Analysis of tangent hyperbolic nanofluid impinging on a stretching cylinder near the stagnation point Academic research paper on "Mathematics"

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

T. Salahuddina'*, M.Y. Malikb, Arif Hussainb, Muhammad Awaisb, Imad Khanb, Mair Khanb

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

ARTICLE INFO ABSTRACT

An analysis is executed to study the influence of heat generation/absorption on tangent hyperbolic nanofluid near the stagnation point over a stretching cylinder. In this study the developed model of a tangent hyperbolic nanofluid in boundary layer flow with Brownian motion and thermophoresis effects are discussed. The governing partial differential equations in terms of continuity, momentum, temperature and concentration are rehabilitated into ordinary differential form and then solved numerically using shooting method. The results specify that the addition of nanoparticles into the tangent hyperbolic fluid yields an increment in the skin friction coefficient and the heat transfer rate at the surface. Comparison of the present results with previously published literature is specified and found in good agreement. It is noticed that velocity profile reduces by enhancing Weissenberg number k and power law index n. The skin friction coefficient, local Nusselt number and local Sherwood number enhances for large values of stretching 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/).

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Article history:

Received 14 October 2016

Received in revised form 26 November 2016

Accepted 23 December 2016

Available online 05 January 2017

Keywords:

Stagnation point flow Tangent hyperbolic nanofluid Stretching cylinder Heat generation/absorption Boundary layer Shooting method

Introduction

The layer in which the effects of viscosity are significant in the close neighborhood of the surface is titled as boundary layer. Ludwig Prandtl presented the concept of boundary layer on August 12, 1904 at the third international conference in Germany. He divided the fluid equations into two categories, one inside the boundary layer where the influence of viscosity is maximum and second outside the boundary where the influence of boundary can be ignored. Boundary layers are pervasive in a great number of natural flows and fluid dynamics engineering applications, such as solicitations in liquid films in condensation procedure, polymer, extrusion of plastic sheets, glass blowing, spinning of fibers, cooling of elastic sheets, the aerodynamic extrusion of plastic sheets, paper manufacturing, the earth's atmosphere, the surface of car, ship and air vehicles, etc. Sakiadis [1] was the forerunner who elaborated the concept of two dimensional boundary layer of Newtonian fluid towards a continuous moving solid surface. Crane [2] extended this concept into two-dimensional steady flow of a linearly stretching sheet and got its exact solution in quiescent fluid. The boundary layer mixed convection flow of heated stretching surface was discussed by Chen [3]. He inspected the behavior different physical

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

parameters and plotted graphs for local Nusselt number and skin friction coefficient. The effect of variable heat flux on two dimensional steady boundary layer flow was deliberated by Lin et al. [4]. They generate the solution in terms of hypergeometric functions and concluded that for smaller Prandtl number the boundary layer is larger. Ali [5] investigated the boundary layer flow of a continuously stretched surface with buoyancy effects. The boundary layer stagnation point flow towards a stretching sheet was discussed by Nadeem et al. [6]. Again, Nadeem et al. [7] investigated the boundary layer flow of second grade fluid towards a stretching sheet with temperature dependent viscosity. Rangi et al. [8] studied the heat transfer and boundary layer flow towards a stretching cylinder with variable thermal conductivity.

The boundary layer flow of pseudoplastic fluids has great importance due to its extensive uses in solutions and melts of high molecular weight polymers, emulsion coated sheets like photographic films, polymer sheets, etc. Nadeem et al. [9] examined the influence of magnetic field and temperature dependent viscosity 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 asymmetric channel under long wavelength and low Reynolds number approximation. Akbar et al. [11] analyzed the effects of chemical reactions and heat transfer on tangent hyperbolic fluid treated through a tapered artery. Nadeem et al. [12] examined the peristaltic motion

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

A base fluid comprises of nanoparticles deferred in conventional heat transfer basic fluid with the length scales of 1-100 nm is called nanofluid. The nanofluid enhances the convective heat transfer coefficient and thermal conductivity of the base fluids. Oil, water and ethylene glycol are poor conductors of heat. In order to increase the thermal conductivity of these fluids many techniques have been taken in description. One of these techniques is the addition of nano-sized material particles in the liquid. Choi et al. [14] found that with the addition of the nanoparticles the thermal conductivity of the base fluid enhances twice. Das [15] concluded that with the inclusion of nanoparticles the thermal conductivity becomes temperature dependent. Buongiorno et al. [16] evaluated the nanofluid coolants for advanced nuclear power plants. Malik et al. [17] analyzed the boundary-layer mixed convective flow of a nanofluid over a stretching plate. Khan et al. [18] investigated the numerical solution of nanofluid over a flat stretching 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. Vajravelu et al. [20] calculated the effects of the nanoparticle volume fraction and heat transfer characteristics with combined effects of temperature dependent internal heat generation/absorption and thermal buoyancy. Noghrehabadi et al. [21] examined the boundary layer and heat transfer flow of nanoparticles volume fraction with slip effects. Akbar et al. [22] Analyzed the magnetic field analysis past over a stretching sheet in a suspension of gyrotactic microorganisms and nanoparticles. Akbar et al. [23] investigated the Double-diffusive natural convective MHD boundary-layer flow of a nanofluid over a stretching sheet.

Heat generation/absorption is reflected as a significant factor in numerous physical glitches such as fluids having endothermic and exothermic chemical reactions. Heat generation/absorption is assumed to be temperature-dependent or space dependent. Lawrence et al. [24] investigated the boundary layer flow of viscoelastic fluid over a stretching sheet with internal heat generation/absorption. Pavithra et al. [25] analyzed the heat transfer boundary layer flow of a dusty fluid over an exponentially stretching surface with combined effects of internal heat generation/absorption and viscous dissipation. Noghrehabadi et al. [26] studied the entropy generation of a nanofluid over an isothermal linear stretching plate with heat generation/absorption. Dessie et al. [27] examined the MHD boundary layer flow of Newtonian fluid with variable viscosity through a porous medium over a stretching sheet with viscous dissipation, 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 surface with temperature and space dependent internal heat generation/absorption. Akbar et al. [29] examined the effects of thermal radiation and variable thermal conductivity on the flow of CNTS over a stretching sheet with convective slip boundary conditions. Akbar et al. [30] examined the two dimensional Magneto-hydrodynamics flow of Eyring-Powell fluid. They noticed that by enhancing the intensity of the magnatic field as well as Eyring-Powell fluid parameter y decrease the velocity profile. Mehmood et al. [31] depicted the Non Aligned point flow and heat transfer of an Ethylene-Glycol and nanofluid towards stretching sheet. They noticed that Ethylene-based nanofluids have higher local heat flux than water-based nanofluids. Rana et al. [32] invistigated the mixed 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, the stagnation point flow of tangent hyperbolic nanofluid over a stretching cylinder. Buongiorno model is used to examine the heat

transfer due to nanoparticles. Moreover, heat generation/absorption effects are encountered for tangent hyperbolic nanofluid. For this persistence proper similarity transformations are used to diminish governing equations to ordinary differential equations. The effect of thermophoresis, Brownian motion and Lewis number for nanofluid are deliberated through graphs. The near wall quantities, such as Sherwood number due to nanoparticle concentration, Nusselt number due to heat transfer and local skin friction for velocity profile are discussed.

Mathematical formulation

Consider steady two-dimensional stagnation point flow of optically dense tangent hyperbolic nanofluid towards a stretching cylinder located at r = R, where r is the coordinate normal to cylinder. The cylinder is stretched with two equivalent and conflicting forces along x-axis with the velocity u = y, by keeping the origin fixed as shown in Fig. 1. It is assumed that concentration and temperature at the wall is maintained at constant concentration Cw and temperature Tw. Where and Tm are ambient concentration and temperature respectively. Under the boundary layer approximation, the governing equations of the mass, momentum, energy and nanoparticles in cylindrical coordinates x and r are written as Ref. [12]

d(ru) + dirv) = 0;

du du . .,

UdX + V~dr = Vl (1

,d2u 1 du du d2u nr (du\2

~n) ~drl+(1 " n)~+nV2r——+_(_)

@r @r2 2r @r

+ Ue dx '

. dT dT\ K d ( dT\ „ (dC dT\ Dt -c>(u dx + v Tr) = 7 d~r{r Tr) + x{ DBKTr Yr) + T1

+ (T - Ti);

dC dC DB d ( dC

u--h v— = — — r—

dx dr r dry dr

Dt 1 d ( dT

T r dr dr

the associated boundary conditions are ax

u = —, v = 0, T = T„, C = Cw at r = R, bx

u = Ue(x)=-J, v = 0, T = TC = Ci as r n.

Here u and v are the velocity components along x and r directions respectively, Q is the temperature dependent volumetric rate of heat source when Q > 0 and heat sink when Q < 0, dealing with the situation of exothermic and endothermic chemical reactions respectively, V is the kinematic viscosity, - is the density, A is the

Fig. 1. Geometry of the problem.

stretching ratio parameter, r is the Williamson parameter, n the power law index, DB is the Brownian diffusion coefficient, cp is

the specific heat, s = (qck is the ratio between the effective heat capacity of the nanoparticles material and heat capacity of the fluid, DT is the thermophoresis diffusion coefficient and C is resealed nanoparticles volume fraction. Using the similarity transformations

g = vffiV( r2#) ; W (g),

0(g) =

h(g) = ;

u i aw v _ i aw

r ar ' ^ r ax :

Eqs. (1)-(5) takes the form

(1 - n)(1 + 2Kg)f ''' + ff''' - f')2 + 2K(1 - nf

+ 2kn(1 + 2K g)f ''f ''' + 3kK (1 + 2Kg)V '')2 + A2 = 0. (7)

(1 + 2Kg)0'' + 2K0'

+ Pr(0'f +(1 + 2Kg)0'h'Nb + N (1 + 2Kg)0'2) + Prße = 0,

NN (1 + 2Kg)h '' + 2Kh' + PrLefh' + N (1 + 2Kg)0'' + 2K^ h = 0, (9)

f (0) = 0, f' (0) = 1, 0(0) = 1, h(0) = 1 f' (i) = A, 0(i) = 0, h(i) = 0.

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

pr=a,k=fv^mrx,

K = A = b,

N — sDb (Cw-Ci) N — sDt (Tw-Ti) 1Wb pcpV ' 1 pcpVTi '

L^ = -a ß =

Db ' ß qcpa.

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

C _ sw Nu _ xqw

Cf = ^ ' NUx = k(Tf - Ti) '

DB(Cw - Ci) '

(,. ,du nr (du\2\

= H(1 - n) au+pf(Jr=R,

^ = - S

, hm = -k(@C

@r 7 r=R '

while the dimensionless forms of skin friction coefficient, local Nusselt number and Sherwood number are

fP =(1 - n)/"(0) + nk/"2(0), = — (0);

= -h' (0).

where Rex = a1/2x/v1/2i1/2.

Numerical solutions

The nonlinear coupled ordinary differential equations (7)-(9) are solved by using shooting method. The step size Dg = 0.1 is chosen to obtain numerical solution. As the energy, momentum and concentration equations are of order second, third and second respectively. After converting Eqs. (7)-(9) into first-order equations takes the form

y\ = y2,

y2 = y3,

((y2)2 - y1y3 - 2K(1 - n)y3 - 3Kk(1 + 2Kg)1/2 - A2)

y3 = A-3/2-, (17)

3 (1 - n)(1 + 2Kg) + 2nk(1 + 2Kg)3/2y3

y4 = y5, (18)

' -2Ky4 - Pr(y,y5 - Nb(1 + 2Kg)y5y7 + Nt(1 + 2Kg)(y5)2) - Prßy(4))

y6 = y7,

(1 + 2Kg)

' (-2Ky7 - PrLey^ - N (1 + 2Kg)y5 - 2^)

(1 + 2Kg)

The boundary conditions become

y1 = 0, y2 = 1, y4 = 1, y6 = 1; at g = 0, y2 ! A, y4 ! 0, y6 ! 0 as g ! 1.

Here four conditions are known and three of them are unknown. Donating the unknown initial conditions by U1, U2 and U3.

y1 = 0, y2 = 1, y3 = U1, y4 = 1, ys = U2, 6 = 1, y7 = U3 at g = 0.

Eqs. (15)-(21) are solved with the help of initial conditions defined in Eq. (23). The computed boundary values at g = 4 depends on the choice of U1, U2 and U3. The accurate choice of U1, U2 and U3 delivers the given boundary conditions at g = 4 that is it satisfies the Eq. (24)(boundary residual).

©1(U1, U2, U3) = 0.1, ®2(U1, U2, U3) = 0, ®3(U1, U2, U3) = 0.

If 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 differential equations (7)-(9) with boundary conditions Eq. (10) and the behavior of curvature parameter K, stretching ratio parameter A, Brownian motion parameter Nb, thermophoresis parameter Nt, heat generation/absorption b, Prandtl number Pr, power law index n, Weissenberg number k and Lewis number Le on velocity, temperature and concentration profiles are illustrated through graphs. In order to check the accuracy of the solution, the values of /"(0) are 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. The results are compared with the values calculated by Mahapatra

Table 1

Comparison of f '(0) with the previous existing literature when n = 0,

k = 0, b = 0, Nt = 0, Nb ! 0, Le = 0, k = : 0 and K = 0.

A Mahapatra [27] Ibrahim [28] Present results

0.01 -0.9980 -0.9980 -0.9980

0.1 -0.9694 -0.9694 -0.9694

0.2 -0.9181 -0.9181 -0.9181

0.5 -0.6673 -0.6673 -0.6673

2 2.0175 2.0175 2.0175

3 4.7292 4.7292 4.7292

Table 2

Comparison of results local Nusselt number -0' (0) for several values of Pr and A when

n = 0, k = 0, Nt = 0, Nb ! 0, Le = 0, b = 0, k = 0 and K = 0.

Pr A Mahapatra [27] Ibrahim [28 Present results

1 0.1 0.603 0.6022 0.6022

0.2 0.625 0.6245 0.6247

0.5 0.692 0.6924 0.6927

1.5 0.1 0.777 0.7768 0.7776

0.2 0.797 0.7971 0.7975

0.5 0.8648 0.8648 0.8648

Table 3

Values of skin friction coefficient f \ with respect to A, k, n and K.

A K k n (1-n) f '(0) + nkf '2(0)

0.1 0.1 0.1 0.1 -0.9546

0.2 -0.9020

0.3 -0.8333

0.1 0.1 -0.9546

0.2 -0.9894

0.3 -1.0243

0.1 0.1 -0.9546

0.2 -0.9540

0.3 -0.9535

0.1 0.1 -0.9546

0.2 -0.8967

0.3 -0.8355

Table 5

Values of local Sherwood number -S^ with respect to Nb,Nt,Le and Pr.

Table 4

Values of local Nusselt number -e^ with respect to Nb, Nt, b and Pr.

Nb Nt Pr b -h (0)

0.1 0.1 1.2 0.1 0.5565

0.2 0.5205

0.3 0.4858

0.1 0.1 0.5565

0.2 0.5296

0.3 0.5034

0.1 1.2 0.5565

1.3 0.5745

1.4 0.5921

0.1 0.5565

0.2 0.4467

0.3 0.3172

[27] and Ibrahim [28] as shown in Table 1. Table 2 shows the comparison of local Nusselt number -0' (0) for different values of Prandtl number Pr, calculated by Mahapatra [27] and Ibrahim

[28]. These tables show that the solution obtained by present paper and the solution obtained by Mahapatra [27] and Ibrahim [28] are approximately equal.

Nb Le Nt Pr -h' (0)

0.1 1.2 0.1 1.2 0.2514

0.2 0.3943

0.3 0.4415

0.1 1.2 0.2514

1.3 0.2980

1.4 0.3436

1.2 1.2 0.2514

1.3 0.2784

1.4 0.3058

0.9 0.8 0.7 ,0.6 0.5 0.4 0.3 0.2 0.1

.....K = 0.1

— K = 0.6

— K= 1.2

\ Pr = 1.2, Le = 1.2, A = 0.1, Nb = 0.1,

' \ N( = 0.1, p = 0.1, X = 0.1, n = 0.1

0 1 2 1

Fig. 2. Influence of K on f (g).

A = 1.2

/K = 0.1,Pr / N. = 0.1, N / = 1.2, Le = = 0.1, p = 1.2, À = 0.1, n = 0.1, ' 0.1

L A = 1.05

A= 0.95

A = 0.8

Fig. 3. Influence of A on f (g).

Tables 3-5 shows the behavior of the skin friction coefficient, local Nusselt number and local Sherwood number respectively for different values of parameters. It is observed that local Nusselt number is decreasing function while local Sherwood number is an increasing function for unlike values of dimensionless parameters Nb, Nt, Le and Pr.

Fig. 2 presents the behavior of curvature parameter K on velocity profile. It is evident from figure that velocity increases with increasing values of curvature parameter K. Near the surface of cylinder velocity reduces due to friction of the wall to the fluid

particles. The augmentation in velocity is due to the fact that after increasing curvature parameter K, radius of curvature decreases which causes area of the cylinder to reduce. Hence less resistance is offered by the cylinder to the fluid, so velocity increases. Fig. 3 shows 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 boundary layer thickness reduces with increase in stretching ratio parameter A. Moreover, when free stream velocity is less than the stretching velocity of the cylinder, the fluid velocity decreases and boundary layer thickness increases monotonically. Fig. 4 depicts the influence of Weissenberg number k on velocity profile. It is observed

•s \ .....p = 0.1

. \ \ '. \ \ % \ \ ™p = 0.3

*. \ \ \ \ \ —p = 0.5

\ \ \ \ \

\ \ K = 0.1, Pr = 1.2 Le = 1.2,

\ \ \ A = 0.1, N. =0.1 •. \ \ b , N( = 0.1, .

\ \ VX = 0.1, n = 0.1

' ""'^Nw

Fig. 11. Influence of ß on 6(g).

Fig. 12. Influence of Pr on 6(g).

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

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

from figure, reduction in velocity is noticed for each increment in value of Weissenberg number k, because after increasing Weissenberg number k the relaxation time increases which offers more resistance to flow and hence velocity reduces. Fig. 5 shows the influence of power law index n on velocity profile. It is quite interesting to observe that for large values of power law index n, the

boundary layer thickness reduces. The effects of Brownian motion parameter Nb on temperature profile is presented in Fig. 6. It is noticed from figure that, as the values of Brownian motion parameter Nb increases, the random motion of fluid particles accelerates the collision between them and hence the temperature of the fluid increases. Moreover, the boundary layer thickness increases by

A =0.1

Fig. 16. Streamlines for different values of A.

A = 0.9

Fig. 17. Streamlines for different values of A.

A = 1.7

T-'-1-1-1-1-1-1-1-1-1-'-r

-3-2-10 1 2 3

Fig. 18. Streamlines for different values of A.

A = 0.1

Fig. 19. Three dimensional graphs of for different values of A.

A = 0.9

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

increasing the Brownian motion parameter Nb. The behavior of thermophoresis parameter Nt on temperature profile may be examined from Fig. 7. As the values of thermophoresis parameter Nt increases, the fluid particles move from hot region to cold region and hence temperature of the fluid increases. Moreover, the boundary layer thickness increases by increasing the ther-mophoresis parameter Nt. Fig. 8 demonstrates the variation in Brownian motion parameter Nb on concentration profile. It is observed that by increasing the value of Brownian motion parameter Nb, the concentration boundary layer thickness reduces. Fig. 9 depicts the variation in thermophoresis parameter Nt on concentration profile. It is observed that by increasing the values of ther-mophoresis parameter Nt, the concentration boundary layer thickness decreases. Fig. 10 shows the behavior of Lewis number

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

Le on concentration profile. It is observed that by increasing the Lewis number Le the concentration profile reduces. Because by increasing the Lewis number Le the mass diffusivity reduces, so concentration profile decreases. Fig. 11 shows the behavior of heat generation/absorption b on temperature profile. It is observed that by increasing heat generation/absorption b kinetic energy of the fluid particles increases, so temperature and boundary layer thickness increases. Fig. 12 is plotted to see the effect of Prandtl number Pr on temperature field. It is noticed that temperature field decreases with an increase in Prandtl number Pr. Because by increasing Prandtl number Pr, the thermal diffusivity of the fluid decreases, which causes decrease in temperature and boundary layer thickness. Fig. 13 shows the behavior of curvature parameter K and velocity ratio parameter A on skin friction coefficient. It is observed that the skin friction coefficient increases, as both parameters increases. Fig. 14 depicts the influence of Prandtl number Pr and velocity ratio parameter A on local Nusselt number. As the velocity ratio parameter A and Prandtl number Pr increases the heat transfer rate on the surface of cylinder increases. Fig. 15 shows the influence of Lewis number Le and velocity ratio parameter A on local Sherwood number. It is observed from figure that local Sherwood number increases as both parameters increases. Figs. 16-18 are plotted to see the flow pattern of fluid particles for different values of stretching ratio parameter A. It can be seen from these figures that by increasing stretching ratio parameter A the stagnation region (the region where fluid velocity is zero) increases. Figs. 19-21 show the three dimensional graphs of function f (g) for different values of stretching ratio parameter A.

Concluding remarks

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

■ A qualitatively different behavior was seen in the velocity profile for different values of stretching ratio parameter (A > 1 and A < 1).

Thickening of the thermal boundary layer was observed for higher values of heat generation/absorption b. The thermal boundary layer thickness reduces for increasing values of Prandtl number Pr.

■ Influence of thermophoresis parameter Nt on nanoparticles concentration is similar as compared to the temperature.

■ Velocity profile reduces for increasing values of Weissenberg number k and power law index n.

■ The skin friction coefficient, local Nusselt number and local Sherwood number increases for increasing values of stretching ratio parameter A.

Concentration profile reduces for large values of Lewis number Le but shows increasing behavior in the case of local Sherwood number.

References

[1] Sakiadis BC. Boundary layer behavior on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axisymmetric flow. AIChEJ 1961;7:26-8.

[2] Crane LJ. Flow past a stretching plate. Z Angew Math Phy 1970;21:645-7.

[3] Chen CH. Mixed convection cooling of a heated, continuously stretching surface. Heat Mass Transfer 2000;36:79-86.

[4] Lin CR, Chen CK. Exact solution of heat transfer from a stretching surface with variable heat flux. Heat Mass Transfer 1998;33:477-80.

[5] Ali ME. The Buoyancy Effect on the boundary layers Induced by continuous surfaces stretched with rapidly decreasing velocities. Heat Mass transfer 2004;40:285-91.

[6] Nadeem S, Hussain A. HAM solutions for boundary layer flow in the region of the stagnation point towards a stretching sheet. Commun Nonlinear Sci Numer Simul 2010;15:475-81.

[7] Nadeem S, Faraz N. Thin film flow of a second grade fluid over astretching/ shrinking sheet with variable temperature dependent viscosity. Chin Phys Lett 2010;27:034704.

[8] Rangi RR, Ahmad N. Boundary layer flow past a stretching cylinder and heat transfer with variable thermal conductivity. Appl Math 2012;3:205-9.

[9] Nadeem S, Akbar NS, Hameed M. Peristaltic transport and heat transfer of a MHD Newtonian fluid with variable viscosity. Int J Numer Methods Fluids 2010;63:1375-93.

10] Nadeem S, Akram S. Magnetohydrodynamic peristaltic flow of a hyperbolic tangent fluid in a vertical asymmetric channel with heat transfer. Acta Mech Sin 2011;27:237-50.

11] Akbar NS, Nadeem S, Ali M. Influence of heat and chemical reactions on hyperbolic tangent fluid model for blood flow through a tapered artery with a stensosis. Heat Transfer Res 2012;43:69-94.

12] Salahuddin T, Malik MY, Hussain A, Bilal S, Awais M. Effects of transverse magnetic field with variable thermal conductivity on tangent hyperbolic fluid with exponentially varying viscosity. AIP Adv 2015;5:127103.

13] Naseer M, Malik MY, Nadeem S, Rehman A. The boundary layer flow of hyperbolic tangent fluid over a vertical exponentially stretching cylinder. Alexandria EngJ 2014;53:747-50.

14] Choi SUS, Zhang ZG, Yu W, Lockwood FE, Grulke EA. Anomalously thermal conductivity enhancement in nanotube suspensions. Appl Phys Lett 2001;79:2252-4.

15] Das S. Temperature dependence of thermal conductivity enhancement for nano fluids. J Heat Transfer 2003;125:567-74.

16] Buongiorno J, Hu W. Nanofluid coolants for advanced nuclear power plants. Proceedings of ICAPP'05, Seoul, vol. 5705. p. 15-9.

17] Malik MY, Khan I, Hussain A, Salahuddin T. Mixed convection flow of MHD Eyring-Powell nanofluid over a stretching sheet: a numerical study. AIP Adv 2015;5:117-8.

18] Khan WA, Pop I. Boundary-layer flow of a nanofluid past a stretching sheet. Int J Heat Mass Transfer 2010;53:2477-83.

19] Akbar NS, Tripathi D, Khan ZH, Beg OA. A numerical study of magnetohydrodynamic transport of nanofluids over a vertical stretching sheet with exponential temperature-dependent viscosity and buoyancy effects. Chem Phys Lett 2016;661:20-30.

20] Vajravelu K, Prasad KV, Lee J, Lee C, Pop I, Gorder RAV. Convective heat transfer in the flow of viscous Ag-water and Cu-water nanofluids over a stretching surface. Int J Therm Sci 2011;50:843-51.

21] Noghrehabadi A, Pourrajab R, Ghalambaz M. Effect of partial slip boundary condition on the flow and heat transfer of nanofluids past stretching sheet prescribed constant wall temperature. Int J Therm Sci 2012;54:253-61.

22] Akbar NS, Khan ZH. Magnetic field analysis in a suspension of gyrotactic microorganisms and nanoparticles over a stretching surface. J Magn Magn Mater 2016;378:320-6.

[23] Akbar NS, Khan Z, Nadeem S. W. khan, Double-diffusive natural convective boundary-layer flow of a nanofluid over a stretching sheet with magnetic field. Int J Numer Methods Heat Fluid Flow 2016;26(1):108-21.

[24] Lawrence PS, Rao BN. Heat transfer in a viscoelastic boundary layer flow over a stretching sheet. J Phys D Appl Phys 1994;2:1323-7.

[25] Pavithra GM, Gireesha BJ. Effect of internal heat generation/absorption on dusty fluid flow over an exponentially stretching sheet with viscous dissipation. J Math 2013. http://dx.doi.org/10.1155/2013/583615.

[26] Noghrehabadi A, Saffarian MR, Pourrajab R, Ghalambaz M. Entropy analysis for nanofluid flow over a stretching sheet in the presence of heat generation/ absorption and partial slip. J Mech Sci Technol 2013:927-37.

[27] Dessie H, Kishan N. mHd effects on heat transfer over stretching sheet embedded in porous medium with variable viscosity, viscous dissipation and heat source/sink. Ain Shams Eng J 2014;5:967-77.

[28] Hakeem AKA, Kalaivanan R, Ganesh NV, Ganga B. Effect of partial slip on hydromagnetic flow over a porous stretching sheetwith non-uniform heat

source/sink, thermal radiation and wall mass transfer. Ain Shams Eng J 2014;5:913-22.

[29] Akbar NS, Khan ZH. Effect of variable thermal conductivity and thermal radiation on the flow of CNTS over a stretching sheet with convective slip boundary conditions: numerical study. J Mol Liq 2016;222:279-86.

[30] Akbar NS, Ebaid A, Khan ZH. Numerical analysis of magnetic field on Eyring-Powell fluid flow towards a stretching sheet. J Magn Magn Mater 2015;382:355-8.

[31] Mehmood R, Nadeem S, Akbar NS. Non aligned Ethylene-Glycol 30% based stagnation point fluid over a stretching surface with hematite aano particles. J Appl Fluid Mech 2016;9:1359-66.

[32] Rana S, Mehmood R, Akbar NS. Mixed convective oblique flow of a Casson fluid with partial slip, internal heating homogeneous-heterogeneous reactions. J Mol Liq 2016;222:1010-9.