM technique. Effects of various embedding parameters on the flow and heat transfer are discussed. It was found that couple stress parameter results in decay of velocity profile. While the heat transfer rate increases by increasing Biot number. Detailed comparison of the results are provided with already published data for limited

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cases. The new findings are in excellent agreement with the existing literature. Finally, the local Nusselt number and other physical quantities are analyzed for various pertinent parameters.

Corresponding Author: Ramzan Ali ([email protected])

Introduction and problem formulation

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Keywords: Heat transfer, convective boundary conditions, couple stress fluid.

The study of flows by stretching surfaces is important in several practical applications especially in the fields of chemical engineering, biological membrane, physiology and metallurgy. These

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includes the extrusion processes in polymer industry, fibers spinning, rubber sheet, glass blowing, manufacturing of plastic sheets, membrane interface, cell growth, just to mention few. Various researchers discussed the topic on two-dimensional flows over stretching surface subjects to various aspects of flow behavior and flow models. However not that much attention has been accorded to the three-dimensional flows induced by stretching surface due to the complexity of

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the three-dimensional flow models. These attempts further narrowed down whenever more realistic non-Newtonian fluids with thermos-physical characteristics [1 5] over a surface with

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convective boundary conditions are considered. These properties include Sorret and Dofour effects [3], heat transfer with vast variety of characteristics and mass transfer. The unsteady helical flow of Oldroyd-B by Jamil et al. [1] and further extension of the work to get results for similar but

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generalized model by Haitao and Jin [2]. The boundary conditions play key role in the understanding and modeling of the physical phenomena. In the recent attempts, we considered the convective boundary conditions. Few recent articles for three-dimensional flows in the absence of convective conditions can be found in the studies [6-8], where Gorla et al. [6] discussed the behavior of power law fluid over a stretching flat sheet. Ariel [7] studied and provided foundation

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with a basic three-dimensional flow past a stretching sheet, his attempt was to portray the perturb solution using homotropy perturbation method. For last couple of decades, his pioneering work on stretching surfaces attracted many researchers from diverse fields such as biomedical science,

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engineering and computational fluid dynamics.

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In this process a series wavelets solution is attempted [15] for MHD 3-D fluid flow in the presence of thermal radiation effects including slip across the boundary. The provided solutions are converging using similar nature of mathematical method. Khan et al. [16-17] proposed boundary layer solution past stretching sheet for nano-fluid model with variety of heat induction in flow

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behavior including heat generation and heat-absorption.

The couple stress fluid theory was initially developed by Stokes [9]. In his ground-breaking work, he presented the simplest generalization of the classical viscous fluid theory. Couple stress fluids consisting of rigid randomly oriented particles suspended in a viscous medium. For-instance blood and lubricants mixed with polymer solutions. Our interest in this article is to proceed further for

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the couple stress fluid model and the three-dimensional flows over stretched bodies. The lower dimensional studies provided the foundation for the work, application in 3-dimension will make it

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worthy to investigate real-life problems using reliable computational methods. Thus, we have analyzed the heat transfer analysis in three-dimensional flow of a couple stress fluid over a stretching sheet. Convective boundary conditions have been utilized in the present

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investigation to make it more practically applicable. Various limiting results have been presented in the form of tables to study emerging physical phenomena. Moreover, homotopy analysis method (HAM) imbedded with numerical [10 14] is employed in computation of series solutions.

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HAM already employed for the series solutions of 3-D and 2-D models with different flow characteristics. Ali et al. [10] compared results obtained from numerical scheme and analytic approach of homotopy-type. The investigation shows that the results are at great agreement.

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However, HAM is assumed to be strong method with excellent convergent behavior for solving differential equations with complicated non-linear behavior and boundary conditions. In addition

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to convergence analysis, in the current work the effects of different embedding parameters on velocity components and temperature distribution are studied and shown graphically. We consider the steady and incompressible three-dimensional flow of couple stress fluid. Flow is

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induced by the stretching of the surface at z  0 whereas fluid occupies the space z  0 . Heat transfer analysis has been considered in the presence of convective boundary conditions. The governing problems can be written as u v w    0, x y z u

u u u  2u  4u v w   2   4 , x y z z z

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(2) (3)

v v v  2v  4v   v  w   2  , u z x y z z 4

T T T  2T u v w  m 2 , x y z z T u  ax, v  by, w  0,  k  hs (T f  T ) z u  0, v  0, T  T as z  .

(1)

(4)

at z  0,

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where u, v and w are the components of velocity along x, y and z directions, respectively. Moreover    /  denotes the kinematic viscosity,     /  the couple stress viscosity,  the density, T the temperature of fluid, T the temperature far away from the surface, T f is the temperature of hot fluid,  m the thermal diffusivity of the fluid and hs the heat transfer parameter.

(5)

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Using a



z , u  ax f ( ), v  ay g ( ), w   a  f ( )  g ( ),  ( ) 

T  T , T f  T

(6)

of



f   f 2  ( f  g ) f   Kf v  0, g   g  ( f  g ) g   Kg  0,    Pr  f  g    0, 2

v

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the continuity equation (1) is identically satisfied and Eqs. (2)  (6) become

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f (0)  0, g (0)  0, f (0)  1, g (0)  c, f ()  0, g ()  0,  (0)   (1   (0)),  ()  0,

(7) (8) (9) (10)

in which prime denotes the differentiation with respect to  and the constants a  0 and b  0. The ratio c, the Prandtl number Pr, the Biot number  and the couple stress parameter K are

c  b / a, K   a / 2 , Pr 

 , m



hs  . k a

(11)

xq w , k (T f  T )

qw  k (

T ) z 0 z

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Nu x 

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Having the local Nusselt number and heat transfer q w as follows (12)

the dimensionless expression of local Nusselt number is

Nu   (0), Re1x/ 2

(13)

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where Re1x/ 2  ux / v is the local Reynolds number. Series solutions

We choose the initial guesses and the auxiliary linear operators in the following forms









f 0 ( )  1  e  , g 0 ( )  c 1  e  ,  0 ( ) 

 exp( ) , 1 

(14)

Journal Pre-proof 6    (15)

L f  f   f , L g  g   g , L      . The auxiliary linear operators satidfies the following properties 







(16)

of





L f (C1  C 2 e  C3e )  0, L g (C 4  C5 e  C 6 e )  0, L (C 7 e  C8 e )  0,

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in which Ci (i  1  8) are the arbitrary constants. The zeroth order deformation problems are given as follows:

1  p  L f  fˆ ( ; p) 







(17)

1  p  L g gˆ ( ; p)  g 0 ( )  p g N g  fˆ ( ; p), gˆ ( ; p),

(18)







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f 0 ( )  p f N f fˆ ( ; p ), gˆ ( ; p ) ,



1  p  L ˆ( ; p)   0 ( )  p  N fˆ ( ; p), gˆ ( ; p), ˆ( , p) ,

(20)

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fˆ (0; p)  0, fˆ  (0; p)  1, fˆ  (; p)  0, gˆ (0; p)  0, gˆ  (0; p)  c, gˆ  (; p)  0,

(19)

fˆ  (; p)  0, gˆ  (; p)  0, ˆ (0, p)   [1   (0, p)], ˆ(, p)  0,

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 3 fˆ ( , p )  fˆ ( , p )  N f [ fˆ ( , p ), gˆ ( , p )]      3     2 fˆ ( , p ) ( fˆ ( , p )  gˆ ( , p ))  K fˆ  ( , p ), 2 

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(21)

(22)

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



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 3 gˆ ( , p )  gˆ ( , p )    N g [ fˆ ( , p ), gˆ ( , p )]     3     2 gˆ ( , p ) ( fˆ ( , p )  gˆ ( , p ))  K gˆ  ( , p ),  2

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ˆ  2ˆ( , p ) ˆ ( , p )  gˆ ( , p )  ( , p ) N [ˆ( , p), fˆ ( , p), gˆ ( , p )]  f  Pr   2

(23)

(24)

where p is an embedding parameter,  f ,  g and   are the non-zero auxiliary parameters and

fˆ ( ; 0)  f 0 ( ), fˆ ( ;1)  f ( ),

gˆ ( ; 0)  g 0 ( ), gˆ ( ;1)  gˆ ( )

ˆ( , 0)   0 ( ), ˆ( , 1)   ( )

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N f , N g and N the nonlinear operators. For p  0 and p  1 we have

(25) (26) (27)

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and fˆ ( , p ), gˆ ( , p) and ˆ( , p) vary from f 0 ( ), g 0 ( ),  0 ( ) to f ( ), g ( ) and  ( ) when

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p varies from 0 to 1 . By Taylor's series expansion, we obtain

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 1  m fˆ ( ; p ) fˆ ( , p)  f 0 ( )   f m ( ) p m , f m ( )  m 1 m!  m



gˆ ( , p )  g 0 ( )   g m ( ) p m , g m ( )  m 1

1  m gˆ ( ; p ) m!  m

,

(28)

,

(29)

p 0

p 0

Journal Pre-proof 8    

ˆ( , p)   0 ( )    m ( ) p m ,  m ( )  m 1

1  mˆ( ; p ) m!  m

,

(30)

p 0

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where the convergence of above series strongly depends upon h f , hg and h . Considering that

h f , hg and h are selected properly so that Eqs. (28)  (30) converge at p  1 therefore 

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fˆ ( )  f 0 ( )   f m ( ), m 1



gˆ ( )  g 0 ( )   g m ( ),



ˆ( )   0 ( )    m ( ). m1

mth-order deformation problems

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m 1

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The subjected problems at the mth order are given by

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L f  f m ( )  f m1 ( )   f R f , m ( ),

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L g g m ( )   m g m1 ( )   g Rg , m ( ),

L  m ( )   m  m1 ( )    R , m ( ),

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f m (0)  f m (0)  f m ()  f m ()  g m (0)  g m ()  g m ()  0,





m 1 k 0



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Rmf ( )  f m1  Kf m1    f m1k  g m1k  f k  f m 1k f k ,

of

 m (0)   m (0)  0,  m ()  0,



m 1

Rmg ( )  g m1  Kg m1    f m1k  g m1k  g k  g m 1k g k , k 0





m 1

Rm ( )   m 1  Pr   f m1k  g m1k  k

0 1

m  

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k 0

m  1, m  1,

Convergence analysis and discussion

To find the convergence of homotopy solutions we have plotted the   curves for the functions

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f , g and  respectively. It is found that range for admissible values of  f ,  g and   are  1.7  ( f ,  g ,   )  0.2 . Further the series solutions converge in the whole region of 

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(0    ) for  f   g  0.7    .

The velocity fields f , g  and temperature profile  are presented to see the influence of various emerging parameters. Fig 1. shows the variation of couple stress parameter K on component of

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velocity f . It is observed that an increase in K results a decrease in f . The influence of couple stress parameter on second component of velocity g  is plotted in Fig. 2. It is clear from this Fig that the behavior of K on g  is similar to that of K on f . Since an increase in K means an increase in extra stresses which adds an extra force to retard the flow. Fig. 3 depicts the variation

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of Biot number  on temperature profile  ( ). From this Fig. we see that the temperature and the thermal boundary layer thickness are increasing functions of  Because an increase in  results an increase in heat transfer rate from the surface which finally rise the temperature. Variation of

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Prandtl number Pr on dimensionless temperature profile  ( ) is plotted in Fig. 4. Here we see

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that for large values of Prandtl number, both the temperature profile and thermal boundary layer thickness decreases.

Table 1 shows that 20th order approximations are sufficient for the velocity fields whereas 25th order of approximations is required for the temperature field. In order to validate our results, we

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have given a comparative study of presents results with previous published results in Table 2. for viscous flow. It is found that HAM solution in a limiting case of present study has a good agreement with the exact and HPM solutions given in ref. [7] . Table 3 is constructed to present the numerical values of local Nusselt number for various values of Pr and  . Clearly an increase

Concluding remarks

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in Prandtl number and Biot number cause an increase in the magnitude of local Nusselt number.

In this paper we focus the flow and heat transfer in a couple-stress fluid over a stretching surface

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in the presence of convective boundary conditions. The system of nonlinear partial differential equations together with respective boundary conditions are transformed to system of coupled nonlinear ordinary differential equations. An analytic scheme was developed to find convergent

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series solutions. The obtained numerical solutions are further verified for limited number of special cases, they result are quite promising and with good agreement with existing literature. Few of the main findings of present attempt are as follows: 

Velocity components f  and g  are decreasing functions of K .

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

Both the temperature field  and thermal boundary layer thickness are decreased by increasing Pr .

Both the temperature field  and thermal boundary layer thickness increase by increasing  .



Magnitude of local Nusselt number increases by increasing Pr and  .

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

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Acknowledgements

The authors would like to acknowledge the reviewers for their valuable comments, which substantially improve the quality of work.

Ramzan Ali would like to acknowledge the School of Arts and Science, University of Central

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Asia for providing an opportunity to visit collaborators.

References

[1] M. Jamil, C. Fetecau and M. Imran, Unsteady helical flows of Oldroyd-B fluids, Comm. Nonlinear Sci. Numer. Simulat., 16 (2011) 1378  1386.

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[2] Q. Haitao and H. Jin, Unsteady helical flows of a generalized Oldroyd-B fluid with fractional derivative, Nonlinear Analysis: Real World Applications 10 (2009) 2700  2708.

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[3] S. Wang and W. C. Tan, Stability analysis of Soret-driven double-diffusive convection of Maxwell fluid in a porous medium, Int. J. Heat Fluid Flow, 32 2011  88  94. [4] M. M. Rashidi, S.A. M. Pour and S. Abbasbandy, Analytic approximate solutions for heat

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transfer of a micropolar fluid through a porous medium with radiation, Comm. Nonlinear Sci. Numer. Simulat., 16 (2011) 1874  1889. [5] R. Ellahi, E. Shivanian, S. Abbasbandy, S. U. Rahman and T. Hayat, Analysis of steady flows in viscous fluid with heat/mass transfer and slip effects, Int. J. Heat Mass Transfer, 55 (2012)

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6384  6390. [6] R. S. R. Gorla, V. Dakappagari and I. Pop, Three-Dimensional Flow of a Power-Law Fluid Due to a Stretching Flat Surface, ZAMM J. Applied Math. Mech., 75 (1995) 389  394.

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method, Comp. and Math. with App. 54 (2007) 920  925.

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[7] P. D. Ariel, Three-dimensional flow past a stretching sheet and the homotopy perturbation

[8] T. Hayat and M. Awais, Three-dimensional flow of an upper-convected Maxwell (UCM) fluid, Int. J. Num. Methods Fluids, 66 (2011) 875  884.

[9] V. K. Stokes, Couple stress in fluids, Phys. Fluids 9 (1966) 1709  1715.

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[10] R. Ali, A Shahzad, M. Khan, M. Ayub, Analytic and numerical solutions for axisymmetric flow with partial slip. Engineering with Computers, 2016, 32 (1), 149-154. [11] A. Farooq, R. Ali, A.C. Benim, Soret and Dufour effects on three dimensional Oldroyd-B fluid. Physica A: Statistical Mechanics and its Applications 503, 345-354

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[12] A. Shahzad, R Ali, M Hussain, M Kamran, Unsteady axisymmetric flow and heat transfer

41.

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over time-dependent radially stretching sheet, Alexandria Engineering Journal, 2017 56 (1), 35-

[13] T. Hayat, S. Ali and H. H. Alsulami, Influence of thermal radiation and Joule heating in the flow of Eyring-Powell fluid with Soret and Dufour effects, J. Appl. Math. Tech. Phys., 57(6)

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(2016) 1051-1060.

[14] T. Hayat, S. Ali, M. Awais and A. Alsaedi, Joule heating effects in MHD flow of Burger’s fluid, Heat Transfer Research, 47(12)(2016) 1083-1092. [15] M. Usman, T. Zubair, M. Hamid, R.U. Haq, W. Wang, Wavelets solution of MHD 3-D fluid

Journal Pre-proof 13   

flow in the presence of slip and thermal radiation effects Physics of Fluids 30 (2), 023104 [16] W.A. Khan, and I. Pop, Boundary layer flow of a nanofluid past a stretching sheet,

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International Journal of Heat and Mass Transfer, 53, 2010, 2477-2483

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[17] W.A. Khan, M. Khan, and R. Malik, Three dimensional flow of an Oldroyd B nanofluid

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towards a stretching surface with heat generation/ absorption, Plos One, 9, 2014, e105107.

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Figure 1: Influence of K on f .

Figure 2: Influence of K on g  .

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Figure 3: Influence of  on  ( ) .

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Figure 4: Influence of Pr. on  ( ) .

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Table 1: Convergence of the HAM solutions for different order of approximations when c  0.5,

𝑔″ 0 0.476500000 0.466965750 0.463416006 0.462701549 0.462701661 0.462701881 0.462701881 0.462701881

𝜃′ 0 0.316666667 0.307822222 0.298205077 0.297574624 0.297532205 0.297529804 0.297529660 0.297529660

Pr e-

p ro

𝑓″ 0 1.053000000 1.076631500 1.093470088 1.094696432 1.094701412 1.094701868 1.094701868 1.094701868

order of approximation 1 2 5 10 15 20 25 30

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K  0.1, Pr  1,   0.5.

Table 2: Comparison of HAM solution with HPM (Ariel [7] ) and exact solution (Ariel [7] ) for different values of c when K  0.

0.2 0.4 0.6 0.8

1.039495 1.075788 1.109946 1.142488

1.034587 1.070529 1.106797 1.142879

Exact[7] 1

HAM 0

 g (0) HPM[7] 0

1.039495 1.075788 1.109946 1.142488

0.148736 0.349208 0.590528 0.866682

0.158231 0.360599 0.600833 0.874551

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al

0.0

HAM 1

 f (0) HPM[7] 1

c

Exact[7] 0 0.148736 0.349208 0.590528 0.866682

Jo

1.0 1.173720 1.178511 1.173720 1.173720 1.178511 1.173720

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Table 3: Local Nusselt Number for various values of Pr and  .

0 .1

  (0) 0.0824965678

1 .0 1 .5

0.0880211802 0.0905334331

2 .0 1 .0

0 .1 0 .2

0.0920046972 0.0880211802 0.1572113575

0 .3 0 .4

0.2130318802 0.2590114942

of



p ro

Pr 0 .5

Pr e-

 

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al

 

Journal Pre-proof

Highlight The article, “Computational approach on three-dimensional flow of couple stress fluid with convective boundary condition” addresses 3-D real life application oriented physical model over a sheet. The revised version improve the quality of text, including the literature review, mathematical formulation and discussion.

urn

al

Pr e-

p ro

Mathematical model for 3-D flow Coupled Stress Fluid Convective boundary condition and boundary layer flow Accuracy of numerical results in comparison with exiting work Graphical discussion

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1. 2. 3. 4. 5.

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The following research findings can be highlighted: