Journal of Engineering

Volume 2016 (2016), Article ID 2689493, 9 pages

http://dx.doi.org/10.1155/2016/2689493

## Slip Effect on MHD Chemically Reacting Convictive Boundary Layer Flow with Heat Absorption

^{1}Department of Mathematics, University of Swaziland, P. Bag 4, Kwaluseni M201, Swaziland^{2}Mechanical Engineering Department, Arba Minch University, B.O. Box 21, Arba Minch, Ethiopia

Received 16 November 2015; Accepted 20 March 2016

Academic Editor: Junling Hu

Copyright © 2016 Mekonnen Shifferaw Ayano and Negussie Tadege Demeke. 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 aim of this paper is to investigate steady magneto micropolar fluid past a stretched semi-infinite vertical and permeable surface taking into account heat absorption, hall and ion-slip effect, first-order chemical reaction, and radiation effects. The system of coupled nonlinear equations is solved numerically and the effects of various parameters on the velocity, the microrotation, the temperature, and the concentration field are calculated. The following discovery was made: an increase in the hall parameter strongly enhances the fluid velocity, an increase in the heat absorption parameter increases the temperature, and an increase in the slip parameter decreases the velocity. Additionally, the local skin friction, Nusselt number, and Sherwood number are also analyzed for various parameters and presented in table form.

#### 1. Introduction

The micropolar fluids are non-Newtonian fluids with microstructures such as polymeric additives, colloidal suspensions, and liquid crystals. Erigen [1, 2] first developed the theory of micropolar fluids and thermomicropolar fluids to describe such characteristics. A thorough review of the subject and applications of micropolar fluid mechanics was provided by Ariman et al. [3, 4]. These fluids are realistic and important from a technological point of view.

Due to the vast applications of the heat and mass transfer over a vertical channel in geophysics and engineering, a lot of authors have studied and published about this subject. Considering the chemical reaction and radiation effects, the common applications that can be found are the combustion of fossil fuels, atmospheric reentry with suborbital velocities, plasma wind tunnels, electric spacecraft propulsion, hypersonic flight through planetary atmosphere photo-dissociation, and photo ionization. Articles have been published based on the study of heat generation or absorption, chemical reaction, and radiation effects. Studies have been done on the individual or collective effects. For instance, Muthucumaraswamy and Ganesan [5] studied numerically natural convection on a moving isothermal vertical plate with chemical reaction. In their study, they have shown that velocity increases during generative reaction and decreases in destructive reaction. Vajravelu and Hadjinicolaou [6] studied the heat transfer in viscous fluid over a stretching sheet with viscous dissipation and internal heat generation. Kelson and Desseaux [7] examined the effect of surface conditions on the flow of a micropolar fluid driven by a porous stretching sheet. Takhar et al. [8] have investigated the flow and mass diffusion of chemical species with first-order and higher order reactions over a continuously stretching sheet with the magnetic field effect. Influence of chemical reaction on heat and mass transfer by natural convection from vertical surfaces in porous media considering soret and dufour effects was analyzed by Postelnicu [9]. El-Arabawy [10] analyzed the problem of the effect of suction/injection on the flow of a micropolar fluid past a continuously moving plate in the presence of radiation.

The effect of magnetic field with or without hall and ion-slip effects has been studied by several researchers. On uniformly stretched vertical permeable surface in the presence of heat generation/absorption and a chemical reaction, Chamkha [11] studied analytically the magnetic effects. Mahmoud [12] considered thermal radiation effects on MHD flow of a micropolar fluid over a stretching surface with variable thermal conductivity. Chemical reaction and heat and mass transfer on MHD flow over a vertical stretching surface with heat source and thermal stratification effects were examined by Kandasamy et al. [13]. Khedr et al. [14] examined numerically MHD flow of a micropolar fluid past a stretched permeable surface with heat generation or absorption neglecting the hall and ion-slip effects.

The abovementioned studies addressed flows when one or more of the effects of chemical reaction, radiation, heat source, and MHD are absent. The objective of this paper is to analyze hall and ion-slip effects of micropolar fluid along a vertical semi-infinite permeable plate taking into account the slip boundary conditions with the combined effects of chemical reaction, radiation, and heat source with wall suction or injection effects. The governing systems of partial differential equations have been solved numerically by the Keller box method. The effects of the various parameters on the velocity, micrototation, temperature, and concentration field will be investigated.

#### 2. Mathematical Formulation of Problem

Let us consider a steady, laminar, incompressible micropolar fluid flowing past a porous surface along a semi-infinite vertical plate in the presence of heat generation or absorption and thermal radiation effect. Further, it is assumed that there exists a first-order homogeneous chemical reaction between the species and the fluid, a magnetic field of a uniform strength is applied transversely to the plate, and the magnetic Reynolds number is assumed to be small so that the induced magnetic field is neglected. When the fluid is assumed to be viscous and has constant properties retaining the hall and ion-slip effect, the equation governing the flow is given bywhere is chemical reaction parameter, is magnetic field intensity, is the effective diffusive coefficient, is density, and are fluid velocities, is microrotation component, is temperature, is concentration, is the dynamic viscosity, is hall parameter, is ion-slip parameter, is the fluid thermal diffusivity, is fluid electrical conductivity, is dimensional heat generation (>0) or absorption (<0) coefficient, and is the radiative heat flux.

The boundary conditions are as follows: where is the proportionality constant of the velocity. For microrotation boundary condition, corresponds to no slip, the concentrated particle where the microelements close to the plate surface are unable to rotate. And the case represents the fact that antisymmetric part of the stress is zero on the surface and weak concentration. The case represents turbulent boundary layer flow.

In (5), the radiative heat flux under Rosseland approximation is given by where is Stephan-Boltzmann constant and is mean absorption coefficient. If the temperature differences within the flow are sufficiently small, may be expanded by Taylor series. Hence, expanding about and neglecting higher order terms, we get Introduce the nondimensional variables through Substituting (8)–(10) into (1)–(6), we get the following nondimensional equations: where primes denote differentiation with respect to the variable , is coupling number, is internal heat generation/absorption parameter, is the Eckert number, is the Prandtl number, is Hartmann number, is microrotation parameter, and is radiation parameter.

The corresponding boundary conditions in dimensionless form are as follows: where is slip parameter and is suction/injection velocity.

The shearing stress component at the plate can be calculated in nondimensional form:

The local mass flux and the local Sherwood number are, respectively, given bywhere is the Reynolds number.

#### 3. Numerical Scheme

The numerical scheme to solve (11)–(16) is outlined based on Keller [15] as follows: (11)–(15) are written as a first-order system of equation. For this, we define new variables and asThe boundary condition becomesWe write the difference equation: where is spacing. Therefore,Linearizing the resulting system of equations by Newton method, we will get a block tridiagonal structured matrix and then solve the system using Keller box method. This method has been proven to be adequate and give accurate results for boundary layer equations. The step size is chosen after testing with different step sizes. The calculations are repeated until some convergence criterion is satisfied and the calculations were stopped when , , , , and . In the present study, the boundary conditions for at are replaced by a sufficiently large value of where the velocity, microrotation, temperature, and concentration profiles approach zero. After some trials, we imposed a maximal value of at of .

#### 4. Results and Discussion

Computations have been performed using the following parameters: , , , , very large number, and , which represents the presence of microelements; the fluid is micropolar (in (2), when and when in (11), equations (11) and (13) decouple and the fluid behaves as Newtonian). To validate the code, results are compared with those of El-Arabawy [10] and it was found that they are in good agreement as shown in Table 1.