Abstract and Applied Analysis

Volume 2015 (2015), Article ID 946350, 8 pages

http://dx.doi.org/10.1155/2015/946350

## Exact Solutions for Unsteady Free Convection Flow of Casson Fluid over an Oscillating Vertical Plate with Constant Wall Temperature

^{1}Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia (UTM), 81310 Skudai, Malaysia^{2}Department of Mathematics, SBK Women’s University, Quetta 87300, Pakistan^{3}College of Engineering, Majmaah University, Majmaah 11952, Saudi Arabia

Received 17 July 2014; Accepted 23 October 2014

Academic Editor: Yasir Khan

Copyright © 2015 Asma Khalid 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

The unsteady free flow of a Casson fluid past an oscillating vertical plate with constant wall temperature has been studied. The Casson fluid model is used to distinguish the non-Newtonian fluid behaviour. The governing partial differential equations corresponding to the momentum and energy equations are transformed into linear ordinary differential equations by using nondimensional variables. Laplace transform method is used to find the exact solutions of these equations. Expressions for shear stress in terms of skin friction and the rate of heat transfer in terms of Nusselt number are also obtained. Numerical results of velocity and temperature profiles with various values of embedded flow parameters are shown graphically and their effects are discussed in detail.

#### 1. Introduction

Newtonian fluids described by Navier-Stokes equations are extensively studied in the literature for the past few decades. Largely, this is due to the fact that they are relatively simple and their solutions are convenient [1–10]. However, Newtonian fluids which have a linear relationship between the stress and the rate of strain are limited in view of their applications. They do not explain several phenomena observed for the fluids in industry and other technological applications. For example, many complex fluids such as blood, soap, clay coating, certain oils and greases, elastomers, suspensions, and many emulsions are noteworthy due to their various applications in industry. Unfortunately, Navier-Stokes equations are no more convincing to describe such fluids. In the literature, they are known as non-Newtonian fluids. These fluids are described by a nonlinear relationship between the stress and the rate of strain. The understanding of flow characteristics of non-Newtonian fluids is very important because they play a significant role in industry and engineering. Such interest is motivated because of extensive applications in diverse areas of biorheology, geophysics, and chemical and petroleum industries [11, 12]. In view of these particular applications, the study and understanding of non-Newtonian fluids have now become an increasingly appealing topic of current research in this field.

Rheological properties of non-Newtonian fluids are described by their so-called constitutive equations. Due to complexity of fluids, several non-Newtonian fluid models/constitutive equations based on their empirical observations have been proposed. Amongst the different non-Newtonian fluids there is one known as Casson fluid which was originally introduced by Casson for the prediction of the flow behavior of pigment-oil suspensions [13]. The Casson model is based on a structure model of the interactive behavior of solid and liquid phases of a two-phase suspension. Casson fluid exhibits yield stress. It is a shear thinning liquid which has an infinite viscosity at zero rates of shear, a yield stress below which no flow occurs, and a zero viscosity at an infinite rate of shear [14]. More exactly, if a shear stress less than the yield stress is applied to the fluid, it behaves like a solid, whereas if a shear stress greater than yield stress is applied, it starts to move. Some famous examples of Casson fluid include jelly, tomato sauce, honey, soup, and concentrated fruit juices. Human blood can also be treated as Casson fluid due to the presence of several substances such as protein, fibrinogen, and globulin in aqueous base plasma and human red blood cells [15, 16].

In the earlier studies on Casson fluid, Fredrickson [17] investigated its steady flow in a tube whereas Boyd et al. [18] described the steady and oscillatory flow of blood flow by taking into account Casson fluid. Mernone et al. [19] discussed the peristaltic flow of a Casson fluid in a two-dimensional channel. Mustafa et al. [20] studied the unsteady boundary layer flow and heat transfer of a Casson fluid over a moving flat plate with a parallel free stream using homotopy analysis method (HAM). Mixed convection stagnation-point flow of Casson fluid with convective boundary conditions is examined by Hayat et al. [21]. Mukhopadhyay [22] described the effects of thermal radiation on Casson fluid flow and heat transfer over an unsteady stretching surface subjected to suction/blowing. Mukhopadhyay et al. [23] also analyzed the Casson fluid flow over an unsteady stretching surface. Bhattacharyya [24] investigated the boundary layer stagnation-point flow of Casson fluid and heat transfer towards a shrinking/stretching sheet and Pramanik [25] studied the Casson fluid flow and heat transfer past an exponentially porous stretching surface in presence of thermal radiation.

In all of the above studies the solutions of Casson fluid are obtained by using either approximate method or any numerical scheme. There are very few cases in which the exact analytical solutions of Casson fluid are obtained. These solutions are even rare when Casson fluid in free convection flow with constant wall temperature is considered. On the other hand, the flow of Casson fluids (such as drilling muds, clay coatings and other suspensions, certain oils and greases, polymer melts, blood, and many emulsions), in the presence of heat transfer, is an important research area due to its relevance to the optimized processing of chocolate, toffee, and other foodstuffs [26].

Motivated by the above investigations, the present analysis is focused on the study of unsteady boundary layer flow of a Casson fluid past an oscillating vertical plate with constant wall temperature. Exact solutions are obtained by using the Laplace transform technique. Analytical as well as numerical results for skin friction and Nusselt number are provided. Graphical results are presented and discussed for various physical parameters. Exact solutions obtained in this paper are important; not only do they correspond to some fundamental flow situations, but also they are useful for explaining the flow physics in detail as well as for being used as a benchmark for validation of other solutions obtained via approximate or numerical schemes.

#### 2. Formulation of the Problem

Let us consider the effect of heat transfer on unsteady boundary layer flow of an incompressible Casson fluid past an infinite vertical flat plate situated at the flow being confined to , where is the coordinate measured in the normal direction to the surface. It is assumed that, at the initial moment , both the plate and fluid are at rest with constant temperature . At time the plate begins to oscillate in its plane according towhere the constant is the amplitude of the plate oscillations, is the unit step function, is the unit vector in the vertical flow direction, and is the frequency of oscillation of the plate. At the same time, the plate temperature is raised to which is thereafter maintained constant (Figure 1).