Journal of Applied Mathematics

Volume 2015, Article ID 741352, 6 pages

http://dx.doi.org/10.1155/2015/741352

## Lie Group Analysis on Brownian Motion and Thermophoresis Effect on Free Convective Boundary-Layer Flow on a Vertical Cylinder Embedded in a Nanofluid-Saturated Porous Medium

^{1}Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh^{2}Mathematics Department, Faculty of Science, Assiut University, Assiut 71516, Egypt^{3}Mechanical Engineering Department, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia

Received 21 January 2014; Revised 23 June 2014; Accepted 6 August 2014

Academic Editor: Gongnan Xie

Copyright © 2015 Mohammad Ferdows 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

Natural convective boundary-layer flow of a nanofluid on a heated vertical cylinder embedded in a nanofluid-saturated porous medium is studied. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. Lie groups analysis is used to get the similarity transformations, which transform the governing partial differential equations to a system of ordinary differential equations. Two groups of similarity transformations are obtained. Numerical solutions of the resulting ordinary differential systems are obtained and discussed for various values of the governing parameters.

#### 1. Introduction

A nanofluid is a colloidal mixture of nanosized (<100 nm) particles in a base fluid. It is known that nanofluids can tremendously enhance the heat transfer characteristics of the original (base) fluid. Thus nanofluids have many applications in industry such as coolants, lubricants, heat exchangers, and microchannel heat sinks [1]. Nanoparticles have been made of various materials such as oxide ceramics and nitride ceramics. The objective of nanofluids is to achieve the best possible thermal properties with the least possible (<1%) volume fraction of nanoparticles in the base fluid [1]. There have been many studies in the literature to better understand the mechanism behind the enhanced heat transfer characteristics.

There have been several recent experimental studies to better understand the mechanism of heat transfer enhancement during natural convection heat transfer in nanofluids ([2, 3]), and a summary can be found in Godson et al. [1]. Godson et al. [1] conclude that firm conclusions cannot be made from the limited number of studies and further experimental and theoretical work is required.

There have been several recent studies on the mathematical and numerical modelling of natural convection heat transfer in nanofluids by Congedo et al. [4]. Mathematical and numerical models have some advantages over experimental studies due to the many factors that influence nanofluid properties. Congedo et al. have used the computer code Fluent (release 6.3) to conduct numerical simulation of the water-Al_{2}O_{3} nanofluid. Natural convection in a horizontal tube heat exchanger was studied for a wide range of conditions and comparison made with available experimental data. Santra et al. [5] have used Artificial Neural Networks (ANN) to study heat transfer due to laminar natural convection of copper-water nanofluid in a differentially heated square cavity. The ANN was trained by resilient propagation algorithm. It was observed that the ANN was able to predict the heat transfer correctly within the given range of training data and is an efficient tool to predict heat transfer with reasonable computational time. Ho et al. [6] have used the finite volume approach with QUICK and central differencing for the convection and diffusion terms, respectively, to model natural convection of a nanofluid (alumina-water) in a square enclosure. They studied the effect of using two different formulas from the literature for effective viscosity and thermal conductivity of the nanofluid.

Lie group analysis, also called symmetry analysis, can be used to obtain similarity transformations that can be used to reduce the system of governing partial differential equations and associates boundary conditions to a system of ordinary differential equations. Ibrahim et al. [7] investigated the similarity reductions for problems of radiative and magnetic field effects on free convection and mass transfer flow past a semi-infinite flat plate. They obtained new similarity reductions and found an analytical solution for the uniform magnetic field case using Lie group method. They also presented numerical results for the nonuniform magnetic field case. Kalpakides and Balassas [8] studied the free convective boundary-layer problem of an electrically conducting fluid over an elastic surface by group theoretic method. Their results agree with the existing result for the group of scaling symmetry. They found that the numerical solution also does so. Megahed et al. [9] investigated convective heat and mass transfer along a semi-infinite vertical flat plate in the presence of a strong nonuniform magnetic field and the effects of Hall currents using the scaling group of transformations. They found that the temperature and concentration increase with an increase in magnetic parameter. Ibrahim et al. [10] presented symmetry group and similarity solutions for the steady laminar boundary-layer flow due to a rotating frustum of a cone in a viscous fluid. They found new four groups of similarity transformations. Hassanien and Hamad [11] introduced new similarity solutions of flow and heat transfer of a micropolar fluid along a vertical plate in a thermally stratified medium. The general analysis is developed in their study for the case of ambient temperature that varies exponentially with time, varies with the position, or has a uniform value. Some examples of the use of Lie group analysis to obtain similarity transformations for three-dimensional Euler equations were presented by Sekhar and Sharma [12]. Bhuvaneswari et al. [13] presented similarity solutions of natural convection heat and mass transfer in an inclined surface with chemical reaction via Lie group analysis. Using symmetry analysis, Sahin et al. [14] studied the self-similarity solutions of the one-layer shallow-water equations representing gravity currents.

In this paper, the similarity solution of natural convective boundary-layer flow on a vertical cylinder embedded in a nanofluid-saturated porous medium is studied. Two groups of similarity transformations are presented.

#### 2. Analysis

Consider steady two-dimensional free convection boundary-layer flow past a vertical circular cylinder of radius placed in a nanofluid-saturated porous medium. Cylindrical system of coordinates are used in which the -axis is aligned vertically upwards. The temperature and the nanoparticle fraction take constant values and at the surface of the cylinder, . The ambient values, attained as tends to infinity, are denoted by and , respectively.

The Boussinesq approximation is employed. Homogeneity and local thermal equilibrium in the porous medium are assumed with porosity denoted by *ε*. The field variables are the velocity , the temperature , and the nanoparticle volume fraction . The following four field equations (in steady case) embody the conservation of total mass, momentum, thermal energy, and nanoparticles, respectively,

Here is the density of the base fluid and , , and are the viscosity, thermal conductivity, and volumetric volume expansion coefficient of the nanofluid, respectively, while is the density of the particles. The porosity *ε*, the effective heat capacity , and the effective thermal conductivity of the porous medium are introduced. The gravitational acceleration is denoted by . The coefficients that appear in (3) and (4) are the Brownian diffusion coefficient and the thermophoretic diffusion coefficient . We consider a cylindrical polar coordinate system corresponding to the axial, azimuthal, and radial directions, respectively, and denote the associated fluid velocities as ; the configuration is shown in Figure 1.