International Journal of Engineering Mathematics

Volume 2015, Article ID 287623, 15 pages

http://dx.doi.org/10.1155/2015/287623

## Mixed Convection Flow of Magnetic Viscoelastic Polymer from a Nonisothermal Wedge with Biot Number Effects

^{1}Department of Mathematics, Jawaharlal Nehru Technological University Anantapur, Anantapur 515002, India^{2}Department of Mathematics, Madanapalle Institute of Technology and Science, Madanapalle 517325, India^{3}Gort Engovation Research (Aerospace and Medical Engineering), 11 Rooley Corft, Bradford BD6 1FA, UK

Received 4 May 2015; Revised 5 September 2015; Accepted 6 September 2015

Academic Editor: Josè A. Tenereiro Machado

Copyright © 2015 S. Abdul Gaffar 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

Magnetic polymers are finding increasing applications in diverse fields of chemical and mechanical engineering. In this paper, we investigate the nonlinear steady boundary layer flow and heat transfer of such fluids from a nonisothermal wedge. The incompressible Eyring-Powell non-Newtonian fluid model is employed and a magnetohydrodynamic body force is included in the simulation. The transformed conservation equations are solved numerically subject to physically appropriate boundary conditions using a second-order accurate implicit finite difference Keller Box technique. The numerical code is validated with previous studies. The influence of a number of emerging nondimensional parameters, namely, the Eyring-Powell rheological fluid parameter (), local non-Newtonian parameter based on length scale (), Prandtl number (Pr), Biot number (), pressure gradient parameter (), magnetic parameter (), mixed convection parameter (), and dimensionless tangential coordinate (), on velocity and temperature evolution in the boundary layer regime is examined in detail. Furthermore, the effects of these parameters on surface heat transfer rate and local skin friction are also investigated.

#### 1. Introduction

The development of modern functional materials which can be manipulated using electromagnetic fields has stimulated great attention in polymer engineering sciences in recent years. These materials include ferromagnetic polymers [1], magnetotropic fluids [2], and liquid crystalline electrically conducting polymers [3]. Such fluids exhibit many complex characteristics including non-Newtonian behavior. For better understanding of the manufacture of such materials,* magnetorheological fluid mechanics* plays a central role. Although many different magnetic properties may arise in such materials, a fundamental methodology for their synthesis involves the application of transverse magnetic fields which alters momentum transfer and therefore also influences coupled processes such as heat transfer and mass transfer (species diffusion). Many constitutive material models have been developed to simulate the departure of such fluids from Newtonian viscous behavior. In these fluids, the constitutive relationship between stress and rate of strain is nonlinear in comparison to the Navier-Stokes equations which are generally good for Newtonian fluids. Most non-Newtonian models involve some form of modification to the momentum conservation equations. A comprehensive summary of such models which include the Maxwell model, power-law model, and also Eyring-Powell and Giesekus viscoelastic models, is provided in the treatise by Shaw [4]. Heat transfer is often also frequently present in polymer dynamics [5]. Furthermore, in many polymer fabrication flows,* boundary layer phenomena* arise. The* convective boundary condition* has therefore also attracted some interest and this is usually simulated via a Biot number in the wall thermal boundary condition. Recently, Ishak [6] discussed the similarity solutions for flow and heat transfer over a permeable surface with convective boundary condition. Aziz [7] provided a similarity solution for laminar thermal boundary layer over a flat surface with a convective surface boundary condition. Aziz [8] further studied hydrodynamic and thermal slip flow boundary layers with an isoflux thermal boundary condition. Buoyancy effects on thermal boundary layers subject to a convective surface boundary condition were examined by Makinde and Olanrewaju [9]. Gupta et al. [10] used a variational finite element to simulate mixed convective-radiative non-Newtonian shrinking sheet flow with a convective boundary condition and Eringen’s micropolar material model. Swapna et al. [11] studied convective wall heating effects on hydromagnetic flow of a micropolar fluid. Makinde et al. [12] studied cross diffusion effects and Biot number influence on hydromagnetic Newtonian boundary layer flow with homogenous chemical reactions and MAPLE quadrature routines. Bég et al. [13] analyzed Biot number and buoyancy effects on magnetohydrodynamic thermal slip flows. Subhashini et al. [14] studied wall transpiration and cross diffusion effects on free convection boundary layers with a convective boundary condition. In many of these studies, magnetohydrodynamic effects were also considered generally via the introduction of a Lorentzian magnetic drag force and consideration of the Hartmann number. Relatively few studies however have considered magnetic field effects on* Eyring-Powell* viscoelastic polymer flows. This rheological model has certain advantages over the other non-Newtonian formulations, including simplicity, ease of computation, and physical robustness. Furthermore, it is deduced from kinetic theory of liquids rather than the empirical relation. Additionally, it correctly reduces to Newtonian behavior for low and high shear rates [15]. It is a three-constant model which displays a nonzero bounded viscosity at both the upper and the lower limits. Distinct from the popular Ostwald-De Waele power-law model, the Eyring-Powell model does not demonstrate infinite effective viscosities for low shear rates. Investigations employing this model of relevance to polymeric transport processes include Hayat et al. [16] in radiative magnetic convection, Sirohi et al. [17] for wedge flows, and Adesanya and Gbadeyan [18] for channel flows. Very recently, Prasad et al. [19] studied the thermal convection flow in permeable materials with Biot number effects, observing that increasing rheological effect accelerates the flow and heats the boundary layer.

Very few of the above studies have considered* Falkner-Skan flows* [20]. This family of boundary layer flows is associated with the two-dimensional wedge configuration. Non-Newtonian flows from wedge bodies arise in a number of chemical engineering systems which have been described in detail by Peddieson [21] employing the second-order Reiner-Rivlin model. The mixed convection boundary layer flow from a heated wedge plate has also drawn some interest. The combined forced and free convection flow and heat transfer about a nonisothermal wedge subject to a nonuniform free stream velocity were first considered by Sparrow et al. [22]. Watanabe et al. [23] analyzed theoretically mixed convection flow over a perforated wedge with uniform suction or injection. Kafoussias and Nanousis [24] and Nanousis [25] studied the effect of suction or injection on MHD mixed convection flow past a wedge. Gorla [26] used a power-law model to study heat transfer in polymer flow past a wedge. Yih [27] evaluated radiation effects on mixed convection flow about an isothermal wedge embedded in a saturated porous medium. Rashidi et al. [28] developed homotopy solutions for third grade viscoelastic flow from a nonisothermal wedge. Chamkha et al. [29] presented computational solutions for MHD forced convection flow from a nonisothermal wedge in the presence of a heat source or sink with a finite difference method. Hsiao [30] reported on MHD convection of viscoelastic fluid past a porous wedge, observing that the elastic effect increases the local heat transfer coefficient and heat transfer rates at the wedge surface. Ishak et al. [31] obtained a self-similar solution for a moving wedge in a micropolar fluid. Ishak et al. [32] further studied numerically steady two-dimensional laminar flow past a moving wedge in non-Newtonian fluid.

The objective of the present study is to investigate the laminar boundary layer flow and heat transfer of an* Eyring-Powell* non-Newtonian fluid from a nonisothermal wedge. Such a study has not appeared in the literature to the knowledge of the authors. The nondimensional equations with associated dimensionless boundary conditions are solved with the Keller implicit finite difference “Box” scheme [33]. The effects of the emerging thermophysical parameters, namely,* the rheological parameters *,* Biot number *,* mixed convection parameter *,* pressure gradient parameter *,* magnetic parameter *,* and Prandtl number *, on the velocity, temperature, local skin friction, and heat transfer rate (local Nusselt number) characteristics are studied. The present problem is relevant to the simulation of magnetized polymer materials fabrication processes.

#### 2. Non-Newtonian Constitutive Eyring-Powell Fluid Model

In the present study, a subclass of non-Newtonian fluids known as the* Eyring-Powell fluid* is employed. The Cauchy stress tensor for Eyring-Powell fluids [15] takes the hyperbolic form: where is dynamic viscosity and and are the rheological fluid parameters of the Eyring-Powell fluid model. Consider the second-order approximation of the function as follows:When the Eyring-Powell formulation is introduced into the subsequent model, the momentum conservation equation is significantly modified with numerous velocity gradient terms including mixed derivatives. Strong nonlinearity therefore results in necessitating numerical solutions. The resulting boundary value problem is found to be well posed and permits an excellent mechanism for the assessment of rheological characteristics on the flow behaviour.

#### 3. Mathematical Thermofluid Polymer Boundary Layer Model

Steady, laminar, and incompressible flow and heat transfer of an Eyring-Powell polymeric fluid from a nonisothermal horizontal wedge are considered, as illustrated in Figure 1.