Mathematical Problems in Engineering

Volume 2012 (2012), Article ID 640289, 13 pages

http://dx.doi.org/10.1155/2012/640289

## Boundary Layer Flow of Second Grade Fluid in a Cylinder with Heat Transfer

^{1}Department of Mathematics, Quaid-i-Azam University, Islamabad 45320, Pakistan^{2}Department of Mathematics, University of Balochistan, Quetta, Pakistan^{3}Department of Computational Science and Engineering, Yonsei University, Seoul, Republic of Korea^{4}Department of Mechanical Engineering, Yonsei University, Seoul, Republic of Korea

Received 28 March 2012; Accepted 13 May 2012

Academic Editor: Xing-Gang Yan

Copyright © 2012 S. Nadeem et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

An analysis is carried out to obtain the similarity solution of the steady boundary layer flow and heat transfer of a second grade through a horizontal cylinder. The governing partial differential equations along with the boundary conditions are reduced to dimensionless form by using the boundary layer approximation and applying suitable similarity transformation. The resulting nonlinear coupled system of ordinary differential equations subject to the appropriate boundary conditions is solved by homotopy analysis method (HAM). The effects of the physical parameters on the flow and heat transfer characteristics of the model are presented. The behavior of skin friction coefficient and Nusselt numbers is studied for different parameters.

#### 1. Introduction

Due to wide range of applications in coating of wires and polymer fiber spinning, the concept of heat convection in the cylinders has been a field of interest for many theoretical and experimental researchers. Buchlin [1] examined the natural and forced convective heat transfer along vertical slender cylinder and also the case of two cylinders. In his analysis, obtained results indicated that the convective heat transfer has a strong dependence on the curvature of the cylinder and its misalignment with the main flow. The natural convection flow from an isothermal circular cylinder of viscous fluid having temperature-dependent viscosity has been tackled by Molla et al. [2]. The problem of mixed convection boundary layer flow over a vertical circular cylinder with prescribed surface heat flux has been studied by Bachok and Ishak [3]. They focused on the suction/injection effects on the flow field and the heat transfer rate at the surface by considering the free stream velocity and the surface heat flux to be linear functions of the distance from the leading edge. Further, Heckel et al. [4] have examined the problem of mixed convective flow of a viscous fluid along a vertical slender cylinder with variable surface temperature. They assumed the temperature to be varying arbitrarily with the axial coordinate. Later on Na [5] has investigated the effect of wall conduction on the heat transfer of the natural convection over a vertical slender hollow circular cylinder. Recently Ahmed et al. [6] have numerically examined the problem of laminar free convection boundary layer flow over horizontal cylinders of elliptic cross-section having constant surface heat flux for both the blunt (major axis of the cylinder is in the horizontal direction) and slender (major axis of the cylinder is in the vertical direction) orientations. They showed that the local skin friction coefficient is an increasing function of the ratio of the length of the major axis to that of the minor axis of the elliptic cylinder. Moreover, the combined heat and mass transfer along a vertical moving cylinder with a free stream for uniform wall temperature and heat flux was countered by Takhar et al. [7]. Later on, in his paper Cheng [8] studied the natural convection heat transfer problem through a horizontal elliptical cylinder in a Newtonian fluid with constant heat flux and temperature-dependent internal heat generation. He concluded that an increase in the aspect ratio decreases (increases) the average surface temperature of the elliptical cylinder with blunt (slender) orientation and that the average surface temperature of the elliptical cylinder with slender orientation is less than that due to the blunt orientation.

The works cited above are all carried out for Newtonian fluids; however, some literature concerning non-Newtonian fluids is also available. In a very recent paper Chang [9] has presented the numerical treatment of the flow and heat transfer characteristics of forced convection in a micropolar fluid flow along a vertical slender hollow circular cylinder with wall conduction and buoyancy effects. The heat transfer to non-Newtonian flows over a cylinder in cross flow is experimentally studied by Rao [10] for different non-Newtonian fluids and a comparison with the water (viscous fluid) flow is also experimentally examined. Amoura et al. [11] provided a finite element solution of the Carreau model mixed convection of non-Newtonian fluids between two coaxial rotating cylinders. Moreover, the effects of transpiration on the boundary layer flow and heat transfer over a vertical slender cylinder are addressed by Ishak et al. [12]. Some interesting and important works concerning boundary layer flow of viscous and non-Newtonian fluids are listed in [13–19]. In the present work we have examined the boundary layer flow and heat transfer of a second grade fluid flow through a horizontal cylinder. The solutions are obtained by implying the analytical technique homotopy analysis method (HAM). A discussion is provided to study the influence of the physical parameters on velocity, the skin friction coefficient, and the local Nusselt number.

#### 2. Formulation

Let us consider the problem of mixed convection boundary layer flow of second grade fluid along a horizontal circular cylinder having radius . The temperature at the surface of the cylinder is assumed to be a constant and the uniform ambient temperature is taken to be such that the quantity for the case of assisting flow, while , in case of the opposing flow, respectively. The viscous dissipation effects are also taken into account. Under these assumptions the boundary layer equations of motion and heat transfer are where the velocity components along the axes are , is density, is the kinematic viscosity, is pressure, and is the free stream velocity and is defined as .

The corresponding boundary conditions for the problem are Introduce the following similarity transformations: where the characteristic temperature is calculated from the relations . With the help of transformations (2.3), (2.1) take the form in which is the curvature parameter, is the dimensionless viscoelastic parameter, is the Prandtl number, is the Eckert number.

The boundary conditions in nondimensional form are defined as where is any constant. The dimensionless coefficient of skin friction and the Nusselt number are defined as where is the local Reynolds number.

#### 3. Solution of the Problem

The solution of the present problem is obtained by using the powerful analytical technique homotopy analysis method (HAM). In the present case we seek the initial guesses to be [20–33]
The corresponding auxiliary linear operators are
satisfying
where are arbitrary constants. The zeroth-order deformation equations are
where the auxiliary convergence parameters and both are taken to be and
The appropriate boundary conditions for the zeroth order system are
The *m*th order deformation is
where
With the help of MATHEMATICA, the solutions of (2.4) can be expressed as
in which the constants and can be computed through any mathematics software, and here we have shown and discussed the complete results through graphs.

#### 4. Results and Discussion

In this section, we have discussed the analytical solutions of the highly nonlinear equations (2.4) subject to boundary conditions (2.5). The analytical solutions has been calculated with the help of homotopy analysis method. The solutions are finally presented in the form of general series. The convergence of these series solutions have been discussed through plotting the graphs of -curves (see Figures 1 and 2). It is seen that the admissible values of in which our solutions are convergent are and . The variation of velocity profile for pertinent parameters is sketched in Figures 3 to 5. Figure 3 is plotted to see the variation of curvature parameter on the nondimensional velocity. It is observed that with the increase in the velocity profile increases; however, the boundary layer thickness reduces. The maximum velocity is achieved very near to the sheet. The variation of viscoelastic parameter (second grade) is given in Figure 4. It is depicted here that, with the increase in , velocity increases and the boundary layer reduces. Thus almost similar effects appear for both and . Almost a similar behavior occurs for the variation of (see Figure 5). The nondimensional temperature for various values of , , , and is displayed in Figures 6 to 10. It is seen that temperature profile increases with the increases of both and , also the thermal boundary layer reduces with the increase in and (see Figures 6 and 7). It is also observed that when temperature is given near the sheet, the temperature is maximum. The effects of Prandtle number and Eckert number are displayed in Figures 8 and 9. It is observed that temperature profile decreases with the increase in and increases with the increase in . The thermal boundary layer reduces for both the case; however, the behavior of temperature for each case is opposite. The temperature profile decreases as well as the thermal boundary layer thickness reduces with the increase in (see Figure 10).

The values of skin friction coefficient against for different values of are plotted in Figure 11. It is seen that with the increase in , decreases for all values of . In Figure 12 we have sketched against for various values of . It is observed that almost similar effects occur as in case of . However, the increases with increase in (see Figure 13). The local Nusselt number against and for different values of is plotted in Figures 14 and 15. In both the figures Nusselt number gives almost similar behavior.

Table 1 is included to check the behavior of boundary derivative (shear rate at the surface), for different values of curvature parameter and the second grade parameter . From Table 1 it is observed that increase in increases shear rate at the surface.

#### Acknowledgments

Professor Lee acknowledges the support by WCU (World Class University) Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology R31-2008-000-10049-0. The first and second authors acknowledge the support of Higher Education Commission (HEC) of Pakistan.

#### References

- J. M. Buchlin, “Natural and forced convective heat transfer on slender cylinders,”
*Revue Generale de Thermique*, vol. 37, no. 8, pp. 653–660, 1998. View at Publisher · View at Google Scholar · View at Scopus - M. M. Molla, M. A. Hossain, and R. S. R. Gorla, “Natural convection flow from an isothermal horizontal circular cylinder with temperature dependent viscosity,”
*Heat and Mass Transfer*, vol. 41, no. 7, pp. 594–598, 2005. View at Publisher · View at Google Scholar · View at Scopus - N. Bachok and A. Ishak, “Mixed convection boundary layer flow over a permeable vertical cylinder with prescribed surface heat flux,”
*European Journal of Scientific Research*, vol. 34, no. 1, pp. 46–54, 2009. View at Google Scholar · View at Scopus - J. J. Heckel, T. S. Chen, and B. F. Armaly, “Mixed convection along slender vertical cylinders with variable surface temperature,”
*International Journal of Heat and Mass Transfer*, vol. 32, no. 8, pp. 1431–1442, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - T. Y. Na, “Effect of wall conduction on natural convection over a vertical slender hollow circular cylinder,”
*Applied Scientific Research*, vol. 54, no. 1, pp. 39–50, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - S. Ahmad, N. M. Arifin, R. Nazar, and I. Pop, “Free convection boundary layer flow over cylinders of elliptic cross section with constant surface heat flux,”
*European Journal of Scientific Research*, vol. 23, no. 4, pp. 613–625, 2008. View at Google Scholar · View at Scopus - H. S. Takhar, A. J. Chamkha, and G. Nath, “Combined heat and mass transfer along a vertical moving cylinder with a free stream,”
*Heat and Mass Transfer*, vol. 36, no. 3, pp. 237–246, 2000. View at Google Scholar · View at Scopus - C. Y. Cheng, “Natural convection boundary layer on a horizontal elliptical cylinder with constant heat flux and internal heat generation,”
*International Communications in Heat and Mass Transfer*, vol. 36, no. 10, pp. 1025–1029, 2009. View at Publisher · View at Google Scholar · View at Scopus - C. L. Chang, “Buoyancy and wall conduction effects on forced convection of micropolar fluid flow along a vertical slender hollow circular cylinder,”
*International Journal of Heat and Mass Transfer*, vol. 49, no. 25-26, pp. 4932–4942, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - B. K. Rao, “Heat transfer to non-Newtonian flows over a cylinder in cross flow,”
*International Journal of Heat and Fluid Flow*, vol. 21, no. 6, pp. 693–700, 2000. View at Publisher · View at Google Scholar · View at Scopus - M. Amoura, N. Zeraibi, A. Smati, and M. Gareche, “Finite element study of mixed convection for non-Newtonian fluid between two coaxial rotating cylinders,”
*International Communications in Heat and Mass Transfer*, vol. 33, no. 6, pp. 780–789, 2006. View at Publisher · View at Google Scholar · View at Scopus - A. Ishak, R. Nazar, and I. Pop, “The effects of transpiration on the boundary layer flow and heat transfer over a vertical slender cylinder,”
*International Journal of Non-Linear Mechanics*, vol. 42, no. 8, pp. 1010–1017, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - K. L. Hsiao, “Viscoelastic fluid over a stretching sheet with electromagnetic effects and nonuniform heat source/sink,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 740943, 14 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - K.-L. Hsiao, “Heat and mass mixed convection for MHD visco-elastic fluid past a stretching sheet with ohmic dissipation,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 15, no. 7, pp. 1803–1812, 2010. View at Publisher · View at Google Scholar - D. Vieru, I. Siddique, M. Kamran, and C. Fetecau, “Energetic balance for the flow of a second-grade fluid due to a plate subject to a shear stress,”
*Computers & Mathematics with Applications*, vol. 56, no. 4, pp. 1128–1137, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - N. Aksel, “A brief note from the editor on the second-order fluid,”
*Acta Mechanica*, vol. 157, no. 1–4, pp. 235–236, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - M. M. Rashidi and M. Keimanesh, “Using differential transform method and padé approximant for solving mhd flow in a laminar liquid film from a horizontal stretching surface,”
*Mathematical Problems in Engineering*, vol. 2010, Article ID 491319, 14 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - K. L. Hsiao, “MHD mixed convection for viscoelastic fluid past a porous wedge,”
*International Journal of Non-Linear Mechanics*, vol. 46, no. 1, pp. 1–8, 2011. View at Publisher · View at Google Scholar · View at Scopus - K.-L. Hsiao, “Multimedia feature for unsteady fluid flow over a non-uniform heat source stretching sheet with magnetic radiation physical effects,”
*Applied Mathematics & Information Sciences*, vol. 6, pp. 59S–65S, 2012. View at Google Scholar - S. Liao,
*Beyond Perturbation: Introduction to the Homotopy Analysis Method*, vol. 2 of*CRC Series: Modern Mechanics and Mathematics*, Chapman & Hall/CRC, Boca Raton, Fla, USA, 2004. - S. Nadeem and M. Ali, “Analytical solutions for pipe flow of a fourth grade fluid with Reynold and Vogel's models of viscosities,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 5, pp. 2073–2090, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Nadeem, A. Rehman, K. Vajravelu, J. Lee, and C. Lee, “Axisymmetric stagnation flow of a micropolar nanofluid in a moving cylinder,”
*Mathematical Problems in Engineering*, vol. 2012, Article ID 378259, 17 pages, 2012. View at Publisher · View at Google Scholar - S. J. Liao, “An approximate solution technique not depending on small parameters: a special example,”
*International Journal of Non-Linear Mechanics*, vol. 30, no. 3, pp. 371–380, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. K. Alomari, M. S. M. Noorani, and R. Nazar, “On the homotopy analysis method for the exact solutions of Helmholtz equation,”
*Chaos, Solitons and Fractals*, vol. 41, no. 4, pp. 1873–1879, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus - A. K. Alomari, M. S. M. Noorani, and R. Nazar, “Explicit series solutions of some linear and nonlinear Schrodinger equations via the homotopy analysis method,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 14, no. 4, pp. 1196–1207, 2009. View at Publisher · View at Google Scholar - S. Nadeem and N. S. Akbar, “Influence of heat transfer on a peristaltic flow of Johnson Segalman fluid in a non uniform tube,”
*International Communications in Heat and Mass Transfer*, vol. 36, no. 10, pp. 1050–1059, 2009. View at Publisher · View at Google Scholar · View at Scopus - S. Nadeem and N. S. Akbar, “Effects of temperature dependent viscosity on peristaltic flow of a Jeffrey-six constant fluid in a non-uniform vertical tube,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 15, no. 12, pp. 3950–3964, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - A. S. Bataineh, M. S. M. Noorani, and I. Hashim, “Approximate analytical solutions of systems of PDEs by homotopy analysis method,”
*Computers & Mathematics with Applications*, vol. 55, no. 12, pp. 2913–2923, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Abbasbandy, “The application of homotopy analysis method to nonlinear equations arising in heat transfer,”
*Physics Letters A*, vol. 360, no. 1, pp. 109–113, 2006. View at Publisher · View at Google Scholar - S. Abbasbandy, “Homotopy analysis method for heat radiation equations,”
*International Communications in Heat and Mass Transfer*, vol. 34, no. 3, pp. 380–387, 2007. View at Publisher · View at Google Scholar · View at Scopus - S. Abbasbandy, Y. Tan, and S. J. Liao, “Newton-homotopy analysis method for nonlinear equations,”
*Applied Mathematics and Computation*, vol. 188, no. 2, pp. 1794–1800, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - S. Nadeem, S. Zaheer, and T. Fang, “Effects of thermal radiation on the boundary layer flow of a Jeffrey fluid over an exponentially stretching surface,”
*Numerical Algorithms*, vol. 57, no. 2, pp. 187–205, 2011. View at Publisher · View at Google Scholar - R. Ellahi, A. Zeeshan, K. Vafai, and H. U. Rehman, “Series solutions for magnetohydrodynamic flow of non Newtonian nanofluid and heat transfer in coaxial porous cylinder with slip conditions,”
*Journal of Nanoengineering and Nanosystems*, vol. 225, no. 3, pp. 123–132, 2011. View at Publisher · View at Google Scholar