Results in Physics 7 (2017) 426–434

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Results in Physics

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Analysis of tangent hyperbolic nanofluid impinging on a stretchingcylinder near the stagnation point

http://dx.doi.org/10.1016/j.rinp.2016.12.0332211-3797/� 2016 Published by Elsevier B.V.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

⇑ Corresponding author.E-mail address: taimoor.salahuddin@math.qau.edu.pk (T. Salahuddin).

T. Salahuddin a,⇑, M.Y. Malik b, Arif Hussain b, Muhammad Awais b, Imad Khan b, Mair Khan b

aMirpur University of Science and Technology (MUST), Mirpur 10250 (AJK), PakistanbDepartment of Mathematics, Quaid-i-Azam University, Islamabad 44000, Pakistan

a r t i c l e i n f o

Article history:Received 14 October 2016Received in revised form 26 November 2016Accepted 23 December 2016Available online 05 January 2017

Keywords:Stagnation point flowTangent hyperbolic nanofluidStretching cylinderHeat generation/absorptionBoundary layerShooting method

a b s t r a c t

An analysis is executed to study the influence of heat generation/absorption on tangent hyperbolic nano-fluid near the stagnation point over a stretching cylinder. In this study the developed model of a tangenthyperbolic nanofluid in boundary layer flow with Brownian motion and thermophoresis effects are dis-cussed. The governing partial differential equations in terms of continuity, momentum, temperature andconcentration are rehabilitated into ordinary differential form and then solved numerically using shoot-ing method. The results specify that the addition of nanoparticles into the tangent hyperbolic fluid yieldsan increment in the skin friction coefficient and the heat transfer rate at the surface. Comparison of thepresent results with previously published literature is specified and found in good agreement. It isnoticed that velocity profile reduces by enhancing Weissenberg number k and power law index n. Theskin friction coefficient, local Nusselt number and local Sherwood number enhances for large values ofstretching ratio parameter A.� 2016 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://

creativecommons.org/licenses/by-nc-nd/4.0/).

Introduction

The layer in which the effects of viscosity are significant in theclose neighborhood of the surface is titled as boundary layer. Lud-wig Prandtl presented the concept of boundary layer on August 12,1904 at the third international conference in Germany. He dividedthe fluid equations into two categories, one inside the boundarylayer where the influence of viscosity is maximum and second out-side the boundary where the influence of boundary can be ignored.Boundary layers are pervasive in a great number of natural flowsand fluid dynamics engineering applications, such as solicitationsin liquid films in condensation procedure, polymer, extrusion ofplastic sheets, glass blowing, spinning of fibers, cooling of elasticsheets, the aerodynamic extrusion of plastic sheets, paper manu-facturing, the earth’s atmosphere, the surface of car, ship and airvehicles, etc. Sakiadis [1] was the forerunner who elaborated theconcept of two dimensional boundary layer of Newtonian fluidtowards a continuous moving solid surface. Crane [2] extended thisconcept into two-dimensional steady flow of a linearly stretchingsheet and got its exact solution in quiescent fluid. The boundarylayer mixed convection flow of heated stretching surface was dis-cussed by Chen [3]. He inspected the behavior different physical

parameters and plotted graphs for local Nusselt number and skinfriction coefficient. The effect of variable heat flux on two dimen-sional steady boundary layer flow was deliberated by Lin et al.[4]. They generate the solution in terms of hypergeometric func-tions and concluded that for smaller Prandtl number the boundarylayer is larger. Ali [5] investigated the boundary layer flow of a con-tinuously stretched surface with buoyancy effects. The boundarylayer stagnation point flow towards a stretching sheet was dis-cussed by Nadeem et al. [6]. Again, Nadeem et al. [7] investigatedthe boundary layer flow of second grade fluid towards a stretchingsheet with temperature dependent viscosity. Rangi et al. [8] stud-ied the heat transfer and boundary layer flow towards a stretchingcylinder with variable thermal conductivity.

The boundary layer flow of pseudoplastic fluids has greatimportance due to its extensive uses in solutions and melts of highmolecular weight polymers, emulsion coated sheets like photo-graphic films, polymer sheets, etc. Nadeem et al. [9] examinedthe influence of magnetic field and temperature dependent viscos-ity on the peristaltic flow of Newtonian incompressible fluid.Again, Nadeem et al. [10] studied the peristaltic flow of a magneto-hydrodynamic tangent hyperbolic fluid in a vertical asymmetricchannel under long wavelength and low Reynolds number approx-imation. Akbar et al. [11] analyzed the effects of chemical reactionsand heat transfer on tangent hyperbolic fluid treated through atapered artery. Nadeem et al. [12] examined the peristaltic motion

Fig. 1. Geometry of the problem.

T. Salahuddin et al. / Results in Physics 7 (2017) 426–434 427

of tangent hyperbolic fluid in a curved channel. Naseer et al. [13]evaluated heat transfer and boundary layer flow of tangent hyper-bolic fluid over a vertical exponentially stretching cylinder.

A base fluid comprises of nanoparticles deferred in conventionalheat transfer basic fluid with the length scales of 1–100 nm iscalled nanofluid. The nanofluid enhances the convective heattransfer coefficient and thermal conductivity of the base fluids.Oil, water and ethylene glycol are poor conductors of heat. In orderto increase the thermal conductivity of these fluids many tech-niques have been taken in description. One of these techniques isthe addition of nano-sized material particles in the liquid. Choiet al. [14] found that with the addition of the nanoparticles thethermal conductivity of the base fluid enhances twice. Das [15]concluded that with the inclusion of nanoparticles the thermalconductivity becomes temperature dependent. Buongiorno et al.[16] evaluated the nanofluid coolants for advanced nuclear powerplants. Malik et al. [17] analyzed the boundary-layer mixed con-vective flow of a nanofluid over a stretching plate. Khan et al.[18] investigated the numerical solution of nanofluid over a flatstretching surface. Akbar et al. [19] studied the MHD flow of nano-fluid over a vertical stretching plate with exponential temperature-dependent viscosity and also considerd buoyancy effects. Vajraveluet al. [20] calculated the effects of the nanoparticle volume fractionand heat transfer characteristics with combined effects of temper-ature dependent internal heat generation/absorption and thermalbuoyancy. Noghrehabadi et al. [21] examined the boundary layerand heat transfer flow of nanoparticles volume fraction with slipeffects. Akbar et al. [22] Analyzed the magnetic field analysis pastover a stretching sheet in a suspension of gyrotactic microorgan-isms and nanoparticles. Akbar et al. [23] investigated theDouble-diffusive natural convective MHD boundary-layer flow ofa nanofluid over a stretching sheet.

Heat generation/absorption is reflected as a significant factor innumerous physical glitches such as fluids having endothermic andexothermic chemical reactions. Heat generation/absorption isassumed to be temperature-dependent or space dependent. Lawr-ence et al. [24] investigated the boundary layer flow of viscoelasticfluid over a stretching sheet with internal heat generation/absorp-tion. Pavithra et al. [25] analyzed the heat transfer boundary layerflow of a dusty fluid over an exponentially stretching surface withcombined effects of internal heat generation/absorption and vis-cous dissipation. Noghrehabadi et al. [26] studied the entropy gen-eration of a nanofluid over an isothermal linear stretching platewith heat generation/absorption. Dessie et al. [27] examined theMHD boundary layer flow of Newtonian fluid with variable viscos-ity through a porous medium over a stretching sheet with viscousdissipation, heat source/sink and heat generation/absorption.Hakeem et al. [28] studied the effect of partial slip on hydromag-netic boundary layer flow in porous medium over a stretching sur-face with temperature and space dependent internal heatgeneration/absorption. Akbar et al. [29] examined the effects ofthermal radiation and variable thermal conductivity on the flowof CNTS over a stretching sheet with convective slip boundary con-ditions. Akbar et al. [30] examined the two dimensional Magneto-hydrodynamics flow of Eyring-Powell fluid. They noticed that byenhancing the intensity of the magnatic field as well as Eyring-Powell fluid parameter c decrease the velocity profile. Mehmoodet al. [31] depicted the Non Aligned point flow and heat transferof an Ethylene–Glycol and nanofluid towards stretching sheet.They noticed that Ethylene-based nanofluids have higher local heatflux than water-based nanofluids. Rana et al. [32] invistigated themixed convective oblique flow of a casson fluid with partial slip,internal heating and homogeneous–heterogeneous reactions.

The aim of the present analysis is to examine numerically, thestagnation point flow of tangent hyperbolic nanofluid over astretching cylinder. Buongiorno model is used to examine the heat

transfer due to nanoparticles. Moreover, heat generation/absorp-tion effects are encountered for tangent hyperbolic nanofluid. Forthis persistence proper similarity transformations are used todiminish governing equations to ordinary differential equations.The effect of thermophoresis, Brownian motion and Lewis numberfor nanofluid are deliberated through graphs. The near wall quan-tities, such as Sherwood number due to nanoparticle concentra-tion, Nusselt number due to heat transfer and local skin frictionfor velocity profile are discussed.

Mathematical formulation

Consider steady two-dimensional stagnation point flow of opti-cally dense tangent hyperbolic nanofluid towards a stretchingcylinder located at r ¼ R, where r is the coordinate normal to cylin-der. The cylinder is stretched with two equivalent and conflictingforces along x-axis with the velocity u ¼ ax

l , by keeping the originfixed as shown in Fig. 1. It is assumed that concentration and tem-perature at the wall is maintained at constant concentration Cw

and temperature Tw. Where C1 and T1 are ambient concentrationand temperature respectively. Under the boundary layer approxi-mation, the governing equations of the mass, momentum, energyand nanoparticles in cylindrical coordinates x and r are writtenas Ref. [12]

@ðruÞ@x

þ @ðrvÞ@r

¼ 0; ð1Þ

u@u@x

þv @u@r

¼ m ð1�nÞ@2u

@r2þð1�nÞ1

r@u@r

þnffiffiffi2

pC@u@r

@2u@r2

þ nCffiffiffi2

pr

@u@r

� �2 !

þUedUe

dx; ð2Þ

qcp u@T@x

þ v @T@r

� �¼ K

r@

@rr@T@r

� �þ s DB

@C@r

@T@r

� �þ DT

T1

@T@r

� �2" #( )

þ Qqcp

ðT � T1Þ; ð3Þ

u@C@x

þ v @C@r

¼ DB

r@

@rr@C@r

� �þ DT

T1

1r

@

@rr@T@r

� �; ð4Þ

the associated boundary conditions are

u ¼ axl; v ¼ 0; T ¼ Tw; C ¼ Cw at r ¼ R;

u ¼ UeðxÞ ¼ bxl; v ¼ 0; T ¼ T1; C ¼ C1 as r ! 1:

ð5Þ

Here u and v are the velocity components along x and r direc-tions respectively, Q is the temperature dependent volumetric rateof heat source when Q > 0 and heat sink when Q < 0, dealing withthe situation of exothermic and endothermic chemical reactionsrespectively, m is the kinematic viscosity, q is the density, A is the

428 T. Salahuddin et al. / Results in Physics 7 (2017) 426–434

stretching ratio parameter, C is the Williamson parameter, n thepower law index, DB is the Brownian diffusion coefficient, cp is

the specific heat, s ¼ ðqcÞpðqcÞf is the ratio between the effective heat

capacity of the nanoparticles material and heat capacity of thefluid, DT is the thermophoresis diffusion coefficient and C isresealed nanoparticles volume fraction.

Using the similarity transformations

g ¼ ffiffiffialm

pr2�R2

2R

� �; w ¼ ffiffiffiffima

l

pxRf ðgÞ;

hðgÞ ¼ T�T1Tw�T1

; hðgÞ ¼ C�C1Cw�C1

;

u ¼ 1r

@w@r ; v ¼ � 1

r@w@x :

ð6Þ

Eqs. (1)–(5) takes the form

ð1� nÞð1þ 2KgÞf 000 þ ff 000 � ðf 0Þ2 þ 2Kð1� nÞf 00

þ 2knð1þ 2KgÞ32f 00f 000 þ 3kKð1þ 2KgÞ12ðf 00Þ2 þ A2 ¼ 0; ð7Þ

ð1þ 2KgÞh00 þ 2Kh0

þ Pr h0f þ ð1þ 2KgÞh0h0Nb þ Ntð1þ 2KgÞh02� �þ Prbh ¼ 0; ð8Þ

ð1þ 2KgÞh00 þ 2Kh0 þ PrLefh0 þ Nt

Nbð1þ 2KgÞh00 þ 2K

Nt

Nbh0 ¼ 0; ð9Þ

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

ð10Þ

where K; k;b;Nb;Nt ; Le;A and Pr denotes curvature parameter,dimensionless Weissenberg number, heat generation/absorption,Brownian motion parameter, thermophoresis parameter, Lewisnumber, stretching ratio parameter and Prandtl number which aregiven by

Pr ¼ ma ; k ¼ a

l32

ffiffiffiffia2m

pCx;

K ¼ 1R

ffiffiffimla

q; A ¼ b

a ;

Nb ¼ sDBðCw�C1Þqcpm ; Nt ¼ sDtðTw�T1Þ

qcpmT1 ;

Le ¼ aDB; b ¼ Q

qcpa:

ð11Þ

Skin friction coefficient Cf , local Nusselt number Nux and thelocal Sherwood number Shx are defined as:

Cf ¼ swqa2x2

2l2

; Nux ¼ xqw

kðTf � T1Þ ; Shx ¼ xhm

DBðCw � C1Þ ; ð12Þ

sw ¼ l ð1� nÞ @u@r

þ nCffiffiffi2

p @u@r

� �2 !

r¼R

;

qw ¼ �k@T@r

� �r¼R

; hm ¼ �k@C@r

� �r¼R

; ð13Þ

while the dimensionless forms of skin friction coefficient, local Nus-selt number and Sherwood number are

CfRe1=2x

2¼ ð1� nÞf 00ð0Þ þ nkf 002ð0Þ; Nux

Re1=2x

¼ �h0ð0Þ;

Shx

Re1=2x

¼ �h0ð0Þ: ð14Þ

where Rex ¼ a1=2x=m1=2l1=2.

Numerical solutions

The nonlinear coupled ordinary differential equations (7)–(9)are solved by using shooting method. The step size Dg ¼ 0:1 is cho-sen to obtain numerical solution. As the energy, momentum andconcentration equations are of order second, third and secondrespectively. After converting Eqs. (7)–(9) into first-order equa-tions takes the form

y01 ¼ y2; ð15Þ

y02 ¼ y3; ð16Þ

y03 ¼ðy2Þ2 � y1y3 � 2Kð1� nÞy3 � 3Kkð1þ 2KgÞ1=2 � A2� �

ð1� nÞð1þ 2KgÞ þ 2nkð1þ 2KgÞ3=2y3; ð17Þ

y04 ¼ y5; ð18Þ

y05 ¼�2Ky4 � Prðy1y5 � Nbð1þ 2KgÞy5y7 þ Ntð1þ 2KgÞðy5Þ2Þ � Prbyð4Þ� �

ð1þ 2KgÞ ;

ð19Þ

y06 ¼ y7; ð20Þ

y07 ¼�2Ky7 � PrLey1y7 � Nt

Nbð1þ 2KgÞy05 � 2K Nt

Nby5

� �ð1þ 2KgÞ : ð21Þ

The boundary conditions become

y1 ¼ 0; y2 ¼ 1; y4 ¼ 1; y6 ¼ 1; at g ¼ 0;y2 ! A; y4 ! 0; y6 ! 0 as g ! 1:

ð22Þ

Here four conditions are known and three of them areunknown. Donating the unknown initial conditions by U1;U2 andU3.

y1 ¼ 0; y2 ¼ 1; y3 ¼ U1; y4 ¼ 1; y5 ¼ U2; 6 ¼ 1;y7 ¼ U3 at g ¼ 0: ð23Þ

Eqs. (15)–(21) are solved with the help of initial conditionsdefined in Eq. (23). The computed boundary values at g ¼ 4depends on the choice of U1;U2 and U3. The accurate choice ofU1;U2 and U3 delivers the given boundary conditions at g ¼ 4 thatis it satisfies the Eq. (24)(boundary residual).

U1ðU1;U2;U3Þ ¼ 0:1; U2ðU1;U2;U3Þ ¼ 0; U3ðU1;U2;U3Þ ¼ 0:

ð24ÞIf U1;U2 and U3 does not satisfy the boundary residual then

their values will be refined by using Newton–Raphson method.

Discussion

Shooting method is used to solve the nonlinear ordinary differ-ential equations (7)–(9) with boundary conditions Eq. (10) and thebehavior of curvature parameter K, stretching ratio parameter A,Brownian motion parameter Nb, thermophoresis parameter Nt ,heat generation/absorption b, Prandtl number Pr, power law indexn, Weissenberg number k and Lewis number Le on velocity, tem-perature and concentration profiles are illustrated through graphs.In order to check the accuracy of the solution, the values of f 00ð0Þare calculated for unlike values of stretching ratio parameter A,by ignoring the effects of Lewis number, curvature parameter,thermophoresis parameter, dimensionless Weissenberg number,Brownian motion parameter and heat generation/absorption. Theresults are compared with the values calculated by Mahapatra

Table 1Comparison of f 00 ð0Þ with the previous existing literature when n = 0,k ¼ 0; b ¼ 0;Nt ¼ 0;Nb ! 0; Le ¼ 0; k ¼ 0 and K ¼ 0.

A Mahapatra [27] Ibrahim [28] Present results

0.01 �0.9980 �0.9980 �0.99800.1 �0.9694 �0.9694 �0.96940.2 �0.9181 �0.9181 �0.91810.5 �0.6673 �0.6673 �0.66732 2.0175 2.0175 2.01753 4.7292 4.7292 4.7292

Table 2Comparison of results local Nusselt number �h0ð0Þ for several values of Pr and Awhenn ¼ 0; k ¼ 0;Nt ¼ 0;Nb ! 0; Le ¼ 0; b ¼ 0; k ¼ 0 and K ¼ 0.

Pr A Mahapatra [27] Ibrahim [28] Present results

1 0.1 0.603 0.6022 0.60220.2 0.625 0.6245 0.62470.5 0.692 0.6924 0.6927

1.5 0.1 0.777 0.7768 0.77760.2 0.797 0.7971 0.79750.5 0.8648 0.8648 0.8648

Table 3Values of skin friction coefficient Cf

ffiffiffiffiffiffiRex

p2 with respect to A; k; n and K.

A K k n (1�n) f 00ð0Þ þ nkf 002ð0Þ0.1 0.1 0.1 0.1 �0.95460.2 �0.90200.3 �0.83330.1 0.1 �0.9546

0.2 �0.98940.3 �1.02430.1 0.1 �0.9546

0.2 �0.95400.3 �0.95350.1 0.1 �0.9546

0.2 �0.89670.3 �0.8355

Table 4Values of local Nusselt number Nux

Re1=2xwith respect to Nb ;Nt ; b and Pr.

Nb Nt Pr b �h0ð0Þ0.1 0.1 1.2 0.1 0.55650.2 0.52050.3 0.48580.1 0.1 0.5565

0.2 0.52960.3 0.50340.1 1.2 0.5565

1.3 0.57451.4 0.5921

0.1 0.55650.2 0.44670.3 0.3172

Table 5Values of local Sherwood number Shx

Re1=2xwith respect to Nb;Nt ; Le and Pr.

Nb Le Nt Pr �h0ð0Þ0.1 1.2 0.1 1.2 0.25140.2 0.39430.3 0.44150.1 1.2 0.2514

1.3 0.29801.4 0.34361.2 1.2 0.2514

1.3 0.27841.4 0.3058

Fig. 2. Influence of K on f 0ðgÞ.

Fig. 3. Influence of A on f 0ðgÞ.

T. Salahuddin et al. / Results in Physics 7 (2017) 426–434 429

[27] and Ibrahim [28] as shown in Table 1. Table 2 shows the com-parison of local Nusselt number �h0ð0Þ for different values ofPrandtl number Pr, calculated by Mahapatra [27] and Ibrahim[28]. These tables show that the solution obtained by present paperand the solution obtained by Mahapatra [27] and Ibrahim [28] areapproximately equal.

Tables 3–5 shows the behavior of the skin friction coefficient,local Nusselt number and local Sherwood number respectivelyfor different values of parameters. It is observed that local Nusseltnumber is decreasing function while local Sherwood number is anincreasing function for unlike values of dimensionless parametersNb;Nt ; Le and Pr.

Fig. 2 presents the behavior of curvature parameter K on veloc-ity profile. It is evident from figure that velocity increases withincreasing values of curvature parameter K. Near the surface ofcylinder velocity reduces due to friction of the wall to the fluid

Fig. 4. Influence of k on f 0ðgÞ.

Fig. 5. Influence of n on f 0ðgÞ.

Fig. 6. Influence of Nb on hðgÞ.

Fig. 7. Influence of Nt on hðgÞ.

Fig. 8. Influence of Nb on hðgÞ.

Fig. 9. Influence of Nt on hðgÞ.

430 T. Salahuddin et al. / Results in Physics 7 (2017) 426–434

particles. The augmentation in velocity is due to the fact that afterincreasing curvature parameter K, radius of curvature decreaseswhich causes area of the cylinder to reduce. Hence less resistanceis offered by the cylinder to the fluid, so velocity increases. Fig. 3shows how stretching ratio parameter A affects the velocity profile.It is noticed that when free stream velocity exceeds the stretching

velocity of the cylinder, the fluid velocity increases and boundarylayer thickness reduces with increase in stretching ratio parameterA. Moreover, when free stream velocity is less than the stretchingvelocity of the cylinder, the fluid velocity decreases and boundarylayer thickness increases monotonically. Fig. 4 depicts the influ-ence of Weissenberg number k on velocity profile. It is observed

Fig. 10. Influence of Le on hðgÞ.

Fig. 11. Influence of b on hðgÞ.

Fig. 12. Influence of Pr on hðgÞ.

Fig. 13. Influence of skin friction coefficient on K and A.

Fig. 14. Influence of Pr and A on local Nusselt number.

Fig. 15. Influence of A and Le on local Sherwood number.

T. Salahuddin et al. / Results in Physics 7 (2017) 426–434 431

from figure, reduction in velocity is noticed for each increment invalue of Weissenberg number k, because after increasing Weis-senberg number k the relaxation time increases which offers moreresistance to flow and hence velocity reduces. Fig. 5 shows theinfluence of power law index n on velocity profile. It is quite inter-esting to observe that for large values of power law index n, the

boundary layer thickness reduces. The effects of Brownian motionparameter Nb on temperature profile is presented in Fig. 6. It isnoticed from figure that, as the values of Brownian motion param-eter Nb increases, the random motion of fluid particles acceleratesthe collision between them and hence the temperature of the fluidincreases. Moreover, the boundary layer thickness increases by

Fig. 16. Streamlines for different values of A.

Fig. 17. Streamlines for different values of A.

Fig. 18. Streamlines for different values of A.

Fig. 19. Three dimensional graphs of f(g) for different values of A.

Fig. 20. Three dimensional graphs of f(g) for different values of A.

432 T. Salahuddin et al. / Results in Physics 7 (2017) 426–434

increasing the Brownian motion parameter Nb. The behavior ofthermophoresis parameter Nt on temperature profile may beexamined from Fig. 7. As the values of thermophoresis parameterNt increases, the fluid particles move from hot region to cold regionand hence temperature of the fluid increases. Moreover, theboundary layer thickness increases by increasing the ther-mophoresis parameter Nt . Fig. 8 demonstrates the variation inBrownian motion parameter Nb on concentration profile. It isobserved that by increasing the value of Brownian motion param-eter Nb, the concentration boundary layer thickness reduces. Fig. 9depicts the variation in thermophoresis parameter Nt on concen-tration profile. It is observed that by increasing the values of ther-mophoresis parameter Nt , the concentration boundary layerthickness decreases. Fig. 10 shows the behavior of Lewis number

Fig. 21. Three dimensional graphs of f(g) for different values of A.

T. Salahuddin et al. / Results in Physics 7 (2017) 426–434 433

Le on concentration profile. It is observed that by increasing theLewis number Le the concentration profile reduces. Because byincreasing the Lewis number Le the mass diffusivity reduces, soconcentration profile decreases. Fig. 11 shows the behavior of heatgeneration/absorption b on temperature profile. It is observed thatby increasing heat generation/absorption b kinetic energy of thefluid particles increases, so temperature and boundary layer thick-ness increases. Fig. 12 is plotted to see the effect of Prandtl numberPr on temperature field. It is noticed that temperature fielddecreases with an increase in Prandtl number Pr. Because byincreasing Prandtl number Pr, the thermal diffusivity of the fluiddecreases, which causes decrease in temperature and boundarylayer thickness. Fig. 13 shows the behavior of curvature parameterK and velocity ratio parameter A on skin friction coefficient. It isobserved that the skin friction coefficient increases, as both param-eters increases. Fig. 14 depicts the influence of Prandtl number Prand velocity ratio parameter A on local Nusselt number. As thevelocity ratio parameter A and Prandtl number Pr increases theheat transfer rate on the surface of cylinder increases. Fig. 15shows the influence of Lewis number Le and velocity ratio param-eter A on local Sherwood number. It is observed from figure thatlocal Sherwood number increases as both parameters increases.Figs. 16–18 are plotted to see the flow pattern of fluid particlesfor different values of stretching ratio parameter A. It can be seenfrom these figures that by increasing stretching ratio parameter Athe stagnation region (the region where fluid velocity is zero)increases. Figs. 19–21 show the three dimensional graphs of func-tion f ðgÞ for different values of stretching ratio parameter A.

Concluding remarks

The effects of nanofluid and stagnation point flow over astretching cylinder are investigated numerically. Tangent hyper-bolic fluid is considered as a base fluid. In order to check theendothermic and exothermic chemical reactions, heat generation/absorption effect is also included. A well-known technique (shoot-ing method) is used to calculate the comparison, graphical and tab-ular results for the governing system of equations. In the light ofpresent analysis following deductions may be drawn.

� A qualitatively different behavior was seen in the velocity pro-file for different values of stretching ratio parameter (A > 1 andA < 1).

� Thickening of the thermal boundary layer was observed forhigher values of heat generation/absorption b.

� The thermal boundary layer thickness reduces for increasingvalues of Prandtl number Pr.

� Influence of thermophoresis parameter Nt on nanoparticles con-centration is similar as compared to the temperature.

� Velocity profile reduces for increasing values of Weissenbergnumber k and power law index n.

� The skin friction coefficient, local Nusselt number and localSherwood number increases for increasing values of stretchingratio parameter A.

� Concentration profile reduces for large values of Lewis numberLe but shows increasing behavior in the case of local Sherwoodnumber.

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