Scholarly article on topic 'Magnetohydrodynamics (MHD) flow of a tangent hyperbolic fluid with nanoparticles past a stretching sheet with second order slip and convective boundary condition'

Magnetohydrodynamics (MHD) flow of a tangent hyperbolic fluid with nanoparticles past a stretching sheet with second order slip and convective boundary condition Academic research paper on "Mathematics"

CC BY-NC-ND
0
0
Share paper
Academic journal
Results in Physics
OECD Field of science
Keywords
{"Tangent hyperbolic fluid" / "Second order slip flow" / MHD / "Convective boundary condition" / "Radiation effect" / "Passive control of nanoparticles"}

Abstract of research paper on Mathematics, author of scientific article — Wubshet Ibrahim

Abstract This article presents the effect of thermal radiation on magnetohydrodynamic flow of tangent hyperbolic fluid with nanoparticle past an enlarging sheet with second order slip and convective boundary condition. Condition of zero normal flux of nanoparticles at the wall is used for the concentration boundary condition, which is the current topic that have yet to be studied extensively. The solution for the velocity, temperature and nanoparticle concentration is governed by parameters viz. power-law index (n), Weissenberg number We, Biot number Bi, Prandtl number Pr, velocity slip parameters δ and γ , Lewis number Le, Brownian motion parameter Nb and the thermophoresis parameter Nt. Similarity transformation is used to metamorphosed the governing non-linear boundary-value problem into coupled higher order non-linear ordinary differential equation. The succeeding equations were numerically solved using the function bvp4c from the matlab for different values of emerging parameters. Numerical results are deliberated through graphs and tables for velocity, temperature, concentration, the skin friction coefficient and local Nusselt number. The results designate that the skin friction coefficient C f deplete as the values of Weissenberg number We, slip parameters γ and δ upturn and it rises as the values of power-law index n increase. The local Nusselt number - θ ′ ( 0 ) decreases as slip parameters γ and δ , radiation parameter Nr, Weissenberg number We, thermophoresis parameter Nt and power-law index n increase. However, the local Nusselt number increases as the Biot number Bi increase.

Academic research paper on topic "Magnetohydrodynamics (MHD) flow of a tangent hyperbolic fluid with nanoparticles past a stretching sheet with second order slip and convective boundary condition"

results in

PHYSICS

Accepted Manuscript

Magnetohydrodynamics (MHD) Flow of a Tangent hyperbolic fluid with nano-particles past a Stretching Sheet with Second order Slip and Convective boundary condition

Wubshet Ibrahim

PII: DOI:

Reference:

S2211-3797(17)30865-3 https://doi.org/10.1016/j.rinp.2017.09.041 RINP 959

To appear in:

Results in Physics

Please cite this article as: Ibrahim, W., Magnetohydrodynamics (MHD) Flow of a Tangent hyperbolic fluid with nanoparticles past a Stretching Sheet with Second order Slip and Convective boundary condition, Results in Physics (2017), doi: https://doi.org/10.1016/j-rinp.2017.09.041

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Wubshet Ibrahim

Department of Mathematics,Ambo University, Ambo, Ethiopia E-mail: wubshetib@yahoo.com,Tel +25191189249

Magnetohydrodynamics (MHD) Flow of a Tangent hyperbolic fluid with nanoparticles past a Stretching Sheet with Second order Slip and Convective boundary condition

Abstract

This article presents the effect of thermal radiation on magnetohydrodynamic flow of tangent hyperbolic fluid with nanoparticle past an enlarging sheet with second order slip and convective boundary condition. Condition of zero normal flux of nanoparticles at the wall is used for the concentration boundary condition, which is the current topic that have yet to be studied extensively. The solution for the velocity, temperature and nanoparticle concentration is governed by parameters viz. power-law index (n), Weissenberg number We, Biot number Bi, Prandtl number Pr, velocity slip parameters 8 and Y, Lewis number Le, Brownian motion parameter Nb and the thermophoresis parameter Nt. Similarity transformation is used to metamorphosed the governing non-linear boundary-value problem into coupled higher order non-linear ordinary differential equation. The succeeding equations were numerically solved using the function bvp4c from the matlab for different values of emerging parameters. Numerical results are deliberated through graphs and tables for velocity, temperature, concentration, the skin friction coefficient and local Nusselt number. The results designate that the skin friction coefficient Cf deplete as the values of Weissenberg number We, slip parameters Y and 8 upturn and it rises as the values of power-law index n increase. The local Nusselt number -d'(0) decreases as slip parameters Y and 8, radiation parameter Nr, Weissenberg number We, thermophoresis parameter Nt and power-law index n increase. However, the local Nusselt number increases as the Biot number Bi increase.

Keywords: Tangent hyperbolic fluid, Second order slip flow, MHD, Convective boundary condition, Radiation effect, passive control of nanoparticles.

Preprint submitted to Elsevier

September 22, 2017

Nomenclature

Cf Cw C Db DT f k k* Le M n

Nb Nr Nt

Nux Pr

u, v We

Biot number Magnetic field strength Skin friction coefficient Concentration at the surface of the sheet Ambient concentration Brownian diffusion coefficient Thermophoresis diffusion coefficient Dimensionless stream function Thermal conductivity Mean absorption coefficient Lewis number Magnetic parameter Power-law index Brownian motion parameter Radiation parameter Thermophoresis parameter Local Nusselt number Prandtl number Local Reynolds number Temperature of the fluid inside the boundary layer Temperature at the surface of the sheet Ambient temperature Velocity component along x- and y-direction Weissnberg number

Greeks

(P)f (Pc) f (pc) p

The molecular mean free path Dimensionless similarity variable Dimensionless concentration function Dynamic viscosity of the fluid Kinematic viscosity of the fluid Density of the basefluid Heat capacity of the base fluid Effective heat capacity of

a nanop First ore

subscripts

order slip condition cond order slip condition Stream function Thermal diffusivity Electrical conductivity Stefan Boltzmann constant Dimensionless tempreture The extra stress tensor Time constant

Parameter defined by

(pc)_p

(PC) f

Condition at the free stream Condition at the surface

1. Introduction

The boundary layer flow of non-Newtonian fluids past a stretching sheet has been examined for many years because of their huge applications in different manufacturing process such as in polymer, textile, food processing industries etc. In modern time industries the applications of non-Newtonian fluids excel that of Newtonian one. The utilization of such fluids can be boosted by using some additives. In any manufacturing process the excellence of the end products of such industries measured on the rate of cooling in the heat transfer procedures. The magnetohydrody-namic(MHD) is one of the factors by which the cooling rate can be measured and the out come of the preferred feature can resulted in. The analysis of the effects of MHD on stretching sheet was conducted by the researchers such as Fadzilah et al [1], Ishak et al [2, 3, 4], and Mahapatra et al [5, 6].

Research in nanofluids flow has gained remarkable attention for the last three decades because of their multipurpose in modern high technology based industries. They have studied

nanofluid both experimentally and theoretical and come up with the conclusion that nanofluids lift up the thermal conductivity of convectional base fluids, since,the convectional heat transport fluids such as water and oil etc. have a limited thermal conductivity property. Hence, a nowadays technology intensive industries seek, a fluid with titanic thermal conductivity coefficient which facilitate the cooling processes of its products and machineries.

It is known that metals have high thermal conductivity coefficient than the convectional heat transfer fluids. Therefore, transforming the thermal conductivity coefficient of a convectional fluid is essential for lifting up the thermal conductivity. Buongiorno [7] examined the convective transport of nanofluid by developing a model which incorporating the influences o f Brownian diffusion and thermophoretic force. Using Buongiorno model, Kuznetsov and Nield [8], Khan and Pop [9] and Makinde and Aziz [10] have studied the natural convective boundary layer flow of nanofluid past a surface. Furthermore, Wubshet and Shanker [11, 12, 13] examined magne-tohydrodynamic boundary layer flow of nanofluid past a stretching surface. Moreover, Wubshet and Makinde [14, 15] numerically examined the effect of double stratification and diffusion on MHD boundary layer flow of nanofluid past stretching sheet. More studies on nanofluid boundary layer are given in the references [16, 17, 18, 17, 19, 20].

At present day, the study of slip flow in nanofluids has received a remarkable attention. Accordingly, Noghrehabadi et al.[21] succinctly explained the effect of partial slip on boundary layer flow and heat transfer of nanofluid past stretching sheet with prescribed constant wall temperature. Similarly, Wubshet and Shanker [22] scrutinize velocity, thermal and concentration slip flow and heat transfer of a nanofluid past stretching sheet . A very comprehensive literature on slip boundary conditions can be found in the references listed under [23, 24, 25, 26, 27, 28, 29].

The aforementioned studies considered the effect of the first order slip flow condition past a stretching sheet. However, the second order slip flow occurs and influences many flow dynamics. Therefore, this study has given attention to the study of second order slip flow past a stretching surface. Accordingly, Fang et al.[30, 31] and Mahantesh et al.[32] explored second order slip flow and heat transfer past a stretching sheet. Further, Rosca and Pop [33, 34] have scrutinize flow and heat transfer past a vertical permeable stretching/shrinking surface with a second order slip. Furthermore, Turkyilmazoglu [35] and Sing and Chamkha [36] have inspected the effects of second order slip flow on heat and mass transfer of MHD past a stretching sheet. The study indicated that increasing values of magnetic parameter and second order slip parameter considerably reduce the magnitude of shear stress at the wall for the stretching sheet problem.

All the above studies discussed the second slip flow towards stretching sheet without nanopar-ticles. None of them discussed the effect of second order slip and convective boundary condition on MHD boundary layer flow and heat transfer of tangent hyperbolic nanofluid over a stretching sheet.

Currently, the study of boundary layer flow and heat transfer of non-Newtonian fluids has been a prime interest of many researchers due to their variety of engineering and industrial applications. Specifically, these fluids are well-matched for industrial sectors such as chemical, food, biological and pharmaceutical and widely observed in polymer and food processing, ice and magma flow etc. Since, the Unprecedented work of Prandtl, various aspects of the boundary layer flow of a non-Newtonian fluids have been studied by different researchers. Due to non-existence of single constitutive relationship between stress and rate of strain of non-Newtonian

fluid, different types of constitutive equations are available in the literature. Accordingly, tangent hyperbolic fluid is one of the non-Newtonian fluid in which constitutive equation is valid for low and high shear rates. The fluid model under investigation falls under the category of rate type and it shows the characteristics of the relaxation time and the retardation time. Literature about boundary layer flow of tangent hyperbolic fluid is scares and few published articles are available in open literature. Thus, Akbar et al.[37] examined the magnetohydrodynamic boundary layer flow and heat transfer of tangent hyperbolic fluid towards stretching sheet. They reduced the boundary layer equations to ordinary differential equations using similarity transformation and then computed the numerical solutions. Furthermore, Hayat et al.[38] have discussed the effect of thermal radiation in the two-dimensional mixed convection flow of a tangent hyperbolic fluid near a stagnation point. Still further, Malik et al.[39] have examined the MHD flow of tangent hyperbolic fluid past a stretching cylinder. Naseer et al. [40] also have explored the boundary layer flow of tangent hyperbolic fluid past a vertical stretching cylinder. Salahuddin et al.[41]have examined the effect of magnetic field and variable thermal conductivity on heat transfer of Tangent hyperbolic fluid past a stretching cylinder. Recently Salahuddin et al. [42] have given an analysis on the effect of heat source/sink on Tangent hyperbolic nanofluid past a stretching cylinder.

The main objective of this analysis is to examine the effect of nanoparticles with respect to a non-Newtonian fluid called tangent hyperbolic fluid. To the best of the author knowledge the boundary layer flow of tangent hyperbolic fluid with second order slip flow and passive control of nanoparticles have never been studied . Therefore, this article scrutinize the aggregate effects of thermal radiation, first, second order slip flow, convective boundary condition and magnetic parameter on boundary layer flow towards stretching sheet.The governing boundary layer equations were transformed into a two-point boundary value problem using similarity variables and numerically solved using bvp4c from matlab. The effects of governing parameters on fluid velocity, temperature and particle concentration were discussed and shown in graphs and tables as well.

2. Mathematical Formulation

In this analysis, a two-dimensional, steady, incompressible, viscous flow of a tangent hyperbolic fluid with nanoparticles past a stretching sheet with second order slip and convective boundary condition is examined. The stretching surface is along x-axis which have a stretching velocity uw(x) = ax and the flow is restricted to a region y > 0. At its lower surface, the sheet heated convectively with temperature Tf and a heat transfer coefficient hf. The uniform ambient temperature and concentration respectively, are T» and C». It is assumed that there is no nanopar-ticle flux at the surface and the effect of thermophoresis is taken into account in the boundary condition. The flow is subjected to a constant transverse magnetic field of strength B = B0 which is assumed to be applied in the positive y-direction, normal to the surface. The induced magnetic field is assumed to be small compared to the applied magnetic field and is neglected. It is further assumed that the base fluid and the suspended nanoparticles are in thermal equilibrium. Where Tw, T», Cw, C» and B0 are temperature at the surface of the sheet, ambient temperature of the

fluid, concentration at the surface of the sheet, ambient concentration and magnetic field strength respectively.

The coordinate system has been chosen that, x-axis is along the stretching sheet and y-axis is normal to the sheet.

The constitutive equation for a tangent hyperbolic fluid as given by Akbar et al.[37] is

t = + (no + V«>)tanh(rQ. )n] ¿1

where t is the extra stress tensor, n°° is the infinity shear rate viscosity, no is viscosity, r is time constant, n is the dimensionless power law index and ¿1 is

" = \ß 11 " ij ¿1

he zero shear rate

ned as:

Here n is the second invariant of strain-rate tensor and n = 2tr^gradV + (gradV)T) . In this study the constitutive equation considered when = 0. It has been considered the tangent hyperbolic fluid which represent shear thinning characteristics when Til < 1. Therefore, Eq.(1)

becomes:

t = no [(rö)n] ¿1 = no [(1 + rö - If] í¿ = no[l + n(m - 1)]ß

racteristi

Under the above assumptions and boundary layer approximations, the governing equation of tangent hyperbolic fluid model with nanoparticle can be expressed as:

du dv dx dy

Where,

i\ . 2

d T dT id 2 T\ I (dC d T\

dx+vdy = + A\ DßUay) +

u — + v — x

a + v 3- = Dß x y

1 dqr (pc) f dy

Dt (d2T\

, n is the power-law index, r is Time constant, A is a Parameter defined by

(Pc) f'

possible to see that for power-law index (n = 0) this problem is reduced to Newtonian fluid while for n = 1 the fluid is Non-Newtonian. The boundary conditions are:

u = uw + Usiip, v = o, u — UU = o, v = o, T — TU, C — C

k d T h T T, D dC Dß d T o

-kdy = hf (Tf - ^ Dßdy + TU dy = o

at y = o

as y -

Uslip is the slip velocity at the surface; Wu's[43] slip velocity equation (valid for arbitrary Knudsen number, Kn) and used by researchers such as [30, 31, 32], is given by

_ _ 2/3 - al Uslip _ 3 i a

. du d2u Usiip _ Ady + dy2

3 1 -12

JPdy 4

+K2 (i -12)

,d2u dy2

Kr, ■ 1

'ree p;

is the momentum

path. Based on the

where A and B are constant, Kn is Knudsen number, l = min accommodation coefficient with 0 < a < 1, and ¡5 is the molecular mean definition of l, it is noticed that for any given value of Kn, we have 0 < l < 1. The molecular mean free path is always positive. Thus we know that B < 0 , and hence the second term in the right hand side of Eq. (9) is a positive number. Moreover, y is the first order velocity slip parameter with y = , which is positive, 5 is the second order velocity slip parameter with 5 = V is negative. x and y represent coordinate axes along the continuous surface in the direction of motion and normal to it, respectively. The velocity components along x and y-axis are u and v respectively. u is the kinematics viscosity, T is the temperature inside the boundary layer, (pc)p effective heat capacity of a nanoparticle, p is the density, Tx is ambient the temperature far away from the sheet.

By introducing the following similarity transforms and dimensionless quantities as:

n _\luy, V _Vâûxf (n)

0(1) _

T - Tx

Tf - T™

y/(x, y) represents the streim function and is defined as,

v _ —■

so that the Using

satisfied identically. nd approximation for thermal radiation

4a* dT4

~3k* dT

k* is the mean absorption coefficient and a* is the Stefan Boltzmann constant and T4

is linear temperature function which is expanded by using Taylor series expansion about Tx as T4 _ 4T3T 3T3 .

Using boundary layer approximation and similarity transform Eqs.(5)- (7) become

( 1 - n)f" + ff - f'2 - Mf + nWef f' _ 0

( 1 + 3Nr)0'' + Pr[f 0 + Nb$'0' + Nt0'2] _ 0

n ' Nt ''

$ + PrLefQ +— 0 _ 0 Nb

With boundary conditions

f (0) = 0, f (0) = 1 + Yf '(0) + 8/"(0), 6'(0)= Bi(6(0) - 1), Nb^'(0)+ Nt6'(0) = 0, at n = 0,

f H ^ 0, 6(~) ^ 0, ^ 0, as n ^ ~(16)

where the governing parameters are defined by:

«= hf ß, Pr = -U, Nr =

k \ a aD k*k

oB0 T an V2a3 xT

= 7T' ^ =-

(pc)pDb(Cw - C.) (pc) f v

(pc)pDT(Tw - TM) (pc) / uTx

> Gove

g paran

meters

where f', d and 0 are the dimensionless velocity, temperature and particle concentration respectively. n is the similarity variable, the prime denotes differentiation with respect to n. Y, <5, We, n, Bi, Pr, M, Nb, Nt, Le denotes first order slip parameter, second order slip parameter, Weissenber number, power-law index, Biot number, Prandtl number, a magnetic parameter, a Brownian motion parameter, a thermophoresis parameter, and a Lewis number, respectively.

The skin friction coefficient Cf and local Nusselt number Nux of the study are defined as:

C __Nu _ xqw

k(Tw - - T„)

Where the wall shear stress Tw, and wall heat flux qw are given by

idu\ 2

Tw = (1 - n) dy + ,

d u nr d y V2'

qw — k I _

n„. „//,_,. \ Nux 4

By using the above equations, we get

CfVReX_ ((1 -n) + 2wef(0))f(o),

Where Rex _ and Nux are local Reynolds number and local Nusselt number, respectively.

= -(1 + 3 Nr)6' (0) y/Rex 3

3. Numerical Technique

System of coupled nonlinear ordinary differential equations (13)-(15) subjected to boundary conditions equation(16) are solved numerically using bvp4c from a matlab. Numerical method bvp4c is BVP solver from matlab which is a finite difference code that implements the 3-stage Lobatto Ilia formula. To use bvp4c from matlab, first, Eqs.(13)-(15) are transformed into a set of coupled first-order system of equations. This transformation is used to set up the system of equations as a boundary value problem(bvp) and use the bvp solver in matlab to numerically

solve this system, with the above boundary condition and assumed a suitable finite value for the far field boundary condition, i.e, n ^ say n°° = 40. In solving the present problem using bvp4c from matlab, bvp4c has three arguments: a function odes for evaluating the ODEs, a function bcs for evaluating the residual in the boundary conditions, and a structure solinit that provides a guess for a mesh and the solution on this mesh. The ODEs are handled exactly as in the Matlab IVP solvers. The numerical results are obtained for different values of governing parameters viz. slip parameters y and 5, power-law index n, Weissenberg number We, Prandtl number Pr, Biot number Bi, magnetic parameter M, a Brownian motion parameter Nb, a thermophoresis parameter Nt and a Lewis number Le.

4. Results and Discussion

randtl nu a thermo

This section presents the graphical and tabular results of the effects of different physical parameters on dimensionless velocity, temperature and concentration which are depicted through figs. 1-14 and tables 1-3.

Fig.1 and fig.2 depict the impact of different physical parameters on velocity graphs, for different values of power-law index n, Weissenberg number We, first order slip parameter y and second order slip parameter 5. Fig.1 shows the effect of power-law index n and Weissenberg number We on f '(n). An increase in the values of power-law index n correspond to depletion in velocity and thinning of hydrodynamic boundary layer. Moreover the velocity graph tends to zero value as n ^ ^ Fig. 1 also demonstrates the influences of Weissenberg number We on velocity graph. The Weissenberg number We is the ratio of the relaxation time of the fluid and a specific process time as it is given by Akbar et al.[37]. An increase in the values of Weissenberg number We correspond to the reduction in velocity of the flow and thinning of hydrodynamic boundary layer. Furthermore, fig.2 show the influences of the first order and second order slip parameters on velocity field. The higher the absolute values of y and 5 the smaller the velocity field and thinner is the hydrodynamic boundary layer thickness. In addition, the velocity graphs tend to zero as n ^ ^ Physically, an increase in power-law index n, Weissenberg number We, slip parameters 5 and y increases the resistance of fluid motion, which reduces the flow field and velocity boundary layer thickness.

Figs.3-5 show the influence of power-law index n, radiation parameter Nr, first order slip parameter y, second order slip parameter 5 and convective parameter Bi on temperature graphs. Fig.3 describe the variation of temperature graphs with respect to power-law index n and radiation parameter Nr. An increase in the values of both parameters n and Nr correspond to an increase in temperature and thermal boundary layer thickness. Physically, an increase in thermal radiation boosts the fluid temperature, which results in an increase in the thermal boundary layer thickness.

The effects of slip parameters y and 5 on temperature profile are given in fig.4. From the figure, we can observe that the temperature of flow field is a decreasing function of both parameters; however, its thermal boundary thickness is an increasing function of both parameters. Fig.5 shows the impact of convective parameter called Biot number Bi and the Weissenberg number We on temperature profile. Physically Biot number is the ratio of convection at the surface to conduction within the surface of a body. As the values of both parameters Biot number and the

Weissenberg number We increase, temperature at the surface increases, this result in an increase in the thermal boundary layer thickness.

The influences of Prandtl number parameter Pr on temperature profile is given by figure 6. It is observable that temperature is an decreasing function of Prandtl number Pr. The larger values of Prandtl number at the sheet surface permit the thermal effect to go deeper into the fluid. This results in the larger temperature and thinner the boundary layer.

The concentration profile graphs for both slip parameters and n are displayed in figures 7-9. As shown in fig.7 and fig.8, the concentration graph is increasing function of both parameters up to some value of n and then asymptotic to the value zero as n ^ ^ The concentration boundary layer thickness is an increasing function of both parameters Y and 8. Fig9 shows the variation of concentration graph with respect to power-law index n. The concentration graph increases for some values of n and then approaches to the value zero.

Fig.10-13 show the variation of the coefficient of skin friction with respect to Magnetic field parameter M for different values of Weisseberg number We, power-law index n and first order slip parameter Y. It is observed that the magnitude of skin friction coefficient increases up to some value of n and after that decreases as the values of M increase. Furthermore, as the values of Weissenberg number We, the graph of skin friction coefficient is a parabolic which open downward. Quantitatively, as the values of Weissenberg number increases, the skin friction coefficient decreases. Moreover, as the values of power-law index n increase, the graph is increases up to some value of n and then it start to decrease. While the values of first slip parameter increase, the graph of skin friction coefficient decreases.

Fig.14-15 show the variation of local Nusselt number -0'(0) with respect to Biot number Bi and thermophoresis parameter Nt for different values of the slip parameters Y and 8 and power-law index n. The graphs show that as the values of Nt, power-law index n and slip parameters

Y and 8 increase, the local Nusselt number -0'(0) decrease, however, the local Nusselt number increases as the values of convective parameter Bi increase. The local Nusselt number at the surface decreases as the values of n increase. From the figures we can see that the values of in local Nusselt number is higher for both slip parameters Y and 8.

In order to validate the accuracy of the method used, comparison with previously reported data available in the literature has been made. From table 1 it can be seen that the numerical values of the skin friction coefficient - f''(0) in this paper for different values of magnetic parameter M when We _ n _ Y _ 8 _ 0 are in an excellent agreement with the results published in [37].

Furthermore, table 2 and 3 shows the computed values of the skin friction coefficient - f''(0) and local Nusselt number -0' (0) for different values of the governing parameters such as We, n, M, Nt, Y and 8. It is found that the skin friction coefficient is a decreasing function of We, n, M,

Y and 8. Moreover, local Nusselt number -0'(0) is a decreasing function of the parameters Nt, Nr, We and n.

5. Conclusions

In general, the effect of thermal radiation, second order slip, convective heating, Weissenberg number, power-law index parameter and magnetic field on boundary layer flow and heat transfer of tangent hyperbolic fluid with nanoparticles past a stretching sheet is discussed. The boundary

Present Result Akbar et al.[37] Fathizadeh et al. [1]

0.0 0.25 1 5 10 50 100 500 1000

1.0000

1.1180

1.4142

2.4495

3.3166

7.1414

10.0499

22.3830

31.6386

1.0000

1.11803

1.41421

2.44948

3.31662

7.14142

10.0499

22.3830

31.6386

1.41421 2.44948 3.31662 7.14142 10.0499

1.0000

Table 1: Comparison of values of - f'' (0) with magnetic field M whei

8 = y = 0

layer equations governing the flow problem are reduced into a couple of high order non-linear ordinary differential equations using the similarity transformation. The obtained differential equations are solved numerically using bvp4c from Matlab software. The effects of various governing parameters such as slip parameters y and 8, magnetic parameter M, Prandtl number Pr, Brow-nian motion parameter Nb, thermophoresis parameter Nt and Lewis number parameter Le on momentum, energy and concentration equation are analyzed using figures and tables.

Moreover, the study shows that the flow velocity and the skin friction coefficient on stretching sheet are strongly influenced by parameters power-law index n and Weissenberg number We . It is also observed that the velocity at the surface decreases as the values of slip parameter increase, but, the thermal boundary layer thickness increases as the values of the thermal radiation Nr and Biot number Bi increase. Furthermore, the local Nusselt number — 6' (0) decreases as the absolute values of slip parameters and power-law index n increase.

The fundamental points of the investigation are summarized as:

1. The Weissenberg number We and power-law index n diminishes the velocity boundary layer thickness.

2. The thermal radiation upturns thermal boundary layer thickness.

3. Power-law index n and Biot number Bi parameters increase the temperature field.

4. Hydrodynamic boundary layer thickness decreases with an increase in slip parameters.

5. The skin friction coefficient is a decreasing function of the slip parameter.

6. The effect Weissenber number is to reduce the skin friction coefficient.

7. The Weissenber number has the characteristic of reducing the local Nusselt number.

8. The effect of convective parameter and power-law index on the local Nusselt number is opposite.

9. Slip parameters reduce the local Nusselt number.

10. Heat transfer at the surface linearly decrease with an increase in thermophoresis parameter

11. Heat transfer at the surface of a sheet nonlinearly increase as the values of Biot number

incases.

We n M Y 8 - / (0)

0.1 0.3 0.2 1 -1 0.3317

0.2 0.3312

0.3 0.3308

0.3 0.1 0.3286

0.2 0.3300

0.3 0.3308

0.3 0.1 0.3288

0.2 0.3308

0.3 0.3316

0.5 á 0.3307

0.2 1 l 0.3308

2 0.2477

3 0.1978

4 0.1647 2 -1 0.2477

-2 0.2053

3 0.1770

4 0.1565

-■f '(0) with (

References

Table 2: computed of values of - f' '(0) with different values of We, n, M, y, 8

[1] Ali FM, Nazar R, Arifin NM, Pop I. MHD boundary layer flow and heat transfer over a stretching sheet with induced magnetic field. Heat and mass transfer. 2011;47(2):155-162.

[2] Ishak A, Nazar R, Pop I. Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet. Heat and Mass Transfer. 2008;44(8):921.

[3] Ishak A, Jafar K, Nazar R, Pop I. MHD stagnation point flow towards a stretching sheet. Physica A: Statistical Mechanics and its Applications. 2009;388(17):3377-3383.

[4] Ishak A, Nazar R, Bachok N, Pop I. MHD mixed convection flow near the stagnation-point on a vertical permeable surface. Physica A: Statistical Mechanics and its Applications. 2010;389(1):40-46.

[5] Mahapatra TR, Gupta A. Magnetohydrodynamic stagnation-point flow towards a stretching sheet. Acta Mechanica. 2001;152(1-4):191-196.

[6] Mahapatra TR, Nandy S, Gupta A. Magnetohydrodynamic stagnation-point flow of a power-law fluid towards a stretching surface. International Journal of Non-Linear Mechanics. 2009;44(2):124-129.

[7] Buongiorno J. Convective transport in nanofluids. Journal of Heat Transfer. 2006;128(3):240-250.

Nt Nr We n —9' (0)

0.1 0.5 0.3 0.3 0.1750

0.2 0.1720

0.3 0.1689

0.4 0.1660

0.5 0.1630

1 0.1490

5 0.0753

0.5 0.1 0.2105

0.3 0.1834

0.8 0.1404

.1606 0.1404 0.1153 0.0659 0.1517 0.1511 0.1500

Table 3: computed of values of —9'(0) with different values of Nt, Nr, We and n

[8] Kuznetsov A, Nield D. Natural convective boundary-layer flow of a nanofluid past a vertical plate. International Journal of Thermal Sciences. 2010;49(2):243-247.

[9] Khan W, Pop I. Boundary-layer flow of a nanofluid past a stretching sheet. International journal of heat and mass transfer. 2010;53(11):2477-2483.

[10] Makinde OD, Aziz A. Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition. International Journal of Thermal Sciences. 2011;50(7):1326-1332

[11] Ibrahim W, Shanker B. Boundary-layer flow and heat transfer of nanofluid over a verplate with convective surface boundary condition. Journal of Fluids Engineering. )12;134(8):081203.

2] Ibrahim W, Shanker B. Magnetohydrodynamic boundary layer flow and heat transfer of a nanofluid over non-isothermal stretching sheet. Journal of Heat Transfer. 2014;136(5):051701.

[13] Ibrahim W, Shanker B. MHD boundary layer flow and heat transfer due to a nanofluid over an exponentially stretching non-isothermal sheet. Journal of Nanofluids. 2015;4(1):16-27.

[14] Ibrahim W, Makinde O. The effect of double stratification on boundary-layer flow and heat transfer of nanofluid over a vertical plate. Computers & Fluids. 2013;86:433-441.

[15] Ibrahim W, Makinde O. Double-diffusive in mixed convection and MHD stagnation point flow of nanofluid over a stretching sheet. Journal of Nanofluids. 2015;4(1):28-37.

[16] Ibrahim W. The Effect of Induced Magnetic Field and Convective Boundary Condition on MHD Stagnation Point Flow and Heat Transfer of Nanofluid Past a Stretching Sheet. IEEE Transactions on Nanotechnology. 2015;14(1):178-186.

[17] Ibrahim W, Haq RU. Magnetohydrodynamic (MHD) stagnation point flow of nanofluid past a stretching sheet with convective boundary condition. Journal of the Brazilian Society of Mechanical Sciences and Engineering. 2016;38(4):1155-1164.

[18] Ibrahim W. MHD boundary layer flow and heat transfer of micropolar fluid past a stretching sheet with second order slip. Journal of the Brazilian Society of Mechanical Sciences and Engineering. 2016;p. 1-9.

[19] Ibrahim W, Makinde O. Magnetohydrodynamic stagnation point flow and heat transfer of Casson nanofluid past a stretching sheet with slip and convective boundary condition. Journal of Aerospace Engineering. 2015;29(2):04015037.

[20] Rauf A, Shehzad S, Hayat T, Meraj M, Alsaedi A. MHD stagnation point flow of micro nanofluid towards a shrinking sheet with convective and zero mass flux conditions. Bulletin of the Polish Academy of Sciences Technical Sciences. 2017;65(2):155-162.

[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. International Journal of Thermal Sciences. 2012;54:253-261.

[22] Ibrahim W, Shankar B, Nandeppanavar MM. MHD stagnation point flow and heat transfer due to nanofluid towards a stretching sheet. International Journal of Heat and Mass Transfer. 2013;56(1):1-9.

[23] Hayat T, Qasim M, Mesloub S. MHD flow and heat transfer over permeable stretching sheet with slip conditions. International Journal for Numerical Methods in Fluids. 2011;66(8):963-975.

[24] Aziz A. Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition. Communications in Nonlinear Science and Numerical Simulation. 2010;15(3):573-580.

[25] Bhattacharyya K, Layek G. Slip effect on diffusion of chemically reactive species in boundary layer flow over a vertical stretching sheet with suction or blowing. Chemical Engineering Communications. 2011;198(11):1354-1365.

[26] Halim N, Haq RU, Noor N. Active and passive controls of nanoparticles in Maxwell stagnation point flow over a slipped stretched surface. Meccanica. 2016;p. 1-13.

[27] Krishnamurthy M, Gireesha B, Prasannakumara B, Gorla RSR. Thermal radiation and chemical reaction effects on boundary layer slip flow and melting heat transfer of nanofluid induced by a nonlinear stretching sheet. Nonlinear Engineering. 2016;5(

of Maxwell fluid with suspended nanoparticles past a bidirectional porous stretching surface with thermal radiation. Thermal Science and Engineering Progress. 2017;1:6-14.

[30] Fang T, Yao S, Zhang J, Aziz A. Viscous flow over a shrinking sheet with a second order slip flow model. Communications in Nonlinear Science and Numerical Simulation. 2010;15(7):1831-1842.

[31] Fang T, Aziz A. Viscous flow with second-order slip velocity over a stretching sheet. Zeitschrift fur Naturforschung A. 2010;65(12):1087-1092.

[32] Nandeppanavar MM, Vajravelu K, Abel MS, Siddalingappa M. Second order slip flow and heat transfer over a stretching sheet with non-linear Navier boundary condition. International Journal of Thermal Sciences. 2012;58:143-150.

[33] Rogca AV, Pop I. Flow and heat transfer over a vertical permeable stretching/shrinking sheet with a second order slip. International Journal of Heat and Mass Transfer. 2013;60:355-

[34] Rogca NC, Pop I. Mixed convection stagnation point flow past a vertical flat plate with a second order slip: heat flux case. International Journal of Heat and Mass Transfer. 2013;65:102-109.

[35] Turkyilmazoglu M. Heat and mass transfer of MHD second order slip flow. Computers & Fluids. 2013;71:426-434.

[36] Singh G, Chamkha A. Dual solutions for second-order slip flow and heat transfer on a vertical permeable shrinking sheet. Ain Shams Engineering Journal. 2013;4(4):911-917.

[37] Akbar NS, Nadeem S, Haq RU, Khan Z. Numerical solutions of Magnetohydrodynamic boundary layer flow of tangent hyperbolic fluid towards a stretching sheet. Indian journal of Physics. 2013;87(11):1121-1124.

[38] Hayat T, Qayyum S, Ahmad B, Waqas M. Radiative flow of a tangent hyperbolic fluid with convective conditions and chemical reaction. The European Physical Journal Plus. 2016;131(12):422.

[29] Ramesh G, Prasannakumara B, Gireesha B, Shehzad S, Abbasi F. Three

[28] Prasannakumara B, Gireesha B, Gorla RS, Krishnamurthy M. Effects of and nonlinear thermal radiation on Williamson nanofluid slip flow over embedded in a porous medium. Journal of Aerospace Engineering. 2016

[39] Malik M, Salahuddin T, Hussain A, Bilal S. MHD flow of tangent hyperbolic fluid over a stretching cylinder: Using Keller box method. Journal of magnetism and magnetic materials. 2015;395:271-276.

[40] Naseer M, Malik MY, Nadeem S, Rehman A. The boundary layer flow of hyper gent fluid over a vertical exponentially stretching cylinder. Alexandria engineering j 2014;53(3):747-750.

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

[42] Salahuddin T, Malik M, Hussain A, Awais M, Khan I, Khan M. Analysis of tangent hyperbolic nanofluid impinging on a stretching cylinder near the stagnation point. Results in Physics. 2017;.

[43] Wu L. A slip model for rarefied gas flows at arbitrary Knudsen number. Applied Physics Letters. 2008;93(25):253103.

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

01 2345678

Figure 1: Velocity graph for different values of n and we when Nb = Nt = 0.5, Bi = 2, M = 0.2, Le = 5, 8 = -1, Y = 1, Nr = 0.8

0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

Y = 1 0.4

Y = 5 0.3

M— 0.1

<5 = -1 5 = -3 5 = -5 -5 = -7

Figure 2: Velocity profile for different values of yand 8 when Nb = Nt = 0.5, Bi = 2, M = 0.2, Le = 5, We = n = 0.3, Nr = 0.8

0.9 0.8 0.7 0.6 ^0.5 0.4 0.3 0.2 0.1 0

Figure 3: Temperature graph for different values o We = 0.3, Y = 1, S = —1

)f n and ]

Nr when Nb = Nt = 0.5, Bi = 2, M = 0.2, Le = 5,

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

s = -1

S = -3 S = -5 -S = -7

Y = 1 1

Y = 5 0.8

Figure 4: Temperature profile for different values of y and S when Nb = Nt = 0.5, Bi = 2, M = 0.2, Le = 5, We = n = 0.3, Nr = 0.8

0.9 0.8 0.7 0.6

0.4 0.3 0.2 0.i

0 5 i0 i5 20 25 30 35 40

Figure 5: Temperature graph for different values of 8 = -1, Nr = 0.8, y = 1

: Bi and W

e when Nb = Nt = 0.5, M = 0.2, Le = 5, n = 0.3,

0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.i 0

H-1-1-T"

Pr = 0.7 Pr = 1 Pr = 5 ■ Pr = 10

0 5 i0 i5 20 25 30 35 40

Figure 6: Temperature graph for different values of Pr when Nb = Nt = 0.5, M = 0.2, Le = 5, n = 0.3, 8 = — 1, Nr = 0.8, Y = 1

0.08 0.06 0.04 0.02 0

Figure 7: Concentration graph for different values of n when Nb = Nt = 0.5, M = 0.2, Le = 5, We = 0.3, S = —1, Nr = 0.8, Y = 1

Figure 8: Concentration profile for different values of y when Nb = Nt = 0.5, M = 0.2, Le = 5, We = n = 0.3, S = — 1, Nr = 0.8

Figure 9: Concentration profile for different values of S when Nb = Nt = 0.5, M = 0.2, Le = 5, We = n = 0.3, Y = 1, Nr = 0.8

0.4 0.6

Figure 10: Variation of the skin friction coefficient - f'(0) with M for different values of We when Nb = Nt = 0.5, M = 0.2, Le = 5, n = 0.3, y = 1, Nr = 0.8, 8 = -1

Figure 11: Variation of the skin friction coefficient M = 0.2, Le = 5, We = 0.3, y = 1, Nr = 0.8, 8 = -1

ith M for different values of n when Nb = Nt = 0.5,

Figure 12: Variation of the skin friction coefficient - f' (0) with M for different values of y when Nb = Nt = 0.5, M = 0.2, Le = 5, We = 0.3, n = 0.1, Nr = 0.8, 8 = -1

Figure 13: Variation of local Nusselt number-9'(0)

Le = 5, We = n = 0.3, Y = 1, Nr = 0.8

or different values of y when Nb = Nt = 0.5, M = 0.2,

Figure 14: Variation of local Nusselt number-9' (0) with Bi for different values of S when Nb = Nt = 0.5, M = 0.2, Le = 5, We = 0.3, Y = 1, Nr = 0.8

0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1 0.09

n = 0.1 n = 0.3 n = 0.5

Variation o

Figure 15: Variation of local Nusselt number -0'(0) with Nt for different values of n when Nb = Nt = 0.5, M = 0.2, Le = 5, We = 0.3, Y = 1, Nr = 0.8