Advances in High Energy Physics

Volume 2015, Article ID 650813, 16 pages

http://dx.doi.org/10.1155/2015/650813

## Similarity Solution for Free Convection Flow of a Micropolar Fluid under Convective Boundary Condition via Lie Scaling Group Transformations

Department of Mathematics, National Institute of Technology, Warangal 506004, India

Received 19 January 2015; Revised 26 April 2015; Accepted 27 April 2015

Academic Editor: Ming Liu

Copyright © 2015 Ch. RamReddy 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. The publication of this article was funded by SCOAP^{3}.

#### Abstract

The free convective flow of an incompressible micropolar fluid along permeable vertical plate under the convective boundary condition is investigated. The Lie scaling group of transformations is applied to get the similarity representation for the system of partial differential equations and then the resulting systems of equations are solved using spectral quasi-linearisation method. A quantitative comparison of the numerical results is made with previously published results for special cases and the results are found to be in good agreement. The results of the physical parameters on the developments of flow, temperature, concentration, skinfriction, wall couple stress, heat transfer, and mass transfer characteristics along vertical plate are given and the salient features are discussed.

#### 1. Introduction

In the past few decades, most of the researchers considered convective heat transfer problems with either constant wall temperature (CWT), constant heat flux (CHF), or Newtonian heating (NH) in a Newtonian and/or non-Newtonian fluid. Recently, a novel mechanism for the heating process has drawn the involvement of many researchers, namely,* convective boundary condition* (CBC), where the heat is supplied to the convecting fluid through a bounding surface with a finite heat capacity. Further, this results in the heat transfer rate through the surface being proportional to the local difference in temperature with the ambient conditions (Merkin [1]). Besides, it is more general and realistic, particularly in various technologies and industrial operations such as transpiration cooling process, textile drying, and laser pulse heating. Aziz [2] reported similarity solution for thermal boundary layer flow over a flat plate in a uniform stream of fluid with the convective boundary condition and he concluded that a similarity solution is possible if the convective heat transfer related to hot fluid on the lower surface of the plate is proportional to the inverse square root of the axial length. In the presence of an internal heat generation local similarity solution for free convection heat transfer from a moving vertical plate with the convective boundary condition is discussed by Makinde [3]. The laminar natural convection flow over a semi-infinite moving vertical plate under the convective boundary condition is examined by Ibrahim and Bhashar Reddy [4]. RamReddy et al. [5] investigated the influence of the prominent Soret effect on mixed convection in a nanofluid under the convective boundary conditions. The nonsimilar result has been presented for the free convection boundary layer flow along a solid sphere under the convective boundary conditions by Alkasasbeh et al. [6]. More recently, a note on the natural convection along convectively heated vertical plate is given by Pantokratoras [7].

One of the best established theories of fluids with microstructure is the theory of micropolar fluids and this theory can be found in the books by Lukaszewicz [8] and Eremeyev et al. [9]. It has gathered a good deal of attention due to the obvious reasons that the Navier Stokes equation for Newtonian fluids cannot successfully explain the attributes of fluids with a substructure. Physically, the micropolar fluids may be treated as non-Newtonian fluids consisting of dumb-bell molecules or rigid cylindrical element, polymer fluids, fluid suspension, animal blood, and so forth. Further, the theory of micropolar fluids includes microrotation as well as microinertia effects. This theory studies viscous fluids in which microconstituents are rigid and spherical or randomly oriented as well. The subject of free convection boundary layer flow in a micropolar fluid has been keyed out by several investigators due to its immense applications in the engineering problems such as solar energy collecting devices, air conditioning of a room, material processing, and passive cooling of nuclear reactors. The boundary layer flow over a semi-infinite flat plate is considered for analyzing theory of micropolar fluid and its application to low concentration suspension flow by Ahmadi [10]. Rees and Pop [11] discussed the free convection boundary layer flow of a micropolar fluid from a vertical flat plate. The nonsimilarity transformations are used to analyze the effects of double stratification on free/mixed convective transport in a micropolar fluid by Srinivasacharya and RamReddy [12–14] (also see the references cited therein). The problems of a steady laminar stagnation point flow towards a stretching/shrinking sheet in an incompressible micropolar fluid under the convective surface boundary condition are discussed by Yacob and Ishak [15] and Zaimi and Ishak [16]. Merely from the literature, it is noted that the majority of the researchers have found the local similarity or nonsimilarity solutions for the problems involving* convective boundary conditions*, since most of the researchers have taken a convective heat transfer coefficient as a function for getting the similarity solutions in their problems. Nevertheless, the assumption of a heat transfer coefficient varying along the plate as a function of is not realistic and very difficult to be obtained in practice. For that cause, it could be supposed that the above works have only theoretical value.

In the recent past, several researchers are focused on obtaining the similarity solutions of the convective transport phenomena problems arising in fluid dynamics, aerodynamics, plasma physics, meteorology, and some branches of engineering by using different procedures. One such procedure is* Lie group analysis*. The concept of Lie group analysis also called symmetry analysis is developed by* Sophius Lie* to determine transformations which map a given differential equation to itself and it unifies almost all known exact integration techniques (see [17–19]). It provides a potent, sophisticated, and systematic tool for generating the invariant solutions of the system of nonlinear partial differential equations (PDEs) with relevant initial or boundary conditions. A special form of Lie group transformations, known as the scaling group, has been suggested by various researchers to study convection flows of different flow phenomena (see Tapanidis et al. [20], Hassanien and Hamad [21], Kandasamy et al. [22], Aziz et al. [23], Mutlag et al. [24], etc.; they are worth observing).

From the literature survey, it seems that the problem of the free convective heat and mass transport along permeable vertical plate in a micropolar fluid under the convective boundary condition has not been investigated so far. Motivated by all these works, this paper attempts to present the new similarity transformations and corresponding similarity solution to investigate the free convection flow of a micropolar fluid under the convective boundary condition using the Lie group transformations. The mathematical model involving the convective boundary conditions becomes slightly more complicated leading to the complex interactions of the flow, heat, and mass transfer mechanism. Further, the analytical solution is out of scope in the present set-up and hence a numerical solution is obtained for the current problem. Also, the influence of important parameters, namely, micropolar, suction/injection, and convective heat transfer parameters, on the physical quantities of the flow, heat, and mass transfer rates is analyzed in different flow situations.

#### 2. Mathematical Formulation

Consider the steady, laminar, and free convective flow of an incompressible micropolar fluid with the free stream temperature and concentration, and , respectively. Choose the coordinate system such that the -axis is along the vertical plate and -axis normal to the plate, as shown in Figure 1. The suction/injection velocity distribution is assumed to be . The plate is either heated or cooled from left by convection from a fluid of temperature with corresponding to a heated surface (assisting flow) and corresponding to a cooled surface (opposing flow), respectively. On the wall concentration is taken to be constant and is given by .