Combined effects of viscous dissipation and Joule heating on MHD Sisko nanofluid over a stretching cylinder

Combined effects of viscous dissipation and Joule heating on MHD Sisko nanofluid over a stretching cylinder

Accepted Manuscript Combined effects of viscous dissipation and Joule heating on MHD Sisko nanofluid over a stretching cylinder Arif Hussain, M.Y. Ma...

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Accepted Manuscript Combined effects of viscous dissipation and Joule heating on MHD Sisko nanofluid over a stretching cylinder

Arif Hussain, M.Y. Malik, T. Salahuddin, S. Bilal, M. Awais PII: DOI: Reference:

S0167-7322(16)33790-4 doi: 10.1016/j.molliq.2017.02.030 MOLLIQ 6943

To appear in:

Journal of Molecular Liquids

Received date: Accepted date:

27 November 2016 9 February 2017

Please cite this article as: Arif Hussain, M.Y. Malik, T. Salahuddin, S. Bilal, M. Awais , Combined effects of viscous dissipation and Joule heating on MHD Sisko nanofluid over a stretching cylinder. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Molliq(2017), doi: 10.1016/ j.molliq.2017.02.030

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ACCEPTED MANUSCRIPT 1

Combined effects of viscous dissipation and Joule heating on MHD Sisko nanofluid over a stretching cylinder

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Arif Hussain a ,1 , M.Y. Malik a , T. Salahuddin b , S. Bilal a and M. Awais a a Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan. b Department of Mathematics, Mirpur University of Science & Technology, Mirpur 10250, AJK, Pakistan. Abstract: A mathematical model is presented to discuss magnetohydrodynamic Sisko fluid flow over a stretching cylinder with cumulative effects of viscous dissipation and Joule heating in the presence of nanoparticles. The mathematical modelling of the physical problem produced nonlinear set of partial differential equations which are transfigured into simultaneous system of ordinary differential equations with appropriate boundary conditions via suitable scaling group of transforms. Shooting technique is used to solve nonlinear set of flow govern equations. Comparison of numerical solution is made with previously reported data to validate the accuracy. The influence of governing parameters on the velocity, temperature and concentration is discussed under different parametric conditions. The numerically computed results are deliberated via graphs by selecting suitable values of the relevant physical parameters. Additionally to insight physical phenomenon in the vicinity of surface skin friction coefficient, local Nusselt number and local Sherwood number are computed and explicated through tables as well as graphs. Keywords: Nanoparticles; Sisko fluid model; Boundary layer flow; Stretching cylinder; MHD; Joule heating; Viscous dissipation; Shooting method.

Introduction

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Nanofluid is a modern type fluid which is the combination of base fluid and nano size particles. These nano particles are of different metals like copper, aluminum, silicon etc. The basic purpose of inserting nanoparticles in base fluid is to enhance the thermal conductivity of base fluid, because it is observed that many conventional heat transfer fluids like oil, ethylene glycol, water and engine oil which have naturally poor thermal conductivity. To enhance thermal conductivity of these fluids a lot of experiments have been performed, e.g. geometry of the problem is changed, different size metallic particles (e.g. mili, micro etc.) are droped into base fluid to enhance the thermal conductivity but these experiments did not gave the required results. Two decades ago, Chio [1  2] used nano sized particles in the base fluid and surprisingly found that thermal conductivity of nanofluid is many times greater than the base fluid. After this successful experiment, many theoretical and experimental studied are performed to analyze nanofluids, because nanofluid are utilized in many thermal engineering processes. Thermal conductivity of nanofluids is experimentally estimated by Kang et al. [3] and Yoo et al. [4] . They analyzed the thermal conductivity by considering different base fluids (water, glycol etc.) and nano particles (ultra-dispersed diamond, iron, aluminum, copper etc.). They found that thermal conductivity of nanofluid is much larger than their corresponding base fluid. Maiga et al. [5] studied the forced convection flow of nanofluids(water- Al 2 O 3 and Ethylene Glycol- Al 2 O 3 ) by considering different geometries. The results show that nanofluids have very better heat transfer rate than 1

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their respective base fluid. Also, the comparison of two nanofluids shows that Ethylene Glycol-  Al 2 O 3 has more capability of heat transfer than water-  Al 2 O 3 . Wang and Mujumdar [6] reviewed the work done by many researchers on nanofluid. They compared thermal conductivity, viscosity and heat transfer coefficient of many proposed models. They suggested that many other effects like Brownian motion, size and shape of nanoparticles should be considered to obtain more better results. Finally, they summarized that it is very difficult to predict heat transfer properties of nanofluids theoretically. Till now, for mathematical formulation of nanofluids two different approaches (Two-phase model and Buongiorno model) have been adopted in literature. Two-phase model elaborated the base fluid and nano particles particularly, but this model is not common. On the other hand Buongiorno [7] proposed a model based on continuum assumption of fluid with nanoparticles. He studied the influence of seven factors (inertia, Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effect, fluid drainage and gravity) in nanofluids and found that only Brownian diffusion and thermophoresis effects are significant. Additionally, this model is simpler and efficient in computational point of view, thus in present analysis Buongiorno model is considered to deliberate thermo-physical properties of nanofluids. Bachok et al. [8] discussed the boundary layer flow of nanofluid using Buongiorno model over seim-infinte flat plate. Kellor-Box technique was implemented to find solution of flow govern equations. The nanofluid flow over linearly stretching surface was firstly investigated by Khan and Pop [9]. The solution of similar equations was calculated numerically. Mabood et al. [10] studied MHD boundary layer flow and heat transfer of nanofluids over a nonlinear stretching sheet numerically. Recently, a lot of studies are presented to describe heat transfer characteristics of nanofluids over stretching surfaces. The numerical investigation of non-Newtonian Eyring-Powell nanofluid flow over linearly stretching sheet was reported by Malik et al. [11]. The influence of chemical reaction and applied magnetic field on visco-elastic nanofluid flow over bidirectional stretching sheet was investigated by Ramzan and Bilal [12]. The similarity equations were solved with the help of homotopy analysis method and results were discussed by varying interesting parameters. Narmgari and Sulochna [13] presented the similarity solution of MHD nanofluid flow over a permeable stretching sheet and used numerical method to solve governing equations. Shahmohamadi and Rashidi [14] discussed the squeezed flow of rotating nanofluid through a channel with lower porous stretching surface. The analytical solution was computed via VIM and compared with numerical data. Stagnation point flow of MHD nanofluid was analyzed numerically by Mabood et al. [15]. They studied the effects of chemical reaction, thermal radiations and volumetric fraction. Recently, Khan and Azam [16] described the thermophysical properties of unsteady Carreau nanofluid over permeable stretching surafce. The study of magnetic field in fluid flow gained considerable attention in last few decades, because MHD is frequently utilized in many areas such as petroleum, agriculture engineering and polymer industry. Also, since rate of cooling is an important factor corresponds to quality of product, hence magnetohydrodynamics fluids are uses in many manufacturing process to control rate of cooling. Additionally, cooling the walls of inside nuclear reactor, fusing metals in electric furnace, crystal growth and metal casting are some other applications of magnetohydrodynamics. Recently, MHD was also found very useful in bio engineering, because it is utilized in many disease diagnostic processes. Thus investigation of MHD flows is an important research area. 1

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Alfven [17] is the pioneer of magnetohydrodynamic flow, he discovered the electro-hydrodynamic waves in his experiment. This discovery expands a new field for researchers. After that many studies are performed to analyze MHD flows for different fluid models (Newtonian and non-Newtonian). Analytical solution of the problem addressing MHD flow of power law fluid over vertically stretching surface was found by Shahzad and Ali [18] . Shahzad and Ali [19] prolonged their previous work by considerring convective heat transfer of MHD power law fluid flow over vertically stretching surface. Akram and Nadeem [20] considered heat transfer problem on peristaltic motion of Jaffery fluid in an asymmetric channel with influence of induced magnetic field and found the closed form solution. Khan et al. [21] inspected the heat and mass transfer of MHD Falknar-Skan flow with mixed convection and convective boundary conditions. Akbar et al. [22] described the fluid characteristics of non-Newtonian Eyring-Powell fluid flow over stretching sheet under the impact of normally applied magnetic field. They concluded that intensity of magnetic field provides resistance to fluid motion. Ahmad et al. [23] studied the MHD Jeffrey fluid flow over stretching sheet by assuming convective boundary conditions. Malik et al. [24] investigated the MHD tangent hyperbolic fluid flow over stretched surface. The results were computed via numerical technique Kellor-Box method. The magnetic field dependent viscosity effects are elaborated on MHD nanofluid flow by Sheikholeslami and Rashidi [25]. The finite element technique was implemented to solve modelled equations and solution was discussed by considering different values of Hartmann number. Recently, Ali et al. [26] computed both numerical and analytic solutions of MHD viscous fluid flow over nonlinear stretching sheet. Many fluids like lubricating greases, oil, polymer sheets, rubber sheets, paints and emulsions are important industrial products which are non-Newtonian in nature. Thus investigation of non-Newtonian fluids is an important and essential part of literature. Non-Newtonian fluids are divided in many subclasses according to their behavior, because it is experimentally verified that non-Newtonian fluids have a lot of variations in their thermo-physical properties. Thus, different fluid models are suggested to analyze the thermo-physical properties of these fluids. For investigations of time-independent fluids flow power law model was found very useful, but this model fails to describe flow properties in high shear rate region. This deficiency overcomes in Sisko fluid model by adding high shear rate effects, this model was proposed by Sisko [27] in 1958. He considered three different greases and compared the results of Bingham and Ree-Eyring fluid models with experimental data. But he found that these models are fails to describe flow properties of lubricating greases. Thus he proposed the Sisko model and shows that outcomes of this model are matched with experimental data. Khan and Shehzad [28] studied the flow behavior of boundary layer Sisko fluid flow over stretching surface. The homotopy analysis technique was used to find solution of similarity equations. They found that Sisko fluid parameter accelerates fluid motion significantly. A similar study was performed by Khan and Shahzad [29], in this investigation they analyzed axisymmetric flow of non-Newtonian Sisko fluid. The Sisko fluid over stretching cylinder was firstly investigated by Malik et al. [30]. They discussed Sisko fluid flow over stretching cylinder with variable thermal conductivity and found numerical solution via shooting method. They discussed the problem by varying controlling physical parameters. Malik et al. [31  32] also examined the Sisko fluid flow over stretching cylinder by taking various 1

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thermophysical effects into account. Heat transfer has a remarkable role due to its widespread uses in almost all fields of science e.g. biomedicine, meteorology, plasma physics, astrophysics, physical chemistry, oceanography etc. Thus in literature a lot of work has been performed by considering heat transfer of fluids flow. Many engineering processes like heat exchangers, filtration of liquid metal, cooling of nuclear control etc. are the applications of heat transfer analysis. Also, it has been observed that viscosity of the fluid provides resistance to the motion, this process converted some amount of mechanical energy into thermal energy. This process is called viscous dissipation and it behaves like an energy source. The effects of viscous dissipation were firstly considered by Brinkman [33] in his analysis. He analyzed the heat effects in capillary flows. After this many studies were proposed which consist the effect of viscous dissipation in heat transfer. Vajravelu and Hadjinicolaou [34] studied the laminar flow of viscous fluid over stretching sheet with variable heat flux and heat generation/absorption. The effects of viscous dissipation were discussed in two different cases prescribed surface temperature (PST-Case) and prescribed wall heat flux (PHF-Case). Shigechi et al. [35] investigated the viscous dissipation effects on Coutte-Poiseuille laminar flow of Newtonian fluid between two parallel plates. Alinejad and Samarbakhsh [36] analyzed the viscous fluid flow over nonlinear stretching surface with viscous dissipation. It is observed that when viscous dissipation effects are taken into account, the fluid temperature rises. Recently, Malik et al. [37] examined the effects of viscous dissipation on Sisko fluid flow over stretching cylinder in the presence of transverse magnetic field. Another important factor affecting the heat transfer of hydrodynamics fluid flow is Joule heating. It is experimentally proved that when current passes through the hydrodynamics fluid flow, its temperature rises. This phenomenon occurs due to interaction between the moving particles and atomic ions which make the body like conductor. The charged particle accelerated and produced electrostatic potential energy. These particles strike with ions and release the energy which escalates temperature of the fluid. There are a lot of Joule heating applications such as incandescent light bulb, electric stoves, electric heaters, soldering irons, cartridge heaters, electric fuses, electronic cigarettes, vaporizing propylene glycol and vegetable glycerine. Also, some food processing equipment utilized Joule heating phenomenon. Hence due to such extensive use of Joule heating, researchers took a lot of interest to investigate this effect. Aissa and Mohammadein [38] examined the Joule heating effect on Micropolar fluid flow past a stretching sheet and found numerical solution. They show that Eckert number and magnetic parameter enlarges the temperature. The Joule heating effect on hydrodynamic natural convection viscous flow over flat plate was studied by Alim et al. [39] . They implemented Kellor-Box scheme to find solution and observed the influence of magnetic field and Joule heating parameters on velocity and temperature. Recently, Ref. [40  41] were discussed Joule heating effects on Couple stress and Jeffrey fluids. Additionally to figure out the physical problem in more realistic way many researchers investigated the combined influence of Joule heating along with viscous dissipation in heat transfer problems. The heat transfer of viscous fluid flow with viscous dissipation and Joule heating was firstly investigated by Fillo [42]. The combined impact of viscous dissipation and Joule heating on forced convective flow of viscous fluid was examined by Hossain [43]. El-Amin [44] was also discussed force convection flow of MHD fluid flow over horizontal cylinder along with effects of Joule heating and viscous 1

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dissipation. Recently, Das et al. [45] and Ibrahim and Suneeth [46] also identified the combined effects of viscous dissipation and Joule heating on MHD Newtonian fluid flow with various physical assumptions. In spite of the substantial importance of Joule heating and viscous dissipation in many engineering processes yet no study has been reported which discuss the MHD Sisko fluid flow with before mentioned effects. The motivation of present analysis is to fill this gap, thus in present study MHD Sisko nanofluid flow over a stretching cylinder with combined effects of Joule heating and viscous dissipation is examined. The self-similar transforms are employed to shift governing partial differential equations into ordinary differential equations. The attained equations are solved numerically by using shooting method in combination with Runge-Kutta Fehlberg method. The influences of involving parameters on interested physical quantities are discussed in detail by plotting graphs.

Mathematical formulation

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Let assume the axisymmetric, steady state, two dimensional flow of Sisko nanofluid over a stretching cylinder. The cylinder surface is considered at r  r0 and it is stretching in axial direction linearly with velocity U ( x)  cx , where c is any constant. A transverse magnetic field of strength B02 is applied normally to the surface of cylinder (along r direction). Also, fluid particles heated when current passes through them, so effects of Joule heating are assumed along with viscous dissipation in the energy equation. After imposing the boundary layer approximations the governing equations of the problem are transformed to   ru    rv    0, (1) x r

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T T    T   B02 2 a  u  b  u  v  u   r     x r r r  r   c p  c p  r   c p  r 

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u u a  u b u n nb u n1  2u  B02 u v  ( r ) ( ) ( ) 2 u , x r r  r r r  r  r r 

 DB (

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C T  D t  T2 . ) ( ) , r r T  r

(3)

D  T C C DB  C v  (r ) T (r ), x r r r r rT r r along with the boundary conditions u  U( x)  c x, v 0 a t r 0 r a n du

(4)

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T  Tw a t r  0r a n dT T

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ar t 

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(5)

C  Cw a t r  0r a n dC C ar t  . In Eqs. (1)  (5), u and v denote the axial and radial velocity components respectively, a, b 1

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and n are the material constants of Sisko fluid,  is the electrical conductivity,  is the density of base fluid, T denotes the temperature, C is the concentration of nanoparticles in base fluid,  is the thermal diffusivity, c p is the specific heat at constant pressure, DB is the Brownian motion diffusion coefficient, DT is the thermophoretic diffusion coefficient and  is the ratio of effective heat capacity of the nanoparticle material to heat capacity of the base fluid where  p is the density of nanoparticles. The temperature and volumetric fraction of

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nanoparticles at the surface of cylinder are denoted by Tw and Cw respectively while

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temperature and concentration away from the cylinder are symbolized by T and C . The stream function of velocity is defined as 1  1  (6) u ,v . r r r x To find similar solution of the governing equations i.e. Eqs. (1)  (4) , define the following set of similarity variables 1 1 r 2  r02  Rebn1 ,   xr0U Rebn1 f ( ), 2 xr0 (7) T  T C  C  ( )  ,  ( )  , Tw  T Cw  C where  is the dimensionless length, f is the dimensionless dependent variable,  is the dimensionless temperature,  denotes the concentration of nanoparticles and Reb denotes the 2n

Reynolds number which is defined as Reb   x Ub . After imposing the similarity transformations in Eqs. (1)  (4), the continuity equation vanishes while momentum, energy and nanoparticles concentration equations are modified to the following form n1 n1 A(1  2 ) f   n(1  2 ) 2 ( f )n1 f   (n  1) (1  2 ) 2 ( f ) n 

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n

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2n f f 2 A  f M f 0 , n 1

(1  2 )   2  

(8)

2n Pr f    MEc Pr f 2  n 1

A(1  2 ) Ec Pr   f   Ec Pr(1  2 ) 2   f  2

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n 1

 0,

2n Nt (1 2  )  2   L e P rf   ( 2   (1  2   ) ) n 1 Nb the boundary conditions in dimensionless form are

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f ( 0 ) 0 ,f  ( 0 ) 1,

  0  1, 

f ( )  0 ,     0,  (  )

(0)

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In the above system of simultaneous equations  represents the curvature parameter, M is the magnetic field parameter , Ec is the Eckert number , A denotes the material parameter, Prandtl number is denoted with Pr, Nb is the Brownian motion parameter, Nt is the thermophoresis parameter and Le denotes the Lewis number. These flow parameters are defined below 1  xB02 x Ux n1   R eb , M  , Ra e , r0 U a 2

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 DB (Cw  C )  D (T  T )  , Nt  T w  , Le  .  T DB

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Re n1 U2 xU 12n A  b , Ec  , Pr  Reb , Re a c p (Tw  T ) 

The physical quantities in the vicinity of surface i.e. skin friction coefficient C f , local Nusselt

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number Nu x and local Sherwood number Shx are defined in Eq. (13)  x qw x qm (13) C f  1 w 2 , N ux  a n d S hx  , k( Tw  T DB ( Cw C )  ) 2 U where  w is the stress at the surface, qw and qm are the heat and mass fluxes at the cylinder surface. Mathematically, these quantities are given below u u T C (14)  w  a( )r 0r  b ( n)r 0 r , q w k ( )r0 r a nqd m D B (  0r) r . r r r r Using similarity transformations into Eq. (13)  (14) , C f , Nu x and Shx are transformed into following dimensionless form 1 1  1 1 C f R enb1  A f ( 0 ) [ f ( 0n ) ] N, ux nb1Re  ( 0 ) Sa hxn d n b1  R  e ( 0(15) ). 2

Numerical solution

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The flow govern equations i.e. Eqs. (8)  (10) are highly nonlinear simultaneous ordinary differential equations. Thus, for solution of this system shooting technique with Runge-Kutta fifth order integration scheme is employed and solution is investigated against variations in curvature parameter  , material parameter A , magnetic field parameter M , Prandtl number Pr , Eckert number Ec, Nb the Brownian motion parameter, Nt the thermophoresis parameter and Le the Lewis parameter. The major advantage of shooting method over many numerical techniques is that this method has fifth order truncation error. Also, the process of solution computation is easier as compared to finite difference methods. In present analysis, well-known software Matlab is utilized for computations. Now, as Runge-Kutta Fehlberg method deals only initial value problems, thus governing equations i.e. Eqs. (8)  (10) are reduced to first order. For this purpose, these equations are re-arranged in the following form 1

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n1

(n  1) (1 2 2 ) (f  n ) A2 f   f

f  

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A(1 2  ) n (1  2 )  (f

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(16)

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Pr f    2   Ec Pr( Mf 2  A(1  2 )   f    (1  2 ) 2   f   ) , (1  2 ) n1

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2   n2n1 L e P r f   NN bt ( 2   (1  2  ) ) . (1 2 )

(17)

(18)

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   

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   

2n n 1

f f  M f ,

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Now to transform above defined equations into first order ordinary differential equations, defined new set of variables given below      f  y1, f   y2 , f   y3 , f  y3 ,   y4 ,   y5 ,   y5 ,   y6 ,    y7 and    y7 . (19)

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After inserting Eq. (19) into Eqs. (16)  (18), they are converted to following system of ordinary differential equations (20) y1  y2 , y2  y3 ,

) y22  A2 y3 n2n1 n1 2

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n 1

y1 y3  M y2 ,

(22)

)

y4  y5 ,

(23)

Pr y1 y5  2 y5  Ec Pr( My22  A(1  2 )   y3   (1  2 ) 2   y3  ) , (1  2 )

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y 

2n n 1

n

A(1 2  ) n (1 2 ) (y3

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 5

(n  1) (1 2 2 ) (y3

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y 

M

n1

 3

(21)

 7

y 

2n n 1

n1

2

n 1

y6  y7 ,

(25)

P r L e y1 y5  2 y7 NN bt ( 2 y5 (1  2  )y5 ) . (1 2 )

The subjected boundary conditions are reduce to y1 ( 0 ) 0 ,y2 ( 0) 1, y2  ( ) y40 , ( 0 )y4 1,   ( y)6

(24)

0 ,

( 0 )y6 1 a .n d

(26)

( (27) ) 0

To solve above set of differential equations along with corresponding boundary conditions shooting method is implemented which consists the following steps: 1

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1. First step is to choose appropriate value for the limit  , after choosing some random values 5 is found better limiting value. 2. Second and most important step is to select good initial approximations for y3 (0), y5 (0) and

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y7 (0). Initially, y3 (0)  1, y5 (0)  1 and y7 (0)  1 are chosen. 3. Hence above system of differential equations i.e. Eqs. (20)  (26) is converted into initial value problem. Then, it is solved with the help of Runge-Kutta-Fehlberg technique. 4. The computed solution will converge if absolute difference of given and computed values of y2 (), y4 () and y6 () is absolutely less than tolerance error i.e. 106. 5. If these differences are greater than tolerance error, then guessed values of y3 (0), y5 (0) and y7 (0) are modified by Newton method. This procedure is repeated until the computed solution meets the convergence criteria.

Results and discussion

Wang [47] 0.0656 0.1691 0.4539 0.9114 1.1854 3.3539 6.4622

CE

PT

Khan and Pop [9] 0.0663 0.1691 0.4539 0.9113 1.8954 3.3539 6.4622

AC

Pr 0.07 0.2 0.7 2 7 20 70

ED

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To figure out the physical problem more clearly, the numerical technique is implemented to solve governing differential equations, because it provides freedom to choose suitable values of involving flow parameters. The accuracy of the present computations is shown in table 1 by comparing values of  (0) with previously reported data for different values of Pr and ignoring the effects of all other parameters. It can be observed from table that results of present investigation are same with previous results calculated by Khan and Pop [9] , Wang [47] and Gorla and Sidawi [48] . Table 1: Comparison table of  (0) for different values of Prandtl number Pr , M  0,   0, Ec  0, Nt  0, Nb  0 and n  1. Gorla and Sidawi [48] Present Results 0.0656 0.0656 0.1691 0.1695 0.549 0.4539 0.9114 0.9114 1.1905 1.8954 3.3539 3.3538 6.4622 6.4621

Also, in this section the influence of curvature parameter  , material parameter A , magnetic field parameter M , Prandtl number Pr , Brownian motion parameter Nb , thermophoresis parameter Nt , Eckert number Ec and Lewis number Le on velocity, temperature and concentration profiles is briefly discussed. 1. Velocity profile

1

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Figure 1(b)

Figure 1(a) 1

1 0.9

 = 0.5, Pr = 2, Le = 2, n = 2, Ec = 0.05, Nb = Nt = 0.1.

0.8

A=2 A=5 A=7 A=2 A=5 A=7

0.7

0.6

0.5

0.5 0.4

M=0

0.3

0.2

0.2

0.1

0.1 2



3

4

Figure 1(c)

0

1

2



3

4

5

M

1 0.9 0.8

M M M M M M

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M = 0.2, A = 2, n = 2, Ec = 0.05,Pr = 2, Le = 2.

0.7 0.6 0.5

PT

f'()

0

5

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1

AN

0

Newtonian fluid A = 0, n = 1.

Sisko fluid A = 1, n = 2.

CR

0.3

0

IP

0.4 M = 0.5

 = 0.1  = 0.5  = 0.9  = 0.1  = 0.5  = 0.9

M = 0.2, Pr = 2, Le = 1, Ec = 0.05, Nt= Nb = 0.1.

0.7

f' (  )

f'()

0.6

0.8

T

0.9

0.4

= 0.1 = 0.6 = 1.5 = 0.1 = 0.6 = 1.5

 = 0.5

CE

0.3

=0

0.2

AC

0.1 0

0

1

2



3

4

5

Figures: 1(a), 1(b) and 1(c). Velocity profile variations against fluid momentum govern parameters A,  , M and n. Figure 1(a) exhibits the behavior of velocity profile for different values of material parameter A on MHD Sisko fluid (M  0.6) and Sisko fluid (M  0) . It is observed that Sisko fluid flows much faster than magnetohydrodynamic Sisko fluid, also from the graph it could be noticed that material parameter A enhances fluid velocity but higher values of A have not create considerable disturbance in the fluid movement i.e. the change in velocity profile is small. It holds 1

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physically because larger values of material parameter A reduce the viscosity of the non-Newtonian fluid. A comparison of Sisko fluid velocity is presented against the Newtonian fluid velocity by fluctuating curvature parameter  via Figure 1(b). The figure shows that Sisko fluid moves faster than Newtonian fluid. Also, curvature parameter provides a substantial growth in the velocity for both cases i.e. Sisko fluid as well as Newtonian fluid but in the vicinity of the cylinder surface curvature parameter has reverse effects on flow velocity. Figure 1(c) deliberates the variations in fluid movement for magnetic field parameter M and curvature parameter   0 (stretching plate case),   0.5 (stretching cylinder case). From the figure, it can be analyzed that the motion of Sisko fluid on cylinder surface is more rapid than plate surface. Additionally, this graph prevailed that magnetic field produced an appreciable resistance to fluid on both surfaces, this is due to the fact that magnetic field produced Lorentz force which aligned the particles of non-Newtonian fluid, hence it reduces the fluid velocity. 2. Temperature profile Figure 2(a)

Figure 2(b)

1

1

0.8

Ec = 0.1 Ec = 0.5 Ec = 0.9 Ec = 0.1 Ec = 0.5 Ec = 0.9

M

0.7

()

0.6 0.5

0.1

0.5

1

2



3

0.3

 = 0.5 =0

0.1 4

5

0

0

AC 1

= 0.1 = 0.6 = 1.5 = 0.1 = 0.6 = 1.5

0.2

CE

0

PT

0.2

0

M M M M M M

A = 2, Pr = 2, n = 2, Ec = 0.05, Nb = 0.1, Nt = 0.1, Le = 2.

0.7

Pr = 1

Pr = 2

0.3

0.8

0.4

ED

0.4

0.9

0.6

()

 = 0.5, A = 2, n = 2, Pr = 2, Nb = Nt = 0.1, Le = 2.

AN

0.9

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2



3

4

5

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Figure 2(c)

Figure 2(d)

1

1

0.9

0.9 Nt = 0.1 Nt = 0.4 Nt = 0.9 Nt = 0.1 Nt = 0.4 Nt = 0.9

0.8 0.7

0.7 0.6

()

0.5 0.4

0.5 0.4

Ec = 0

Nb = 0.5

0.3

IP

()

0.6

0.3

0.2

0.2

0.1

0.1 2



3

4

Figure 2(e)

M

1 0.9

CE

0.1 0

1

AC

0

2



3

4

5

Figure 2(f)

Pr = 1 Pr = 2 Pr = 4 Pr = 1 Pr = 2 Pr = 4

M = 0.1, A = 2, n = 2, Nb = Nt = 0.1, Ec = 0.05, Le = 2.

0.5 0.4

=0  = 0.5

0.3

0.2 Newtonian fluid

A = 0, n = 1.

3



0.6

()

Sisko fluid A = 1, n = 2.

0.3

2

1

0.7

PT

()

0.6

0.4

1

0.8

ED

0.7

0.5

0

Ec = 0.5

0.9

 = 0.1  = 0.5  = 0.9  = 0.1  = 0.5  = 0.9

M = 0.2, Pr = 2, Le = 2, Ec = 0.05, Nt = Nb = 0.1.

0.8

0

5

US

1

AN

0

CR

Nb = 0.1

0

Nb = 0.1 Nb = 0.4 Nb = 0.9 Nb = 0.1 Nb = 0.4 Nb = 0.9

M = 0.2, A = 2, n = 2, Pr = 2, Nt = 0.1, Le = 2,  = 0.5.

0.8

T

M = 0.2, A = 2, n = 2,  = 0.5, Ec = 0.05, Pr = 2, Le = 2.

0.2 0.1 4

5

0

0

1

2



3

4

5

Figures: 2(a), 2(b), 2(c), 2(d), 2(e) and 2(f). Fluid temperature behavior against the heat flow responsible parameters Ec, M , Pr,  , Nb, Nt and n. The temperature profile of non-Newtonian Sisko fluid is plotted in Figure 2(a) for different values of Eckert number Ec and Prandtl number (Pr  1, 2). This figure shows that Prandtl number declines the temperature of the fluid, also this figure provides the evident that Eckert number Ec has prominent effects on temperature profile. In addition, it can be seen that higher values of 1

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AC

CE

PT

ED

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Eckert number Ec rises fluid temperature more rapidly, because Eckert number accelerates the fluid particles which collide more frequently with each other and they release the heat, this causes enhancement in fluid temperature. The behavior of magnetic field parameter M on the temperature profile  ( ) is depicted via Figure 2(b) for curvature parameter   0 (stretching plate case) and   0.5 (stretching cylinder case). This figure shows that the fluid temperature in the case of stretching plate surface is less than stretching cylinder surface. Also, magnetic field parameter M increases the temperature, this is due to the fact that when magnetic field is applied to the fluid particles, it heated the surface and fluid particles. The effect of thermophoresis parameter Nt over the dimensionless temperature  ( ) is discussed in Figure 2(c) for Brownian motion parameter ( Nb  0.1, 0.5) . This figure indicates that temperature of the fluid is higher for Nb  0.1 as compared to temperature for Nb  0.5 . Additionally, it is analyzed that thermophoresis parameter Nt rises the thermal energy i.e. fluid temperature. Because thermophoresis phenomenon accelerates fluid and nanoparticles from hot region to clod region which causes escalates in the temperature of the fluid. Figure 2(d) shows the variations of field temperature  ( ) for different values of Brownian motion parameter Nb and ( Ec  0, 0.5). This graph exhibits that effects of Eckert number Ec are noteworthy on temperature profile i.e. when effects of Eckert number are assumed there is noticeable change in temperature is observed. Also, as Brownian motion tells the random movement of nanoparticles in the fluid, thus increasing Brownian motion parameter corresponds to collide particles more frequently, this collision of particles change thermal energy into kinetic energy, hence it enhances the temperature. The influence of curvature parameter  on temperature distribution  ( ) is shown in Figure 2(e) for Newtonian and non-Newtonian fluids respectively. It can be seen from figure that curvature parameter enhances temperature profile for both cases i.e. Newtonian fluid as well as Sisko fluid. Also, current graph shows that Sisko fluid is much hotter than Newtonian fluid. The fluctuations in Sisko nanofluid temperature against variations in Prandtl number Pr is shown via Figure 2(f) by considering   0 (stretching plate case),   0.5 (stretching cylinder case). It can be noticed that the fluid temperature on cylindrical surface is more as compared to flat surface, also in both cases larger values of Prandtl number Pr fall down the temperature substantially. This hold physically because Prandtl number Pr reduced the thermal conductivity i.e. capability of the fluid to transfer heat and hence temperature.

1

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3. Concentration profile Figure 3(a)

Figure 3(b)

1

1

0.9

0.9

0.6

0.5 0.4

0.5 0.4

0.3

0.3

Pr = 1

0.2

 = 0.5

0.2

Pr = 2

0.1 1

2

3



4

0

5

0

1

2

=0



3

4

5

M

AN

0

US

0.1 0

T

0.7

CR

()

0.6

Pr = 1 Pr = 2 Pr = 4 Pr = 1 Pr = 2 Pr = 4

M = 0.2, A = 2, n = 2, Nb = Nt = 0.1, Ec = 0.05, Le = 2.

0.8

()

0.7

Le = 2 Le = 5 Le = 7 Le = 2 Le = 5 Le = 7

IP

M = 0.2, A = 2,  = 0.5, Ec = 0.05, Nt = Nb = 0.1, n = 2.

0.8

Figure 3(c) 1

Figure 3(d) 1 0.9

M = 0.2,  = 0.5, n = 2, A = 2, Ec = 0.05, Pr = 2, Nb = 0.1.

0.8 0.7

Nt = 0.1 Nt = 0.4 Nt = 0.9 Nt = 0.1 Nt = 0.4 Nt = 0.9

0.8

0.4

Nb = 0.1 Nb = 0.4 Nb = 0.9 Nb = 0.1 Nb = 0.4 Nb = 0.9

Le = 4

0.6

()

0.5

M = 0.2, A = 2, n = 2, Nt = 0.1, Ec = 0.05, Pr = 2,  = 0.5.

0.7

PT

0.6

()

ED

0.9

0.5 0.4

CE

Le = 2

0.3

Le = 4 0.2

0

0

AC

0.1 1

2



3

Le = 2

0.3 0.2 0.1 4

5

0

0

1

2



3

4

5

Figures: 3(a), 3(b), 3(c) and 3(d). Variations in concentration of nanoparticles for altering values of involving thermophysical parameters Nb, Nt , Le, Pr and n. Figure 3(a) portraits the nanoparticles concentration  ( ) variations against Lewis number Le by assuming Prandtl number (Pr  1, 2). The concentration of nanoparticles becomes low when Prandtl number increases, also this graph illustrates that Lewis number Le declines the 1

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volumetric fraction of nanoparticles massively. This graph provides evident to the fact that the Lewis number explains the behavior of thermal to mass diffusivity, thus larger values of Lewis number reduces mass diffusivity and hence concentration of nanoparticles. Figure 3(b) examines the influence of Prandtl number Pr on concentration profile  ( ) for   0 (stretching plate case) and   0.5 (stretching cylinder case). This figures exhibits that concentration of nanoparticles is higher in the case   0.5 (stretching cylinder) as compared to   0 (stretching plate) case. Also, this figure deliberates that concentration profile  ( ) decreases considerably against progressing values of Prandtl number Pr in both cases. Figure 3(c) reflects the behavior of thermophoresis parameter Nt and Lewis number ( Le  2, 4) on concentration profile  ( ). It can be observed from the concentration curves that nanoparticles concentration is very high in the case of Lewis number Le  2. Additionally, this graph provides a valuable information that in both cases thermophoresis parameter Nt increases concentration profile  ( ). Figure 3(d) elaborates the impact of Brownian motion parameter Nb on the concentration profile  ( ) for Lewis number ( Le  2, 4) . It can be deduces from the figure that Brownian motion parameter Nb decreases concentration profile  ( ) remarkably in quantitative sense. This result holds practically because Brownian motion enhances the random movement of nanoparticles i.e. they disperse nanoparticles in irregular way and hence concentration becomes low.

ED

4. Wall friction factor (skin friction coefficient) Figure 4(a)

3.2

2.8

AC

2.6 2.4 2.2 2

 = 0.1  = 0.4  = 0.9  = 0.1  = 0.4  = 0.9

4

CE

3

= 0.1 = 0.6 = 1.3 = 0.1 = 0.6 = 1.3

-(A f '' (0)- (-f'' (0))n)

M M M M M M

PT

n=1

3.4

-(A f '' (0)- (- f '' (0))n)

Figure 4(b) 4.5

n=2

3.5

3

n=1 2.5

2

1.8

n=2 1

2

3

4

1.5

1

A

2

3

4

A

Figures: 4(a) and 4(b). Wall fraction factor curves against varying values of flow parameters A, 1

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M ,  and n. The next two graphs depict the impact of physical parameters on the skin friction coefficient. It is 

1

( A  1) f (0) Af (0)  f 2 (0)

M

-1.6455 -1.9163 -2.1818 -2.4416 -1.8265 -2.3528 -2.8454 -3.3201 -1.7640 -1.8265 -1.8872 -1.9461

ED

1 2 3 4 1

0.2

PT

1

M

0.1 0.2 0.3 0.4

-1.8221 -2.0223 -2.2284 -2.4396 -1.9549 -2.3716 -2.8531 -3.3594 -1.8704 -1.9549 -2.0374 -2.1182

AC

CE

0.1 0.4 0.7 1 0.3

A

n2

AN

n 1



US

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defined in Eq. 15, it can be seen that C f  Reb n1 . Thus for larger values of Reb the skin friction coefficient reduces in absolute sense. In this problem the skin friction coefficient depends upon three parameters A,  and M . Figure 4(a) depicts the influence of material parameter A and magnetic field parameter M on wall friction factor. This graph exhibits that wall friction factor enlarges against larger values of both parameters, while its values are decline verses power law index n. Figure 4(b) shows variations of skin friction coefficient due to progressing values of A and  . Current figure shows that curvature parameter escalates the factor of wall friction substantially in magnitude sense and similar results can be observed for larger values of material parameter. On the hand power law index decreases the wall friction factor slightly. Table 2: Values of skin friction coefficient for different values of parameters  , M and A for n  1 and 2.

Table 2 shows the impact of flow parameters A, M and  on skin friction coefficient for n  1 and 2 . It can be seen that numeric values of skin friction coefficient are larger for n  2. In addition, this table gives the information that larger values of all three parameters A, M and  enhances the skin friction coefficient in absolute sense. But effects of material parameter are prominent on wall friction factor as compared to others effects i.e. skin friction coefficient grows more rapidly against material parameter.

1

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5. Wall heat flux (Local Nusselt number) Figure 5(b)

Figure 5(a) 1.2

1.1

1

-'(0)

0.9 0.8 0.7

0.5

US

0.7

0.4 0.3

0.9

0.8

0.6

n=1 1

2

3

0.6

4

1

n=1

2

3

4

Pr

M

AN

Pr

n=2

= 0.1 = 0.6 = 1.3 = 0.1 = 0.6 = 1.3

IP

1

M M M M M M

CR

1.1

-'(0)

n=2

Ec = 0.1 Ec = 0.3 Ec = 0.5 Ec = 0.1 Ec = 0.3 Ec = 0.5

1.2

T

1.3

Figure 5(c)

Figure 5(d) 1.2

1.05 1

-'(0)

0.95 0.9

0.75 0.7 0.65 0.6

1

1

AC

0.8

0.55

1.15

1.05 1 0.95

n=2

0.9

0.85 0.8

0.75 0.7 0.65 0.6

n=1

2

Nb = 0.1 Nb = 0.2 Nb = 0.3 Nb = 0.1 Nb = 0.2 Nb = 0.3

1.1

CE

0.85

n=2

-'(0)

Nt = 0.1 Nt = 0.2 Nt = 0.3 Nt = 0.1 Nt = 0.2 Nt = 0.3

1.1

PT

1.15

ED

1.2

3

0.55 4

0.5

n=1 1

Pr

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3

Pr

4

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Figure 5(e) 1.5

1.3 1.2

T

1.1 1

IP

-'(0)

n=2

 = 0.1  = 0.4  = 0.9  = 0.1  = 0.4  = 0.9

1.4

0.9

CR

0.8 0.7

n=1 0.6 1

2

3

4

US

Pr

AC

CE

PT

ED

M

AN

Figures: 5(a), 5(b), 5(c), 5(d) and 5(e). Influence of pertinent parameters on wall heat flux Ec, M , Pr,  , Nb, Nt and n. Figures 5(a)-5(e) depict the variations of local Nusselt number i.e.  (0) for different values of existing parameters  , Ec , Pr , Nb and Nt. As Nusselt number is ratio of convective to conductive heat transfer, thus by increasing conductivity the values of Nusselt number decreases. Figure 5(a) explains the behavior of Ec and Pr on Nusselt number for power law index (n  1, 2). This graph reveals that the effects of Eckert number and Prandtl number are reverse on local Nusselt number, also Eckert number affected Nusselt number more significantly rather than Prandtl number. The reason behind that is the larger values of Eckert number decrease the enthalpy i.e. Tw  T , so it reduces heat transfer rate and hence Nusselt number. Also, Pr increases the values of Nusselt number due to fact that for larger values of Prandtl number Pr thermal conductivity of the fluid decreases. The behavior of magnetic field parameter M and Prandtl number Pr on  (0) is illustrated in Figure 5(b). This graph shows that the Pr enhances the rate of heat transfer while magnetic field parameter M causes reduction in the values of local Nusselt number. Also, one can see clearly that impact of magnetic parameter is not significant on surface heat transfer coefficient. Figure 5(c) displays the deviations in the local Nusselt number against thermopheroses parameter Nt , power law index n and Prandtl number Pr. Thermopheroses parameter influenced the local Nusselt number substantially which can be observed from the figure. Also, it can be deducted from the current figure that wall heat flux diminishes for larger values of thermopheroses parameter. Figure 5(d) illustrates the behavior of Brownian motion parameter N b and Prandtl number Pr on  (0) by selecting power law index (n  1, 2). The consequences of Brownian motion parameter N b on wall heat flux are quantitatively significant i.e. it reduces the local Nusselt number considerably (it can be observed from the figure). 1

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The combined effects of Pr and  on local Nusselt number are shown in Figure 5(e) by choosing power law index (n  1, 2). This figure depicts that power law index declines the local Nusselt number, also both curvature parameter and Prandtl number inclines wall heat flux in absolute sense. Table 3: Nusselt number table for different values of parameters  , Ec, M , Nt , Nb and Pr for n  1 , 2.

US

1 2 3 4 1

PT CE AC

T

IP

Nt 0.1

n 1  (0) 0.5749 0.6824 0.7976 0.9126 0.6022 0.4313 0.2602 0.0891 0.6449 0.8061 0.9021 0.9492 0.6526 0.6449 0.6375 0.6303 0.6449 0.5780 0.5160 0.4588 0.5695 0.5458 0.5229 0.5001

CR

1

Nb 0.1

M 0.2

AN

0.1 0.3 0.5 0.7 0.05

Pr

M

Ec 0.05

0.1 0.2 0.3 0.4 0.2

ED

 0.1 0.4 0.7 1 0.3

0.1 0.3 0.5 0.7 0.1

0.4 0.5 0.6 0.7

n2  (0) 0.6520 0.7646 0.8792 0.9900 0.6898 0.5446 0.3992 0.2537 0.7262 0.9420 1.0672 1.1306 0.7443 0.7262 0.7183 0.7108 0.7262 0.6517 0.5827 0.5190 0.6461 0.6210 0.5966 0.5729

Table 3 depicts the variations in Nusselt number for various values of involving physical parameters  , Ec , Pr , M , Nb and Nt. This table demonstrates that curvature parameters  and Prandtl number Pr have produced similar results i.e. both parameters increases the local Nusselt number, on the other hand Eckert number Ec, magnetic parameter M , Brownian motion parameter Nb, power law index n and thermopheroses parameter Nt causes reduction in local Nusselt number. This table also discloses that effects of Eckert number Ec, Brownian motion parameter Nb and thermopheroses parameter are prominent on coefficient of surface heat transfer. 1

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6. Wall mass flux (Local Sherwood number) Figure 6(a)

Figure 6(b)

2

1.6

Pr = 1 Pr = 2 Pr = 3 Pr = 1 Pr = 2 Pr = 3

1.8 1.7 1.6 1.5 1.4

-  ' (0)

1.4 1.3 1.2 1.1

1.3 1.2 1.1

1

1

0.9

0.9

0.8

0.8

0.7

0.7

0.6

0.6

n=2

0.5 0.1

0.2

0.3

0.4

n=1

0.5 0.1

n=2

0.2

0.3

0.4

Nb

AN

Nb

US

-'(0)

1.5

1.9

n=1

T

1.7

Le = 1 Le = 2 Le = 3 Le = 1 Le = 2 Le = 3

IP

1.8

CR

1.9

2

Figures: 6(a) and 6(b). Wall mass flux verses involving physical parameters Nb, Nt , Pr and

M

n.

AC

CE

PT

ED

The variations of physical parameters Nb , Pr and Le on local Sherwood number i.e.  (0) are illustrated in the Figures 6(a)-6(b). Sherwood number is the ratio of convective to diffusive mass transfer. Figure 6(a) manifests the combined consequences of power law index n, Brownian motion parameter Nb and Lewis number Le on local Sherwood number Shx . Both parameters Nb and Le accelerates the mass transfer but their effects on wall mass flux are not prominent which can be noticed from the figure. Finally, this figure displays that local Sherwood number is less in the case of n  2. The influences of Brownian motion parameter Nb and Prandtl number Pr on wall mass flux i.e.  (0) is shown by Figure 6(b) for power law index (n  1, 2). It can be experiences from the figure that local Sherwood number curves grows rapidly for larger values of Prandtl number Pr. Table 4: Local Sherwood number i.e.  (0) for different values of physical parameters Le, Nb, Nt and Pr for n  1 , 2.

Le

Pr

1 2 3 4 1 1

1

1

Nb 0.1

Nt 0.1

n 1  (0) 0.5353 0.7105 0.9362 1.1745 0.5353

n2  (0) 0.5369 0.7529 1.0248 1.3019 0.5369

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0.1 0.3 0.5 0.7 0.1

0.8513 1.1235 1.3568 0.5353 0.7004 0.7325 0.7456 0.1192 0.1391 0.1478 0.1960

0.4 0.5 0.6 0.7

0.9298 1.2566 1.5317 0.5369 0.7564 0.7993 0.8170 0.1251 0.1592 0.2159 0.2353

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. The variations of local Sherwood number for different values of involving thermophysical parameters n, Le, Nb, Nt and Pr are represented in table 4. This table explicates that the rate of mass transfer at the surface enhances against progressing values of all involving parameters n, Le, Nb, Nt and Pr .

Concluding remarks

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Present communication carried out speculative investigation of MHD Sisko nanofluid over axially stretching cylinder. In addition, combined effects of Joule heating and viscous dissipation are also factored into the analysis. The computational solution of the governing set of differential equations is attained by employing well-known numerical approach shooting method. To insight problem in the physical sense, feasible values of controlled parameters i.e. curvature parameter  , material parameter A , magnetic field parameter M , Eckert number Ec, Brownian motion parameter Nb, thermophoresis parameter Nt , Prandtl number Pr and Lewis number Le are selected. The solutions of dimensionless velocity, temperature and concentration have asymptotic behavior (it can be seen from the graphs). The main outcomes of the problem are:  The computed results revealed that curvature parameter  and material parameter A both have qualitively similar effects on dimensionless velocity f ( ), but in quantitative sense material parameter A is highly significant impact on fluid movement. On the other hand magnetic field parameter M retarded the fluid acceleration very rapidly. Also, momentum transport of Sisko fluid is much faster than Newtonian fluid.  An appreciable growth has been noticed in wall friction factor against all three momentum controlling parameters i.e. curvature parameter  , material parameter A and magnetic field parameter M .  Both Eckert number (viscous dissipation effect) and magnetic field parameter (Joule heating) substantially enlarges the temperature, a qualitively similar consequences were observed against Brownian motion parameter Nb, thermophoresis parameter Nt and curvature parameter  . But in quantifying sense, Eckert number has more prominent effects than resting parameters. On the other hand heat transfer trends verses Prandtl number Pr are different i.e. progressing values of Prandtl number corresponds to low temperature.  Conventional heat transfer coefficient progresses positively verses both curvature parameter 1

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 and Prandtl number Pr while Eckert number Ec, Brownian motion parameter Nb, thermophoresis parameter Nt and magnetic field parameter M have opposite effects. A relatively small growing behavior of nanoparticles concentration has been noticed for thermophoresis parameter Nt while impact of Brownian motion parameter Nb, Prandtl number Pr and Lewis number Le leads to reduce concentration. Wall mass flux enhances by varying all controlling parameters Nb, Nt , Pr and Le.

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Highlights  

Combined effects of viscous dissipation and Joule heating are discussed. Numerical solution is computed via shooting technique. Accuracy has been proved by comparing results with reported data.

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 

MHD Sisko Nanofluid is analyzed over stretching cylinder.

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