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Results in Physics

journal homepage: www.journals.elsevier.com/results-in-physics

Numerical simulation for nonlinear radiative flow by convective cylinder

Tasawar Hayata,b, Muhammad Tamoorc, Muhammad Ijaz Khana'*, Ahmad Alsaedib

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

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ARTICLE INFO

ABSTRACT

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

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 profile 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 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 heating 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 carbon water nanofluid flow towards a stretched cylinder is analyzed by Hayat. et al. [3]. Pandey and Kumar [4] examined natural convection nanofluid flow by a stretched cylinder with viscous dissipation. 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 generation, 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

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

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 nonlinear 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 radiation. 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 nonlinear thermal radiation is presented for flow due to cylinder.

http://dx.doi.org/10.1016/j.rinp.2016.11.026 2211-3797/® 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/).

Table 1

Numerical values of f '(0) and h (0) due to variation in physical parameters.

n C M p Nr Pr K -f^O)

1.0 1.0 1.0 0.3 0.3 0.3 2.0 1.76404 4.01940

2.0 2.08236 4.04324

3.0 2.35093 4.06404

2.0 1.0 2.08236 4.04324

2.0 2.43465 4.48634

3.0 2.76804 4.80308

1.0 1.0 2.08236 4.04324

2.0 2.85587 4.02992

3.0 3.75147 4.02003

1.0 0.1 2.14610 4.04188

0.7 1.94565 4.04634

1.0 1.83311 4.04911

0.3 0.1 2.08236 6.67717

0.7 2.08239 1.33891

1.0 2.08239 1.02591

0.3 0.1 2.08236 4.00021

0.7 2.08236 4.12951

1.0 2.08236 4.19444

0.3 1.0 2.08236 4.59244

2.0 2.08236 4.04324

3.0 2.08236 3.87436

Fig. 1. Influence of n on velocity distribution.

Therefore an attempt is made to investigate resulting nonlinear problem numerically. Governing partial differential equations have been reduced to ordinary differential equations. Shooting technique 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 further studied. Fluid occupying porous space is conducting via

Fig. 2. Influence of C on velocity distribution.

applied magnetic field. Induced magnetic and electric fields are negligible. Applied magnetic field is taken in the B = BoxV. The following statements lead to the resulting flow and temperature fields.

d(ru) + @(rt) = 0;

du du _ v d ( du\ v aB U dx + V dr-y dr\rdrJ — \Fi + ~p

dT dT- a df dl) - _L @3L

u dx + V dr - r dr \ dr) qCp dr

u - UoXn, t - 0, —kg - hi(Tw - T) at r - R,

Invoking

r2 — R2 / U\ 2

2R \vx One can arrived at

— , W - (Uvx)2Rf (g), 0(g)-

TT 1 1 1

TW — T1

(1 + 2gC)f"' + (^i-1) f + 2Cf ' — nf2 — Pf — Mf - 0, (1 + 2gC)(1 +(1 +(Nr — 1)0)3)0" + Qk +(1 +(Nr — 1)0)3) C0

•> Pr

+ 2(1 + 2gC)(Nr — 1)(1 +(Nr — 1)0)2h '2 + —

0'f — nhf

(1) (2)

f - 1, f - 0, 0' - —a(1 — 0) at g - 0, f ! 0, 0 ! 0 as g

The velocity components parallel to x and r directions are denoted by u and v, qr - —(341) dT4 radiative heat flux, a* Stefan-Boltzman constant, k* mean absorption coefficient, a - thermal diffusivity, j thermal conductivity, cp specific heat, p fluid density, h1 heat transfer coefficient, v - P kinematic viscosity, 1 coefficient of fluid viscosity, a electrical conductivity, B uniform magnetic field strength and k1 permeability of porous medium. Here Tw(x) - T1 + T0xn surface temperature, T0 reference temperature and T1 ambient temperature. Physical parameters under discussion are:

u ! 0, T ! Tœ as r !i.

Fig. 3. Influence of M on velocity distribution. Fig. 6. Influence of C on temperature distribution.

0 2 4 6 S 10

Fig. 4. Influence of P on velocity distribution.

«I-1-' --

0 2 4 6 S 10

Fig. 7. Influence of P on temperature distribution.

C = (§) curvature parameter, M = (pU0)2 magnetic parameter, P = porous medium parameter, Nr = Tw temperature ratio

parameter, Pr = a Prandtl number and K = j1 radiation parameter.

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

2sw Nu — xqw pU2 ' " k(Tw - Ti) '

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

Re-2Nm = —1 + 4K)h(0), /"(0) = 1 RelCfX.

In which Rex = xu denotes the local Reynolds number.

0 2 4 6 S 10

Fig. 9. Influence of Pr on temperature distribution.

f)\-,- i—'---,

0 2 4 6 S 10 1

Fig. 10. Influence of K on temperature distribution.

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 in Fig. 3. Velocity profile and associated layer thickness decay for larger M. Physically Lorentz force enhances resistive forces. Porosity parameter 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 in Fig. 5. Temperature of the fluid particles enhances for larger n. Fig. 6 depicts the behavior of temperature field for larger C. Temperature distribution decays near the surface of cylinder and then shows increasing behavior far away from the surface of cylinder. The radius of cylinder 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. 7 shows the effect of porosity parameter P on temperature distribution. It is noted that for larger P the temperature enhances. Influence of temperature ratio parameter Nr on temperature field is plotted in Fig. 8. Temperature profile and associated boundary layer thickness are enhanced for increasing values of Nr. From Fig. 8, it is clear that an increase in the Nr relates to a higher wall temperature when compared with the surrounding liquid. Behavior of Pr on temperature profile is displayed in Fig. 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 in Fig. 10. Increasing values of thermal radiation K enhance temperature. Thermal layer thickness enhances for larger values of radiation parameter Table 1 shows that local skin friction increases due to P only and local Nusselt number enhances for M, Nr and K.

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 criteria. The inner iteration is executed with convergence criteria of 10~6 in all cases taking step size h = 0.01.

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.

4. Results and discussions

This section provides the graphical and tabular outlook on the effect of various flow related parameters for the velocity and temperature 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 nonlinear radiation parameter K on velocity, temperature, local skin friction 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 in Table 1.

Influence of nonlinearity exponent n on velocity profile is portrayed in Fig. 1. Here we can see that velocity field shows decreasing 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. 2 portrays the effect of curvature parameter

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