## THERMAL SCIENCE

International Scientific Journal

### HOMOTOPY ANALYSIS METHOD FOR MIXED CONVECTIVE BOUNDARY LAYER FLOW OF A NANOFLUID OVER A VERTICAL CIRCULAR CYLINDER

**ABSTRACT**

This article deals with the study of the steady axisymmetric mixed convective boundary layer flow of a nanofluid over a vertical circular cylinder with prescribed external flow and surface temperature. By means of similarity transformation, the governing partial differential equations are reduced into highly non-linear ordinary differential equations. The resulting non-linear system has been solved analytically using an efficient technique namely homotopy analysis method (HAM). Expressions for velocity and temperature fields are developed in series form. In this study, three different types of nanoparticles are considered, namely alumina (), titania (), and copper () with water as the base fluid. For copper-water nanofluid, graphical results are presented to describe the influence of the nanoparticle volume fraction on the velocity and temperature fields for the forced and mixed convection flows. Moreover, the features of the flow and heat transfer characteristics are analyzed and discussed for foregoing nanofluids. It is found that the skin friction coefficient and the heat transfer rate at the surface are highest for copper-water nanofluid compared to the alumina-water and titania-water nanofluids.

**KEYWORDS**

PAPER SUBMITTED: 2012-02-25

PAPER REVISED: 2012-09-02

PAPER ACCEPTED: 2012-09-12

**THERMAL SCIENCE** YEAR

**2015**, VOLUME

**19**, ISSUE

**2**, PAGES [549 - 561]

- Gebhart, B., Jaluria, Y., Mahajan, R.L., Sammakia, B., Buoyancy-Induced Flows and Transport, Hemisphere, New York, 1988.
- Martynenko, O.G., Khramtsov, P.P., Free-Convective Heat Transfer, Springer, Berlin, 2005.
- Chen, T.S., Mucoglu, A., Buoyancy effects on forced convection along a vertical cylinder, J. Heat Transfer, 97 (1975) 198-203.
- Mahmood, T., Merkin, J.H., Mixed convection on a vertical circular cylinder, J. Appl. Math. Phys. (ZAMP) 39 (1988) 186-203.
- Ishak, A., Nazar, R., Pop, I., The effects of transpiration on the boundary layer flow and heat transfer over a vertical slender cylinder, Int. J. Non-Linear Mech. 42 (2007) 1010-1017.
- Liao, S.J., The proposed homotopy analysis technique for the solution of non-linear problems, Ph.D. Thesis, Shanghai Jiao Tong University, 1992.
- Liao, S.J., Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman and Hall/CRC Press, Boca Raton (2003).
- Liao, S.J., A kind of approximate solution technique which does not depend upon small parameters (II): an application in fluid mechanics. Int. J. of Non-Linear Mech. 32 (1997) 815-822.
- Liao, S.J., An explicit, totally analytic approximation of Blasius viscous flow problems, Int. J. Nonlinear Mech. 34(1999)759-778.
- Liao, S.J., A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate, J. Fluid Mech. 385(1999)101-128.
- Liao, S.J., On the homotopy analysis method for nonlinear problems, Appl. Math. Comput. 147 (2004) 499-513.
- Liao, S.J., Tan, Y., A general approach to obtain series solutions of nonlinear differential equations. Studies in Applied Mathematics 119 (2007) 297-355.
- Liao, S.J., Beyond perturbation: a review on the basic ideas of the homotopy analysis method and its applications. Advanced Mechanics 38 (2008) 1-34 (in chinese).
- Abbasbandy, S., The application of homotopy analysis method to nonlinear equations arising in heat transfer, Phys. Lett. A 360 (2006) 109-113.
- Abbasbandy, S., Approximate solution for the non-linear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method, Chemical. Eng. J. 136 (2008) 144-150.
- Allan, F.M., Syam, M.I., On the analytic solution of non-homogeneous Blasius problem, J. Comput. Appl. Math. 182 (2005) 362-371.
- Hayat, T., Javed, T., On analytic solution for generalized three-dimensional MHD flow over a porous stretching sheet, Phys. Lett. A 370 (2007) 243-250.
- Sajid, M., Siddiqui, A.M., Hayat, T., Wire coating analysis using MHD Oldroyd 8-constant fluid, Int. J. Eng. Sci. 45 (2007) 381-392.
- Dinarvand, S., A reliable treatment of the homotopy analysis method for viscous flow over a non-linearly stretching sheet in presence of a chemical reaction and under influence of a magnetic field, Cent. Eur. J. Phys. 7(1) (2009) 114- 122.
- Dinarvand, S., On explicit, purely analytic solutions of off-centered stagnation flow towards a rotating disc by means of HAM, Nonlinear Anal. Real World Appl. 11 (2010) 3389-3398.
- Cheng, J., Liao, S.J., Mohapatra, R.N., Vajravelu, K., Series solutions of nano boundary layer flows by means of the homotopy analysis method, J. Math. Anal. Appl. 343 (2008) 233-245.
- Dinarvand, S., The laminar free-convection boundary-layer flow about a heated and rotating down-pointing vertical cone in the presence of a transverse magnetic field, Int. J. Numer. Meth. Fluids, In Press.
- Dinarvand, S., Rashidi, M.M. , A reliable treatment of homotopy analysis method for two-dimensional viscous flow in a rectangular domain bounded by two moving porous walls, Nonlinear Anal. Real World Appl. 11 (2010) 1502-1512.
- Dinarvand, S., Doosthoseini, A., Doosthoseini, E., Rashidi, M. M. , Series solutions for unsteady laminar MHD flow near forward stagnation point of an impulsively rotating and translating sphere in presence of buoyancy forces, Nonlinear Anal. Real World Appl. 11 (2010) 1159-1169.
- Kousar, N., Liao, S.J., Series solution of non-similarity boundary-layer flows over a porous wedge, Transport in Porous Media, 83 (2) (2010) 397 − 412.
- Khan, M., Munawar, S., Abbasbandy, S., Steady flow and heat transfer of a Sisko fluid in annular pipe, Int. J. Heat Mass Transfer, 53 (2010) 1290−1397.
- Khan, M., Qurrat-ul-Ain, Sajid, M., Heat transfer analysis of the steady flow of an Oldroyd 8-constant fluid due to a suddenly moved plate, Commun. Nonlinear Sci. Numer. Simulat. 16 (2011) 1347−1355.
- Tiwari, R.K., Das, M.K., Heat transfer augmentation in a two-sided lid-driven differentially heated square cavity utilizing nanofluids, Int. J. Heat Mass Transfer 50 (2007) 2002-2018.
- Ellahi, R., Series solutions for magnetohydrodynamic flow of non-Newtonian nanofluid and heat transfer in coaxial porous cylinder with slip conditions, J. Nanoengineering and Nanosystems 225(3) (2011) 123-132.
- Ellahi, R., Raza, M., Vafai, K., Series solutions of non-Newtonian nanofluids with Reynolds' model and Vogel's model by means of the homotopy analysis method, Math. Compu. Modelling 55 (2012) 1876-189.
- Ellahi, R., The effects of MHD and temperature dependent viscosity on the flow of non-Newtonian nanofluid in a pipe: analytical solutions, Appl. Math. Modeling 55 (23-24) (2012) 6384-6390.
- Mahmood, T., Merkin, J.H., Similarity solutions in axisymmetric mixed convection boundary-layer flow, J. Eng. Math. 22 (1988) 73-92.
- Oztop, H.F., Abu-Nada, E., Numerical study of natural convection in partially heated rectangular enclosures filled with nanofluids, Int. J. Heat Fluid Flow 29 (2008) 1326-1336.
- Grosan, T., Pop, I., Axisymmetric mixed convection boundary layer flow past a vertical cylinder in a nanofluid, Int. J. Heat Mass Transfer 54 (2011) 3139-3145.