Numerical solutions for magnetohydrodynamic flow of nanofluid over a bidirectional non-linear stretching surface with prescribed surface heat flux boundary

Numerical solutions for magnetohydrodynamic flow of nanofluid over a bidirectional non-linear stretching surface with prescribed surface heat flux boundary

Author’s Accepted Manuscript Numerical solutions for Magnetohydrodynamic flow of nanofluid over a bidirectional non-linear stretching surface with pre...

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Author’s Accepted Manuscript Numerical solutions for Magnetohydrodynamic flow of nanofluid over a bidirectional non-linear stretching surface with prescribed surface heat flux boundary B. Mahanthesh, B.J. Gireesha, R.S. Reddy Gorla, F.M. Abbasi, S.A. Shehzad www.elsevier.com/locate/jmmm

PII: DOI: Reference:

S0304-8853(16)30692-8 http://dx.doi.org/10.1016/j.jmmm.2016.05.051 MAGMA61465

To appear in: Journal of Magnetism and Magnetic Materials Received date: 4 April 2016 Revised date: 10 May 2016 Accepted date: 17 May 2016 Cite this article as: B. Mahanthesh, B.J. Gireesha, R.S. Reddy Gorla, F.M. Abbasi and S.A. Shehzad, Numerical solutions for Magnetohydrodynamic flow of nanofluid over a bidirectional non-linear stretching surface with prescribed surface heat flux boundary, Journal of Magnetism and Magnetic Materials, http://dx.doi.org/10.1016/j.jmmm.2016.05.051 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 galley proof before it is published in its final citable 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.

Numerical solutions for magnetohydrodynamic flow of nanofluid over a bidirectional non-linear stretching surface with prescribed surface heat flux boundary B. Mahanthesh1,2, B.J. Gireesha2,3, R.S. Reddy Gorla3, F.M. Abbasi4 and S.A. Shehzad5

1

Department of Mathematics, AIMS Institutes, Peenya-560058, Bangalore, INDIA

2

Department of Studies and Research in Mathematics, Kuvempu University Shankaraghatta-577 451, Shimoga, Karnataka, INDIA.

3

Department of Mechanical Engineering, Cleveland State University, Cleveland, OHIO, USA.

4

Department of Mathematics, Comsats Institute of Information Technology, Islamabad 44000,

Pakistan 5

Department of Mathematics, Comsats Institute of Information Technology, Sahiwal 57000, Pakistan

[email protected] [email protected] [email protected] [email protected] [email protected]

Abstract Numerical solutions of three-dimensional flow over a non-linear stretching surface are developed in this article. An electrically conducting flow of viscous nanoliquid is considered. Heat transfer phenomenon is accounted under thermal radiation, Joule heating and viscous dissipation effects. We considered the variable heat flux condition at the surface of sheet. The governing mathematical equations are reduced to nonlinear ordinary differential systems through suitable dimensionless variables. A well-known shooting technique is implemented to obtain the results of dimensionless velocities and temperature. The obtained results are plotted for multiple values of pertinent parameters to discuss the salient features of these parameters on fluid velocity and temperature. The expressions of skin-friction coefficient and Nusselt number are computed and analyzed comprehensively through numerical values. A comparison of present results with the previous results in absence of nanoparticle volume

fraction, mixed convection and magnetic field is computed and an excellent agreement noticed. We also computed the results for both linear and non-linear stretching sheet cases.

Keywords: Nanofluid; Joule heating; Magnetohydrodynamic; bidirectional stretching surface; viscous dissipation.

1. Introduction In the recent world of technology, liquid cooling is the major issue in industrial processes. The liquid cooling methods based on ordinary heat transfer fluid are inefficient and ineffective to fulfil the needs of industrial processes. A new class of nanotechnology has been introduced for the higher cooling efficiency during the industrial manufacturing of products. Nanofluids can be formed by the mixture of solid-liquid which have also capability of enhancing the thermal efficiency of ordinary liquids. The suspension of micrometre and millimetre sized solid particles in ordinary fluids resulted into the models of clogging, sedimentation etc. The nanoparticles are generally made of metals like Au, Ti, Ag, Fe, Cu, Al etc., nitride carbide, and metal oxide CuO, TiO2 and Al2O3.The shape of such particles may be tubular, spherical and rod-like. The production of nanofluids is categorized in two ways namely single step and two steps. The general applications of nanoliquids include generation of new type fuels, industrial and vehicle cooling, reduction of fuel in power generation plant, imaging, sensing, cancer therapy etc. Choi [1] was firstly done the experimental work on nanofluid and deduced that the nanoparticles greatly enhance the thermal conductivity of liquid. After his remarkable work on nanofluid technology, the researchers investigated the mechanism of nanofluids widely which can be noticed in the studies [2-17]. The magneto nanofluids have potential role in the industrial manufacturing and production of biomedical equipments. It is involved in the processes of gastric medications, sterilized devices and wound treatment [18]. The desired effects in industrial applications can be achieved by the use of magnetic field for the manipulation of electrically conducting nanoliquids. Number of diverse investigations has been made in the past to examine the performance

of

magnetic

nanoparticles

based

suspensions.

Nowadays,

magnetic

nanoparticles based suspensions have been implemented in magnetic resonance imaging, targeted drug release, elimination of tumors and many others. Having all such potential roles

of magnetic field in modern industry, Sheikholeslami et al. [19] reported the impact of nonuniform magnetic field in the flow of viscous nanofluid past a lid driven annulus enclosure. Zhang et al. [20] discussed the magneto nanofluid over a radiative sheet in presence of chemical reaction. They imposed the prescribed heat flux condition at the boundary of surface instead of prescribed surface temperature. Abbasi et al. [21] provided a mathematical model for drug delivery system through the peristaltic transport of magnetic particles in water based liquid. Numerical computations for variable magnetic field nanoliquid problem have been made by Sheikholeslami et al. [22]. They used the two-phase model by considering the effects of thermophoresis and Brownian motion. Melting heat transfer phenomenon of Williamson nanofluid in presence of an applied magnetic field has been numerically explored by Krishnamurthy et al. [23]. Impact of constant magnetic field on three-dimensional flow of rate type fluid in presence of Robin’s conditions has been addressed by Shehzad et al. [24]. Khan et al. [25] discussed the nonaligned MHD flow of variable viscosity nanoliquid over a heated surface. The importance of thermal radiation in modern science and technology is very vital and involved in many engineering and industrial processes like agriculture, electric power, food, medical industry, non-destructive testing, solar cell panels and many others. The thermally radiative surface characteristics and surface thermal radiation properties can be changed and affected seriously by the distribution of depositions on the surfaces. It is very essential to understand the mechanism of thermal radiation to obtain the desired highly quality products in the industrial processes. Mahmoud and Waheed [26] analyzed the effects of variable fluid properties in thermally radiative flow of micropolar fluid by imposing the slip velocity condition. Su et al. [27] utilized the thermal radiation effects in convective flow of viscous fluid induced by a permeable wedge. Combined impacts of thermal-diffusion and diffusionthermo on the non-Newtonian liquid over a radiative sheet in presence of porous medium have been explored by Mahmoud and Megahed [28]. Das et al. [29] reported the impact of thermally radiative flow in a vertical channel by considering the prescribed surface temperature and heat flux conditions. They presented the exact solutions of velocity and temperature through Laplace transform. Shehzad et al. [30] elaborated the effects of radiation in three-dimensional flow of Jeffrey nanoliquid past a bidirectional stretching surface. This article is based on the numerical computations for three-dimensional flow of viscous fluid over a nonlinear bidirectional surface. The effects of viscous dissipation, Joule heating, magnetic field and thermal radiation are encountered. We suspended the Cu,Al2O3 and TiO2 types nanoparticles in ordinary based liquid. The governing equations are highly nonlinear

and coupled due to consideration of mixed convection. Shooting technique is utilized to find the numerical solutions of governing expressions. A benchmark is made to validate the present methodology and we noticed the good agreement with the previous results in limiting way. The computed results are plotted for multiple values of arising parameters and discussed physically. 2. Mathematical Modelling The physical regime of the present study is demonstrated in Fig. 1, in Cartesian coordinate system. A steady three-dimensional flow of an electrically conducting fluidover a stretching surface in the presence of nanoparticlesis considered. Let velocity components along directions ( and

(

)

be the

and directions correspondingly. The sheet is stretched in two

) with the non-linear velocities

respectively, in where

and

and

(

) and

(

)

are constants. A variable kind of magnetic field

is applied in the normal direction, i.e., in the direction. The electric field is

neglected. The induced magnetic field is also ignored as of small magnetic Reynolds number. Further, the surface is maintained with power law heat flux

(

) , where

is

constant.

Fig. 1: Geometry of the problem. We have used the Tiwari and das model to simulate the nanofluid. The thermophysical properties of the base fluid water and nanoparticles are taken from [5].It is assumed that, the base fluid and nanoparticles are in thermal equilibrium and no slip occurs between them. The

influences of viscous dissipation, Joule heating and thermal radiation are also present in thermal analysis. In accordance with aforementioned assumptions, the conservation of flow and temperature equations are given by (2.1) (

)

(

)

,

(2.2)

,

(2.3)

,

(

(2.4) (

[

)

(

(( )

) )

(

( ) ) ]

)

(

(2.5)

)

the boundary conditions suggested by the physics of the problem are } Where

the acceleration due to gravity,

(2.6)

the magnetic field,

nanofluid,

the ambient temperature,

the radiative heat flux,

the density,

the thermal conductivity,

the specific heat,

the temperature of the the dynamic viscosity,

the electrical conductivity and

the thermal volumetric coefficient. The subscript ‘ ’ represents the nanofluid. There are several models have reported to characterize the effective properties of nanofluid on basis of particle shape, size and other characteristics. The Maxwell-Garnetts [2] and Brinkman [3] models are utilized and these models are restricted to spherical shaped nanoparticles. The nanofluids effective properties are given by; (

)

(

)

(

)

(

)(

)

(

)

(

)

(

)(

)

(

)

(

[

) (

(

)

(

)

)

(2.7)

] }

were -the volume fraction of nanoparticles, the subscripts ‘ ’ and ‘ ’ represents the base fluid and the nanoparticles respectively. Further, the last term in the right hand side of equation (2.5) represents the heat radiation effect. The radiative heat flux

is simulated by using the Rosseland approximation as

follows; , where

(2.8)

is the Stefan-Boltzmann constant and

is the mean absorption coefficient. It is

assumed that, the temperature difference with in the flow is sufficiently small. So that be expanded in a Taylor series about (

can

as follows;

)

(

)

(2.9)

Now by neglecting higher order terms beyond the first degree in (

) , and by

substituting the same into the equation (2.8) one can we get .

(2.9)

Consequently, the energy equation (2.5), takes the following form; ((

)

(

(

)

)

[

) (

(( ) )

( ) ) ] )

(

(2.10)

Following appropriate similarity transformations are chosen to reduce the system of PDEs to ODEs;



(

)

(

(

)

(

)

(( )

( ))

(

(

))

(2.11)

) ( )

√ (

) }

here

√ . In view above defined variables, the equation (2.1) is automatically

satisfiedand the equations (2.2), (2.3) (2.4), (2.6) and (2.10) can be written in the following manner; (

(

)

(

) )

,

(2.12)

(

(

(

)

(

(

)

)

)

)

,

((

[

)

(

(2.13)

) )

(( )

]

(2.14)

( ) )

With the corresponding boundary conditions are }

(2.15)

where, ( (

(

) (

)

(

)

)

) (

[

(

) (

(

)

(

)

)

]

)

-magnetic parameter, -mixed convection parameter, (

)(

)

-local Grashof number,

-thermal radiation parameter, (

)

- Prandtl number,

(

and

)

- Eckert number

-stretching ratio parameter.

The most important physical quantities for the problem in engineering point of view are the skin-friction coefficients along

and

directions and local Nusselt number. They are

defined as (

)

(

)

(2.16)

( here

)

and

( )

are respectively, the local Reynolds number along the

and

directions. 3. Method of Solution and Validation The two-point boundary value problem defined by the equations (2.11)-(2.14) is highly nonlinear and coupled in nature, thus closed form solution is impracticable. Therefore, the complicated non-linear boundary value problem is numerically solved by converting them to an initial-value problem. To this end consider following equations

(

(

)

(

(

(

(

)

(

)

)

(

(

)( (

)

)

)

)

(

)

))

with the boundary conditions ( ) ( )

( ) ( )

( )

( )

( )

In order to solve initial-value problem, the unknowns

( ) and

are determined by iterative

method called shooting method using algebraic software Maple. The unknowns

and

are chosen such a way that the solutions satisfy the boundary condition at infinity. The shooting technique is based on Maple functioning ‘shoot’ algorithm. A detailed explanation of the Shooting method on maple implementation can be found in Meade et al [31]. Then after, the resultant system of the initial value problem is solved by Runge-Kutta-Fehlberg scheme. Throughout our numerical computation, the step size is taken as the convergence criteria

.

with

Table 1: Comparison of the values of

and

with that of Junaid et al [4] with

for various values of and . Junaid et al [4] Present study

1 1 1 3 3 3

0 0.5 1 0 0.5 1

-1 -1.224745 -1.414214 -1.624356 -1.989422 -2.297186

0 -0.612372 -1.414214 0 -0.994711 -2.297186

-1 -1.22474 -1.41421 -1.62436 -1.98942 -2.29719

0 -0.61237 -1.41421 0 -0.99471 -2.29712

The accuracy of the used numerical method is validated by direct comparison of

and

values with the available published results of Junaid et al [4]. Table 1 shows the comparison results and it is found to be an excellent agreement. This confirms that the numerical method adopted in the present work gives accurate results. Results and discussion Table 2 is computed to explore the properties of multiple values of nanoparticle volume fraction ( ) and magnetic parameter M2 on Rex0.5Cfx by considering different types of nanoparticles. In this Table we considered n  3 and n  1 where n  3 for non-linear stretching case and n  1 is for linear stretching of surface. We have noticed that the values of Rex0.5Cfxare enhanced in the case of Cu , Al 2 O3 and TiO2 for the larger values of 𝜙while decrease when we give rise to the values of M2 by fixing n=3. The reverse trend is examined in the case of linear stretching sheet (n=1). The values of Rex0.5Cfx are higher in the case of hydromagnetic flow in comparison to hydrodynamic case. In Table 3, we investigated the impact of nanoparticle volume fraction ( ) and magnetic parameter (M2) onRex0.5Cfy for Cu ,

Al 2 O3 and TiO2 nanoparticles by fixing n  3 and n  1. The values of Rex0.5Cfyare smaller in presence of nanoparticles and larger values of nanoparticle volume fraction lead to the higher shear stress at the wall. The values of Nusselt number Re-0.5xNu for different nanoparticles by using the various values of nanoparticle volume fraction and magnetic parameter by setting n=3 and n=1. The numerical values of Nusselt number are higher in the case of Cu-water nanoparticles in comparison to Al2O3 and TiO2. An increasing behavior of Nusselt number is noticed when we insert the larger values of magnetic parameter. Further,

we studied that the Nusselt number is higher for n  1 in comparison to n  3. It is also revealed that values of Nusselt number for different nanoparticles are similar when   0 and n=1 and in the case of n=3. Table 5 is computed to analyze the values of Rex0.5Cfx of multiple nanoparticles for different values of λ, c, n, R, Ec and Pr when 𝜙=0.2 and M2=0.5. From this Table, it is revealed that the values of Rex0.5Cfxare decay corresponding to the increasing values of mixed convection parameter λ while these values are boost up for higher stretching ratio parameter c. An increase in radiation parameter R and Eckert number Ec correspond to larger values Rex0.5Cfx for all the cases of nanoparticles. On contrary, these values are risen for the increasing values of Prandtl number Pr. Table 6 is presented to explore the values of Rex0.5Cfyfor multiple values of λ, c, n, R, Ec and Pr by fixing 𝜙=0.2 and M2=0.5.The values of Rex0.5Cfyare larger in the case non-linear stretching surface in comparison to linear stretching of sheet. The values of Nusselt number Re-0.5xNucorresponding to various values of λ, c, n, R, Ec and Pr in Table 7. In this Table, we made the analysis by fixing the values of nanoparticle volume fraction (𝜙) and magnetic parameter (M2). The values of Nusselt number in the case of Cwater, Al2O3and TiO2-water are smaller in presence of mixed convection parameter. The presence of mixed convection parameter involves the buoyancy force that creates a reduction in temperature gradient due to which Nusselt number is reduced for the larger values of λ. The Nusselt number is risen with an increase in the values of Ec and Pr. Fig. 2 is presented to visualize the curves of velocity

df d

,

dg d

and temperature  ( ) for

multiple values of magnetic parameter ( M 2 ). It is evident from this Fig. that the temperature at the wall is much more while the velocity velocity

df d

and

dg d

df d

and

dg d

is lesser. It is noticed that the

is reduced when we use the higher values of M2 . On the other hand

side, the fluid temperature is enhanced with an increase in M 2 . The Lorentz force takes place when we consider the hydromagnetic flow. The larger values of magnetic parameter correspond to stronger Lorentz force and this stronger force produced a resistance to fluid flow due to which the velocity of liquid is decreased while temperature enhances. The curves of velocity

df d

,

dg d

and temperature  ( )

for different values of mixed convection

parameter ( ) are given in Fig. 3. An enhancement in  corresponds to higher velocity and smaller temperature. Buoyancy force is involved due to presence of mixed convection and it is well known that the stronger buoyancy force gives rise to the fluid flow while reduces its temperature. Here the larger values of  imply higher buoyancy force that

corresponds to smaller temperature and higher velocity. Fig. 4 shows the plots of velocity

df d

,

dg d

and temperature  ( ) for various values of

power law index (n). It is worthmentioning to point out here that n  1 is for linear and n  3, 5 corresponds to non-linear stretching surface cases. We examined that the velocity

and temperature are higher in the case of linear stretching surface. It is also observed that the reduction in velocity and temperature is appeared when nanoparticle volume fraction ( ) on velocity

df d

,

dg d

n

increases. The impact of

and temperature  ( ) is analyzed in

Fig. 5. The temperature is enhanced dramatically with an increase in nanoparticle volume fraction. The thermal conductivity of considered liquid is higher in presence of nanoparticles. The liquid of stronger thermal conductivity has higher temperature due to which increasing temperature is observed in Fig. 5. Fig. 6 gives the curves of velocity

df d

,

dg d

and temperature  ( ) for different values of

stretching ratio parameter (c). Here we noticed that the velocity ddf is decreased while the velocity

dg d

enhances when we use the larger values of

c. The velocity

df d

and

temperature  ( ) are boost up when we increase the values of radiation parameter (R) (see Fig. 7). The larger values of R implies to more heat absorbed by the fluid due to which velocity and temperature enhance. Fig. 8 elucidates that the velocity

df d

,

dg d

and temperature

 ( ) are decreased as the value of Prandtl number (Pr) enhances. Prandtl number is strongly depends on the thermal conductivity of liquid. The weaker thermal conductivity is occurred due to larger Prandtl number. Such facts lead to lower temperature and velocity. The curves of velocity

df d

,

dg d

and temperature  ( ) for multiple values of Eckert number

(Ec ) are seen in Fig. 9. This Fig. clearly indicates that velocity

df d

,

dg d

and temperature

 ( ) are enhanced for the larger values of Eckert number. Kinetic energy is involved in the definition of Eckert number. An increase in Eckert number corresponds to higher kinetic energy that gives rise to fluid velocity and temperature. Table 2: The Values

for different nanoparticles for various values

. Non-linear stretching sheet(n=3)

water

water

water

Linear stretching sheet(n=1)

water

water

water

and

-0.74152

-0.74152

-0.74152

1.26678

1.26678

1.26678

0.1

-1.57448

-1.27361

-1.28537

0.76495

0.89980

0.90612

0.2

-2.50614

-1.93187

-1.95494

0.32064

0.51248

0.52915

0.0

-2.51103

-1.94553

-1.96576

-0.05032

0.12320

0.13534

0.5

-2.50614

-1.93187

-1.95494

0.32064

0.51248

0.52915

1.0

-2.50359

-1.92256

-1.94842

0.73582

0.93410

0.95559

0

0.2

0.5

Table 3: The Values

for different nanoparticles for various values

Non-linear stretching sheet (n=3)

and

.

Linear stretching sheet (n=1)

water

water

water

water

water

water

-1.38648

-1.38648

-1.38648

-0.98293

-0.98293

-0.98293

0.1

-2.08849

-1.79674

-1.81389

-1.44337

-1.28450

-1.29995

0.2

-2.89135

-2.30627

-2.45537

-1.97605

-1.92069

-1.98667

0.2 0.0

-2.78546

-2.17334

-2.21084

-1.80628

-1.40979

-1.43402

0.5

-2.89135

-2.30627

-2.34182

-1.97605

-1.61560

-1.63729

1.0

-2.99238

-2.43012

-2.46407

-2.13126

-1.79538

-1.81542

0

0.5

Table 4: The Values

for different nanoparticles for various values

Non-linear stretching sheet(n=3)

and

.

Linear stretching sheet(n=1)

Cu water

water

water

Cu water

water

Water

-2.496579

-2.496579

-2.496579

-1.83892

-1.83892

-1.83892

0.1

-2.207378

-2.388335

-2.337880

-1.81918

-1.83126

-1.83767

0.2

-1.97276

-2.238515

-2.151899

-1.73896

-1.77571

-1.77084

0.0

-2.185714

-2.568473

-2.459575

-2.20285

-2.45808

-2.35645

0.5

-1.97276

-2.238515

-2.151899

-1.73896

-1.83434

-1.77084

0

0.2

0.5

1.0

-1.799064

-1.989285

Table 5: The values and

when

-1.917776

-1.43635

-1.39296

for different nanoparticles for various values and

.

water

water

water

nanofluid

nanofluid

nanofluid

-4.54567

-3.64972

-3.70604

1

-4.01471

-3.23783

-3.28594

2

-3.58106

-2.88316

-2.92384

0.2

-3.14484

-2.52232

-2.55671

0.4

-3.39529

-2.72829

-2.76641

0.6

-3.58106

-2.88316

-2.92384

1

-1.56627

-1.25803

-1.26967

2

-2.72383

-2.19842

-2.22662

3

-3.58107

-2.88316

-2.92384

0

-3.62595

-2.92695

-2.96789

0.4

-3.59011

-2.89203

-2.93277

0.8

-3.55422

-2.85655

-2.89700

0

-4.37595

-3.50200

-3.55691

1

-2.97756

-2.38371

-2.41276

2

-1.99015

-1.53485

-1.54446

7

-3.59398

-2.89583

-2.93659

10

-3.62376

-2.92482

-2.96575

15

-3.67876

-2.94621

-3.01365

0

2

0.6

0.6

3

3

0.5

0.5

0.5

6.2

0.5

Table 6: The values and

-1.40933

when

for different nanoparticles for various values and

.

0

0.6

3

0.5

0.5

6.2

water

water

water

nanofluid

nanofluid

nanofluid

-2.72741

-2.18983

1

-2.77535

-2.22045

2

-2.81168

-2.24571

0.2

-0.82938

-0.66634

0.4

-1.76832

-1.41616

0.6

-2.81168

-2.24571

1

-1.87287

-1.53449

2

-2.39153

-1.92511

3

-2.81168

-2.24571

0

-2.79801

-2.23618

0.4

-2.80921

-2.24392

0.8

-2.81851

-2.25079

0

-2.74207

-2.19985

1

-2.97756

-2.28116

2

-3.09901

-2.33941

7

-2.80811

-2.24314

10

-2.79878

-2.23669

15

-2.79016

-2.23103

2

0.6

3

0.5

0.5

Table 7: The values and

when

0

0.6

1 2 2

0.2 0.4 0.6

-2.25468 -2.28654 -0.67626 -1.43766 -2.28035 -1.55476 -1.95344 -2.28035 -2.27072 -2.27854 -2.28550 -2.23364 -2.31655 -2.37609 -2.27775 -2.27125 -2.26554

for different nanoparticles for various values and

3

-2.22362

0.5

.

0.5

6.2

water

water

water

nanofluid

nanofluid

nanofluid

-1.2908417

-1.5892357

-1.4181972

-1.4425756

-1.727211

-1.5410705

-1.5642853

-1.8434911

-1.5410705

-1.9143628

-2.1725183

-1.9418189

-1.772706

-2.0500513

-1.8307996

-1.5642853

-1.8434911

-1.6449536

0.6

1 2 3 3

0 0.4 0.8 0.5

0 1 2 0.5

7 10 15

Figure 2: Velocities in

and directions (

-1.5714216

-1.7123665

-1.5356672

-1.5754936

-1.8024517

-1.6112418

-1.5642853

-1.8434911

-1.6449536

-1.4358136

-1.719659

-1.5336354

-1.5452251

-1.8262211

-1.6293582

-1.6085355

-1.8809863

-1.6789836

-7.7360526

-7.8897323

-7.0795142

-0.9673386

-1.1410987

-1.0181212

-0.5957946

-0.697687

-0.6229675

-1.5362412

-1.8179072

-1.6218621

-1.4447181

-1.7286773

-1.5417139

-1.3317351

-1.6112677

-1.4367322

and

) and temperature ( ( )) graph for different

values of magnetic parameter (

).

Figure 3: Velocities in

and directions (

and

) and temperature ( ( )) graph for different

values of mixed convection parameter ( ).

Figure 4: Velocities in

and directions (

and

) and temperature ( ( )) graph for different

values of power-law index ( ).

Figure 5: Velocities in

and directions (

and

) and temperature ( ( )) graph for different

values of nanoparticles volume fraction (𝜙).

Figure 6: Velocities in

and directions (

and

) and temperature ( ( )) graph for different

values of stretching ratio parameter ( ).

Figure 7: Velocities in

and directions (

and

) and temperature ( ( )) graph for different

values of radiation parameter ( ).

Figure 8: Velocities in

and directions (

and

) and temperature ( ( )) graph for different

values of Prandtl number (

Figure 9: Velocities in

and directions (

different values of Eckert number (

).

) and temperature ( ( )) graph for

and

).

Conclusions The impact of Cu, Al2O3 and TiO2 nanoparticles in three-dimensional flow of viscous liquid is explored in this research. An electrically conducting flow is considered in presence of thermal radiation and viscous dissipation. The governing equations are coupled due to presence of mixed convection. We examined that the values of skin-friction coefficient are larger in case of nonlinear stretching surface in comparison to linear stretching case. The values of Nusselt number are increased in presence of Cu when we use the larger values of nanoparticle volume fraction. The presence of magnetic field gives rise to the temperature and thermal boundary layer thickness but retards the fluid velocity and momentum boundary layer thickness. The velocity filed and temperature are higher for linear stretching problem when we compare it with the case of non-linear stretching. An enhancement in ratio parameter corresponds to retardation in velocity

df d

but the velocity

dg d

increases. The

presence of thermal radiation parameter gives more heat to fluid due to which higher temperature is achieved. Further, an increase in Eckert number gives rise to kinetic energy due to which higher temperature and thicker thermal boundary layer thickness is noted. Acknowledgment One of the authors (B. J. Gireesha) is thankful to the University Grants Commission, India, for the financial support under the scheme of Raman Fellowship-2014No. F 5-110/2014 (IC), for Post-Doctoral Research for Indian Scholars in USA.

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Highlights    

Hydromagnetic flow of nanofluid over a bidirectional non-linear stretching surface is examined. Cu, Al2O3 and TiO2 types nanoparticles are taken into account. Numerical solutions have been computed and addressed. The values of skin-friction and Nusselt number are presented.