Radiative flow and heat transfer of a fluid along an expandablestretching horizontal cylinder
 75 Downloads
Abstract
The effect of thermal radiation and suction/injection on heat transfer characteristics of an unsteady expandablestretched horizontal cylinder has been investigated. Similarity equations are obtained through the application of similarity transformation techniques. The governing boundary layer equations are reduced to a system of ordinary differential equations. Mathematica has been used to solve such system after obtaining the missed initial conditions. The fluid velocity and temperature, within the boundary layer, are plotted and discussed in details for various values of the different parameters such as the thermal radiation parameter, suction/injection parameter, and unsteadiness parameter. Comparison of obtained numerical results is made with previously published results in some special cases and found to be in a good agreement. The obtained results show that the fluid velocity and temperature are affected by the variation of the parameters included in the study such as the radiation parameter, the unsteadiness parameter, and the suction/injection parameter.
Keywords
Heat transfer Stretching cylinder Thermal radiation Suction and injectionNomenclature
 t
Time [s]
 a(t)
Radius of the cylinder [m]
 U_{w}
Stretching timedependent velocity [m s^{−1}]
 x
Axial direction coordinate [m]
 r
Perpendicular to the axis coordinate [m]
 u
Velocity component in the xdirection [m s^{−1}]
 v
Velocity component in the rdirection [m s^{−1}]
 T_{w}
Temperature of the cylinder surface [K]
 V
Constant of suction [−]
 A
Unsteadiness parameter [−]
 f_{0}
Suction (injection) parameter [−]
 C_{f}
Local skin friction coefficient [−]
 f
Dimensionless stream function [−]
 N_{R}
Radiation parameter [−]
 Nu_{x}
The local Nusselt number coefficient [−]
 Pr
Prandtl number [−]
 Re_{x}
Reynolds number [−]
 q_{r}
Cylinder surface heat flux [kg s^{−3}]
PACS
44.40.+a44.90.+c44.05.+e47.15.xIntroduction
Several applications in engineering and industrial processes arise from the study of the flow of either Newtonian fluid or nonNewtonian fluid. Such fields have been interesting for many authors for the last few decades. The fields of plastic and metallurgy industries, the drawing of wires, and glass fiber production are good examples for the applications of the problem of the flow over a stretching/shrinking cylinder.
The problem of the flow inside a tube that has a timedependent diameter was first presented by Uchida and Aoki [1] and Shalak and Wang [2]. Wang [3] have studied the steady flow of incompressible viscous flow outside a hollow stretching cylinder. Elbashbeshy et al. [4] have investigated the effect of magnetic field on flow and heat transfer over a stretching horizontal cylinder in the presence of a heat source/sink with suction/injection. Hayat et al. [5] have examined the effects of variable thermal conductivity in mixed convection flow of viscoelastic nanofluid due to a stretching cylinder with a heat source/sink. Ishak et al. [6] have studied the MHD flow and heat transfer outside a stretching cylinder. They have got numerical solutions to the problem using the Kellerbox method.
The thermal radiation effect is considerable when the difference between the surface temperature and the ambient temperature is big. Mabood et al. [7] have presented a theoretical investigation of flow and heat transfer of a Casson fluid from a horizontal circular cylinder in a nonDarcy porous medium under the action of slips and thermal radiation parameters.
Zaimi et al. [8] have studied the unsteady flow due to a contracting cylinder in a nanofluid using Buongiorno’s model. Elbashbeshy et al. [9] have studied the effects of thermal radiation, heat generation, and suction/injection on the mechanical properties of the unsteady continuous moving cylinder in a nanofluid.
Fang et al. [10] have recently studied the problem of unsteady viscous flow over an expanding stretching cylinder which gives exact similarity solution to the NavierStokes equations. They found that the reversal flow fluid is strongly affected by the Reynolds number and the unsteadiness parameter. The numerical solution of the unsteady viscous flow outside of an expanding or contracting cylinder has been reported by Fang et al. [11].
The unsteady nature of the fluid flow is very important from a practical point of view. Some unsteady effects arise due to nonuniformities in the surrounding fluid. Other effects arise due to the selfinduction of the body. In fact, there are some devices are designed to execute timedependent motion in order to perform desired functions [12]. The understanding of unsteady flow and hence applying such knowledge to new design techniques enable scientists and engineers to make important improvements in reliability and costs of several fluid dynamics devices.
The problem introduced in this work involves such concept of unsteadiness. In fact, we investigate the case of unsteady viscous flow over a stretching horizontal cylinder with variable radius where the thermal radiation is considered. The understanding of unsteady flow and hence applying such knowledge to new design techniques enable scientists and engineers to make important improvements in reliability and costs of several fluid dynamics devices. Mathematica is used to solve the problem numerically. The obtained results show how the fluid velocity and temperature are affected by the variation of the parameters included in the study such as the radiation parameter, the unsteadiness parameter, and the suction/injection parameter.
Mathematical formulation of the problem
Consider an unsteady axisymmetric boundary layer flow of an incompressible viscous fluid along a horizontal cylinder which is considered to be continuously stretching. The cylinder is contracting or expanding according to the relation \( a(t)={a}_0\sqrt{1\beta t} \), where a(t) is the radius of the cylinder at any time t, a_{0} is the initial value of the cylinder radius, and β is a constant which indicates to contraction (β > 0) or expansion (β < 0).
where u and v are the components of the fluid velocity along x axis and r axis respectively. ν is the fluid kinematic viscosity, α is the fluid thermal diffusivity, κ is the thermal conductivity, V is the constant of suction (V < 0) or injection (V > 0), and \( {q}_r=\frac{4\ \sigma }{3\ {\alpha}^{\ast }}\ \frac{\partial\ {T}^4}{\partial\ r} \) is the radiation heat flux such that σ and α^{∗} are the StefanBoltzman constant and the mean absorption coefficient, respectively.
while primes denote differentiation with respect to η, \( A=\frac{\beta\ {a}_0^2}{4\ \nu } \) is the unsteadiness parameter. Where the negative values of A correspond to contraction and the positive values of A correspond to expansion, \( \mathit{\Pr}=\frac{\nu }{\alpha } \) is the Prandtl number, \( {f}_0=\frac{a_0V}{2\ \nu } \) is the suction (f_{0} < 0) or injection (f_{0} > 0) parameter, and \( {N}_R=\frac{16\ \sigma {T}_{\infty}^3}{3\ \kappa\ {\alpha}^{\ast }} \) is the thermal radiation parameter.
where \( R{e}_x=\frac{x\ {U}_w}{\nu } \) is the Reynolds number.
Method of solutions
where y_{1} = f, y_{2} = f^{′}, y_{3} = f^{′′}, y_{4} = θ, y_{5} = θ^{′}
numerical values are given to U_{0} and f_{0} . m and n are priori unknown to be determined as part of the solution. Mathematica is used to define the function F[m, n] ≕ NDSolve[System(14) − (19)]. The values of m and n are found upon solving the equations y_{2}(η_{max}) = 0, y_{4}(η_{max}) = 0. A suitable value of η is taken and then increased to reach η_{max} such that the difference between successive values of m and those of n is less than 10^{−7}. The problem now is an initial value problem which in turn is solved using the Mathematica function NDSolve.
Special cases
Comparison of −f^{′′}(0) for various values of A and f_{0} given that Pr = 0.7, U_{0} = − 1, N_{R} = 0
A  f _{0}  Ref. [13] OHAM  Ref. [13] Numerical  Ref. [14] OHAM  Ref. [14] Numerical  Present Results 

− 1  − 1  1.0000007  0.9999999  1.0000000  1.0000002  1.0000000 
− 1  − 2  2.5632048  2.5632048  2.5632043  2.5632048  2.5632049 
−2  − 1  2.6012207  2.6012207  2.6012207  2.6012207  2.6012207 
−2  − 2  3.7150911  3.7150910  3.7150910  3.7150910  3.7150910 
From the table, one can find a comparison of the obtained values of −f^{′′}(0) with previously published results in the literature. The comparison is made for various values of A and f_{0} given Pr = 0.7, U_{0} = − 1, N_{R} = 0. The obtained results show good amendment which gives rise to the validation of the used method.
Results and discussions
This section is devoted to the analysis of the behavior of the parameters included in the problem on the fluid velocity f^{′}(η), the fluid temperature θ(η), modified skin friction −f^{′′}(0) and the modified Nusselt number −θ^{′}(0).
Values of −f^{′′}(0) and −θ^{′}(0) for various values of A, Pr , f_{0}, and N_{R} for U_{0} = 1
A  Pr  f _{0}  N _{ R}  −f^{′′}(0)  −θ^{′}(0) 

− 2  3.3170862  1.8884144  
− 1  0.7  − 1  0.5  2.4515742  1.4338464 
1  1.4434620  0.8077094  
− 1  0.7  2.4515742  1.4338464  
1  − 1  0.5  2.4515742  1.9023636  
10  2.4515742  14.3735363  
− 1  − 2  3.3307422  1.7883330  
0.7  − 1  0.5  2.4515742  1.4338464  
1  2.0150457  0.1622559  
− 1  0.5  2.4515742  1.4338464  
0.7  − 1  0.7  2.4515742  1.3019903  
1  2.4515742  1.1515467 
Prandtl number
Thermal radiation
The equations governing the fluid velocity does not include N_{R} so there is no effect on N_{R} on the skin friction or the fluid velocity as recognized from Table 2.
Expandable unsteadiness parameter
Suction/injection velocity
Conclusions

With the decrease of the value of the contraction parameter, the skin friction increases while both of the fluid velocity and temperature decrease.

Increasing the Prandtl number leads to a decrease of the Nusselt number and the fluid temperature as well.

The fluid velocity decreases as suction increases while the increase of injection leads to increasing the fluid velocity

The increase of injection enhances the fluid temperature while the inverse behavior of takes place in the case of suction

The surface flux (Nusselt number) and consequently the fluid temperature increase as the thermal radiation parameter increases.

Introducing the injection may help to reduce the friction at the surface of the cylinder
Notes
Greek symbols
α The thermal diffusivity [m^{2} s^{− 1}]
α^{*} Mean absorption coefficient [−]
β Contraction (expansion) constant [s^{− 1}]
η The dimensionless similarity variable [−]
υ Kinematic viscocity [m^{2} s^{−1}]
μ Dynamic viscocity [m^{2} s^{−1}]
κ Thermal conductivity of the fluid [kg m s^{−3} K^{−1}]
Acknowledgements
The author want to thank the reviewers for their valuable comments which enabled the author to improve the manuscript.
Funding
This work has been accomplished in the faculty of science and arts Khulais, university of Jeddah where I work as associate professor.
Availability of data and materials
All data generated or analysed during this study are included in this published article.
Authors’ contributions
The author read and approved the final manuscript.
Competing interests
The author declares that he has no competing interests.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
References
 1.Uchida, S., Aoki, H.: Unsteady flows in a semiinfinite contracting or expanding pipe. J. Fluid Mech. 82(2), 371–387 (1977)MathSciNetCrossRefGoogle Scholar
 2.Skalak, F.M., Wang, C.Y.: On the unsteady squeezing of viscous fluid from a tube. J. Aust. Math. Soc. B. 21, 65–74 (1979)CrossRefGoogle Scholar
 3.Wang, C.Y.: Fluid flow due to a stretching cylinder. Phys. Fluids. 31, 466–468 (1988)CrossRefGoogle Scholar
 4.Elbashbeshy, E.M.A., Emam, T.G., Elazab, M.S., Abdelgaber, K.M.: Effect of magnetic field on flow and heat transfer over a stretching horizontal cylinder in the presence of a heat source/sink with suction/injection. J. Appl. Mech. Eng. 1(1), 1–5 (2012)CrossRefGoogle Scholar
 5.Hayat, T., Waqas, M., Shehzad, S.A., Alsaedi, A.: Mixed convection flow of viscoelastic nanofluid by a cylinder with variable thermal conductivity and heat source/sink. Int. J. Numer. Methods Heat Fluid Flow. 26(1), 214–234 (2016)MathSciNetCrossRefGoogle Scholar
 6.Ishak, A., Nazar, R., Pop, I.: Magnetohydrodynamic (MHD) flow and heat transfer due to a stretching cylinder. Energ. Convers. Manage. 49(11), 3265–3269 (2008)CrossRefGoogle Scholar
 7.Mabood, F., Shateyi, S., Khan, W.A.: Effect of thermal radiation on Casson flow heat and mass transfer around a circular cylinder in a porus medium. Eur. Phys. J. Plus. 130, 188 (2015)CrossRefGoogle Scholar
 8.Zaimi, K., Ishak, A., Pop, I.: Unsteady flow due to a contracting cylinder in a nanofluid using Buongiorno’s model. Int. J. Heat Mass Transf. 68, 509–513 (2014)CrossRefGoogle Scholar
 9.ElBashbeshy, E.M.A., Emam, T.G., Abdelwahed, M.S.: The effect of thermal radiation, heat generation, and suction/injection on the mechanical properties of unsteady continuous moving cylinder in a nanofluid. Therm. Sci. 19(5), 1591–1601 (2015)CrossRefGoogle Scholar
 10.Fang, T., Zhang, J., Zhong, Y., Tao, H.: Unsteady viscous flow over an expanding stretching cylinder. Chin. Phys. Lett. 28, 124707 (2011)CrossRefGoogle Scholar
 11.Fang, T., Zhang, J., Zhong, Y.: Note on unsteady viscous flow on the outside of an expanding or contracting cylinder. Commun. Nonlinear Sc. Num. Simul. 17, 3124–3128 (2012)MathSciNetCrossRefGoogle Scholar
 12.McCrosky, W.J.: Some current research in unsteady fluid dynamics  the 1976 Freeman scholarship lecture. ASME J. Fluid Eng. 99, 8–39 (1977)CrossRefGoogle Scholar
 13.Marinca, V., Ene, R.D.: Dual approximate solutions of the unsteady viscous flow over a shrinking cylinder with optimal homotopy asymptotic method. Adv. Math. Phys. 2014, 417643, 11 (2014)Google Scholar
 14.E.M.A. Elbashbeshy, T.G. Emam, and K.M. Abdelgaber, Semianalytic and numerical solutions of unsteady flow and heat transfer of a fluid over an expandablestretching horizontal cylinder in the presence of suction/injection, preprint. (2014)Google Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.