Mathematical Problems in Engineering

Volume 2018, Article ID 9840862, 8 pages

https://doi.org/10.1155/2018/9840862

## MHD Mixed Convection Micropolar Fluid Flow through a Rectangular Duct

Correspondence should be addressed to Stanford Shateyi; az.ca.nevinu@iyetahs.drofnats

Received 5 July 2017; Revised 1 December 2017; Accepted 5 December 2017; Published 9 January 2018

Academic Editor: Filippo de Monte

Copyright © 2018 Mekonnen Shiferaw Ayano 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

Mixed convection flow through a rectangular duct with at least one of the sides of the walls of the rectangle being isothermal under the influence of transversely applied magnetic field has been analyzed numerically in this study. The governing differential equations of the problem have been transformed into a system of nondimensional differential equations and then solved numerically. The dimensionless velocity, microrotation components, and temperature profiles are displayed graphically showing the effects of various values of the parameters present in the problem. The results showed that the flow field is notably influenced by the considered parameters. It is found that increasing the aspect ratio increases flow reversal, commencement of the flow reversal is observed after some critical value, and the applied magnetic field increases the flow reversal in addition to flow retardation. The microrotation components flow in opposite direction; also it is found that one component of the microrotation will show no rotational effect around the center of the duct.

#### 1. Introduction

The theory of fluids with microstructures has been the subject of a large number of investigations. These are realistic and important from a technological point of view. The classical theories of continuum mechanics are inadequate to explain the microscopic manifestations of complex hydrodynamic behaviour. Microcontinuum theory or generalized continuum theories incorporate independent deformations of the microstructure inside of a material point. There are a number of microcontinuum theories by Eringen [1, 2], namely, couple stress, micropolar, microstretch, and micromorphic. These theories impose more or less constraints on the motion of microstructure inside of a material point. In microstretch theory, it is assumed that the microstructure of each material point can undergo expansion or contraction independently in addition to translation and rigid rotation. This theory is a generalization of micropolar theory, in which the microstructure can only have translation and rigid rotation. Micromorphic theory constitutes extensions of the classical field theories concerned with the deformations, motions, and electromagnetic interactions of material media, as continua, in microscopic time and space scales. In micromorphic theory, a material body is considered as a continuous collection of deformable particles, each with finite size and inner structure. Axtell et al. [3] discussed blood circulation in tissues as a multistate, multiphase porous media problem by simulating the fluid phase as a micromorphic continuum. While blood is known to be Newtonian, it also has electrically conducting properties. Using the finite element method, Ansari et al. [4] studied micromorphic first-order shear deformable plate element and the microstructure effect on the analysis of small-scale structures. Applications of micromorphic theory in micro-/nanoscale are discussed by formulating generalized micromorphic solid and fluid; see Lee and Wang [5]. Within the framework of micromorphic and elasticity theory, microstructure effect is developed to describe the bending behaviour of microplates by Ansari et al. [6]. Micropolar fluids are a subset of the micromorphic fluid theory introduced in a pioneering paper by Eringen [7]. Micropolar fluids constitute an important branch of non-Newtonian fluid dynamics where microrotation effects as well as microinertia are exhibited. Traditional Newtonian fluids cannot precisely describe the characteristic of fluid with suspended particles. Micropolar fluids formulated by Eringen [8] include certain microscopic effects arising from the local structure and micromotions of the fluid elements and provide a mathematical model for the non-Newtonian fluid flow behaviour such as exotic lubricants, polymers, animal blood, and real fluids with suspensions. Eringen [9] extended the micropolar fluid theory and developed the theory of thermomicropolar fluids. An excellent review of micropolar fluids and their applications was given by Ariman et al. [10]. Extensive review of foundations of the micropolar continuum mechanics is found in the book by Eremeyev et al. [11].

There have been numerous studies for the laminar flow and heat transfer through rectangular ducts [13–19]. Cheng and Ou [20] have also studied the mixed convection heat transfer in the thermal entrance region of horizontal rectangular channels. Aung and Worku [21] analyzed mixed convection flow in ducts with asymmetric wall heat fluxes. Mahaney et al. [22] studied development of laminar mixed convection flow in a horizontal rectangular duct with uniform bottom heating and Abou-Ellail and Morcos [23] also carried out investigations for similar flow geometry buoyancy effects in the entrance region. Huang et al. [24] examined laminar mixed convection heat and mass transfer in vertical rectangular ducts with film evaporation and condensation. Yan and Lin [25, 26] numerically investigated the laminar or turbulent mixed convection flow in a vertical channel under simultaneous influence of the combined buoyancy effects of the thermal and mass diffusion for an air-water vapour system.

To the knowledge of the authors and in all of the above-mentioned studies, the combined effect of an applied magnetic field under strong buoyancy force on mixed convection on the micropolar fluid model which represents realistic non-Newtonian flow in a rectangular duct has not been well evaluated. This motivates the present investigation. In this paper, we study the problem of steady laminar, incompressible mixed convection micropolar fluid flow through a rectangular duct subject to applied magnetic field with at least one side of the rectangle being isothermal. Mathematical formulation has been presented for the governing equations and the solution is found numerically using the finite difference method. Results are discussed through velocity, microrotation, and temperature.

#### 2. Mathematical Formulation

Consider a steady, laminar, incompressible, mixed convective heat transfer micropolar fluid through an infinite vertical rectangular duct given in Figure 1, with sides of length and in the presence of an external applied magnetic field. In the formulation of the problem, the following assumptions are made: Boussinesq approximations are valid, the flow is parallel and hence the only nonvanishing component is the (vertical) direction component, a uniform magnetic field is applied normal to the surfaces, and the induced magnetic field is assumed to be very small. The last assumption is considered to be valid for the case of small magnetic Reynolds number as done in previous works by Hayat et al. [27]. At least one of the duct walls is kept isothermal and the temperature is uniform along the duct vertical direction; hence .