Numerical simulation for nonlinear radiative flow by convective cylinder

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Numerical simulation for nonlinear radiative flow by convective cylinder

Tasawar Hayat

a,b

, Muhammad Tamoor

c

, Muhammad Ijaz Khan

a,⇑

, Ahmad Alsaedi

b

a

Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan

b

Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80257, Jeddah 21589, Saudi Arabia

c

Department of Basic Sciences, University of Engineering and Technology, Taxila 47050, Pakistan

a r t i c l e i n f o

Article history:

Received 30 September 2016

Received in revised form 7 November 2016 Accepted 15 November 2016

Available online 16 November 2016

Keywords:

Nonlinear thermal radiation Nonlinear stretching Porous medium

Convective boundary condition

a b s t r a c t

Present study explores the effect of nonlinear thermal radiation and magnetic field in boundary layer flow of viscous fluid due to nonlinear stretching cylinder. An incompressible fluid occupies the porous medium. Nonlinear differential systems are obtained after invoking appropriate transformations. The problems in hand are solved numerically. Effects of flow controlling parameters on velocity, temperature, local skin friction coefficient and local Nusselt numbers are discussed. It is found that the dimensionless velocity decreases and temperature increases when magnetic parameter is enhanced. Temperature pro-file is also increasing function of thermal radiation.

Ó 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/).

1. Introduction

The flow over a stretched surface has attached much interest of the researchers due to its various applications in the technological processes. Such applications include extrusion, cooling of strips or fibers, paper production, hot rolling, metallurgical procedures, wire drawing, glass fiber and so forth. The problems due to stretched surface have been extended to various flow situations. MHD vis-coelastic fluid flow due to stretched cylinder with Newtonian heat-ing is investigated by Farooq et al.[1]. Hayat et al.[2]worked on Cattaneo-Christov heat flux model with thermal stratification and temperature dependent conductivity. Numerical simulation of car-bon water nanofluid flow towards a stretched cylinder is analyzed by Hayat. et al.[3]. Pandey and Kumar[4]examined natural con-vection nanofluid flow by a stretched cylinder with viscous dissi-pation. MHD axisymmetric flow of third grade fluid by a stretching cylinder is studied by Hayat et al.[5]. Si et al.[6]worked on unsteady viscous fluid flow due to porous stretched cylinder.

Radiation has much significance in atomic reactor, glass gener-ation, heater outline, power plant furthermore in space innovation and many others. In radiation process the electromagnetic waves are responsible for transfer of energy which carries energy from the emanating object. MHD two dimensional unsteady boundary layer flow with thermal radiation is studied by Tian et al.[7]. Hayat et al. [8] investigated boundary layer flow of hydro-magnetic

Williamson liquid with thermal radiation. Further Hayat et al.[9]

analyzed mixed convection flow of an Oldroyd-B fluid bounded by stretching sheet with thermal radiation. Farooq et al.[10]worked on MHD stagnation point flow of viscoelastic nanofluid with non-linear radiation effects. Carbon water nanofluid with Marangoni convection and thermal radiation is examined by Hayat et al.[11]. Maria et al.[12]analyzed thermal radiation effects on convective flow with carbon nanotubes. Khan et al. [13]investigated three dimensional Burgers nanoliquid flow with non-linear thermal radi-ation. Waqar et al.[14]studied characteristics of heterogeneous-homogenous processes in three-dimensional flow of Burgers fluid.

The flow and heat transfer in presence of magnetic field has enormous application in many engineering and technological fields such as MHD power generators, in petroleum process, significant performance in nuclear reactors cooling, studies in the field of plasma, extractions of energy in geothermal field, orientation of the configuration of the boundary layer structure etc. Several methods have been developed in order to control the boundary layer structure. Thus chemically reactive MHD stretched flow due to curved surface is studied by Imtiaz et al. [15]. Waqas et al.

[16]investigated micropolar fluid flow due to nonlinear stretching surface with convective conditions. Magnetic field effects in flow of thixotropic nanofluid is explored by Hayat. et al.[17]. Numerical and analytical solutions for MHD flow of viscous fluid with variable thermal conductivity are studied by Khan et al.[18]. Few other studies related to MHD are examined in the Refs.[19–28].

To the best of author’s knowledge no study for MHD and non-linear thermal radiation is presented for flow due to cylinder. http://dx.doi.org/10.1016/j.rinp.2016.11.026

2211-3797/Ó 2016 Published by Elsevier B.V.

This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

⇑ Corresponding author.

E-mail address:mikhan@math.qau.edu.pk(M.I. Khan).

Contents lists available atScienceDirect

Results in Physics

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Therefore an attempt is made to investigate resulting nonlinear problem numerically. Governing partial differential equations have been reduced to ordinary differential equations. Shooting tech-nique and Runge-Kutta method evaluate the results numerically

[29–31]. Graphical results are also carefully analyzed.

2. Mathematical formulation

We are interested to examine the flow caused by nonlinear stretching phenomenon of cylinder. Permeable cylinder is chosen. Cylinder is convectively heated. Nonlinear radiation effect is fur-ther studied. Fluid occupying porous space is conducting via

CÞðNr 1Þð1 þ ðNr 1ÞhÞ2_h02_þPr 4K nþ 1 2 h0_f_nhf0 ¼ 0; ð7Þ f0¼ 1; f ¼ 0; h0_¼

_a

_{ð1 hÞ at}

_g

_{¼ 0;} f0! 0; h ! 0 as

g

! 1: ð8Þ

The velocity components parallel to x and r directions are denoted by u and v, qr¼ ð3k4Þ@T

4

@r radiative heat flux,

r

⁄

Stefan-Boltzman constant, kmean absorption coefficient,

a

¼ j

qcpthermal

diffusivity,

j

thermal conductivity, cpspecific heat,

q

fluid density,

h1heat transfer coefficient,

m

¼lqkinematic viscosity,

l

coefficient

of fluid viscosity,

r

electrical conductivity, B uniform magnetic field strength and k1 permeability of porous medium. Here

TwðxÞ ¼ T1þ T0xn surface temperature, T0reference temperature

and T1 ambient temperature. Physical parameters under

discus-sion are:

Table 1

Numerical values of f00ð0Þ and h0_{ð0Þ due to variation in}

physical parameters.

Fig. 1. Influence of n on velocity distribution.

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C¼ 2gm r2_R2 _Ux curvature parameter, M¼ rB2 0 qU0 1 2 magnetic parame-ter, P¼ mx

k1U porous medium parameter, Nr¼

Tw

T1temperature ratio

parameter, Pr_¼m_a Prandtl number and K_¼4T3 1

jk radiation

parameter.

The physical quantities like skin friction coefficient and local Nusselt number are defined as:

Cf ¼

2

s

w

q

U2; Nu ¼

xqw

kðTw T1Þ: ð9Þ

where

s

wis the surface shear stress and qwthe surface heat flux. Use

of transformation yields Re12 x Nux¼ 1 þ 4 3K h0_{ð0Þ; f}00_{ð0Þ ¼}1 2Re 1 2 xCfx: ð10Þ

In which Rex¼xU_t denotes the local Reynolds number.

Fig. 3. Influence of M on velocity distribution.

Fig. 4. Influence of P on velocity distribution.

Fig. 5. Influence of n on temperature distribution.

Fig. 6. Influence of C on temperature distribution.

Fig. 7. Influence of P on temperature distribution.

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3. Method for Numerical solution

Since the governing Eqs.(6) and (7)are nonlinear. Hence we intend to solve these by Runge-Kutta method. In the numerical procedure scheme we choose MATLAB software which satisfies our desired RK-4 methodology in conjunction with shooting crite-ria. The inner iteration is executed with convergence criteria of 106in all cases taking step size h = 0.01.

4. Results and discussions

This section provides the graphical and tabular outlook on the effect of various flow related parameters for the velocity and tem-perature profiles. Here our investigation lies on the comparative study of governing parameters namely nonlinearity exponent n, curvature parameter C, magnetic parameter M, porosity parameter P, temperature ratio parameter Nr, Prandtl number Pr and nonlin-ear radiation parameter K on velocity, temperature, local skin fric-tion and local Nusselt number. Graphs in Figs. 1–11 are constructed for fixed values of n = 2.0, C = 1.0, M = 1.0, P = 0.3, Nr = 0.3, Pr = 0.3, K = 2.0. Local skin friction and Nusselt number against different parameters are shown inTable 1.

Influence of nonlinearity exponent n on velocity profile is por-trayed inFig. 1. Here we can see that velocity field shows decreas-ing behavior for larger nonlinearity exponent n. It is due to the fact that fluid particle is disturbed for larger n. Therefore collision between the fluid particles enhances and as a result the velocity profile decreases.Fig. 2portrays the effect of curvature parameter

C on velocity distribution. Velocity field decreases when we increase the values of curvature parameter C. In fact radius of cylinder decreases. Therefore velocity field increases. Effect of magnetic parameter M on velocity distribution is shown inFig. 3. Velocity profile and associated layer thickness decay for larger M. Physically Lorentz force enhances resistive forces. Porosity param-eter P effect on velocity profile is illustrated in Fig. 4. With an increase in (P) the velocity profile enhances because porosity parameter is the capacity of medium to increase the motion of fluid particles.

Influence of n on temperature profile is shown inFig. 5. Temper-ature of the fluid particles enhances for larger n.Fig. 6depicts the behavior of temperature field for larger C. Temperature distribu-tion decays near the surface of cylinder and then shows increasing behavior far away from the surface of cylinder. The radius of cylin-der decreases for higher values of curvature parameter C due to which less particles are sticked to the surface of cylinder. Therefore temperature profile and associated boundary layer thickness are decreaseed.Fig. 7shows the effect of porosity parameter P on tem-perature distribution. It is noted that for larger P the temtem-perature enhances. Influence of temperature ratio parameter Nr on temper-ature field is plotted inFig. 8. Temperature profile and associated boundary layer thickness are enhanced for increasing values of Nr. FromFig. 8, it is clear that an increase in the Nr relates to a higher wall temperature when compared with the surrounding liq-uid. Behavior of Pr on temperature profile is displayed inFig. 9. Temperature of the fluid reduces for larger Pr. It is due to the fact that an increase in Pr reduces the thermal diffusivity. The particles are able to conduct less heat and consequently temperature decreases. The characteristics of thermal radiation on temperature distribution are sketched inFig. 10. Increasing values of thermal radiation K enhance temperature. Thermal layer thickness enhances for larger values of radiation parameterTable 1shows that local skin friction increases due to P only and local Nusselt number enhances for M, Nr and K.

5. Conclusions

In this article the nonlinear radiation in MHD flow by stretching cylinder is explored. Main points in this study include:

Shear stresses are increased for larger porosity parameter. Heat transfer rate is an increasing function of M, Nr and K. Velocity profile is increasing function of C, P.

Temperature profile is decreasing function of n, C and Pr.

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Fig. 9. Influence of Pr on temperature distribution.

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