Results in Physics 6 (2016) 1031–1035

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Results in Physics

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Numerical simulation for nonlinear radiative flow by convective cylinder

http://dx.doi.org/10.1016/j.rinp.2016.11.0262211-3797/� 2016 Published by Elsevier B.V.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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

Tasawar Hayat a,b, Muhammad Tamoor c, Muhammad Ijaz Khan a,⇑, Ahmad Alsaedi b

aDepartment of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, PakistanbNonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80257, Jeddah 21589,Saudi ArabiacDepartment 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 2016Received in revised form 7 November 2016Accepted 15 November 2016Available online 16 November 2016

Keywords:Nonlinear thermal radiationNonlinear stretchingPorous mediumConvective boundary condition

a b s t r a c t

Present study explores the effect of nonlinear thermal radiation and magnetic field in boundary layerflow of viscous fluid due to nonlinear stretching cylinder. An incompressible fluid occupies the porousmedium. Nonlinear differential systems are obtained after invoking appropriate transformations. Theproblems 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 dimensionlessvelocity decreases and temperature increases when magnetic parameter is enhanced. Temperature pro-file is also increasing function of thermal radiation.� 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 ofthe researchers due to its various applications in the technologicalprocesses. Such applications include extrusion, cooling of strips orfibers, paper production, hot rolling, metallurgical procedures, wiredrawing, glass fiber and so forth. The problems due to stretchedsurface 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 onCattaneo-Christov heat flux model with thermal stratification andtemperature dependent conductivity. Numerical simulation of car-bon water nanofluid flow towards a stretched cylinder is analyzedby 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 astretching cylinder is studied by Hayat et al. [5]. Si et al. [6] workedon 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 innovationand many others. In radiation process the electromagnetic wavesare responsible for transfer of energy which carries energy fromthe emanating object. MHD two dimensional unsteady boundarylayer flow with thermal radiation is studied by Tian et al. [7]. Hayatet 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 bystretching sheet with thermal radiation. Farooq et al. [10] workedon MHD stagnation point flow of viscoelastic nanofluid with non-linear radiation effects. Carbon water nanofluid with Marangoniconvection and thermal radiation is examined by Hayat et al. [11].Maria et al. [12] analyzed thermal radiation effects on convectiveflow with carbon nanotubes. Khan et al. [13] investigated threedimensional 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 hasenormous application in many engineering and technological fieldssuch as MHD power generators, in petroleum process, significantperformance in nuclear reactors cooling, studies in the field ofplasma, extractions of energy in geothermal field, orientation ofthe configuration of the boundary layer structure etc. Severalmethods have been developed in order to control the boundarylayer structure. Thus chemically reactive MHD stretched flow dueto curved surface is studied by Imtiaz et al. [15]. Waqas et al.[16] investigated micropolar fluid flow due to nonlinear stretchingsurface with convective conditions. Magnetic field effects in flow ofthixotropic nanofluid is explored by Hayat. et al. [17]. Numericaland analytical solutions for MHD flow of viscous fluid with variablethermal conductivity are studied by Khan et al. [18]. Few otherstudies 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.

Table 1Numerical values of f 00 ð0Þ and h0ð0Þ due to variation inphysical parameters.

Fig. 1. Influence of n on velocity distribution.

Fig. 2. Influence of C on velocity distribution.

1032 T. Hayat et al. / Results in Physics 6 (2016) 1031–1035

Therefore an attempt is made to investigate resulting nonlinearproblem numerically. Governing partial differential equations havebeen 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 nonlinearstretching 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

applied magnetic field. Induced magnetic and electric fields are

negligible. Applied magnetic field is taken in the B ¼ B0xn�12 . The fol-

lowing statements lead to the resulting flow and temperaturefields.

@ðruÞ@x

þ @ðrtÞ@r

¼ 0; ð1Þ

u@u@x

þ t@u@r

¼ mr

@

@rr@u@r

� �� m

k1þ rB2

q

!u; ð2Þ

u@T@x

þ t@T@r

¼ ar

@

@rr@T@r

� �� 1qcp

@qr

@r; ð3Þ

u ¼ U0xn; t ¼ 0; �k @T@r ¼ h1ðTw � TÞ at r ¼ R;

u ! 0; T ! T1 as r ! 1:ð4Þ

Invoking

g ¼ r2 � R2

2RUmx

� �12

;w ¼ ðUmxÞ12Rf ðgÞ; hðgÞ ¼ T � T1Tw � T1

: ð5Þ

One can arrived at

ð1þ 2gCÞf 000 þ nþ 12

� �ff 00 þ 2Cf 00 � nf 02 � Pf 0 �Mf 0 ¼ 0; ð6Þ

ð1þ2gCÞð1þð1þðNr�1ÞhÞ3Þh00 þ 12K

þð1þðNr�1ÞhÞ3� �

Ch0

þ2ð1þ2gCÞðNr�1Þð1þðNr�1ÞhÞ2h02þ Pr4K

nþ12

� �h0f �nhf 0

� �¼0;

ð7Þ

f 0 ¼ 1; f ¼ 0; h0 ¼ �að1� hÞ at g ¼ 0;f 0 ! 0; h ! 0 as g ! 1:

ð8Þ

The velocity components parallel to x and r directions are

denoted by u and v, qr ¼ �ð 43k�Þ @T4

@r radiative heat flux, r⁄ Stefan-Boltzman constant, k� mean absorption coefficient, a ¼ j

qcpthermal

diffusivity, j thermal conductivity, cp specific heat, q fluid density,h1 heat transfer coefficient, m ¼ l

q kinematic viscosity, l coefficient

of fluid viscosity, r electrical conductivity, B uniform magneticfield strength and k1 permeability of porous medium. HereTwðxÞ ¼ T1 þ T0xn surface temperature, T0 reference temperatureand T1 ambient temperature. Physical parameters under discus-sion are:

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.

Fig. 8. Influence of Nr on temperature distribution.

T. Hayat et al. / Results in Physics 6 (2016) 1031–1035 1033

C ¼ 2gmr2�R2

xU

� �curvature parameter,M ¼ rB20

qU0

� 12magnetic parame-

ter, P ¼ mxk1U

porous medium parameter, Nr ¼ TwT1

temperature ratio

parameter, Pr ¼ ma Prandtl number and K ¼ 4�T31

jk� radiationparameter.

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

Cf ¼ 2swqU2 ; Nu ¼ xqw

kðTw � T1Þ : ð9Þ

where sw is the surface shear stress and qw the surface heat flux. Useof transformation yields

Re�1

2x Nux ¼ � 1þ 4

3K

� �h0ð0Þ; f 00ð0Þ ¼ 1

2Re

12xCfx: ð10Þ

In which Rex ¼ xUt denotes the local Reynolds number.

Fig. 9. Influence of Pr on temperature distribution.

Fig. 10. Influence of K on temperature distribution.

1034 T. Hayat et al. / Results in Physics 6 (2016) 1031–1035

3. Method for Numerical solution

Since the governing Eqs. (6) and (7) are nonlinear. Hence weintend to solve these by Runge-Kutta method. In the numericalprocedure scheme we choose MATLAB software which satisfiesour desired RK-4 methodology in conjunction with shooting crite-ria. The inner iteration is executed with convergence criteria of10�6 in all cases taking step size h = 0.01.

4. Results and discussions

This section provides the graphical and tabular outlook on theeffect of various flow related parameters for the velocity and tem-perature profiles. Here our investigation lies on the comparativestudy of governing parameters namely nonlinearity exponent n,curvature parameter C, magnetic parameter M, porosity parameterP, 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 areconstructed 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 numberagainst different parameters are shown in Table 1.

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

C on velocity distribution. Velocity field decreases when weincrease the values of curvature parameter C. In fact radius ofcylinder decreases. Therefore velocity field increases. Effect ofmagnetic parameter M on velocity distribution is shown in Fig. 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 anincrease in (P) the velocity profile enhances because porosityparameter is the capacity of medium to increase the motion of fluidparticles.

Influence of n on temperature profile is shown in Fig. 5. Temper-ature of the fluid particles enhances for larger n. Fig. 6 depicts thebehavior of temperature field for larger C. Temperature distribu-tion decays near the surface of cylinder and then shows increasingbehavior far away from the surface of cylinder. The radius of cylin-der decreases for higher values of curvature parameter C due towhich less particles are sticked to the surface of cylinder. Thereforetemperature profile and associated boundary layer thickness aredecreaseed. Fig. 7 shows the effect of porosity parameter P on tem-perature distribution. It is noted that for larger P the temperatureenhances. Influence of temperature ratio parameter Nr on temper-ature field is plotted in Fig. 8. Temperature profile and associatedboundary layer thickness are enhanced for increasing values ofNr. From Fig. 8, it is clear that an increase in the Nr relates to ahigher wall temperature when compared with the surrounding liq-uid. Behavior of Pr on temperature profile is displayed in Fig. 9.Temperature of the fluid reduces for larger Pr. It is due to the factthat an increase in Pr reduces the thermal diffusivity. The particlesare able to conduct less heat and consequently temperaturedecreases. The characteristics of thermal radiation on temperaturedistribution are sketched in Fig. 10. Increasing values of thermalradiation K enhance temperature. Thermal layer thicknessenhances for larger values of radiation parameter Table 1 showsthat local skin friction increases due to P only and local Nusseltnumber enhances for M, Nr and K.

5. Conclusions

In this article the nonlinear radiation in MHD flow by stretchingcylinder 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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